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Existence of solutions of a system of 3D axisymmetric inviscid stagnation flows

GC Yang*, LF Dang and YZ Xu

Author Affiliations

College of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan, 610225, P.R. China

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Boundary Value Problems 2012, 2012:153  doi:10.1186/1687-2770-2012-153


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/153


Received:7 February 2012
Accepted:6 December 2012
Published:28 December 2012

© 2012 Yang 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

A system of two integral equations is presented to describe the system of 3D axisymmetric inviscid stagnation flows related to Navier-Stokes equations and existence of its solutions is studied. Utilizing it, we construct analytically the similarity solutions of the 3D system. A nonexistence result is obtained. Previous study was only supported by numerical results.

MSC: 34B18.

Keywords:
Navier-Stokes equations; 3D flows; similarity solutions; integral systems; existence results

1 Introduction

The following system of two differential equations arising in the boundary layer problems in fluid mechanics

(1.1)

(1.2)

with boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M3">View MathML</a>

(1.3)

has been used to describe the system of 3D axisymmetric inviscid stagnation flow [1,2], which consists of three partial differential equations [2,3], where λ is a parameter related to the external flow components.

A solution of (1.1)-(1.3) is called a similarity solution and can be used to express the solutions of the 3D system. Regarding the study of (1.1)-(1.3), Howarth [3] presented a numerical study for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M4">View MathML</a> which can be applied to the stagnation region of an ellipsoid. Davey [2] investigated numerically the stagnation region near a saddle point (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M5">View MathML</a>). The two-dimensional cases, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M6">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M8">View MathML</a>, and the special cases of the Falkner-Skan equation were solved by Hiemenz [4] and by Homann [5], respectively. Regarding the Falkner-Skan problems, further analytical study can be found in [6-10]. Also, one may refer to recent review of similarity solutions of the Navier-Stokes equations [11].

However, up to now, there has been very little analytical study on the existence of solutions of (1.1)-(1.3).

The main aim of this paper is to study the existence of solutions of (1.1)-(1.3) analytically for the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M9">View MathML</a>. The method is to present a system of two integral equations and study the existence of its solutions and then use it to construct the solutions of (1.1)-(1.3). Also, a nonexistence result is obtained.

2 A system of two integral equations related to (1.1)-(1.3)

In this section, we present a system of two integral equations to describe a system of (1.1)-(1.3) under suitable conditions, which will be utilized in Section 4.

Let

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M11">View MathML</a>

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12">View MathML</a>is a solution of (1.1)-(1.3), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M13">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M14">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M15">View MathML</a>

(2.1)

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M20">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M21">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M22">View MathML</a>, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M23">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M24">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M25">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M26">View MathML</a> exists. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M27">View MathML</a> by (2.1).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M28">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M29">View MathML</a> by (1.2). By (2.1), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M31">View MathML</a> and then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M32">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M33">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M34">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M35">View MathML</a>. We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M24">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M37">View MathML</a>.

In fact, if there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M38">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M39">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M40">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M41">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M42">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M44">View MathML</a>.

Differentiating (1.2) with η, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M45">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M46">View MathML</a>

a contradiction. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M47">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M48">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M13">View MathML</a>.

This completes the proof. □

Theorem 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12">View MathML</a>is a solution of (1.1)-(1.2), then

(2.2)

(2.3)

has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M53">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M54">View MathML</a>denotes the Green function for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M55">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M57">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M58">View MathML</a>

(2.4)

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M60">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a> be the inverse function to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M62">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M63">View MathML</a> is strictly increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M65">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M67">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M68">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a>, by Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M70">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M71">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a> and x is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M73">View MathML</a>. By Lemma 2.1, we see that x is continuous from the left at 1. Hence, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M75">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M76">View MathML</a>.

Using the chain rule to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M77">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M78">View MathML</a> and by the inverse function theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M79">View MathML</a>

This, together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M80">View MathML</a>, implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M81">View MathML</a>

Integrating the last equality from 0 to t implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M82">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M83">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M84">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M85">View MathML</a>, we know that y is continuous from the left at 1 and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M86">View MathML</a>.

