SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Existence of subharmonic solutions for non-quadratic second-order Hamiltonian systems

Xingyong Zhang1* and Xianhua Tang2

Author Affiliations

1 Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P.R. China

2 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410083, P.R. China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:139  doi:10.1186/1687-2770-2013-139

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/139


Received:23 October 2012
Accepted:10 May 2013
Published:30 May 2013

© 2013 Zhang and Tang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, some existence theorems are obtained for subharmonic solutions of second-order Hamiltonian systems with linear part under non-quadratic conditions. The approach is the minimax principle. We consider some new cases and obtain some new existence results.

MSC: 34C25, 58E50, 70H05.

Keywords:
second-order Hamiltonian systems; subharmonic solution; critical point; linking theorem

1 Introduction and main results

Consider the second-order Hamiltonian system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M1">View MathML</a>

(1.1)

where A is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M2">View MathML</a> symmetric matrix and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M3">View MathML</a> is T-periodic in t and satisfies the following assumption:

Assumption (A)′<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M4">View MathML</a>is measurable intfor every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M5">View MathML</a>and continuously differentiable inxfor a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6">View MathML</a>, and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M7">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M8">View MathML</a>which isT-periodic and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M9">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M10">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M11">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M12">View MathML</a>and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M14">View MathML</a>, system (1.1) reduces to the second-order Hamiltonian system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M15">View MathML</a>

(1.2)

There have been many existence results for system (1.2) (for example, see [1-7] and references therein). In 1978, Rabinowitz [6] obtained the nonconstant periodic solutions for system (1.2) under the following AR-condition: there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M17">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M18">View MathML</a>

From then on, the condition has been used extensively in the literature; see [8-12] and the references therein. In [13], Fei also obtained the existence of nonconstant solutions for system (1.2) under a kind of new superquadratic condition. Subsequently, Tao and Tang [14] gave the following more general one than Fei’s: there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M20">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M21">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M22">View MathML</a>

(1.4)

They also considered the existence of subharmonic solutions and obtained the following result.

Theorem A (See [14], Theorem 2)

Suppose thatFsatisfies

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M4">View MathML</a>is measurable intfor every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M5">View MathML</a>and continuously differentiable inxfor a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6">View MathML</a>, and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M7">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M27">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M28">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M12">View MathML</a>and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6">View MathML</a>. Assume that (1.3), (1.4) and the following conditions hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M31">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M32">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M33">View MathML</a>

(1.7)

Then system (1.2) has a sequence of distinct periodic solutions with period<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M34">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37">View MathML</a>.

Recently, Ma and Zhang [15] considered the following p-Laplacian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M38">View MathML</a>

(1.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M10">View MathML</a>. By using some techniques, they obtained the following more general result than Theorem A.

Theorem B (See [15], Theorem 1)

Suppose thatFsatisfies (A), (1.3) and (1.4) with 2 replaced byp, (1.5) and the following condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M40">View MathML</a>

(1.9)

Then system (1.8) has a sequence of distinct periodic solutions with period<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M34">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M45">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M47">View MathML</a> is the unit matrix of order N. Ye and Tang [16] obtained the following result.

Theorem C (See [16], Theorem 2)

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M48">View MathML</a>, Fsatisfies (A), (1.3), (1.4), (1.5), (1.6) and the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M49">View MathML</a>

Then system (1.1) has a sequence of distinct periodic solutions with period<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M34">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37">View MathML</a>.

Recently, in [17], we considered a more general case than that in [16]. We considered the case that A only has 0 or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M54">View MathML</a> as its eigenvalues, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M58">View MathML</a>. In [17], we used the following condition which presents some advantages over (1.3) and (1.4):

(H) there exist positive constantsm,ζ,ηand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M59">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M60">View MathML</a>

In this paper, we consider some new cases which can be seen as a continuance of our work in [17].

Next, we state our main results. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M62">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M63">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M64">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M66">View MathML</a>) be the positive and negative eigenvalues of A, respectively, where r and s denote the number of positive eigenvalues and zero eigenvalues of A (counted by multiplicity), respectively. Moreover, we denote by q the number of negative eigenvalues of A (counted by multiplicity). We make the following assumption:

Assumption (A0)Ahas at least one nonzero eigenvalue and all positive eigenvalues are not equal to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M67">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M68">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M46">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M70">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M71">View MathML</a>) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M72">View MathML</a>.

