- Open Access
Some existence theorems for fractional integro-differential equations and inclusions with initial and non-separated boundary conditions
© Ahmad et al.; licensee Springer. 2014
Received: 19 September 2014
Accepted: 13 November 2014
Published: 9 December 2014
In this paper, we study the existence of solutions for a new class of boundary value problems of nonlinear fractional integro-differential equations and inclusions of arbitrary order with initial and non-separated boundary conditions. In the case of inclusion, the existence results are obtained for convex as well as non-convex multifunctions. Our results rely on the standard tools of fixed point theory and are well illustrated with the aid of examples.
The subject of fractional calculus has recently been investigated in an extensive manner. The publication of several books, special issues, and a huge number of articles in journals of international repute, exploring numerous aspects of this branch of mathematics, clearly indicates the popularity of the topic. One of the key factors accounting for the utility of the subject is that fractional-order operators are nonlocal in nature in contrast to the integer-order operators and can describe the hereditary properties of many underlying phenomena and processes. Owing to this characteristic, the principles of fractional calculus have played a significant role in improving the modeling techniques for several real world problems –.
Many researchers have focused their attention on fractional differential equations and inclusions, and a variety of interesting and important results concerning existence and uniqueness of solutions, stability properties of solutions, analytic and numerical methods of solutions of these equations have been obtained and the surge for investigating more and more results is still under way. For details and examples, we refer the reader to a series of papers – and the references therein. Anti-periodic boundary value problems occur in the mathematical modeling of a variety of physical processes and some works have been published in this area, for instance; see – and the references therein.
where and are continuous maps, (), (), (), , , and for .
The paper is organized as follows. In Section 2, we recall some preliminary facts that we used in the sequel. Section 3 deals with the existence result for single-valued initial boundary value problem, while the results for multivalued problem are presented in Section 4. We present some examples illustrating the main results in Section 5.
Let be a normed space, the set of all non-empty subsets of X, the set of all non-empty closed subsets of X, the set of all non-empty bounded subsets of X, the set of all non-empty compact subsets of X and the set of all non-empty compact and convex subsets of X. A multivalued map is said to be convex (closed) valued whenever is convex (closed) for all . The multifunction G is called bounded on bounded sets whenever is bounded subset of X for all , that is, for all . Also, the multifunction is called upper semi-continuous whenever for each the set is a non-empty closed subset of X, and for every open set N of X containing , there exists an open neighborhood of such that , . The multifunction is called compact whenever is relatively compact for all and also is called completely continuous whenever G is upper semi-continuous and compact , . It is well known that a compact multifunction G with non-empty compact valued is upper semi-continuous if and only if G has a closed graph, that is, , for all n, and imply . We say that is a fixed point of a multifunction G whenever . Let and a multifunction. We say that G is measurable whenever the function is measurable for all , .
The Riemann-Liouville fractional integral of order with the lower limit zero for a function is defined by for provided the integral exists.
where and .
Obviously is a Banach space.
We need the following result  in the sequel.
a bounded set. Then S has a fixed point in E.
The operatoris completely continuous.
Clearly the right-hand sides of the above inequalities tend to zero as . So T is completely continuous. This completes the proof. □