Solvability of fractional differential inclusions with nonlocal initial conditions via resolvent family of operators

Yong-Kui Chang,Rodrigo Ponce,Xu-Sheng Yang

doi:10.1515/ijnsns-2019-0210

Abstract

Abstract In this paper, we consider mild solutions to fractional differential inclusions with nonlocal initial conditions. The main results are proved under conditions that (i) the multivalued term takes convex values with compactness of resolvent family of operators; (ii) the multivalued term takes nonconvex values with compactness of resolvent family of operators and (iii) the multivalued term takes nonconvex values without compactness of resolvent family of operators, respectively.

Highlights

The main results are proved under conditions that (i) the multivalued term takes convex values with compactness of resolvent family of operators; (ii) the multivalued term takes nonconvex values with compactness of resolvent family of operators and (iii) the multivalued term takes nonconvex values without compactness of resolvent family of operators, respectively
A differential inclusion is a generalization of the notion of an ordinary differential equation, which is often used to deal with differential equations with a discontinuous righthand side or an inaccurately known right-hand side [1, 2]
It is founded that the aforementioned control system has the same trajectories as the differential inclusion x′ ∈f (x, U) ⋃u∈U f (x, u)

Summary

Introduction

A differential inclusion is a generalization of the notion of an ordinary differential equation, which is often used to deal with differential equations with a discontinuous righthand side or an inaccurately known right-hand side [1, 2]. For nonlocal initial conditions of abstract differential inclusions, we can refer to [4, 6, 10, 11] and references therein. Some new properties on the compactness of resolvent family of operators related to fractional differential equations have been established [35]. We consider the following abstract fractional differential inclusions with nonlocal initial conditions: Dαt x(t) ∈ Ax(t) + J1t−αF(t, x(t)), t ∈ J: [0, b] (1.1). Chang et al.: Fractional differential inclusions with nonlocal initial conditions where 0 < α < 1, Jβt v(t). SG, x f ∈ L1(J, X) : f (t) ∈ G(t, x(t)), for a.e. t ∈ J ≠ Ø, and let Γ be a linear continuous mapping from L1(J, X) to C(J, X), the operator Γ ∘ SG : C(J, X) → Pcp, cv(C(J, X)), x ↦ (Γ ∘ SG)(x): Γ SG, x is a closed graph operator in C(J, X) × C(J, X)

Preliminaries

Existence results