The Assumption (A0) implies that one can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M73">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M74">View MathML</a>

(1.10)

For the sake of convenience, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M75">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M76">View MathML</a>

Corresponding to (1.10), we know that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M77">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M78">View MathML</a>

Moreover, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M79">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M80">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M81">View MathML</a>. Corresponding to (1.10), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M82">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M83">View MathML</a>

(1.11)

Theorem 1.1Assume that (A0) holds andFsatisfies (A)′, (1.5) and the following conditions.

(H1) For some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84">View MathML</a>, assume thatksatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M85">View MathML</a>

(1.12)

(H2) There exist positive constantsm, ζ, ηand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M86">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M87">View MathML</a>

(H3) Assume that one of the following cases holds:

(1) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M90">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M92">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M93">View MathML</a>

(1.13)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M94">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M95">View MathML</a>are defined by (1.11);

(2) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M98">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M100">View MathML</a>such that (1.13) holds;

(3) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M102">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M98">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M105">View MathML</a>such that (1.13) holds;

(4) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M102">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M108">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M110">View MathML</a>such that (1.13) holds;

(5) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M113">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M115">View MathML</a>such that (1.13) holds;

(6) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M117">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M118">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M120">View MathML</a>such that (1.13) holds;

(H4) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M121">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M122">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M123">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M124">View MathML</a>

whereσimplies that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M125">View MathML</a>is independent ofk. Then system (1.1) has a nonzerokT-periodic solution. Especially, for cases (H3)(1) and (H3)(4), system (1.1) has a nonconstantkT-periodic solution.

Remark 1.1 For cases (H3)(1)-(H3)(4), from (1.10) and (1.12), it is easy to see that the number of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84">View MathML</a> satisfying (1.12) is finite. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M127">View MathML</a> be the maximum integer satisfying (1.12), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M128">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M129">View MathML</a>. Hence, Theorem 1.1 implies that system (1.1) has nonzero kT-periodic solutions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M130">View MathML</a>). For cases (H3)(5) and (H3)(6), since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M116">View MathML</a>, (1.12) holds for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84">View MathML</a>. Hence, Theorem 1.1 implies that system (1.11) has nonzero kT-periodic solutions for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84">View MathML</a>.

Remark 1.2 In [18], Costa and Magalhães studied the first-order Hamiltonian system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M134">View MathML</a>

(1.14)

They obtained that system (1.14) has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M135">View MathML</a> periodic solution under the following non-quadraticity conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M136">View MathML</a>

(1.15)

and the so-called asymptotic noncrossing conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M137">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M138">View MathML</a> are consecutive eigenvalues of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M139">View MathML</a>. Moreover, they also obtained system (1.14) has a nonzero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M135">View MathML</a> periodic solution under (1.15) and the called crossing conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M141">View MathML</a>

One can also establish the similar results for the second-order Hamiltonian system (1.1). Some related contents can be seen in [19]. It is worth noting that in [18] and [19], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M138">View MathML</a> are consecutive eigenvalues of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M139">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M144">View MathML</a>. In our Theorem 1.1 and Theorem 1.2, we study the existence of subharmonic solutions for system (1.1) from a different perspective. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M145">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M146">View MathML</a>) in our theorems are the eigenvalues of the matrix A. Obviously, it is much easier to seek the eigenvalue of a matrix. In Section 4, we present an interesting example satisfying our Theorem 1.1 but not satisfying the theorem in [19].

Theorem 1.2Suppose that (A0) holds andFsatisfies (A)′, (1.5), (H2) and the following conditions:

(H3)′ when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M113">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M17">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M151">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M152">View MathML</a>

(1.16)

(H4)′

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M153">View MathML</a>

Then system (1.1) has a sequence of distinct periodic solutions with period<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M154">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37">View MathML</a>.

In the final theorem, we present a result about the existence of subharmonic solutions for system (1.8). Using a condition like (H2) and similar to the argument of Remark 1.1 in [17], we can improve Theorem B.

Theorem 1.3Suppose thatFsatisfies (A), (1.5) and the following conditions:

(H5) there exist positive constantsm, ζ, ηand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M158">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M159">View MathML</a>

(H6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M160">View MathML</a>

Then system (1.8) has a sequence of distinct nonconstant periodic solutions with period<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M34">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37">View MathML</a>.

