Open Access Research

Oscillating global continua of positive solutions of second order Neumann problem with a set-valued term

Dongming Yan

Author Affiliations

Department of Mathematics, Sichuan University, Chengdu 610064, China

Boundary Value Problems 2012, 2012:47  doi:10.1186/1687-2770-2012-47


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/47


Received:14 October 2011
Accepted:23 April 2012
Published:23 April 2012

© 2012 Yan; 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 note, we study the oscillating global continua of the differential inclusion of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M1">View MathML</a>

where F is a "set-valued representation" of a function with jump discontinuities along the line segment [0, 1] × {0}, and λ ∈ [0, ∞) is a parameter. The proof of our main result relies on an approximation procedure.

Mathematics Subject Classification 2000: 34B16; 34B18.

Keywords:
climate model; differential inclusion; eigenvalue; positive solutions

1 Introduction

In recent years, nonsmooth analysis has come to play an important role in functional analysis [1], dynamical systems [2], control theory [3], optimization [4], mechanical systems [5], differential equation [6,7] etc. Since many mathematical and physical problems may be reduced to ODES or PDES with discontinuous nonlinearities, the existence of multiple solutions for differential inclusion problems has been widely investigated [8-19].

In this article, we are concerned with the following differential inclusion problem which raises from a Budyko-North type energy balance climate models:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M2">View MathML</a>

(1.1)

(see [20-25] and the references therein). In particular, the set-valued right hand side arise from a jump discontinuity of the albedo at the ice-edge in these models. By filling such a gap, one arrives at the set-valued problem (1.1). As in [25], we are here interested in a considerably simplify version as compared to the situation from climate modeling, e.g. a one-dimensional regular Sturm-Liouville differential operator substitutes for a two-dimensional Laplace-Beltrami operator or a singular Legendre-type operator, and the jump discontinuity is transformed to u = 0 in a way, which resembles only locally the climatological problem.

We are concerned with the set-valued problem (1.1) under the following assumptions

(H1) qC([0, 1],(0,+∞));

(H2) f+C ([0, 1] × [0,+∞), (0,+∞)), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M3">View MathML</a>.

Let the set-valued function F in (1.1) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M4">View MathML</a>

Notice that if f+(x, 0) ≡ 0, x ∈ [0, 1], then the differential inclusion problem (1.1) reduces to the BVP of differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M5">View MathML</a>

(1.2)

In the last 20 years, the positive solutions of (1.2) have been studied by several authors, see Jiang and Liu [26], Chu et al. [27] and Sun et al. [28].

The purpose of this article is to investigate the oscillating global continua of positive solutions of the differential inclusion problem (1.1). The proof of our main result relies on an approximation procedure. The rest of the article is organized as follows. In Section 2, we state some notations and prove some preliminary results. In Section 3, we state and prove our main result. In Section 4, an example is given to illustrate the application of our main result.

2 Notations and preliminaries

Recall Kuratowski's notion of lower and upper limits of sequence of sets.

Definition 2.1. [29]Let X be a metric space and {Zl}l∈ℕbe a sequence of subsets of X. The set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M6">View MathML</a>

is called the upper limit of the sequence {Zl}, whereas

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M7">View MathML</a>

is called the lower limit of the sequence {Zl}.

Definition 2.2. [29]A component of a set M is meant a maximal connected subset of M.

Lemma 2.1. [29]Suppose that Y is a compact metric space, A and B are non-intersecting closed subsets of Y, and no component of Y intersects both A and B. Then there exist two disjoint compact subsets YAand YB, such that Y = YAYB, A YA, B YB.

Using the above Whyburn Lemma, Ma and An [30] proved the following

Lemma 2.2. [30, Lemma 2.1] Let Z be a Banach space and let {An} be a family of closed connected subsets of Z. Assume that

(i) there exist znAn, n = 1, 2, ..., and z* Z, such that znz*;

(ii) rn = sup {∥x∥ | x An} = ∞;

(iii) for every R > 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M8">View MathML</a>is a relatively compact set of Z, where BR= {x Z | ∥x∥ ≤ R}. Then there exists an unbounded component in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M10">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M11">View MathML</a>.

Remark 2.1. The limiting processes for sets go back at least to the work of Kuratowski [31]. Lemma 2.2 is a slight generalization of the following well-know result due to Whyburn [29]:

Proposition 2.1. (Whyburn [29, p. 12]) Let Z be a Banach space and {An} be a family of closed connected subsets of Z. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M12">View MathML</a> and ∪l∈ℕAl is relatively compact. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M10">View MathML</a> is nonempty, compact and connected.

