# Nontrivial solutions for a boundary value problem with integral boundary conditions

- Bingmei Liu
^{1}Email author, - Junling Li
^{1}and - Lishan Liu
^{2}

**2014**:15

https://doi.org/10.1186/1687-2770-2014-15

© Liu et al.; licensee Springer. 2014

**Received: **27 July 2013

**Accepted: **13 November 2013

**Published: **13 January 2014

## Abstract

This paper concerns the existence of nontrivial solutions for a boundary value problem with integral boundary conditions by topological degree theory. Here the nonlinear term is a sign-changing continuous function and may be unbounded from below.

## 1 Introduction

where $(Lu)(t)={(\tilde{p}(t){u}^{\prime}(t))}^{\prime}+q(t)u(t)$, $\tilde{p}(t)\in {C}^{1}[0,1]$, $\tilde{p}(t)>0$, $q(t)\in C[0,1]$, $q(t)<0$, *α* and *β* are right continuous on $[0,1)$, left continuous at $t=1$ and nondecreasing on $[0,1]$ with $\alpha (0)=\beta (0)=0$; ${\gamma}_{0},{\gamma}_{1}\in [0,\pi /2]$, ${\int}_{0}^{1}u(\tau )\phantom{\rule{0.2em}{0ex}}d\alpha (\tau )$ and ${\int}_{0}^{1}u(\tau )\phantom{\rule{0.2em}{0ex}}d\beta (\tau )$ denote the Riemann-Stieltjes integral of *u* with respect to *α* and *β*, respectively. Here the nonlinear term $f:[0,1]\times (-\mathrm{\infty},+\mathrm{\infty})\to (-\mathrm{\infty},+\mathrm{\infty})$ is a continuous sign-changing function and *f* may be unbounded from below, $h:(0,1)\to [0,+\mathrm{\infty})$ with $0<{\int}_{0}^{1}h(s)\phantom{\rule{0.2em}{0ex}}ds<+\mathrm{\infty}$ is continuous and is allowed to be singular at $t=0,1$.

Problems with integral boundary conditions arise naturally in thermal conduction problems [1], semiconductor problems [2], hydrodynamic problems [3]. Integral BCs (BCs denotes boundary conditions) cover multi-point BCs and nonlocal BCs as special cases and have attracted great attention, see [4–14] and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer to the book of Corduneanu [4], Agarwal and O’Regan [5]. Yang [6], Boucherif [8], Chamberlain *et al.* [10], Feng [11], Jiang *et al.* [14] focused on the existence of positive solutions for the cases in which the nonlinear term is nonnegative. Although many papers investigated two-point and multi-point boundary value problems with sign-changing nonlinear terms, for example, [15–20], results for boundary value problems with integral boundary conditions when the nonlinear term is sign-changing are rarely seen except for a few special cases [7, 12, 13].

Inspired by the above papers, the aim of this paper is to establish the existence of nontrivial solutions to BVP (1.1) under weaker conditions. Our findings presented in this paper have the following new features. Firstly, the nonlinear term *f* of BVP (1.1) is allowed to be sign-changing and unbounded from below. Secondly, the boundary conditions in BVP (1.1) are the Riemann-Stieltjes integral, which includes multi-point boundary conditions in BVPs as special cases. Finally, the main technique used here is the topological degree theory, the first eigenvalue and its positive eigenfunction corresponding to a linear operator. This paper employs different conditions and different methods to solve the same BVP (1.1) as [7]; meanwhile, this paper generalizes the result in [17] to boundary value problems with integral boundary conditions. What we obtain here is different from [6–20].

## 2 Preliminaries and lemmas

*P*is a total cone in

*E*, that is, $E=\overline{P-P}$. ${P}^{\ast}$ denotes the dual cone of

*P*, namely, ${P}^{\ast}=\{g\in {E}^{\ast}\mid g(u)\ge 0,\text{for all}u\in P\}$. Let ${E}^{\ast}$ denote the dual space of

*E*, then by Riesz representation theorem, ${E}^{\ast}$ is given by

We assume that the following condition holds throughout this paper.

_{1}) $u(t)\equiv 0$

*is the unique*${C}^{2}$

*solution of the linear boundary value problem*

Let ${\kappa}_{1}=1-{\int}_{0}^{1}\phi (\tau )\phantom{\rule{0.2em}{0ex}}d\alpha (\tau )$, ${\kappa}_{2}={\int}_{0}^{1}\psi (\tau )\phantom{\rule{0.2em}{0ex}}d\alpha (\tau )$, ${\kappa}_{3}={\int}_{0}^{1}\phi (\tau )\phantom{\rule{0.2em}{0ex}}d\beta (\tau )$, ${\kappa}_{4}=1-{\int}_{0}^{1}\psi (\tau )\phantom{\rule{0.2em}{0ex}}d\beta (\tau )$.

(H_{2}) ${\kappa}_{1}>0$, ${\kappa}_{4}>0$, $k={\kappa}_{1}{\kappa}_{4}-{\kappa}_{2}{\kappa}_{3}>0$.

