Abstract
The Cauchy problem for a class of fractional impulsive differential equations with delay
Highlights
IntroductionWe consider the Cauchy initial value problem (IVP for short) of fractional impulsive differential equations with delay of the form
X ∈ C([−τ, T ], R), for any t ∈ [0, T ], define xt by xt(θ) = x(t + θ) for θ ∈ [−τ, 0], here xt represents the history of the state from time t − τ to the present time t. φ ∈ C([−τ, 0], R) and φ(0) = 0
Fractional differential equations serve as an excellent tool for the description of memory and hereditary properties of various materials and processes
Summary
We consider the Cauchy initial value problem (IVP for short) of fractional impulsive differential equations with delay of the form. In [3, 7], Benchohra et al established sufficient conditions for the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the Caputo fractional derivative of order 0 < q < 1 and 1 < q < 2, respectively. Zhou et al [30] studied Cauchy initial value problem of fractional neutral functional differential equations with infinite delay, obtaining various criteria on existence and uniqueness. In consequence, motivated by the works mentioned above, the aim of this paper is to discuss the existence and uniqueness of solutions of fractional differential equations with delay and impulses in (1.1).
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