Abstract
Abstract Our aim in this work is to study the existence and the attractivity of solutions for a system of delay partial integro-differential equations of fractional order. We use the Schauder fixed point theorem for the existence of solutions, and we prove that all solutions are locally asymptotically stable. AMS (MOS) Subject Classifications: 26A33.
Highlights
Fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non-integer order
We use the Schauder fixed point theorem for the existence of solutions, and we prove that all solutions are locally asymptotically stable
Fractional differential and integral equations have recently been applied in various areas of Engineering, Mathematics, Physics and Bio-engineering and so on
Summary
Fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non-integer order. We established sufficient conditions for the existence and the attractivity of solutions of the following system of delay integro-differential equations of fractional order of the form cDrθ u(t, x) = f (t, x, I0r2,xu(t, x), u(t − τ1, x − ξ1), . Definition 2.6 Solutions of Equation (4) are locally attractive if there exists a ball B (u0, h) in the space BC such that, for arbitrary solutions v = v(t, x) and w = w(t, x) of Equation (4) belonging to B(u0, h) ∩ Ω, we have that, for each × Î [0, b], lim (v(t, x) − w(t, x)) = 0. (c) The functions from D are equiconvergent, that is, given ε > 0, x Î [0, b] there corresponds T (ε, x) > 0 such that |u(t, x)-limt®∞u(t, x)|
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