Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions
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...
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2019
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oai:doaj.org-article:d332611b40f545d9a36f79d240ffce682021-12-02T12:13:25ZExistence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions10.1186/s13663-018-0652-11687-1812https://doaj.org/article/d332611b40f545d9a36f79d240ffce682019-01-01T00:00:00Zhttp://link.springer.com/article/10.1186/s13663-018-0652-1https://doaj.org/toc/1687-1812Abstract 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.M. KamenskiiV. ObukhovskiiG. PetrosyanJen-Chih YaoSpringerOpenarticleFractional differential equationSemilinear differential equationCauchy problemApproximationSemidiscretizationIndex of the solution setApplied mathematics. Quantitative methodsT57-57.97AnalysisQA299.6-433ENFixed Point Theory and Applications, Vol 2019, Iss 1, Pp 1-21 (2019) |
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DOAJ |
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EN |
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Fractional differential equation Semilinear differential equation Cauchy problem Approximation Semidiscretization Index of the solution set Applied mathematics. Quantitative methods T57-57.97 Analysis QA299.6-433 |
| spellingShingle |
Fractional differential equation Semilinear differential equation Cauchy problem Approximation Semidiscretization Index of the solution set Applied mathematics. Quantitative methods T57-57.97 Analysis QA299.6-433 M. Kamenskii V. Obukhovskii G. Petrosyan Jen-Chih Yao Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions |
| description |
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. |
| format |
article |
| author |
M. Kamenskii V. Obukhovskii G. Petrosyan Jen-Chih Yao |
| author_facet |
M. Kamenskii V. Obukhovskii G. Petrosyan Jen-Chih Yao |
| author_sort |
M. Kamenskii |
| title |
Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions |
| title_short |
Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions |
| title_full |
Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions |
| title_fullStr |
Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions |
| title_full_unstemmed |
Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions |
| title_sort |
existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions |
| publisher |
SpringerOpen |
| publishDate |
2019 |
| url |
https://doaj.org/article/d332611b40f545d9a36f79d240ffce68 |
| work_keys_str_mv |
AT mkamenskii existenceandapproximationofsolutionstononlocalboundaryvalueproblemsforfractionaldifferentialinclusions AT vobukhovskii existenceandapproximationofsolutionstononlocalboundaryvalueproblemsforfractionaldifferentialinclusions AT gpetrosyan existenceandapproximationofsolutionstononlocalboundaryvalueproblemsforfractionaldifferentialinclusions AT jenchihyao existenceandapproximationofsolutionstononlocalboundaryvalueproblemsforfractionaldifferentialinclusions |
| _version_ |
1718394601998385152 |