**Lemma 2.1** ([7])

*If*(H

_{1})

*and*(H

_{2})

*hold*,

*then BVP*(1.1)

*is equivalent to*

*where* $G(t,s)\in C([0,1]\times [0,1],{\mathbf{R}}^{+})$ *is the Green function for* (1.1).

It is easy to show that $A:E\to E$ is a completely continuous nonlinear operator, and if $u\in E$ is a fixed point of *A*, then *u* is a solution of BVP (1.1) by Lemma 2.1.

*K*is positive. The Krein-Rutman theorem [21] asserts that there are $\varphi \in P\setminus \{0\}$ and $\omega \in {P}^{\ast}\setminus \{0\}$ corresponding to the first eigenvalue ${\lambda}_{1}=1/r(K)$ of

*K*such that

*K*given by:

*G*and the integrability of

*h*imply that $\omega \in {C}^{1}[0,1]$. Let $e(t):={\omega}^{\prime}(t)$. Then $e\in P\setminus \{0\}$, and (2.4) can be rewritten equivalently as

**Lemma 2.2** ([7])

*If* (H_{1}) *holds*, *then there is* $\delta >0$ *such that* ${P}_{0}=\{u\in P\mid {\int}_{0}^{1}u(t)e(t)\phantom{\rule{0.2em}{0ex}}dt\ge \delta \parallel u\parallel \}$ *is a subcone of* *P* *and* $K(P)\subset {P}_{0}$.

**Lemma 2.3** ([22])

*Let* *E* *be a real Banach space and* $\mathrm{\Omega}\subset E$ *be a bounded open set with* $0\in \mathrm{\Omega}$. *Suppose that* $A:\overline{\mathrm{\Omega}}\to E$ *is a completely continuous operator*. (1) *If there is* ${y}_{0}\in E$ *with* ${y}_{0}\ne 0$ *such that* $u\ne Au+\mu {y}_{0}$ *for all* $u\in \partial \mathrm{\Omega}$ *and* $\mu \ge 0$, *then* $deg(I-A,\mathrm{\Omega},0)=0$. (2) *If* $Au\ne \mu u$ *for all* $u\in \partial \mathrm{\Omega}$ *and* $\mu \ge 1$, *then* $deg(I-A,\mathrm{\Omega},0)=1$. *Here* deg *stands for the Leray*-*Schauder topological degree in* *E*.

**Lemma 2.4** *Assume that* (H_{1}), (H_{2}) *and the following assumptions are satisfied*:

(C_{1}) *There exist* $\varphi \in P\setminus \{0\}$, $\omega \in {P}^{\ast}\setminus \{0\}$ *and* $\delta >0$ *such that* (2.3), (2.4) *hold and* *K* *maps* *P* *into* ${P}_{0}$.

_{2})

*There exists a continuous operator*$H:E\to P$

*such that*

(C_{3}) *There exist a bounded continuous operator* $F:E\to E$ *and* ${u}_{0}\in E$ *such that* $Fu+{u}_{0}+Hu\in P$ *for all* $u\in E$.

(C_{4}) *There exist* ${v}_{0}\in E$ *and* $\zeta >0$ *such that* $KFu\ge {\lambda}_{1}(1+\zeta )Ku-KHu-{v}_{0}$ *for all* $u\in E$.

*Let*$A=KF$,

*then there exists*$R>0$

*such that*

*where* ${B}_{R}=\{u\in E\mid \parallel u\parallel <R\}$.

*Proof*Choose a constant ${L}_{0}={(\delta {\lambda}_{1})}^{-1}(1+{\zeta}^{-1})+\parallel K\parallel >0$. From (C

_{2}), for $0<{\epsilon}_{0}<{L}_{0}^{-1}$, there exists ${R}_{1}>0$ such that $\parallel u\parallel >{R}_{1}$ implies

provided that *R* is sufficiently large.

_{4}), (2.5), we get

where ${L}_{1}={\zeta}^{-1}{\int}_{0}^{1}{v}_{0}(t)e(t)\phantom{\rule{0.2em}{0ex}}dt+{\int}_{0}^{1}(K{u}_{0})(t)e(t)\phantom{\rule{0.2em}{0ex}}dt$ is a constant.

_{3}) shows $F{u}_{1}+{u}_{0}+H{u}_{1}\in P$ and (C

_{1}) implies ${\mu}_{1}\varphi ={\mu}_{1}{\lambda}_{1}K{\phi}_{1}\in {P}_{0}$. Then (C

_{1}), (2.8) and Lemma 2.2 tell us that

where ${L}_{2}=\parallel K{u}_{0}\parallel +{L}_{1}{\delta}^{-1}$ is a constant.

*R*is sufficiently large such that $R>max\{{L}_{2}/(1-{\epsilon}_{0}{L}_{0}),{R}_{1}\}$. By (2.13) and Lemma 2.3, we have

□

## 3 Main results

**Theorem 3.1** *Assume that* (H_{1}), (H_{2}) *hold and the following conditions are satisfied*:

(A_{1}) *There exist two nonnegative functions* $b(t),c(t)\in C[0,1]$ *with* $c(t)\not\equiv 0$ *and one continuous even function* $B:\mathbf{R}\to {\mathbf{R}}^{+}$ *such that* $f(t,x)\ge -b(t)-c(t)B(x)$ *for all* $x\in \mathbf{R}$. *Moreover*, *B* *is nondecreasing on* ${\mathbf{R}}^{+}$ *and satisfies* ${lim}_{x\to +\mathrm{\infty}}\frac{B(x)}{x}=0$.

