Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions
| Autor: | Valeri Obukhovskii, Mikhail Kamenskii, Jen-Chih Yao, Garik Petrosyan |
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| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
0211 other engineering and technologies Banach space Fixed-point theorem Index of the solution set 02 engineering and technology 01 natural sciences Separable space Differential inclusion 0101 mathematics Approximation Mathematics Cauchy problem T57-57.97 QA299.6-433 021103 operations research Applied mathematics. Quantitative methods Generator (category theory) Semidiscretization Applied Mathematics Order (ring theory) Multimap Fractional differential equation Fractional calculus 010101 applied mathematics Geometry and Topology Semilinear differential equation Analysis |
| Zdroj: | Fixed Point Theory and Applications, Vol 2019, Iss 1, Pp 1-21 (2019) |
| ISSN: | 1687-1812 |
| Popis: | We study a semilinear fractional order differential inclusion in a separable Banach space E of the form $$ {}^{C}D^{q}x(t)\in Ax(t)+ F\bigl(t,x(t)\bigr),\quad t\in [0,T], $$ where ${}^{C}D^{q}$ is the Caputo fractional derivative of order $0 < q < 1$ , $A \colon D(A) \subset E \rightarrow E$ is a generator of a $C_{0}$ -semigroup, and $F \colon [0,T] \times E \multimap E$ 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: $$ x(0)\in \Delta (x), $$ where $\Delta : C([0,T];E) \multimap E$ is a given multivalued map. The semidiscretization scheme is developed and applied to the approximation of solutions to the considered nonlocal boundary value problem. |
| Databáze: | OpenAIRE |
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