2 Some preliminaries

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M165">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166">View MathML</a> is a Hilbert space with the inner product and the norm defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M167">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M168">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M169">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M170">View MathML</a>

Then one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M171">View MathML</a>

(see Proposition 1.3 in [1]).

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M172">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M173">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M174">View MathML</a>.

Proof Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M175">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M176">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M177">View MathML</a>

(2.1)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M178">View MathML</a>

Integrating (2.1) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M179">View MathML</a> and using the Hölder inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M180">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M181">View MathML</a>

The proof is complete. □

Lemma 2.2 (see [[17], Lemma 2.2])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M182">View MathML</a>isT-periodic in t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M4">View MathML</a>is measurable intfor every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M5">View MathML</a>and continuously differentiable inxfor a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M6">View MathML</a>. If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M186">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M187">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M10">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M189">View MathML</a>

(2.2)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M190">View MathML</a>

is weakly continuous and uniformly differentiable on bounded subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166">View MathML</a>.

Remark 2.1 In [[17], Lemma 2.2], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M192">View MathML</a>. In fact, in its proof, it is not essential that F is continuously differentiable in t.

We use Lemma 2.3 below due to Benci and Rabinowitz [20] to prove our results.

Lemma 2.3 (see [20] or [[5], Theorem 5.29])

LetEbe a real Hilbert space with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M193">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M194">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M195">View MathML</a>satisfies (PS)-condition, and

(I1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M196">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M197">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M198">View MathML</a>bounded and self-adjoint, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M199">View MathML</a>;

(I2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M200">View MathML</a>is compact, and

(I3) there exists a subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M201">View MathML</a>and sets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M203">View MathML</a>and constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M204">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M205">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M206">View MathML</a>,

(ii) Qis bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M207">View MathML</a>,

(iii) Sand∂Qlink.

Thenφpossesses a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M208">View MathML</a>which can be characterized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M209">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M210">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M211">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M212">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M213">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M214">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M215">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M216">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M217">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M218">View MathML</a>andKis compact.

Remark 2.2 As shown in [21], a deformation lemma can be proved with replacing the usual (PS)-condition with condition (C), and it turns out that Lemma 2.3 holds true under condition (C). We say φ satisfies condition (C), i.e., for every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M220">View MathML</a> has a convergent subsequence if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M221">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M222">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M223">View MathML</a>.

3 Proofs of theorems

Proof of Theorem 1.1 It follows from Assumption (A)′ that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M226">View MathML</a>

is continuously differentiable. Moreover, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M227">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M169">View MathML</a> and the solutions of system (1.1) correspond to the critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224">View MathML</a> (see [1]).

Obviously, there exists an orthogonal matrix Q such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M230">View MathML</a>

(3.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M231">View MathML</a>. Then by (1.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M232">View MathML</a>

Furthermore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M233">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M234">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M235">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M236">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M237">View MathML</a>

Then the critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M238">View MathML</a> correspond to solutions of system (3.2). It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M239">View MathML</a> and G satisfies all the conditions of Theorem 1.1 and Theorem 1.2 if F satisfies them. Hence, w is the critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M240">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M231">View MathML</a> is the critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224">View MathML</a>. Therefore, we only need to consider the special case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M243">View MathML</a> is the diagonal matrix defined by (3.1). We divide the proof into six steps.

Step 1: Decompose the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M245">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M246">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M247">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M249">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M250">View MathML</a> are closed subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166">View MathML</a> and

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M252">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M253">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M254">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M255">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M256">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M257">View MathML</a>

Remark 3.1 When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258">View MathML</a>, it is easy to see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M259">View MathML</a>.

Step 2: Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M260">View MathML</a>

Next we consider the relationship between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M262">View MathML</a> on those subspaces defined above. We only consider the case that (H3)(2) holds. For others, the conclusions are easy to be seen from the argument of this case.