(A_{2}) $f:[0,1]\times \mathbf{R}\to \mathbf{R}$ *is continuous*.

(A_{3}) ${lim\hspace{0.17em}inf}_{x\to +\mathrm{\infty}}\frac{f(t,x)}{x}>{\lambda}_{1}$ *uniformly on* $t\in [0,1]$.

(A_{4}) ${lim\hspace{0.17em}sup}_{x\to 0}|\frac{f(t,x)}{x}|<{\lambda}_{1}$ *uniformly on* $t\in [0,1]$.

*Here* ${\lambda}_{1}$ *is the first eigenvalue of the operator* *K* *defined by* (2.2).

*Then BVP* (1.1) *has at least one nontrivial solution*.

*Proof*We first show that all the conditions in Lemma 2.4 are satisfied. By Lemma 2.2, condition (C

_{1}) of Lemma 2.4 is satisfied. Obviously, $B:E\to P$ is a continuous operator. By (A

_{1}), for any $\epsilon >0$, there is $L>0$ such that when $x>L$, $B(x)<\epsilon x$ holds. Thus, for $u\in E$ with $\parallel u\parallel >L$, $B(\parallel u\parallel )<\epsilon \parallel u\parallel $ holds. The fact that

*B*is nondecreasing on ${\mathbf{R}}^{+}$ yields $(Bu)(t)\le B(\parallel u\parallel )$ for any $u\in P$, $t\in [0,1]$. Since $B:\mathbf{R}\to {\mathbf{R}}^{+}$ is an even function, for any $u\in E$ and $t\in [0,1]$, $(Bu)(t)\le B(\parallel u\parallel )$ holds, which implies $\parallel Bu\parallel \le B(\parallel u\parallel )$ for $u\in E$. Therefore,

that is, ${lim}_{\parallel u\parallel \to +\mathrm{\infty}}\frac{\parallel Bu\parallel}{\parallel u\parallel}=0$. Take $Hu={c}_{0}Bu$, for any $u\in E$, where ${c}_{0}={max}_{t\in [0,1]}c(t)>0$. Obviously, ${lim}_{\parallel u\parallel \to +\mathrm{\infty}}\frac{\parallel Hu\parallel}{\parallel u\parallel}=0$ holds. Therefore *H* satisfies condition (C_{2}) in Lemma 2.4.

_{1}) that

which shows that condition (C_{3}) in Lemma 2.4 holds.

_{3}), there exist ${\epsilon}_{1}>0$ and a sufficiently large number ${l}_{1}>0$ such that

_{1}), there exists ${b}_{1}\ge 0$ such that

*K*is a positive linear operator, from (3.2) we have

So condition (C_{4}) in Lemma 2.4 is satisfied.

_{4}) it follows that there exist $0<{\epsilon}_{2}<1$ and $0<r<R$ such that

If there exist ${u}_{1}\in \partial {B}_{r}$ and ${\mu}_{1}\in [0,1]$ such that ${u}_{1}={\mu}_{1}A{u}_{1}$. Let $z(t)=|{u}_{1}(t)|$. Then $z\in P$ and by (3.4), $z\le (1-{\epsilon}_{2}){\lambda}_{1}Kz$. The *n* th iteration of this inequality shows that $z\le {(1-{\epsilon}_{2})}^{n}{\lambda}_{1}^{n}{K}^{n}z$ ($n=1,2,\dots $), so $\parallel z\parallel \le {(1-{\epsilon}_{2})}^{n}{\lambda}_{1}^{n}\parallel {K}^{n}\parallel \cdot \parallel z\parallel $, that is, $1\le {(1-{\epsilon}_{2})}^{n}{\lambda}_{1}^{n}\parallel {K}^{n}\parallel $. This yields $1-{\epsilon}_{2}=(1-{\epsilon}_{2}){\lambda}_{1}r(K)=(1-{\epsilon}_{2}){\lambda}_{1}{lim}_{n\to \mathrm{\infty}}\sqrt[n]{\parallel {K}^{n}\parallel}\ge 1$, which is a contradictory inequality. Hence, (3.5) holds.

So *A* has at least one fixed point on ${B}_{R}\setminus {\overline{B}}_{r}$, namely, BVP (1.1) has at least one nontrivial solution. □

**Corollary 3.1** *Using* (${\mathrm{A}}_{1}^{\ast}$) *instead of* (A_{1}), *the conclusion of Theorem * 3.1 *remains true*.

*There exist three constants*$b>0$, $c>0$

*and*$\alpha \in (0,1)$

*such that*

## Declarations

### Acknowledgements

The first two authors were supported financially by the National Natural Science Foundation of China (11201473, 11271364) and the Fundamental Research Funds for the Central Universities (2013QNA35, 2010LKSX09, 2010QNA42). The third author was supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.

## Authors’ Affiliations

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