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M89">View MathML</a>.

Differentiating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90">View MathML</a> with t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M91">View MathML</a>

From this, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M92">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M94">View MathML</a>.

Differentiating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M95">View MathML</a> with t and utilizing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M96">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M97">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M98">View MathML</a>

Substituting g, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M101">View MathML</a> and f into (1.2) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M102">View MathML</a>

(2.5)

Integrating (2.5) from t to 1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M103">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M75">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M105">View MathML</a>

Substituting f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M108">View MathML</a> and g into (1.1) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M109">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M110">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M111">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M112">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M54">View MathML</a> is defined by (2.4). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114">View MathML</a> is a solution of (2.2)-(2.3) in Q. □

3 Positive solutions of the system (2.2)-(2.3)

In this section, we will use the fixed point theorem to study the existence of positive solutions of the system (2.2)-(2.3).

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M115">View MathML</a>

It is easy to verify

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M116">View MathML</a>

We define some functions

By computation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M119">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M120">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M121">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M123">View MathML</a>.

In order to study the existence of solutions of (2.2)-(2.3) in Q for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M124">View MathML</a>, we denote the norm of the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M126">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M127">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M129">View MathML</a> be a natural number, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M130">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M133">View MathML</a>

(3.1)

Notation

and

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M54">View MathML</a> is defined by (2.4).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M128">View MathML</a>, we define an operator F as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M138">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M139">View MathML</a>

It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M140">View MathML</a>, θ are continuous operators from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M141">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>, we know the following proposition holds:

Lemma 3.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M145">View MathML</a>is a continuous and compact operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125">View MathML</a>.

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M148">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M149">View MathML</a>such that

(3.2)

(3.3)

Then the following assertions hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M152">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M154">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M155">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M156">View MathML</a>is a total variation ofyon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157">View MathML</a>.

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M158">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90">View MathML</a>is increasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M160">View MathML</a>and then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M161">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>.

Proof We shall use the basic fact: let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M164">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M165">View MathML</a>) be local minimum (maximum), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M166">View MathML</a> (≤0).

(i) If there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M167">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M168">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M169">View MathML</a>, we know that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M171">View MathML</a>. Differentiating (3.3) with t twice, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M172">View MathML</a>

(3.4)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M173">View MathML</a> and (3.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M174">View MathML</a>

a contradiction. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M175">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M176">View MathML</a>.

If there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M177">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M178">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M179">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M181">View MathML</a>, we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M183">View MathML</a>. This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M184">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M185">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M186">View MathML</a>. By (3.4) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M187">View MathML</a>, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M188">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M189">View MathML</a>

a contradiction. Hence, (i) holds.

(ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M190">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M192">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M193">View MathML</a>, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90">View MathML</a> is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M195">View MathML</a> and decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M196">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M198">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M199">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M193">View MathML</a>. If there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M201">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M202">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M203">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M204">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M205">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M206">View MathML</a> by (i). From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M208">View MathML</a> and (3.4), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M209">View MathML</a>

a contradiction.

If there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M210">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M202">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M212">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M213">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M214">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M206">View MathML</a> by (i). Analogously, we know easily

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M216">View MathML</a>

a contradiction. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M217">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M218">View MathML</a>, i.e., (ii) holds.

(iii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M158">View MathML</a>. By (i) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M86">View MathML</a>, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M221">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90">View MathML</a> is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M160">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M161">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>. Hence, (iii) holds. □

Lemma 3.3[12]

LetEbe a Banach space, Dbe a bounded open set of E and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M227">View MathML</a>is compact. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M228">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M229">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M230">View MathML</a>, thenFhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M231">View MathML</a>.

Lemma 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M232">View MathML</a>, thenFhas a fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125">View MathML</a>, i.e., there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M235">View MathML</a>such that

(3.5)

(3.6)

hold.

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M238">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M239">View MathML</a>. We prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M240">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M229">View MathML</a> and with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M242">View MathML</a>.