(a) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M263">View MathML</a>, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M264">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M265">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M266">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M267">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M268">View MathML</a>

(3.3)

Remark 3.2 Obviously, if one of (H3)(5) and (H3)(6) holds, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M269">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M270">View MathML</a>

(b) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M271">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M272">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M273">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M274">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M275">View MathML</a>

Since for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M277">View MathML</a>

are strictly increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M278">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M279">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M280">View MathML</a>

Moreover, it is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M281">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M282">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M283">View MathML</a>

(3.4)

Remark 3.3 From the above discussion, it is easy to see the following conclusions:

(i) if (H3)(1) holds, then (3.4) holds with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M284">View MathML</a>

(ii) if (H3)(2) holds, then (3.4) holds with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M285">View MathML</a>

(iii) if (H3)(3) holds, then (3.4) holds with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M286">View MathML</a>

(iv) if (H3)(4) holds, then (3.4) holds with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M287">View MathML</a>

(v) if (H3)(5) holds, then (3.4) holds with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M288">View MathML</a>

(vi) if (H3)(6) holds, then (3.4) holds with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M289">View MathML</a>

(c) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M290">View MathML</a>, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M291">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M292">View MathML</a>

Obviously, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M294">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M295">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M296">View MathML</a>, it follows from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M297">View MathML</a>

that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M298">View MathML</a>

(3.5)

(d) Obviously, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M299">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M300">View MathML</a>

(3.6)

Step 3: Assume that (H3)(2) holds. We prove that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M301">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M302">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M303">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M304">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M305">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M306">View MathML</a>, by Lemma 2.1, (H4) and (3.4), we have, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M307">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M308">View MathML</a>

For cases (H3)(1) and (H3)(3)-(H3)(6), correspondingly, by (H4) and Remark 3.3, similar to the above argument, we can also obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M309">View MathML</a>

Step 4: Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M310">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M311">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M312">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M313">View MathML</a> will be determined later. In this step, we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M314">View MathML</a>. We only consider the case that F satisfies (H3)(2). For other cases, the results can be seen easily from the argument of case (H3)(2).

Assume that F satisfies (H3)(2). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M315">View MathML</a>

Case (i): if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M316">View MathML</a>

then we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M317">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M320">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M321">View MathML</a>

By (H3)(2), (1.5) and the periodicity of F, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M322">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M323">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M324">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M325">View MathML</a> is the finite dimensional space, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M326">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M327">View MathML</a>

(3.8)

By (3.3), (3.5), (3.6), (3.7) and (3.8), we know that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M329">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M330">View MathML</a>

(3.9)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M331">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M332">View MathML</a>

Case (ii): if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M333">View MathML</a>, then we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M334">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M335">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M336">View MathML</a>

(3.10)

By (H3)(2), (1.5) and the periodicity of F, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M337">View MathML</a>

(3.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M324">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M339">View MathML</a>. By (3.3), (3.5), (3.6), (3.8) and (3.11), we know that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M329">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M342">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M343">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M344">View MathML</a>

Case (iii): if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M345">View MathML</a>, then we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M346">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M347">View MathML</a>

By (H3)(2), (1.5) and the periodicity of F, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M348">View MathML</a>

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M349">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M350">View MathML</a>. By (3.3), (3.5), (3.6), (3.8) and (3.12), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M329">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M353">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M354">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M355">View MathML</a>

Combining cases (i), (ii) and (iii), if we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M356">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M357">View MathML</a>

(3.13)

By (1.5), (3.3), (3.5) and (3.6), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M358">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M359">View MathML</a>

(3.14)

Thus, it follows from (3.13) and (3.14) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M360">View MathML</a>.

Step 5: We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224">View MathML</a> satisfies (C)-condition in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166">View MathML</a>. The proof is similar to that in Theorem 1.1 in [17]. We omit it.

Step 6: We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224">View MathML</a> has a nontrivial critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M364">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M365">View MathML</a>. Especially, we claim that, for cases (H3)(1) and (H3)(4), since A is a positive semidefinite matrix, (1.5) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366">View MathML</a> is nonconstant.

In fact, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M367">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M368">View MathML</a> is the linear self-adjoint operator defined, using the Riesz representation theorem, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M369">View MathML</a>

The compact imbedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M371">View MathML</a> implies that K is compact. In order to use Lemma 2.3, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M372">View MathML</a> and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M199">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M375">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M376">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M377">View MathML</a>. Since K is a self-adjoint compact operator, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M378">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M199">View MathML</a>) are bounded and self-adjoint. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M380">View MathML</a>

Assumption (A)′ and Lemma 2.2 imply that b is weakly continuous and is uniformly differentiable on bounded subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M381">View MathML</a>. Furthermore, by standard theorems in [22], we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M200">View MathML</a> is compact. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M383">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M384">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M385">View MathML</a> link. Hence, by Step 1-Step 5, Lemma 2.3 and Remark 2.2, there exists a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M364">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M365">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366">View MathML</a> is nonzero. For cases (H3)(1) and (H3)(4), since A is a positive semidefinite matrix, it follows from (1.5) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366">View MathML</a> is nonconstant. The proof is complete.  □

