Abstract
Abstract The main motive of this research article is to establish the existence, uniqueness and stability results for the non-linear fractional differential equation with impulsive condition on time scales. Banach, Leray-Schauder’s alternative type fixed point theorems are used to examine these results. Further, we give the existence and uniqueness of solution for the corresponding non-local problem. Moreover, to outline the utilization of these outcomes some examples are given.
Highlights
In many physical problems of engineering and science the fractional di erential equations are arise for e.g control problem, image processing, signal identi cation, optical systems and so on, see [1, 2] and references therein
A lot of attention has been given by scholars to it and established the various results on existence and uniqueness of di erent type of fractional di erential equations
Numerous researchers paid much attention to the existence of initial and boundary value problem with fractional di erential equations with di erent-di erent boundary conditions using the various type of xed point theorems like contraction, Schaefer’s, Schauder’s, Krasnoselskii’s and so on, see [3,4,5,6,7,8,9,10] and references therein
Summary
In many physical problems of engineering and science the fractional di erential equations are arise for e.g control problem, image processing, signal identi cation, optical systems and so on, see [1, 2] and references therein. Ahmadkhanlu et al [23], give the necessary and su cient conditions for the existence and uniqueness of solution to the fractional order di erential equations (FODE) on time scales: c∆α z(θ) = F(θ, z(θ)), ∀ θ ∈ J = [θ , θ + a] ⊆ T, α ∈ ( , ], z(θ ) = z , where F : J ×R → R is a right-dense continuous bounded function and c∆α is the Caputo fractional derivative. ≤ θ < θ < θ < · · · < θp < θp+ = T, z(θ−l ) = limh→ + z(θl − h), z(θ+l ) = limh→ + z(θl + h) denotes the left and right limits of z(θ) at θ = θl Another thing of the qualitative principle which could be very important from optimization and numerical point of view is dedicated to stability analysis of the solution to di erential equations of integer as well as fractional order. |Jl(θ, w) − Jl(θ, z)| ≤ LJl |w − z|, ∀ θ ∈ I, w, z ∈ R
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