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

Abstract

Abstract 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.

Keywords

• Fractional differential equation
• Semilinear differential equation
• Cauchy problem
• Approximation
• Semidiscretization
• Index of the solution set