Impulsive functional delay differential inclusions of fractional order at variable times

Hilmi Ergören

doi:10.1186/s13662-016-0770-1

Abstract

We are concerned with some sufficient conditions for the existence of solutions of a class of initial value problems for impulsive fractional differential inclusions with functional delay at variable moments.

Highlights

1 Introduction This work considers the existence of solutions to the following initial value problem (IVP) for a class of impulsive retarded fractional differential inclusions at variable times: CDα CDβ x(t) – g(t, xt) ∈ F(t, xt), t ∈ J, t = τk x(t), ( )
Where CDα and CDβ are Caputo fractional derivatives, < α, β ≤, < α + β
The subject of impulsive fractional differential equations and inclusions has generated a good deal of interest among a good many researchers due to fact that fractional calculus and impulsive theory arise in mathematical modeling of some certain problems in science and engineering [ – ]

Summary

Introduction

Where CDα and CDβ are Caputo fractional derivatives, < α, β ≤ , < α + β < , J = [ , T], < r < ∞, D = {ψ : [–r, ] → R is continuous everywhere except for a finite number of points s at which ψ(s–) and ψ(s+) exist and ψ(s–) = ψ(s)}, and φ ∈ D, F : J × D → P(R) is compact convex valued multivalued map (P(R) is the family of all nonempty subsets of R), g : J × D → R, Ik, Ik∗ : R → R, τk : R → R, k = , , .

Results

Conclusion