# Nontrivial solutions for a fractional boundary value problem

- Keyu Zhang
^{1, 2}Email author and - Jiafa Xu
^{2}

**2013**:171

https://doi.org/10.1186/1687-1847-2013-171

© Zhang and Xu; licensee Springer 2013

**Received: **24 March 2013

**Accepted: **28 May 2013

**Published: **17 June 2013

## Abstract

In this work, we discuss the existence of nontrivial solutions for the fractional boundary value problem

Here $\alpha \in (2,3]$ is a real number, ${D}_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order *α*. By virtue of some inequalities associated with Green’s function, without the assumption of the nonnegativity of *f*, we utilize topological degree theory to establish our main results.

**MSC:**26A33, 34B15, 34B18, 34B27.

## Keywords

## 1 Introduction

where $\alpha \in (2,3]$, $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ ($\mathbb{R}:=(-\mathrm{\infty},+\mathrm{\infty})$) is continuous.

In view of a fractional differential equation’s modeling capabilities in engineering, science, economy and other fields, the last few decades have resulted in a rapid development of the theory of fractional differential equation; see the books [1–3]. This may explain the reason why the last few decades have witnessed an overgrowing interest in the research of such problems, with many papers in this direction published. We refer the interested reader to [4–21] and the references therein.

where $1<\alpha \le 2$ is a real number and $f:[0,1]\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is continuous. They obtained the existence of positive solutions by means of Guo-Krasnosel’skii fixed point theorem and Leggett-Williams fixed point theorem.

In [5], Jiang *et al.* discussed some positive properties of the Green function for boundary value problem (1.2), and as an application, they utilized the Guo-Krasnosel’skii fixed point theorem to obtain the existence of positive solutions for (1.2).

In [6], El-Shahed and Nieto investigated the existence of nontrivial solutions for a multi-point boundary value problem for fractional differential equations. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of a nontrivial solution were obtained by using the Leray-Schauder nonlinear alternative.

*et al.*utilized fixed point theory to consider some fractional differential equations with fractional boundary conditions and obtained some new existence results. In particular, He and his coauthors [10] investigated the existence of solutions for the fractional nonlinear integro-differential equation of mixed type on a semi-infinite interval in a Banach space

Meanwhile, we also note that they developed an explicit iterative sequence for approximating the solution together with an error estimate for the approximation.

In [22, 23], Sun and Zhang discussed a class of singular superlinear and sublinear Sturm-Liouville problems, respectively. In the two papers, the Sturm-Liouville problems are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, and the nonnegativity is not necessary to be nonnegative. The existence results of nontrivial solutions and positive solutions are given by means of topological degree theory.

Motivated by the works mentioned above, in our paper, we adopt the methods of [22, 23] to investigate the fractional problem (1.1). As we know, the eigenvalue and eigenfunction of an integer-order differential equation have been a very perfect theory; however, this work on fractional order differential equation has not appeared in the literature. In order to overcome the difficulty arising from it, we establish some inequalities associated with Green’s function; see Lemma 2.3 in Section 2. With the aid of these inequalities, the nonlinear term *f* can grow superlinearly and sublinearly, and we obtain that problem (1.1) has at least one nontrivial solution by topological degree theory. This means that both our methodology and results in this paper are different from those in [4–7, 11–16].

## 2 Preliminaries

where $n=[\alpha ]+1$, $[\alpha ]$ denotes the integer part of number *α*, provided that the right-hand side is pointwise defined on $(0,+\mathrm{\infty})$. For more details on fractional calculus, we refer the reader to the recent books; see [1–3]. Next, we present Green’s function of fractional differential equation boundary value problem (1.1).

**Lemma 2.1** (See [[16], Lemma 2.7])

*Let*$v\in C[0,1]$

*and*$\alpha \in (2,3]$.

