Abstract degenerate fractional differential inclusions in Banach spaces

Marko Kostic

doi:10.2298/aadm1701039k

Abstract

In the paper under review, we investigate a class of abstract degenerate fractional differential inclusions with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.

Highlights

AND PRELIMINARIESThe main purpose of this paper is to consider a class of abstract degenerate fractional differential inclusions with multivalued linear operators satisfying the following condition (QP): There exist finite numbers 0 < β ≤ 1, 0 < d ≤ 1, M > 0 and 0 < η < η ≤ 1 such that Ψd,πη /2 := λ ∈ C : |λ| ≤ d or λ ∈ Σπη /2 ⊆ ρ(A)and R(λ : A) ≤ M 1 + |λ| −β, λ ∈ Ψd,πη /2. we continue our previous research study [19], where we have recently considered fractional resolvent families subordinated to infinitely differentible semigroups generated by the multivalued linear operators satisfying the following condition
It is almost straightforward to extend the results from [19] concerning subordinated degenerate fractional resolvent families and semilinear fractional Cauchy inclusion (DFP)f,s,η : Dηt u(t) ∈ Au(t) + f (t, u(t)), t ∈ (0, T ], u(0) = u0, where 0 < T < ∞, 0 < η < η and Dηt denotes the Caputo fractional derivative operator of order η ([2]); cf
The situation is slightly different with the assertion of [33, Theorem 5.3], where the authors have considered the existence of mild solutions of semilinear degenerate fractional Cauchy inclusion (DFP)f,s,η, provided that the resolvent of A is compact

Summary

INTRODUCTION

The main purpose of this paper is to consider a class of abstract degenerate fractional differential inclusions with multivalued linear operators satisfying the following condition (QP): There exist finite numbers 0 < β ≤ 1, 0 < d ≤ 1, M > 0 and 0 < η < η ≤ 1 such that Ψd,πη /2 := λ ∈ C : |λ| ≤ d or λ ∈ Σπη /2 ⊆ ρ(A). The main contributions of paper are contained, where we transfer the assertions of [11, Theorem 3.1, Theorem 3.3, Theorem 3.5; Proposition 3.4] from semigroup case to fractional relaxation case (we have faced ourselves with some serious difficulties concerning the fractional analogue of the assertion [11, Proposition 3.2], when we are no longer in a position to conclude that the operator Tη ,r+θ(z) defined below is a bounded linear section of the operator (−A)θTη ,r(z)) Having this done, it is almost straightforward to extend the results from [19] concerning subordinated degenerate fractional resolvent families and semilinear fractional Cauchy inclusion (DFP)f,s,η : Dηt u(t) ∈ Au(t) + f (t, u(t)), t ∈ (0, T ], u(0) = u0, where 0 < T < ∞, 0 < η < η and Dηt denotes the Caputo fractional derivative operator of order η ([2]); cf Section 4 for more details.

MULTIVALUED LINEAR OPERATORS

SUBORDINATED FRACTIONAL RESOLVENT FAMILIES

SUBORDINATION PRINCIPLES