Proof of Theorem 1.2 Obviously, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M113">View MathML</a>, (H1) holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84">View MathML</a>. Moreover, since (H3)′ implies that (H3)(5) and (H4)′ implies that (H4), system (1.1) has kT-periodic solution for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M395">View MathML</a>. Like the argument of case (ii) in the proof of Theorem 1.1, choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M396">View MathML</a>

By (H3)′, (1.5) and the T-periodicity of F, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M397">View MathML</a>

(3.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M398">View MathML</a>. In the proof of Theorem 1.1, if we replace (3.15) with (3.11), then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M399">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M400">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M312">View MathML</a> is independent of k. Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366">View MathML</a> is the critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M224">View MathML</a>, then it follows from (3.3), (3.5), (3.6), the definitions of critical value c in Lemma 2.3 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M404">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M405">View MathML</a>

(3.16)

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M406">View MathML</a> is bounded for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M84">View MathML</a>.

Obviously, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M408">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M409">View MathML</a>, then we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M366">View MathML</a> is distinct from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M411">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M412">View MathML</a>. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M413">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M412">View MathML</a>, it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M415">View MathML</a>

Then by (3.16), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M416">View MathML</a>, a contradiction. We also can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M417">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M418">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M419">View MathML</a>. Otherwise, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M420">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M412">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M422">View MathML</a>. Then by (3.16), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M423">View MathML</a>, a contradiction. In the same way, we can obtain that system (1.1) has a sequence of distinct periodic solutions with period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M154">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M425">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37">View MathML</a>. The proof is complete. □

Proof of Theorem 1.3 Except for verifying (C) condition, the proof is the same as in Theorem B (that is Theorem 1 in [15]). To verify (C) condition, we only need to prove the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M428">View MathML</a> is bounded if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M221">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M430">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M431">View MathML</a>. Other proofs are the same as in [15]. The proof of boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M428">View MathML</a> is essentially the same as in Theorem 1.1 in [17] except that 2 is replaced by p, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M166">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M434">View MathML</a>

equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M435">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M436">View MathML</a>

by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M437">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M438">View MathML</a>. So, we omit the details. □

4 Examples

Example 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M135">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M440">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M441">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M443">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M444">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M445">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M446">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M447">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M448">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M449">View MathML</a>. Obviously, the matrix A satisfies Assumption (A0) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M450">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M451">View MathML</a>

It is easy to verify that (H1) holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M452">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M453">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M454">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M5">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M456">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M457">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M458">View MathML</a>

(4.2)

It is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M459">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M460">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M461">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M462">View MathML</a>. Moreover, obviously, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M463">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M464">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M465">View MathML</a>

Hence, (H2) holds.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M467">View MathML</a>

By (4.2), we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M468">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M469">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M470">View MathML</a>. Then (H3)(2) holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258">View MathML</a>. Moreover, by (4.1), we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M472">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M473">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M474">View MathML</a>. Then (H4) holds. By Theorem 1.1, we obtain that system (1.1) has a T-periodic solution.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M475">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M476">View MathML</a>

By (4.2), we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M477">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M478">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M479">View MathML</a>. Then (H3)(2) holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M475">View MathML</a>. Moreover, by (4.1), we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M481">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M482">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M483">View MathML</a>. Then (H4) holds. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M484">View MathML</a>. So, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M475">View MathML</a>, by Theorem 1.1, we cannot judge that system (1.1) has a T-periodic solution. However, we can obtain that system (1.1) has a 2T-periodic solution.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M486">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M487">View MathML</a>

By (4.2), we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M488">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M489">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M490">View MathML</a>. Then (H3)(2) holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M486">View MathML</a>. Moreover, by (4.1), we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M492">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M493">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M494">View MathML</a>. Then (H4) holds. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M495">View MathML</a>. So, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M486">View MathML</a>, by Theorem 1.1, we cannot judge that system (1.1) has T-periodic solution and 2T-periodic solution. However, we can obtain that system (1.1) has a 3T-periodic solution. It is easy to verify that Example 4.1 does not satisfy the theorem in [19] even if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M258">View MathML</a>.

