Faedo-Galerkin approximations to fractional integro-differential equation of order $\alpha \in (1, 2]$ with deviated argument

In this paper, we consider a fractional integro-differential equation of order $\alpha \in (1, 2]$ with deviated argument in a separable Hilbert space $X$. We used the $\alpha$-order cosine family of linear operators and Banach fixed point theorem to study the existence and uniqueness of approximate solutions. We define the fractional power of the closed linear operator and used it to prove the convergence of the approximate solutions. Also, we prove the existence and convergence of the Faedo–Galerkin approximate solutions. Finally, an example is provided to illustrate the application of these abstract results.

Keywords

fractional integro-differential equation with deviated argument, $\alpha \in$-order cosine family, Faedo–Galerkin Approximation, Banach fixed point theorem

2010 Mathematics Subject Classification

34A08, 34K30, 93B05, 93C25

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Published 16 December 2016