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Nontrivial solutions for a fractional boundary value problem
Advances in Difference Equations volume 2013, Article number: 171 (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 RiemannLiouville 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.
1 Introduction
In this paper, we investigate nontrivial solutions for the boundary value problem of fractional order involving RiemannLiouville’s derivative
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.
In [4], Bai and Lü studied the existence and multiplicity of positive solutions for the nonlinear fractional differential equation
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 GuoKrasnosel’skii fixed point theorem and LeggettWilliams 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 GuoKrasnosel’skii fixed point theorem to obtain the existence of positive solutions for (1.2).
In [6], ElShahed and Nieto investigated the existence of nontrivial solutions for a multipoint 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 LeraySchauder nonlinear alternative.
In [7], Wang and Liu adopted the same methods in [6] to discuss the existence of solutions for nonlinear fractional differential equations with fractional antiperiodic boundary conditions
In [8–10], Ahmad 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 integrodifferential equation of mixed type on a semiinfinite 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 SturmLiouville problems, respectively. In the two papers, the SturmLiouville 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 integerorder 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
The RiemannLiouville fractional derivative of order \alpha >0 of a continuous function f:(0,+\mathrm{\infty})\to (\mathrm{\infty},+\mathrm{\infty}) is given by
where n=[\alpha ]+1, [\alpha ] denotes the integer part of number α, provided that the righthand 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{(1t)}^{\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. □
By simple computation, we have {max}_{t\in [0,1]}{\int}_{0}^{1}G(t,s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s={\int}_{0}^{1}\phi (t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t=\frac{1}{\alpha (\alpha 1)}={\mathcal{K}}_{2}\mathrm{\Gamma}(\alpha ). Let
Then (E,\parallel \cdot \parallel ) becomes a real Banach space and 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
Clearly, f\in C([0,1]\times \mathbb{R},\mathbb{R}) implies A:E\to E is a completely continuous operator. Denote
Then L:E\to E is a completely continuous linear operator, satisfying L(P)\subset P. That is, 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
On the other hand, from (2.2) and (2.3), we find
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(IA,\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(IA,\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
Take R>b{\mathcal{K}}_{2}\mathrm{\Gamma}(\alpha )+{\epsilon}^{1}b({\mathcal{K}}_{1}^{3}{\mathcal{K}}_{2}^{3}{\mathcal{K}}_{1}^{1}{\mathcal{K}}_{2})+b{\mathcal{K}}_{1}^{2}{\mathcal{K}}_{2}^{3}+{\epsilon}^{1}{b}_{1}{\mathcal{K}}_{1}^{2}{\mathcal{K}}_{2}^{2}. In what follows, we shall prove that
where {u}^{\ast}\in {P}_{0}. Indeed, if {u}_{0}\in E, \parallel {u}_{0}\parallel =R, and {\mu}_{0}\ge 0 such that
Let \tilde{u}(t)=b{\int}_{0}^{1}G(t,s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s, then we have
which leads to {u}_{0}+\tilde{u}\in {P}_{0} by Lemma 2.4. Combining this with (3.2), we find
On the other hand, we have by (3.4)
That is a contradiction. As a result of this, (3.3) holds. Lemma 2.5 gives
It follows from the third inequality of (3.1) that there exists 0<r<R such that f(t,u)\le {\lambda}_{2}u, \mathrm{\forall}u\le r, t\in [0,1]. In the following, we prove
In fact, suppose that there exist {u}_{1}\in \partial {B}_{r}, {\mu}_{1}\ge 1 such that A{u}_{1}={\mu}_{1}{u}_{1}. We may suppose that {\mu}_{1}>1 (otherwise we are done). Thus