Example 4.2 Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M498">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M499">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M500">View MathML</a>

Obviously, (A0), (A)′, (1.5), (H3)′ and (H4)′ hold. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M460">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M461">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M503">View MathML</a>. Similar to the argument in Example 4.1, we obtain (H2) also holds. Then by Theorem 1.2, system (1.1) has a sequence of distinct periodic solutions with period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M154">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37">View MathML</a>.

Example 4.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M508">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M509">View MathML</a>

Then (1.5) holds and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M510">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M460">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M461">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M513">View MathML</a>. Then it is easy to obtain that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M514">View MathML</a> such that (H5) holds. By Theorem 1.3, system (1.8) has a sequence of distinct periodic solutions with period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M154">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M36">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/139/mathml/M37">View MathML</a>. It is easy to see that Example 4.3 does not satisfy (1.3). Hence, Theorem 1.3 improved Theorem B.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

XZ proposed the idea of the paper and finished the main proofs. XT provided some important techniques in the process of proofs.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions.

References

  1. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems, Springer, New York (1989)

  2. Tang, CL: Periodic solutions of nonautonomous second order systems with sublinear nonlinearity. Proc. Amer. Math. Soc.. 126, 3263–3270 (1998). Publisher Full Text OpenURL

  3. Ailva, EAB: Subharmonic solutions for subquadratic Hamiltonian systems. J. Differ. Equ.. 115, 120–145 (1995). Publisher Full Text OpenURL

  4. Schechter, M: Periodic non-autonomous second-order dynamical systems. J. Differ. Equ.. 223, 290–302 (2006). Publisher Full Text OpenURL

  5. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)

  6. Rabinowitz, PH: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math.. 31, 157–184 (1978). Publisher Full Text OpenURL

  7. Tao, ZL, Yan, S, Wu, SL: Periodic solutions for a class of superquadratic Hamiltonian systems. J. Math. Anal. Appl.. 331, 152–158 (2007). Publisher Full Text OpenURL

  8. Chang, KC: Infinite Dimensional Morse Theory and Multiple Solution Problems (1993)

  9. Ekeland, I: Convexity Method in Hamiltonian Mechanics, Springer, Berlin (1990)

  10. Ekeland, I, Hofer, H: Periodic solutions with prescribed period for convex autonomous Hamiltonian systems. Invent. Math.. 81, 155–188 (1985). Publisher Full Text OpenURL

  11. Fei, G, Qiu, Q: Minimal periodic solutions of nonlinear Hamiltonian systems. Nonlinear Anal.. 27, 821–839 (1996). Publisher Full Text OpenURL

  12. Fei, G, Kim, S, Wang, T: Minimal period estimates of periodic solutions for superquadratic Hamiltonian systems. J. Math. Anal. Appl.. 238, 216–233 (1999). PubMed Abstract | Publisher Full Text OpenURL

  13. Fei, G: On periodic solutions of superquadratic Hamiltonian systems. Electron. J. Differ. Equ.. 2002, 1–12 (2002)

  14. Tao, ZL, Tang, CL: Periodic and subharmonic solutions of second order Hamiltonian systems. J. Math. Anal. Appl.. 293, 435–445 (2004). Publisher Full Text OpenURL

  15. Ma, S, Zhang, Y: Existence of infinitely many periodic solutions for ordinary p-Laplacian systems. J. Math. Anal. Appl.. 351, 469–479 (2009). Publisher Full Text OpenURL

  16. Ye, YW, Tang, CL: Periodic and subharmonic solutions for a class of superquadratic second order Hamiltonian systems. Nonlinear Anal.. 71, 2298–2307 (2009). Publisher Full Text OpenURL

  17. Zhang, X, Tang, X: Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems. Nonlinear Anal., Real World Appl.. 13, 113–130 (2012). Publisher Full Text OpenURL

  18. Costa, DG, Magalhães, CA: A unified approach to a class of strongly indefinite functions. J. Differ. Equ.. 125, 521–547 (1996). Publisher Full Text OpenURL

  19. Kyristi, ST, Papageorgiou, NS: On superquadratic periodic systems with indefinite linear part. Nonlinear Anal.. 72, 946–954 (2010). Publisher Full Text OpenURL

  20. Benci, V, Rabinowitz, PH: Critical point theorems for indefinite functions. Invent. Math.. 52, 241–273 (1979). Publisher Full Text OpenURL

  21. Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal.. 7, 241–273 (1983)

  22. Krosnoselski, MA: Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan Co., New York (1964)