In fact, if there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114">View MathML</a> and μ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M245">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M240">View MathML</a>, by Lemma 3.2(i) and (iii), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M247">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M249">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M250">View MathML</a>, this, together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M251">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M143">View MathML</a>, implies

And then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M255">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M256">View MathML</a>.

By (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M257">View MathML</a>

(3.7)

Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M258">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M259">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M250">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M261">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M262">View MathML</a>. This, together with (3.7) and Lemma 3.2(ii), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M263">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M264">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M265">View MathML</a>

a contradiction.

By Lemmas 3.1 and 3.3, F has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M125">View MathML</a>. □

Lemma 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233">View MathML</a>be in Lemma 3.4, then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269">View MathML</a>is bounded on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M271">View MathML</a>is bounded on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M273">View MathML</a>.

Proof By Lemma 3.3(i), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M274">View MathML</a>. By (3.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M275">View MathML</a>

(3.8)

(i) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M22">View MathML</a>, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M277">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M279">View MathML</a> is decreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M282">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M284">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a> and (3.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M286">View MathML</a>

(3.9)

And then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M287">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M289">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M290">View MathML</a>. This, together with the decrease in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M291">View MathML</a>, implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M292">View MathML</a>

(3.10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M293">View MathML</a>, μ defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M294">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M295">View MathML</a> defined in (3.1).

It is easy to verify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M296">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>. And then

The last two inequalities imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269">View MathML</a> is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157">View MathML</a>.

(ii) By (3.8),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M301">View MathML</a>

we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M271">View MathML</a> is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M304">View MathML</a>. □

Lemma 3.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233">View MathML</a>be in Lemma 3.4, then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M306">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M307">View MathML</a>is increasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M309">View MathML</a>is bounded and equicontinuous in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M311">View MathML</a>.

Proof

(i) Lemma 3.2(i) and (iii) imply the desired results.

(ii) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M273">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M313">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M314">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M315">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157">View MathML</a>, by Lemma 3.2(ii), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M317">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M318">View MathML</a>.

Differentiating (3.6) with t twice, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M319">View MathML</a>. Integrating this equality from 0 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M320">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M321">View MathML</a>

Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M322">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M323">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M324">View MathML</a> and Lemma 3.2(ii), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M325">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M309">View MathML</a> is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M328">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M329">View MathML</a>), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M330">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M309">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272">View MathML</a>. □

Theorem 3.1There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M333">View MathML</a>such that

(3.11)

(3.12)

hold, where

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M233">View MathML</a> be in Lemma 3.4, by Lemma 3.5(ii) and (iii), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269">View MathML</a> is bounded and equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M273">View MathML</a>. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M341">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M342">View MathML</a>), utilizing the diagonal principle and the Arzela-Ascoli theorem, we know that there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M343">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M345">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M346">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M347">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a>. Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M343">View MathML</a> is itself of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M269">View MathML</a>.

By Lemma 3.6, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M309">View MathML</a> is bounded and equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M273">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M354">View MathML</a> is bounded and equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M272">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M356">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M342">View MathML</a>), the diagonal principle and the Arzela-Ascoli theorem imply that there exist y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M358">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M359">View MathML</a> and two subsequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M360">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M361">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M362">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M363">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M367">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M368">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a>. For the sake of convenience, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M370">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M371">View MathML</a> are itself of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M354">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M373">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M374">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M375">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M378">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M379">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M380">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M381">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M61">View MathML</a>, by the Lebesgue dominated theorem (the dominated function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M383">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M384">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114">View MathML</a> satisfies (3.11) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M386">View MathML</a>.

