Abstract
This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and Krasnosel’skii’s fixed point theorem. To illustrate the results, an example is provided.
Highlights
Fractional calculus is that branch of classical analysis that generalizes derivatives and integrals of integer order to non-integer orders [1,2,3]
In [15], by means of Monch’s fixed point theorem, Subashini et al consider the existence of mild solutions to a class of evolution equations involving the Hilfer derivative
Differential equations with impulses often serve as models in studying the dynamics of processes that are subject to sudden changes in their states
Summary
Fractional calculus is that branch of classical analysis that generalizes derivatives and integrals of integer order to non-integer orders [1,2,3]. In [3], the authors studied some new classes of abstract impulsive differential equations with instantaneous impulses; for some very interesting results on equations with non-instantaneous impulses, we refer the reader to [16,17,18]. Motivated by the results in the above mentioned papers, here we establish some new existence and stability results for the boundary value problem with nonlinear implicit generalized Hilfer-type fractional differential equations with non-instantaneous impulses α θ,r α θ,r. 1− ξ where α Dθ,r and α Ja+ are the generalized Hilfer-type fractional derivative of order θ ∈. We discuss the Ulam-Hyers-Rassias stability of our problem in Section 4, and in Section 5 we give an example to illustrate the applicability of our main results
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