- Open Access
Multiple positive solutions of boundary value problems for fractional order integro-differential equations in a Banach space
© Liu et al.; licensee Springer. 2013
Received: 25 December 2012
Accepted: 18 March 2013
Published: 8 April 2013
In this paper, we obtain the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space by means of fixed point index theory of completely continuous operators.
Fractional differential equations (FDEs) have been of great interest for the last three decades [1–11]. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity , electrochemistry , control, porous media , etc. Therefore, the theory of FDEs has been developed very quickly. Many qualitative theories of FDEs have been obtained. Many important results can be found in [15–19] and references cited therein.
In this paper, we shall use the fixed point index theory of completely continuous operators to investigate the multiple positive solutions of a boundary value problem for a class of α order nonlinear integro-differential equations in a Banach space.
Let E be a real Banach space, P be a cone in E and denote the interior points of P. A partial ordering in E is introduced by if and only if . P is said to be normal if there exists a positive constant N such that implies , where θ denotes the zero element of E, and the smallest constant N is called the normal constant of P. P is called solid if is nonempty. If and , we write . If P is solid and , we write . For details on cone theory, see .
where , .
provided the right-hand side is pointwise defined on .
where , provided the right-hand side is pointwise defined on .
for some , , .
, , , denotes the set of all nonnegative real numbers.
2 Several lemmas
To establish the existence of multiple positive solutions in of (1), let us list the following assumptions.
() , , , as ().
() For any and , is relatively compact in E, where .
where , .
where , .
Remark 2.1 It is clear that () is satisfied automatically when E is finite dimensional.
Remark 2.2 It is clear that assumption () is weaker than assumption ().
In our main results, we make use of the following lemmas.
and (5) is obvious. □
Lemma 2.2 Let assumptions (), () and () be satisfied, then the operator A defined by (3) is a continuous operator from into .
where λ is defined in the operator A.
Thus, we have .
It follows from (14), (17) and (18) that (), and the continuity of A is proved. □
i.e., u is a fixed point of the operator A defined by (3) in .
Now, substituting (22) and (23) into (21), we see that satisfies integral equation (19).
Consequently, , and by (19), (24) and (25), it is easy to see that satisfies BVP (1). □
By simple calculation, we can prove the rest of the lemma. □
uniformly with respect to , as .
This, together with (9) and (10), implies that are equicontinuous on any finite subinterval of J.
for all and .
Consequently, the proof is complete. □
First, we claim that .
uniformly with respect to and .
where denote the diameters of bounded subsets of .
Consequently, the proof is complete. □
3 Main results
In this section, we give and prove our main results.
Theorem 3.1 Let ()-() be satisfied. Then BVP (1) has at least two positive solutions such that for .
Proof By Lemma 2.2 and Lemma 2.4, the operator A defined by (3) is continuous from into , and by Lemma 2.3, we need only to show that A has two positive fixed points such that for .
First, we shall prove A is compact.
where , , , .
Thus, we can conclude that AU is relatively compact in , i.e., A is compact.
Thus, we have proved that is open in .
Finally, (53), (54) and (55) imply that A has two fixed points and . We have, by (51), for . The proof is complete. □
Remark 3.1 Assumption () and the continuity of f imply that for . Hence, under the assumptions of the theorem, BVP (1) has the trivial solution besides two positive solutions and .
Theorem 3.2 Let ()-() and () be satisfied. Then BVP (1) has at least one positive solution such that for .
Proof By Lemma 2.2, Lemma 2.4 and the proof of Theorem 3.1, the operator A defined by (3) is completely continuous from into , and by Lemma 2.3, we need only to show that A has one positive fixed point such that for .
Hence, , and therefore . Thus, the Schauder fixed point theorem implies that A has a fixed point , and by (56) for . The proof is complete. □
In this paper, the issue on the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space has been addressed for the first time. Taking advantage of the fixed point index theory of completely continuous operators, the existence conditions for such boundary value problems have been established.
This work was supported by the Natural Science Foundation of China under grant No. 11271248 and the Science and Technology Research Program of Zhejiang Province under grant No. 2011C21036.
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.Google Scholar
- Guo D, Lakshmikantham V, Liu XZ: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht; 1996.View ArticleGoogle Scholar
- Arara A, Benchohra M, Hamidi N, Nieto JJ: Fractional order differential equations on an unbounded domain. Nonlinear Anal. 2010, 72: 580-586. 10.1016/j.na.2009.06.106MathSciNetView ArticleGoogle Scholar
- Babakhani A, Gejji VD: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 2003, 278: 434-442. 10.1016/S0022-247X(02)00716-3MathSciNetView ArticleGoogle Scholar
- Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 1996, 204: 609-625. 10.1006/jmaa.1996.0456MathSciNetView ArticleGoogle Scholar
- Diethlm K, Ford NJ: Analysis of fractional differential equations. J. Math. Anal. Appl. 2002, 265: 229-248. 10.1006/jmaa.2000.7194MathSciNetView ArticleGoogle Scholar
- Sayed WGE, Sayed AMAE: On the functional integral equations of mixed type and integro-differential equations of fractional orders. Appl. Math. Comput. 2004, 154: 461-467. 10.1016/S0096-3003(03)00727-6MathSciNetView ArticleGoogle Scholar
- Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.View ArticleGoogle Scholar
- El-Sayed AMA: On the fractional differential equation. Appl. Math. Comput. 1992, 49: 205-213. 10.1016/0096-3003(92)90024-UMathSciNetView ArticleGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems I. Appl. Anal. 2001, 78: 153-192. 10.1080/00036810108840931MathSciNetView ArticleGoogle Scholar
- Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems II. Appl. Anal. 2002, 81: 435-493. 10.1080/0003681021000022032MathSciNetView ArticleGoogle Scholar
- Kosmatov N: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal. 2009, 70: 2521-2529. 10.1016/j.na.2008.03.037MathSciNetView ArticleGoogle Scholar
- Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Anal. 2008, 69: 3337-3343. 10.1016/j.na.2007.09.025MathSciNetView ArticleGoogle Scholar
- Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Anal. 2008, 69: 2677-2682. 10.1016/j.na.2007.08.042MathSciNetView ArticleGoogle Scholar
- Muslim M, Conca C, Nandakumaran AK: Approximate of solutions to fractional integral equation. Comput. Math. Appl. 2010, 59: 1236-1244. 10.1016/j.camwa.2009.06.028MathSciNetView ArticleGoogle Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.Google Scholar
- Podlubny I: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego; 1999.Google Scholar
- Stojanović M: Existence-uniqueness result for a nonlinear n -term fractional equation. J. Math. Anal. Appl. 2009, 353: 244-245. 10.1016/j.jmaa.2008.11.056MathSciNetView ArticleGoogle Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, Yverdon; 1993.Google Scholar
This article is published under license to BioMed Central Ltd. 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.