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M176">View MathML</a> and choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M388">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M320">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M390">View MathML</a>

Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M391">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M392">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M393">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M394">View MathML</a>, by the Lebesgue dominated theorem (the dominated function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M395">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M396">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M397">View MathML</a>

Differentiating the last equality twice, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M398">View MathML</a>

By (i), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M399">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M400">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M401">View MathML</a>. This, together with (2.4), implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M90">View MathML</a> satisfies (3.12). Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M403">View MathML</a>. □

Theorem 3.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M124">View MathML</a>, the system (2.2)-(2.3) has at least a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114">View MathML</a>inQ.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114">View MathML</a> in Theorem 3.1. It is clear that we only prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M407">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M22">View MathML</a>, by (3.10), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M409">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M407">View MathML</a>. Next, we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M412">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M414">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M415">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M416">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M417">View MathML</a>

From this and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M418">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M419">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M420">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M421">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M422">View MathML</a>, we prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M423">View MathML</a>

(3.13)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M424">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M425">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M426">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M427">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M428">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M132">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M379">View MathML</a> is concave down on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M157">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M432">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M433">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M434">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M435">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M436">View MathML</a>.

By (3.11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M437">View MathML</a>

we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M438">View MathML</a>

Integrating the last inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M439">View MathML</a> to 1 and utilizing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M440">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M441">View MathML</a>

Hence, (3.13) holds.

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M443">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M75">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M445">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M446">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M447">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M448">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M449">View MathML</a>

This, together with (3.13), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M450">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M451">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M452">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M453">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M454">View MathML</a>

And then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M455">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M456">View MathML</a>.

Finally, we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M457">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M458">View MathML</a>.

In fact, if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M459">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M460">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M461">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M462">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M463">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M464">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M465">View MathML</a>.

From

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M466">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M467">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M468">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M469">View MathML</a>

By (3.11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M470">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M471">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M472">View MathML</a>. Integrating this inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M473">View MathML</a> to δ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M474">View MathML</a>

and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M475">View MathML</a>, a contradiction.

This completes the proof. □

4 Existence of solutions of (1.1)-(1.3)

In this section, we use positive solutions obtained in Theorem 3.2 to construct the solutions of (1.1)-(1.3) in Γ.

Theorem 4.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M124">View MathML</a>, the system (1.1)-(1.3) has at least a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M12">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M124">View MathML</a>, by Theorem 3.2, the system (2.2)-(2.3) has at least a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M114">View MathML</a> in Q. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M480">View MathML</a> and (2.2), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M481">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M482">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M483">View MathML</a> and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M484">View MathML</a>

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M485">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M486">View MathML</a>

(4.1)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M487">View MathML</a> is strictly increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M73">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M489">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M490">View MathML</a> be the inverse function to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M491">View MathML</a>, we define the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M492">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M493">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M494">View MathML</a>

From (4.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M495">View MathML</a>

(4.2)

Differentiating (4.2) with respect to η, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M496">View MathML</a>

(4.3)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M497">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M498">View MathML</a>.

Differentiating (4.3) with respect to η, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M499">View MathML</a>

(4.4)

Differentiating (2.2) with respect to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M500">View MathML</a>

(4.5)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M501">View MathML</a> and utilizing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M502">View MathML</a> and (4.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M503">View MathML</a>

(4.6)

By (4.3), (4.4), (4.5) and (4.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M504">View MathML</a>

By (4.1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M505">View MathML</a>. Differentiating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M506">View MathML</a> with respect to η, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M507">View MathML</a>

Differentiating (2.3) with t twice and combining (4.5) and (4.6), we obtain

This completes the proof. □

Remark 4.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M509">View MathML</a>, by Theorem 1 [2], (1.1)-(1.3) has no solution such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M510">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M511">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M512">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M513">View MathML</a> is a constant.

Utilizing the system (2.2)-(2.3), we know easily that (1.1)-(1.3) has no solution in Γ for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M514">View MathML</a>.

In fact, if (1.1)-(1.3) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M515">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M514">View MathML</a>, by Theorem 2.1, then (1.1)-(1.3) has a solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M403">View MathML</a>. Noticing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M518">View MathML</a>

we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/153/mathml/M519">View MathML</a>

a contradiction.

This research uses integrals of equations to investigate the existence of solutions of the 3D axisymmetric inviscid stagnation flows related to Navier-Stokes equations and supplies a gap of analytical study in this field.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments. This research was supported by the National Natural Science Foundation of China (Grant No. 11171046) and the Scientific Research Foundation of the Education Department of Sichuan Province, China.

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