An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces
© The Author(s) 2009
Received: 30 January 2009
Accepted: 15 May 2009
Published: 22 June 2009
The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness.
The theory of fractional differential equations has been emerging as an important area of investigation in recent years. Let us mention that this theory has many applications in describing numerous events and problems of the real world. For example, fractional differential equations are often applicable in engineering, physics, chemistry, and biology. See Hilfer, Glockle and Nonnenmacher , Metzler et al. , Podlubny , Gaul et al. , among others. Fractionaldifferential equations are also often an object of mathematical investigations; see the papers of Agarwal et al. , Ahmad and Nieto , Ahmad and Otero-Espinar , Belarbi et al. , Belmekki et al , Benchohra et al. [11–13], Chang and Nieto , Daftardar-Gejji and Bhalekar , Figueiredo Camargo et al. , and the monographs of Kilbas et al.  and Podlubny .
Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain , and so forth. the same requirements of boundary conditions. Caputo's fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types, see [18, 19].
Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics  and cellular systems .
Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors such as, for instance, Arara and Benchohra , Benchohra et al. [23, 24], Infante , Peciulyte et al. , and the references therein.
In our investigation we apply the method associated with the technique of measures of noncompactness and the fixed point theorem of Mönch type. This technique was mainly initiated in the monograph of Bana and Goebel  and subsequently developed and used in many papers; see, for example, Bana and Sadarangoni , Guo et al. , Lakshmikantham and Leela , Mönch , and Szufla .
In this section, we present some definitions and auxiliary results which will be needed in the sequel.
Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.
Definition 2.1 (see ).
The Kuratowski measure of noncompactness satisfies some properties (for more details see ).
For completeness we recall the definition of Caputo derivative of fractional order.
Definition 2.2 (see ).
Definition 2.3 (see ).
For our purpose we will only need the following fixed point theorem and the important Lemma.
Lemma 2.6 (see ).
3. Existence of Solutions
Let us start by defining what we mean by a solution of the problem (1.1).
Lemma 3.2 (see ).
For the forthcoming analysis, we introduce the following assumptions
then the boundary value problem (1.1) has at least one solution.
By (3.8) it follows that , that is, for each , and then is relatively compact in . In view of the Ascoli-Arzelà theorem, is relatively compact in . Applying now Theorem 2.5 we conclude that has a fixed point which is a solution of the problem (1.1).
4. An Example
Clearly, conditions (H1),(H2) hold with
The authors thank the referees for their remarks. The research of A. Cabada has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.
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