*Then*$-{D}_{0+}^{\alpha}u:=v$,

*together with the boundary conditions*$u(0)={u}^{\prime}(0)={u}^{\prime}(1)=0$,

*is equivalent to*$u(t)={\int}_{0}^{1}G(t,s)v(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$,

*where*

**Lemma 2.2** (See [[16], Lemma 2.8])

*The functions*$G(t,s)\in C([0,1]\times [0,1],[0,+\mathrm{\infty}))$.

*Moreover*, $G(t,s)$

*satisfies the following inequalities*:

**Lemma 2.3**

*Let*$\phi (t)=t{(1-t)}^{\alpha -2}$, $\mathrm{\forall}t\in [0,1]$,

*and*$\frac{\alpha \mathrm{\Gamma}(\alpha -1)}{\mathrm{\Gamma}(2\alpha )}\phantom{\rule{0.25em}{0ex}}(:={\mathcal{K}}_{1})\le \frac{1}{\alpha (\alpha -1)\mathrm{\Gamma}(\alpha )}\phantom{\rule{0.25em}{0ex}}(:={\mathcal{K}}_{2})$.

*Then*

*Proof* By (2.2), we arrive at the inequality (2.3) immediately. The proof is completed. □

*P*is a cone on

*E*. Now, note that

*u*solves (1.1) if and only if

*u*is a fixed point of the operator

*L*is a positive, completely continuous, linear operator. Let

where $\phi (t)$ is defined by Lemma 2.3 and $\omega :={\mathcal{K}}_{1}\mathrm{\Gamma}(\alpha )>0$. By (2.2) and (2.3), we easily have the following result.

**Lemma 2.4** $L(P)\subset {P}_{0}$.

*Proof*From (2.2), for $u\in P$, we have

Therefore, $L(P)\subset {P}_{0}$. This completes the proof. □

**Lemma 2.5** (See [24])

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

**Lemma 2.6** (See [24])

*Let* *E* *be a Banach space and let* $\mathrm{\Omega}\subset E$ *be a bounded open set with* $0\in \mathrm{\Omega}$. *Suppose that* $A:\overline{\mathrm{\Omega}}\to E$ *is a completely continuous operator*. *If* $Au\ne \mu u$, $\mathrm{\forall}u\in \partial \mathrm{\Omega}$ *and* $\mu \ge 1$, *then the topological degree* $deg(I-A,\mathrm{\Omega},0)=1$.

## 3 Main results

We denote ${\lambda}_{1}:={\mathcal{K}}_{1}^{-1}>0$, ${\lambda}_{2}:={\mathcal{K}}_{2}^{-1}>0$ and ${B}_{\rho}:=\{u\in E:\parallel u\parallel <\rho \}$ for $\rho >0$.

**Theorem 3.1**

*If there exists a constant*$b\ge 0$

*such that*

*then* (1.1) *has at least one nontrivial solution*.

*Proof*The first two inequalities of (3.1) imply that there are $\epsilon >0$ and ${b}_{1}>0$ such that

By (3.5) and (3.8), we have $deg(I-A,{B}_{R}\setminus {\overline{B}}_{r},0)=deg(I-A,{B}_{R},0)-deg(I-A,{B}_{r},0)=0-1=-1$. Then *A* has at least one fixed point on ${B}_{R}\setminus {\overline{B}}_{r}$. This means that problem (1.1) has at least one nontrivial solution. □

In order to prove Theorem 3.2, we need the following result involving the spectral radius of *L*, denoted by $r(L)$.

**Lemma 3.1** $0<r(L)\le {\mathcal{K}}_{2}$.

*Proof* We easily obtain the result by Gelfand’s theorem and (2.2). This completes the proof. □

**Theorem 3.2**

*If there exists a constant*$b\ge 0$

*such that*

*then* (1.1) *has at least one nontrivial solution*.

