Some existence theorems for fractional integro-differential equations and inclusions with initial and non-separated boundary conditions

Bashir Ahmad,Ahmed Alsaedi,Shahram Rezapour,Sayyedeh Zahra Nazemi

doi:10.1186/s13661-014-0249-5

Abstract

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.

Highlights

1 Introduction The subject of fractional calculus has recently been investigated in an extensive manner
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

Summary

Introduction

The subject of fractional calculus has recently been investigated in an extensive manner. ([ ]) Let E be a Banach space, G : I × E → Pcp,c(E) an L -Carathéodory multifunction, and θ a linear continuous mapping from L (I, E) to C(I, E). If F : U → Pcp,c(C) is an upper semi-continuous compact map, either F has a fixed point in U or there is a x ∈ ∂U and λ ∈ ( , ) such that x ∈ λF(x). Following the procedure employed in the last result, one can show that is continuous and completely continuous and satisfies all conditions of the nonlinear alternative of Leary-Schauder type for single-valued maps. Since the multivalued operator t −→ V (t) ∩ F(t, z(t)) is measurable (Proposition III- in [ ]), there exists a function f ∈ SF,zsuch that f (t) – f (t).

Here α θ

Here α

As in the