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Periodic solutions for N + 2 -body problems with N + 1 fixed centers

Furong Zhao12*, Fengying Li2 and Jian Chen23

Author Affiliations

1 Department of Mathematics and Computer Science, Mianyang Normal University, Mianyang, Sichuan, 621000, P.R. China

2 Department of Mathematics, Sichuan University, Chengdu, 610064, P.R. China

3 School of Science, Southwest University of Science and Technology, Mianyang, Sichuan, 621000, P.R. China

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Boundary Value Problems 2013, 2013:129  doi:10.1186/1687-2770-2013-129

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/129


Received:26 January 2013
Accepted:2 May 2013
Published:17 May 2013

© 2013 Zhao et al.; 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, we prove the existence of a new periodic solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M1">View MathML</a>-body problems with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M2">View MathML</a> fixed centers and strong-force potentials. In this model, N particles with equal masses are fixed at the vertices of a regular N-gon and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M5">View MathML</a>th particle is fixed at the center of the N-gon, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M6">View MathML</a>th particle winding around N particles.

MSC: 34C15, 34C25, 70F10.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M1">View MathML</a>-body problems with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M2">View MathML</a>-fixed centers; minimizing variational methods; strong-force potentials

1 Introduction and main results

In the eighteenth century, the 2-fixed center problem was studied by Euler [1-3]. Here, let us consider the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M2">View MathML</a>-fixed center problem: We assume N particles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M10">View MathML</a> with equal masses 1 are fixed at the vertices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M11">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M12">View MathML</a>) of a regular polygon and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M5">View MathML</a>th particle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M14">View MathML</a> is fixed at the origin <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M15">View MathML</a>, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M6">View MathML</a>th particle with mass <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M17">View MathML</a> is attracted by the other particles, and moves according to Newton’s second law and a more general power law than the Newton’s universal gravitational square law. In this system, the position <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M18">View MathML</a> for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M6">View MathML</a>th particle satisfies the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M20">View MathML</a>

(1.1)

Equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M21">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M22">View MathML</a>

(1.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M23">View MathML</a>

The type of system (1.2) is called a singular Hamiltonian system which attracts many researchers (see [1-10] and [11-16]).

Specially, Gordon [10] proved the Keplerian elliptical orbits are the minimizers of Lagrangian action defined on the space for non-zero winding numbers.

In this paper, we use a variational minimizing method to look for a periodic solution for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M6">View MathML</a>th particle which winds around the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M25">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M26">View MathML</a>).

Definition 1.1[10]

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M27">View MathML</a> be a given oriented closed curve, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M28">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M29">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M30">View MathML</a>

When some point on C goes around the curve once, its image point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M31">View MathML</a> will go around <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M32">View MathML</a> a number of times. This number is defined as the winding number of the curve C relative to the point p and is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M33">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M34">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M35">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M36">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M37">View MathML</a>

(1.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M38">View MathML</a>

(1.8)

We have the following theorem.

Theorem 1.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M39">View MathML</a>, the minimizer of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M40">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M41">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M42">View MathML</a>) exists and it is a non-collision periodic solution of (1.1) or (1.2)-(1.3) (please see Figures1-4for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M43">View MathML</a>).

2 The proof of Theorem 1.1

We recall the following famous lemmas, which we need to prove Theorem 1.1.

Lemma 2.1[9]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M50">View MathML</a>, and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M51">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M52">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M53">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M54">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M55">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M56">View MathML</a>, s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M39">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M59">View MathML</a>.

Lemma 2.2 (Palais’s symmetry principle [17])

Letσbe an orthogonal representation of a finite or compact groupGon a real Hilbert spaceH, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M60">View MathML</a>be such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M62">View MathML</a>. Set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M63">View MathML</a>. Then the critical point offin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M64">View MathML</a>is also a critical point offinH.

Lemma 2.3[5]

IfXis a reflexive Banach space, Mis a weakly closed subset ofX, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M66">View MathML</a>is weakly lower semi-continuous and coercive, thenfattains its infimum onM.

Lemma 2.4 (Poincare-Wirtinger inequality)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M67">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M68">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M69">View MathML</a>. And the inequality takes the equality if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M71">View MathML</a>.

We now prove Theorem 1.1.

Proof By the symmetry of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M72">View MathML</a>, we know for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M73">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M74">View MathML</a>

(2.1)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M75">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M41">View MathML</a>, then by Sobolev’s compact embedding theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M77">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M78">View MathML</a>.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M79">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M80">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M82">View MathML</a> can be regarded as the square of an equivalent norm for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M83">View MathML</a>, so it is weakly lower semi-continuous, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M84">View MathML</a>.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M85">View MathML</a>, then by Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M86">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M87">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M88">View MathML</a>. Hence f is w.l.s.c.

Using (2.1), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M40">View MathML</a> is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M41">View MathML</a>. Lemma 2.3 guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M40">View MathML</a> attains its infimum on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M41">View MathML</a>. Let the minimizer be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M93">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M94">View MathML</a>

(2.2)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M93">View MathML</a> is a collision periodic solution, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M97">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M98">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M99">View MathML</a> and note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M52">View MathML</a>. By Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M101">View MathML</a>

(2.3)

which contradicts the inequality in (2.2). By Lemma 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M102">View MathML</a> is the critical point of f in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M103">View MathML</a>; therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/129/mathml/M102">View MathML</a> is a non-collision periodic solution.

This completes the proof of Theorem 1.1. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the final manuscript.

Acknowledgements

The authors sincerely thank the referees for their many helpful comments and suggestions and also express their sincere gratitude to Professor Zhang Shiqing for his discussions and corrections. This work is supported by NSF of China and Youth Fund of Mianyang Normal University.

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