Next, we introduce the result of global solution behavior of the bifurcation branches of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M13">View MathML</a>

(2.1)

to wit the following lemma.

Lemma 2.3. [32] (Dancer (1974)) Assume that

(C1) The operators L, N: X X are compact on the Banach space X over R. Furthermore, L is linear and Nx∥/∥x∥ → 0 as x∥ → 0;

(C2) The real number μ0is a characteristic number of L of odd algebraic multiplicity;

(C1+) The real Banach space X has an order cone K with X = K-K, i.e., every x X can be represented as x = x1 - x2, where x1, x2K. Furthermore, L + N is positive, i.e., L + N maps K into K;

(C2+) The spectral radius r(L) of L is positive. We set μ0 = (r(L))-1.

Then (μ0, 0) is a bifurcation point of equation (2.1) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M14">View MathML</a>

contains an unbounded solution component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15">View MathML</a>which passes through (μ0, 0).

If additionally

(C3+) The linear operator L is strongly positive, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M16">View MathML</a>and μ μ0always implies x > 0 and μ > 0.

Remark 2.2. This result is often called the nonlinear Krein-Rutman theorem. It will play an important role in the proof of our main result.

Let φ and ψ be the unique solution of the problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M17">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M18">View MathML</a>

respectively. Then it is easy to check φ(·) is nondecreasing on (0,1), ψ(·) is nonincreasing on (0,1), and the Green's function G(x, s) of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M19">View MathML</a>

is explicitly given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M20">View MathML</a>

(2.2)

Moreover, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M21">View MathML</a>

(2.3)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M22">View MathML</a>.

3 The main result

Let Σ be the closure of the set of positive solutions of (1.1) in [0, ∞) × C1[0, 1], and ℕ* := {1, 2,..., N}. The main result of this article is the following theorem.

Theorem 3.1. Assume that (H1)-(H2) hold. If

(H3) there is an increasing sequence of positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M23">View MathML</a>and a small enough constant δ such that ξ1 < σ(ξ2- δ) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M24">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M25">View MathML</a>

then there exits an unbounded component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15">View MathML</a>in Σ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M26">View MathML</a>. Moreover,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a>with u = ξ2j-1for some j ∈ ℕ* implies that λ ≥ 2;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a>with u = ξ2jfor some j ∈ ℕ* implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M28">View MathML</a>.

Actually, such continua <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15">View MathML</a> can be obtained as upper limits in the sense of Kura-towski of sequence of solution continua from associated continuous problems. To this end one sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M29">View MathML</a>

(3.1)

fixes l0 ∈ ℕ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M30">View MathML</a>, and selects an approximation sequence {fl} ⊂ C ([0, 1] × ℝ, ℝ) (l > l0) of F satisfying:

(A1) fl (x, y) = ly for x ∈ [0, 1] and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M31">View MathML</a>;

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M32">View MathML</a> for x ∈ [0, 1] and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M33">View MathML</a>;

(A3) fl(x,y) = f+(x, y) for x ∈ [0, 1] and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M34">View MathML</a>;

(A4) {fl (x, y)}l∈ℕ is nondecreasing in l for (x, y) ∈ [0, 1] × (0,∞).

Next, we show that the continuous problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M35">View MathML</a>

(3.2l)

has an unbounded closed subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M36">View MathML</a> of the positive solutions set of (3.2l) with

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M37">View MathML</a> is the bifurcation point contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M36">View MathML</a>;

(b) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M38">View MathML</a> and ϑ ≢ 0, then ϑ is positive on (0,1).

It is easy to see that (3.2l) equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M39">View MathML</a>

(3.3)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M40">View MathML</a>

Then according to (3.3), (3.2l) can be written as the following operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M41">View MathML</a>

Clearly, the operators L, N : C[0, 1] → C[0, 1] are compact on the Banach space

C[0, 1]. Furthermore, L is linear and thanks to (2.3)(A1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M42">View MathML</a>

which implies that the condition (C1) of Lemma 2.3 is satisfied.

Denote the principal eigenvalue of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M43">View MathML</a>

(3.4)

by λ1, then we know that λ1> 0 (see [33]). Since (3.4) is equivalent to operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M44">View MathML</a>

we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M45">View MathML</a>. Therefore, the conditions (C2)(C2+) of Lemma 2.3 are satisfied.

