- Open Access
Nontrivial solutions for a fractional boundary value problem
© Zhang and Xu; licensee Springer 2013
- Received: 24 March 2013
- Accepted: 28 May 2013
- Published: 17 June 2013
In this work, we discuss the existence of nontrivial solutions for the fractional boundary value problem
Here is a real number, 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.
- fractional boundary value problem
- nontrivial solution
- topological degree
- Riemann-Liouville derivative
where , () 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 is a real number and 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 , 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 , 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.
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].
where , denotes the integer part of number α, provided that the right-hand side is pointwise defined on . 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 [, Lemma 2.7])
Lemma 2.2 (See [, Lemma 2.8])
Proof By (2.2), we arrive at the inequality (2.3) immediately. The proof is completed. □
where is defined by Lemma 2.3 and . By (2.2) and (2.3), we easily have the following result.
Lemma 2.4 .
Therefore, . This completes the proof. □
Lemma 2.5 (See )
Let E be a Banach space and let be a bounded open set. Suppose that is a completely continuous operator. If there is such that , and , then the topological degree .
Lemma 2.6 (See )
Let E be a Banach space and let be a bounded open set with . Suppose that is a completely continuous operator. If , and , then the topological degree .
We denote , and for .
then (1.1) has at least one nontrivial solution.
By (3.5) and (3.8), we have . Then A has at least one fixed point on . 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 .
Lemma 3.1 .
Proof We easily obtain the result by Gelfand’s theorem and (2.2). This completes the proof. □
then (1.1) has at least one nontrivial solution.
and then , . By Lemma 3.1 and , . Therefore, the inverse operator exists and . It follows from that . So, we have , and W is bounded.
By (3.12) and (3.15), we get , which implies that A has at least one fixed point on . This means that the problem (1.1) has at least one nontrivial solution. □
It is easy to see that is bounded below and usually sign-changing for . In addition, and . Thus, by Theorem 3.2, we can obtain the existence of a nontrivial solution of (1.1).
Research is supported by the NNSF-China (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), IIFSDU (2012TS020).
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
- Kilbas A, Srivastava H, Trujillo J: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATHGoogle Scholar
- Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.MATHGoogle Scholar
- Bai Z, Lü H: Positive solutions for boundary-value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar
- Jiang D, Yuan C: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. 2010, 72: 710-719. 10.1016/j.na.2009.07.012MathSciNetView ArticleMATHGoogle Scholar
- El-Shahed M, Nieto J: Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order. Comput. Math. Appl. 2010, 59: 3438-3443. 10.1016/j.camwa.2010.03.031MathSciNetView ArticleMATHGoogle Scholar
- Wang F, Liu ZH: Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. Adv. Differ. Equ. 2012., 2012: Article ID 116Google Scholar
- Ahmad B, Nieto J: Riemann-Liouville fractional differential equations with fractional boundary conditions. Fixed Point Theory 2012, 13: 329-336.MathSciNetMATHGoogle Scholar
- Zhang L, Ahmad B, Wang G, Agarwal RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56.MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Ntouyas S, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multi-strip boundary conditions. Math. Probl. Eng. 2013., 2013: Article ID 320415Google Scholar
- Guo Y: Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations. Bull. Korean Math. Soc. 2010, 47: 81-87. 10.4134/BKMS.2010.47.1.081MathSciNetView ArticleMATHGoogle Scholar
- Ferreira R: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 70Google Scholar
- Jia M, Zhang X, Gu X: Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions. Bound. Value Probl. 2012., 2012: Article ID 70. doi:10.1186/1687-2770-2012-70Google Scholar
- Yang L, Chen HB: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011., 2011: Article ID 404917Google Scholar
- Sudsutad W, Tariboon J: Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions. Adv. Differ. Equ. 2012., 2012: Article ID 93Google Scholar
- El-Shahed M: Positive solutions for boundary value problems of nonlinear fractional differential equation. Abstr. Appl. Anal. 2007., 2007: Article ID 10368Google Scholar
- Xu JF, Wei ZL, Dong W: Uniqueness of positive solutions for a class of fractional boundary value problems. Appl. Math. Lett. 2012, 25: 590-593. 10.1016/j.aml.2011.09.065MathSciNetView ArticleMATHGoogle Scholar
- Xu JF, Yang ZL: Multiple positive solutions of a singular fractional boundary value problem. Appl. Math. E-Notes 2010, 10: 259-267.MathSciNetMATHGoogle Scholar
- Xu JF, Wei ZL, Ding YZ: Positive solutions for a boundary-value problem with Riemann-Liouville’s fractional derivative. Lith. Math. J. 2012, 52: 462-476. 10.1007/s10986-012-9187-zMathSciNetView ArticleMATHGoogle Scholar
- Wei ZL, Li Q, Che J: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 2010, 367: 260-272. 10.1016/j.jmaa.2010.01.023MathSciNetView ArticleMATHGoogle Scholar
- Wei ZL, Dong W, Che J: Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Anal. 2010, 73: 3232-3238. 10.1016/j.na.2010.07.003MathSciNetView ArticleMATHGoogle Scholar
- Sun J, Zhang G: Nontrivial solutions of singular superlinear Sturm-Liouville problems. J. Math. Anal. Appl. 2006, 313: 518-536. 10.1016/j.jmaa.2005.06.087MathSciNetView ArticleMATHGoogle Scholar
- Sun J, Zhang G: Nontrivial solutions of singular sublinear Sturm-Liouville problems. J. Math. Anal. Appl. 2007, 326: 242-251. 10.1016/j.jmaa.2006.03.003MathSciNetView ArticleMATHGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando; 1988.MATHGoogle Scholar
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