Existence and approximation of solutions to fractional differential equations. (English) Zbl 1165.34304
Summary: We study a fractional order semilinear differential equation in an arbitrary Banach space \(X\). We used the analytic semigroup theory of linear operators and fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of a global solution. Existence and convergence of an approximate solution to the given problem is also proved in a separable Hilbert space. Finally, we give an example to illustrate the applications of the abstract results.
MSC:
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 34K30 | Functional-differential equations in abstract spaces |
Keywords:
fractional differential equations; mild solution; Banach fixed point theorem; analytic semigroup; faedo-Galerkin approximationsReferences:
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