Let the cone K in C[0, 1] is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M46">View MathML</a>

It is easy to see thanks to (A1)-(A4) and (2.3) that the (C1+)(C3+) conditions of Lemma 2.3 are satisfied.

According to Lemma 2.3, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M37">View MathML</a> is a bifurcation point of the positive solutions set of (3.2l) for every l ∈ {l0 + 1, l0 + 2, ...} =: ℕ0, and for each l ∈ ℕ0 there exits an unbounded closed subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M36">View MathML</a> of the positive solutions set of (3.2l) with (a) and (b).

Combining the above with the fact

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M47">View MathML</a>

and utilizing Lemma 2.2, it concludes that there exits an unbounded component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M48">View MathML</a>

(3.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M49">View MathML</a>

(3.6)

Denote the cone P in C[0, 1] by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M50">View MathML</a>

Define an operator Tλ : P C[0, 1] by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M51">View MathML</a>

It is easy to get the following lemma.

Lemma 3.1. Assume that (H1), (H2) and (A1)-(A4) hold. Then Tλ: P P is completely continuous.

Lemma 3.2. Assume that (H1), (H2) and (A1)-(A4) hold. If 0 ≤ u(x) ≤ r, r > 0, for x ∈ [0, 1], then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M52">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M53">View MathML</a>.

Proof. Since fl(x, u(x)) ≤ Mr for x ∈ [0, 1], it follows from (2.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M54">View MathML</a>

Lemma 3.3. Assume that (H1), (H2) and (A1)-(A4) hold. If σ(r - δ) ≤ u(x) ≤ r + δ, r > δ, for x ∈ [0, 1], then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M55">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M56">View MathML</a>.

Proof. Since fl(x, u(x)) ≥ mr for x ∈ [0, 1], it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M57">View MathML</a>

Lemma 3.4. Assume that (H1), (H2), (H3) and (A1)-(A4) hold. then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M58">View MathML</a>with u ∈ (ξ2j-1 - δ2j-1 + δ) for some j ∈ ℕ* implies that λ > 2;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M58">View MathML</a>, with u ∈ ( ξ2j- δ,ξ2j + δ) for some j ∈ ℕ* implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M59">View MathML</a>.

Proof. (i) Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M58">View MathML</a> with ∥u ∈ (ξ2j-1 - δ, ξ2j-1 + δ) for some j ∈ ℕ*, then u = Tλu and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M60">View MathML</a>

By Lemma 3.2 and (H3), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M61">View MathML</a>

Thus λ > 2.

(ii) Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M58">View MathML</a> with ∥u ∈ (ξ2j - δ, ξ2j + δ) for some j ∈ ℕ*, then u = Tλu and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M62">View MathML</a>

By Lemma 3.3 and the assumption (H3), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M63">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M59">View MathML</a>.

Lemma 3.5. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a>, then (λ, u) is a solution of (1.1) and u W2,∞(0, 1).

Proof. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M15">View MathML</a> there exists a sequence {lk} ∈ ℕ0 strictly increasing, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M64">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M65">View MathML</a> for k ∈ ℕ and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M66">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M67">View MathML</a> is uniformly bounded, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M68">View MathML</a>

(3.7)

we can assume after passing to a subsequence, if necessary, that it converges weekly in L2(0, 1) to some ϕ. We claim that ϕ(x) ∈ F(x, u(x)) a.e. on (0, 1).

Let x0 ∈ (0, 1) with u(x0) > 0. Then there exist ρ > 0 and τ ∈ (0, min{x0, 1-x0}) with u(x) > ρ for all x ∈ (x0 - τ, x0 + τ), hence there is a k0 ∈ ℕ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M69">View MathML</a> for all k > k0 and x ∈ (x0 - τ, x0 + τ). Choose k1 > k0 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M70">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M71">View MathML</a> for all k k1 and x ∈ (x0 - τ, x0 + τ), which yields ϕ(x) = f+(x, u(x)) for x ∈ (x0 - τ, x0 + τ) a.e.

Next, if u ≡ 0, let K: = {x ∈ (0, 1) : ϕ(x) > f+(x, 0)}. We claim that meas(K) = 0. Suppose that meas(K) > 0. Then ε := ∫K [ϕ(x) - f+(x, 0)] dx > 0, and one finds η ∈ (0, ∞) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M72">View MathML</a> for x ∈ [0, 1] and y ∈ [0, η]. Choosing k2 ∈ ℕ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M73">View MathML</a> for k k2. One obtains for k k2:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M74">View MathML</a>

which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M75">View MathML</a>. Thus, meas(K) = 0.

