Abstract
This paper deals with the existence of mild solutions of a class of impulsive fractional partial neutral semilinear differential equations. A series of analytical results about the mild solutions are obtained by using fixed-point methods. Then, we present an example to further illustrate the applications of these results.
Highlights
Great progress in fractional differential equations has been achieved in recent years
In order to further investigate these models, it is essential to study the fractional differential equations analytically. ough, mathematically, a fractional differential equation is closely related to its corresponding ordinary differential equation or partial differential equation, that is, the ordinary differential equation or the partial differential equation can be obtained by letting αα = 1, 2, ... in its corresponding fractional differential equation, many mathematical methods which can be used in investigating ordinary differential equations or partial differential equations fail in analyzing fractional differential equations since fractional differential equations usually have more complicated structures and different properties. us, it is essential to develop novel methods to study fractional differential equations analytically
Impulsive differential equations can be used to investigate natural phenomena or dynamical processes which are subject to great changes in a short period of time [16,17,18]
Summary
Great progress in fractional differential equations has been achieved in recent years. One of the important techniques for analyzing impulsive differential equations is the semigroup theory, which has been successfully used in investigating the existence, uniqueness, and continuous dependence of the solutions of impulsive differential equations It has wide applications in the study of periodic and almost periodic solutions of different kinds of differential equations. Semigroup theory was used inappropriately to study the existence and uniqueness of mild solutions to impulsive fractional differential equations [13] and impulsive partial neutral functional differential equation [39]. Rough analyzing operator SSαα(tt), TTαα(tt), and the corresponding semigroup TT(tt), a sufficient condition, which guarantees the existence and uniqueness of solutions to the following system, is obtained: DDαttα (xx (tt) + FF (ttttttt)) + AAAA (tt) = GG (ttttttt) , tt t tt = [0, TT] , tt ≠ ttkk, (1).
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