Weak solutions of a fractional orders quadratic functional integro-differential inclusion in a reflexive Banach space

Abstract

Let E be a reflexive Banach space. In this paper we are concerned with the existence of weak solutions x∈C(I,E) of the quadratic functional integro-differential inclusion of fractional orders, α,β∈(0,1),

^RD^α(x(t)−x(0))∈G(t,g₁(t,x(t))I^βg₂(t,^RD^βx(t)), t∈[0,T ]

with the initial condition, x(0)=x₀, x₀∈E , in the reflexive Banach space E under the assumption that the multivalued function G satisfy Lipschitz condition in E . The main tool applied in this work is O’Regan fixed point theorem. We investigate qualitative properties of the solution of this inclusion such as the continuous dependence on the set of selections SG and the continuous dependence on the data x0. Here, we prove two new theorems on the mentioned properties of the solution of the considered quadratic functional integro-differential inclusion of fractional orders. We additionally provide an example given as numerical application to illustrate our main result.