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.
1. Introduction
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[1], Glockle and Nonnenmacher [2], Metzler et al. [3], Podlubny [4], Gaul et al. [5], among others. Fractionaldifferential equations are also often an object of mathematical investigations; see the papers of Agarwal et al. [6], Ahmad and Nieto [7], Ahmad and OteroEspinar [8], Belarbi et al. [9], Belmekki et al [10], Benchohra et al. [11–13], Chang and Nieto [14], DaftardarGejji and Bhalekar [15], Figueiredo Camargo et al. [16], and the monographs of Kilbas et al. [17] and Podlubny [4].
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 RiemannLiouville and Caputo types, see [18, 19].
In this paper we investigate the existence of solutions for boundary value problems with fractional order differential equations and nonlinear integral conditions of the form
where is the Caputo fractional derivative, , , and are given functions satisfying some assumptions that will be specified later, and is a Banach space with norm .
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 [20] and cellular systems [21].
Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors such as, for instance, Arara and Benchohra [22], Benchohra et al. [23, 24], Infante [25], Peciulyte et al. [26], 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 [27] and subsequently developed and used in many papers; see, for example, Bana and Sadarangoni [28], Guo et al. [29], Lakshmikantham and Leela [30], Mönch [31], and Szufla [32].
2. Preliminaries
In this section, we present some definitions and auxiliary results which will be needed in the sequel.
Denote by the Banach space of continuous functions , with the usual supremum norm
Let be the Banach space of measurable functions which are Bochner integrable, equipped with the norm
Let be the Banachspace of measurable functions which are bounded, equipped with the norm
Let be the space of functions , whose first derivative is absolutely continuous.
Moreover, for a given set of functions let us denote by
Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.
Definition 2.1 (see [27]).
Let be a Banach space and the bounded subsets of . The Kuratowski measure of noncompactness is the map defined by
Properties
The Kuratowski measure of noncompactness satisfies some properties (for more details see [27]).
(a) is compact ( is relatively compact).
(b).
(c).
(d)
(e)
(f).
Here and denote the closure and the convex hull of the bounded set , respectively.
For completeness we recall the definition of Caputo derivative of fractional order.
Definition 2.2 (see [17]).
The fractional order integral of the function of order is defined by
where is the gamma function. When , we write where
for , and as .
Here is the delta function.
Definition 2.3 (see [17]).
For a function given on the interval , the Caputo fractionalorder derivative of , of order is defined by
Here and denotes the integer part of .
Definition 2.4.
A map is said to be Carathéodory if
(i) is measurable for each
(ii) is continuous for almost each
For our purpose we will only need the following fixed point theorem and the important Lemma.
Let be a bounded, closed and convex subset of a Banach space such that , and let be a continuous mapping of into itself. If the implication
holds for every subset of , then has a fixed point.
Lemma 2.6 (see [32]).
Let be a bounded, closed, and convex subset of the Banach space , G a continuous function on and a function satisfies the Carathéodory conditions, and there exists such that for each and each bounded set one has
If is an equicontinuous subset of , then
3. Existence of Solutions
Let us start by defining what we mean by a solution of the problem (1.1).
Definition 3.1.
A function is said to be a solution of (1.1) if it satisfies (1.1).
Let be continuous functions and consider the linear boundary value problem
Lemma 3.2 (see [11]).
Let and let be continuous. A function is a solution of the fractional integral equation
with
if and only if is a solution of the fractional boundary value problem (3.1).
Remark 3.3.
It is clear that the function is continuous on , and hence is bounded. Let
For the forthcoming analysis, we introduce the following assumptions
(H1)The functions satisfy the Carathéodory conditions.
(H2)There exist , such that
(H3)For almost each and each bounded set we have
Theorem 3.4.
Assume that assumptions hold. If
then the boundary value problem (1.1) has at least one solution.
Proof.
We transform the problem (1.1) into a fixed point problem by defining an operator as
where
and the function is given by (3.4). Clearly, the fixed points of the operator are solution of the problem (1.1). Let and consider the set
Clearly, the subset is closed, bounded, and convex. We will show that satisfies the assumptions of Theorem 2.5. The proof will be given in three steps.
Step 1.
is continuous.
Let be a sequence such that in . Then, for each ,
Let be such that
By (H2) we have
Since and are Carathéodory functions, the Lebesgue dominated convergence theorem implies that
Step 2.
maps into itself.
For each , by and (3.8) we have for each
Step 3.
is bounded and equicontinuous.
By Step 2, it is obvious that is bounded.
For the equicontinuity of . Let , and . Then
As , the righthand side of the above inequality tends to zero.
Now let be a subset of such that .
is bounded and equicontinuous, and therefore the function is continuous on . By (H3), Lemma 2.6, and the properties of the measure we have for each
This means that
By (3.8) it follows that , that is, for each , and then is relatively compact in . In view of the AscoliArzelà 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
In this section we give an example to illustrate the usefulness of our main results. Let us consider the following fractional boundary value problem:
Set
Clearly, conditions (H1),(H2) hold with
From (3.4) the function is given by
From (4.4), we have
A simple computation gives
Condition (3.8) is satisfied with . Indeed
which is satisfied for each . Then by Theorem 3.4 the problem (4.1) has a solution on .
Acknowledgments
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 MTM200761724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.
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