Now, let A be the closed linear operator in L2(0, 1) defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M76">View MathML</a>

and := -φ" + . Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M77">View MathML</a>

(3.8)

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M78">View MathML</a> and the fact that A is weakly closed yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M79">View MathML</a>

(3.9)

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M80">View MathML</a>

Finally, we show that u W2,∞(0,1). In fact, from (3.9) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M81">View MathML</a>

(3.10)

According to (H1) and the boundedness of u we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M82">View MathML</a>

(3.11)

We claim that ϕ L(0,1). Suppose on the contrary that there exists a set E ⊂ [0, 1], meas(E) > 0 such that |ϕ| is unbounded on E. Without loss of generality, we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M83">View MathML</a>

(3.12)

where M is given by (3.7) and w L2(0,1). On the one hand, for k larger enough from (3.7), (3.8) and (H2) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M84">View MathML</a>

(3.13)

On the other hand, from (3.12) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M85">View MathML</a>

which contradicts (3.13). Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M86">View MathML</a>

(3.14)

Therefore, from (3.10), (3.11) and (3.14) we obtain u W2,∞(0,1).

Now we are in the position to prove Theorem 3.1.

Proof of Theorem 3.1.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a>. We divide the proof into two cases.

Case l. If ∥u = ξ2j-1 for some j ∈ ℕ*, then λ ≥ 2.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a>, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M87">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M88">View MathML</a>

Hence, for δ > 0 there exists i0 ∈ ℕ, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M89">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M90">View MathML</a>

By using Lemma 3.4, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M91">View MathML</a>

Hence, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M92">View MathML</a>

Case 2. If ∥u = ξ2j for some j ∈ ℕ*, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M28">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a>, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M87">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M93">View MathML</a>

Hence, for δ > 0 there exists i0 ∈ ℕ, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M89">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M94">View MathML</a>

By using lemma 3.4, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M95">View MathML</a>

Hence, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M96">View MathML</a>

Corollary 3.1. Assume that (H1)-(H3) hold. Then

(i) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M97">View MathML</a>, (1.1) has at least one positive solution: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M98">View MathML</a>;

(ii) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M99">View MathML</a>, (1.1) has N positive solutions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M100">View MathML</a>

which satisfy that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M101">View MathML</a>.

Proof. According to Theorem 3.1, the boundary value problem (1.1) has an unbounded component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M102">View MathML</a> in Σ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M48">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M103">View MathML</a> with ∥u = ξ2j-1 for some j ∈ ℕ* implies that λ ≥ 2;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M104">View MathML</a> with ∥u = ξ2j for some j ∈ ℕ* implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M28">View MathML</a>.

From the facts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M48">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a> with ∥u= ξ1 implies that λ ≥ 2 and the connectivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M102">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M105">View MathML</a>

which implies for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M97">View MathML</a>, (1.1) has at least one positive solution: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M98">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M106">View MathML</a>

where ξ0 = 0, ξk (k = 1, 2,..., N) is given by (H3). Then according to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M107">View MathML</a> and the connectivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M102">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M108">View MathML</a>

which implies for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M109">View MathML</a>, (1.1) has N positive solutions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M100">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M110">View MathML</a>.

4 Example

In this section, an example is given to illustrate the application of our main result (Theorem 3.1). Consider second order Neumann differential inclusion problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M111">View MathML</a>

(4.1)

where the set-valued function F in (4.1) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M112">View MathML</a>

Obviously, (H1), (H2) conditions of Theorem 3.1 are satisfied. Moreover, Green's function of the associated linear problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M113">View MathML</a>

can be explicitly expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M114">View MathML</a>

By calculation we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M116">View MathML</a>.

Let ξ1 = 3, ξ2 = 11, δ = 1, then we can check that ξ1 = 3 < 5 < σ(ξ2- δ), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M117">View MathML</a>

So that (H3) condition of Theorem 3.1 is satisfied. Therefore, according to Theorem 3.1 the differential inclusion problem (4.1) has an unbounded component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M102">View MathML</a> in Σ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M48">View MathML</a>. Moreover,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a> with ∥u = 3 implies that λ ≥ 2;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M27">View MathML</a> with ∥u = 11 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/47/mathml/M28">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

The authors express their gratitude to Professors Ma Tian and Ma Ruyun for their guidance and encouragement, also to an anonymous referee for a number of valuable comments and suggestions.

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