Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions

We study a semilinear fractional order differential inclusion in a separable Banach space E of the form DqCx(t)∈Ax(t)+F(t,x(t)),t∈[0,T], where DqC is the Caputo fractional derivative of order 0 < q< 1 , A: D(A) ⊂ E→ E is a generator of a C-semigroup, and F: [ 0 , T] × E⊸ E is a nonlinear multivalued map. By using the method of the generalized translation multivalued operator and a fixed point theorem for condensing multivalued maps, we prove the existence of a mild solution to this inclusion satisfying the nonlocal boundary value condition: x(0) ∈ Δ (x) , where Δ : C([ 0 , T] ; E) ⊸ E is a given multivalued map. The semidiscretization scheme is developed and applied to the approximation of solutions to the considered nonlocal boundary value problem. © 2019, The Author(s).