Abstract
Abstract In this manuscript, we investigate the existence, uniqueness, Hyer-Ulam stability and controllability analysis for a fractional dynamic system on time scales. Mainly, this manuscript has three segments: In the first segment, we give the existence of solutions. The second segment is devoted to the study of stability analysis while in the last segment, we establish the controllability results. We use the Banach and nonlinear alternative Lery-Schauder–type fixed point theorem to establish these results. Also, we give some numerical examples for different time scales. Moreover, we give two applications to outline the effectiveness of these obtained results.
Highlights
The theory of fractional calculus began with a correspondence between Leibniz and L’Hospitalin in 1695
In this manuscript, we investigate the existence, uniqueness, Hyer-Ulam stability and controllability analysis for a fractional dynamic system on time scales
One of the qualitative principles which could be very significant from the numerical factor, and optimization view is committed to stability analysis of the solution to dynamical systems of integer as well as fractional order
Summary
The theory of fractional calculus began with a correspondence between Leibniz and L’Hospitalin in 1695. Inspired by the above works, for the existence and stability results, we consider the fractional dynamic system on time scales of the form c,TDκp(t) = Φ(t, p(t), q(t), r(t)), t ∈ I = [0, T]T, κ ∈ (0, 1) , c,TDνq(t) = Ψ(t, p(t), q(t), r(t)), t ∈ I, ν ∈ (0, 1) , c,TDωr(t) = Θ(t, p(t), q(t), r(t)), t ∈ I, ω ∈ (0, 1) ,. We considered a more general form of the fractional dynamic system on time scales which includes these mathematical models, as well as many more models for the continuous, discrete and combination of these two In this setting, our results are new and contribute significantly to the existing literature on the topic.
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