Multi-term fractional differential equations in a nonreflexive Banach space

Ravi P Agarwal,Donal O’Regan,Vasile Lupulescu,Ghaus Ur Rahman

doi:10.1186/1687-1847-2013-302

Abstract

In this paper we establish an existence result for the multi-term fractional differential equation 1 where and are fractional pseudo-derivatives of a weakly absolutely continuous and pseudo-differentiable function of order and , , respectively, the function is weakly-weakly sequentially continuous for every and is Pettis integrable for every weakly absolutely continuous function , T is a bounded interval of real numbers and E is a nonreflexive Banach space, and are real numbers such that .

Highlights

1 Introduction The mathematical field that deals with derivatives of any real order is called fractional calculus
We need to solve fractional differential equations containing more than one differential operator, and this type of fractional differential equation is called a multi-term fractional differential equation
The existence of solutions of multi-term fractional differential equations was studied by many authors [ – ]

Summary

Introduction

The mathematical field that deals with derivatives of any real order is called fractional calculus. (b) A function x(·) : T → E is said to be weakly absolutely continuous (wAC) on T if for every x∗ ∈ E∗, the real-valued function t → x∗, x(t) is AC on T. In Kadets [ ] proved that there exists a strongly measurable and Pettis integrable function x(·) : T → E such that the indefinite Pettis integral t y(t) = x(s) ds, t ∈ T, is not weakly differentiable on a set of positive Lebesgue measures (see [ , ]). Let us denote by P∞(T, E) the space of all weakly measurable and Pettis integrable functions x(·) : T → E with the property that x∗, x(·) ∈ L∞(T) for every x∗ ∈ E∗. If y(·) : T → E is a pseudo-differentiable function with a pseudo-derivative x(·) ∈ P∞(T, E) on T, the following fractional Pettis integral.

Along with

If we let