Multiply by \phi (t) both sides of the preceding inequality and integrate over [0,1], and use (2.3) to obtain
This, together with {\int}_{0}^{1}{u}_{1}(t)\phi (t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t>0, leads to {\mu}_{1}\le 1, which is a contradiction. So, (3.6) holds. Lemma 2.6 implies
By (3.5) and (3.8), we have deg(IA,{B}_{R}\setminus {\overline{B}}_{r},0)=deg(IA,{B}_{R},0)deg(IA,{B}_{r},0)=01=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
For every u\in {\overline{B}}_{{r}_{1}}, we have from (3.10) that
and thus A({\overline{B}}_{{r}_{1}})\subset P. For all u\in \partial {B}_{{r}_{1}}\cap P, from (3.10), we know
We may suppose that A has no fixed point on \partial {B}_{{r}_{1}} (otherwise, the proof is completed). Now we show that
where {u}^{\ast}\in P. Otherwise, there exist {u}_{0}\in \partial {B}_{{r}_{1}}\cap P, {\mu}_{0}\ge 0 such that {u}_{0}=A{u}_{0}+{\mu}_{0}{u}^{\ast}\ge A{u}_{0}. Consequently,
Multiply by \phi (t) both sides of the preceding inequality and integrate over [0,1], and use (2.3) to obtain
which implies {\int}_{0}^{1}{u}_{0}(t)\phi (t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t=0, and then {u}_{0}(t)\equiv 0, \mathrm{\forall}t\in [0,1]. It contradicts {u}_{0}\in \partial {B}_{{r}_{1}}\cap P. Hence (3.11) is true. Since A({\overline{B}}_{{r}_{1}})\subset P, we have, from the permanence property of fixed point index and Lemma 2.5, that
where 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
Let {L}_{1}u=\sigma {\lambda}_{2}Lu, \mathrm{\forall}u\in C[0,1]. Then {L}_{1}:C[0,1]\to C[0,1] is a bounded linear operator and {L}_{1}(P)\subset P. Let
and W:=\{u\in P:u=\mu \tilde{A}u,0\le \mu \le 1\}. In what follows, we will show 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.
Select {r}_{3}>max\{{r}_{2},supW+b{\mathcal{K}}_{2}\mathrm{\Gamma}(\alpha )\} and thus \tilde{A} has no fixed point on \partial {B}_{{r}_{3}}. Indeed, if there exists {u}_{1}\in \partial {B}_{{r}_{3}} such that \tilde{A}{u}_{1}={u}_{1}, then {u}_{1}\in W and \parallel {u}_{1}\parallel ={r}_{3}>supW, which is a contradiction. Then we have from the permanence property and the homotopy invariance property of fixed point index that
Set the completely continuous homotopy H(t,u)=A(ut\tilde{u})+t\tilde{u}, (t,u)\in [0,1]\times {\overline{B}}_{{r}_{3}}. If there exists ({t}_{0},{u}_{2})\in [0,1]\times \partial {B}_{{r}_{3}} such that H({t}_{0},{u}_{2})={u}_{2}, and then A({u}_{2}{t}_{0}\tilde{u})={u}_{2}{t}_{0}\tilde{u} and \tilde{A}({u}_{2}{t}_{0}\tilde{u}+\tilde{u})={u}_{2}{t}_{0}\tilde{u}+\tilde{u}. Thus {u}_{2}{t}_{0}\tilde{u}+\tilde{u}\in W and \parallel {u}_{2}{t}_{0}\tilde{u}+\tilde{u}\parallel \ge \parallel {u}_{2}\parallel (1{t}_{0})\parallel \tilde{u}\parallel \ge {r}_{3}\parallel \tilde{u}\parallel >supW, which is a contradiction. From the homotopy invariance of topological degree and (3.14), we have
By (3.12) and (3.15), we get deg(IA,{B}_{{r}_{3}}\setminus {\overline{B}}_{{r}_{1}},0)=deg(IA,{B}_{{r}_{3}},0)deg(IA,{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
where n is a positive even number, {a}_{i}\in \mathbb{R} (i=1,2,\dots ,n1), {a}_{1}<{\lambda}_{2}, {a}_{n}>0. It is easy to see that f(t,u) is bounded below and usually signchanging 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 signchanging 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).
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Acknowledgements
Research is supported by the NNSFChina (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), IIFSDU (2012TS020).
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KZ and JX gave the proof for the main result together. All authors read and approved the final manuscript.
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Zhang, K., Xu, J. Nontrivial solutions for a fractional boundary value problem. Adv Differ Equ 2013, 171 (2013). https://doi.org/10.1186/168718472013171
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DOI: https://doi.org/10.1186/168718472013171