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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],$$ {}^{C}D^{q}x(t)\in Ax(t)+ F\bigl(t,x(t)\bigr),\quad t\in [0,T], $$ where DqC${}^{C}D^{q}$ is the Caputo fractional derivative of order 0<q<1$0 < q < 1$, A:D(A)⊂E→E$A \colon D(A) \subset E \rightarrow E$ is a generator of a C0$C_{0}$-semigroup, and F:[0,T]×E⊸E$F \colon [0,T] \times E \multimap E$ 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),$$ x(0)\in \Delta (x), $$ where Δ:C([0,T];E)⊸E$\Delta : C([0,T];E) \multimap E$ is a given multivalued map. The semidiscretization scheme is developed and applied to the approximation of solutions to the considered nonlocal boundary value problem.

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