*Proof*By the second inequality of (3.9), there exist $\epsilon >0$ and ${r}_{1}>0$ such that

*A*has no fixed point on $\partial {B}_{{r}_{1}}$ (otherwise, the proof is completed). Now we show that

*i*denotes fixed point index on

*P*. Recall the definition of $\tilde{u}$. Clearly, $\tilde{u}\in P$ and $A:C[0,1]\to P-\tilde{u}$ by (3.9). Define $\tilde{A}u=A(u-\tilde{u})+\tilde{u}$, $u\in C[0,1]$. We easily find $\tilde{A}:C[0,1]\to P$. By the third inequality of (3.9), there exist ${r}_{2}>{r}_{1}+\parallel \tilde{u}\parallel ={r}_{1}+b{\mathcal{K}}_{2}\mathrm{\Gamma}(\alpha )$ and $0<\sigma <1$ such that

*W*is bounded. For all $u\in W$, let $\tilde{\psi}(t)=min\{u(t)-\tilde{u}(t),{r}_{2}\}$ and $e(u)=\{t\in [0,1]:u(t)-\tilde{u}(t)>{r}_{2}\}$. When $u(t)-\tilde{u}(t)<0$, $\tilde{\psi}(t)=u(t)-\tilde{u}(t)\ge u(t)-{r}_{2}\ge -{r}_{2}$, and so $\parallel \tilde{\psi}\parallel \le {r}_{2}$. Consequently, for $u\in W$, we have from (3.13)

and then $((I-{L}_{1})u)(t)\le M$, $t\in [0,1]$. By Lemma 3.1 and $0<\sigma <1$, $r({L}_{1})=\sigma {\lambda}_{2}r(L)\le \sigma {\lambda}_{2}{\mathcal{K}}_{2}<1$. Therefore, the inverse operator ${(I-{L}_{1})}^{-1}$ exists and ${(I-{L}_{1})}^{-1}=I+{L}_{1}+{L}_{1}^{2}+\cdots +{L}_{1}^{n}+\cdots $ . It follows from ${L}_{1}(P)\subset P$ that ${(I-{L}_{1})}^{-1}(P)\subset P$. So, we have $u(t)\le {(I-{L}_{1})}^{-1}M$, $t\in [0,1]$ and *W* is bounded.

By (3.12) and (3.15), we get $deg(I-A,{B}_{{r}_{3}}\setminus {\overline{B}}_{{r}_{1}},0)=deg(I-A,{B}_{{r}_{3}},0)-deg(I-A,{B}_{{r}_{1}},0)=1$, which implies that *A* has at least one fixed point on ${B}_{{r}_{3}}\setminus {\overline{B}}_{{r}_{1}}$. This means that the problem (1.1) has at least one nontrivial solution. □

**Two examples**1. Let

*n*is a positive even number, ${a}_{i}\in \mathbb{R}$ ($i=1,2,\dots ,n-1$), $|{a}_{1}|<{\lambda}_{2}$, ${a}_{n}>0$. It is easy to see that $f(t,u)$ is bounded below and usually sign-changing for $u\ge 0$. In addition, ${lim\hspace{0.17em}sup}_{u\to 0}|\frac{f(t,u)}{u}|=|{a}_{1}|<{\lambda}_{2}$ and ${lim\hspace{0.17em}inf}_{u\to +\mathrm{\infty}}\frac{f(t,u)}{u}=+\mathrm{\infty}$. Thus by Theorem 3.1, we can obtain the existence of a nontrivial solution of (1.1).

- 2.Let$f(t,u)=\frac{1-{u}^{2}}{1+{u}^{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}(t,u)\in [0,1]\times \mathbb{R}.$

It is easy to see that $f(t,u)$ is bounded below and usually sign-changing for $u\ge 0$. In addition, ${lim\hspace{0.17em}sup}_{u\to +\mathrm{\infty}}\frac{f(t,u)}{u}=0<{\lambda}_{2}$ and ${lim\hspace{0.17em}inf}_{u\to 0}\frac{f(t,u)}{|u|}=+\mathrm{\infty}$. Thus, by Theorem 3.2, we can obtain the existence of a nontrivial solution of (1.1).

## Declarations

### Acknowledgements

Research is supported by the NNSF-China (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), IIFSDU (2012TS020).

## Authors’ Affiliations

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