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.
MSC: 26A33, 34B15.
Keywords:fractional order; integro-differential equation; measure of noncompactness; fixed point index; boundary value problem
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 .
Lemma 1.1LetPbe a cone in a real Banach spaceE, and let Ω be a nonempty bounded open convex subset ofP. Suppose thatis completely continuous and, wheredenotes the closure of Ω inP. Then the fixed point index
Consider the boundary value problem (BVP) for a fractional nonlinear integro-differential equation of mixed type in E:
2 Several lemmas
We shall reduce BVP (1) to an integral equation in E. To this end, we first consider the operator A defined by
In our main results, we make use of the following lemmas.
Proof Inequalities (4) follow from two simple inequalities:
and (5) is obvious. □
where λ is defined in the operator A.
which implies the convergence of the infinite integral
Thus, we have, by (3), (9) and (10),
It follows from (11) that
It is clear that
and by (9),
It follows from (15) and (16) and the dominated convergence theorem that
Conversely, if u is a solution of (19), the direct differentiation of (19) gives
Lemma 2.4Integral equation (19) can be expressed as
By simple calculation, we can prove the rest of the lemma. □
Therefore, from (29), (30) and (31) we have
Consequently, the proof is complete. □
It follows from (33) and (34) that
Because ε is arbitrary, we obtain
Consequently, the proof is complete. □
3 Main results
In this section, we give and prove our main results.
First, we shall prove A is compact.
It follows from Lemma 2.5 that
It follows from (38) and (39) that
which implies, by virtue of the arbitrariness of ε, that
Using Lemma 2.6, we have
As in the proof of Lemma 2.2, (12) holds. Choose
It follows from (3), (43) and (44) that
Let , and we are going to show that is an open set of . It is clear that we need only to show the following: for any , there exists such that , implies that for . We have for . So, for any , there exists a such that
which implies by virtue of (48) that
By (53), (54) and the additivity of the fixed point index (Lemma 1.2), we can obtain
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 .
As in the proof of Lemma 2.2, (12) holds. Choose R satisfying (40) and let , where is given by assumption (). It is clear that U is a nonempty bounded closed convex subset in ( because ). Let , by (40), we have . On the other hand, as in the proof of Theorem 3.1, Lemma 2.4 and assumption () imply
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.
The authors declare that they have no competing interests.
RL completed the proof and wrote the initial draft. CK provided the problem and gave some suggestions for amendment. RL then finalized the manuscript. All authors read and approved the final manuscript.
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.
Arara, A, Benchohra, M, Hamidi, N, Nieto, JJ: Fractional order differential equations on an unbounded domain. Nonlinear Anal.. 72, 580–586 (2010). Publisher Full Text
Babakhani, A, Gejji, VD: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl.. 278, 434–442 (2003). Publisher Full Text
Delbosco, D, Rodino, L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl.. 204, 609–625 (1996). Publisher Full Text
Diethlm, K, Ford, NJ: Analysis of fractional differential equations. J. Math. Anal. Appl.. 265, 229–248 (2002). Publisher Full Text
Sayed, WGE, Sayed, AMAE: On the functional integral equations of mixed type and integro-differential equations of fractional orders. Appl. Math. Comput.. 154, 461–467 (2004). Publisher Full Text
El-Sayed, AMA: On the fractional differential equation. Appl. Math. Comput.. 49, 205–213 (1992). Publisher Full Text
Kilbas, AA, Trujillo, JJ: Differential equations of fractional order: methods, results and problems I. Appl. Anal.. 78, 153–192 (2001). Publisher Full Text
Kilbas, AA, Trujillo, JJ: Differential equations of fractional order: methods, results and problems II. Appl. Anal.. 81, 435–493 (2002). Publisher Full Text
Kosmatov, N: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal.. 70, 2521–2529 (2009). Publisher Full Text
Lakshmikantham, V: Theory of fractional functional differential equations. Nonlinear Anal.. 69, 3337–3343 (2008). Publisher Full Text
Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal.. 69, 2677–2682 (2008). Publisher Full Text
Muslim, M, Conca, C, Nandakumaran, AK: Approximate of solutions to fractional integral equation. Comput. Math. Appl.. 59, 1236–1244 (2010). Publisher Full Text
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)
Stojanović, M: Existence-uniqueness result for a nonlinear n-term fractional equation. J. Math. Anal. Appl.. 353, 244–245 (2009). Publisher Full Text