Boundary Value Problem for Fractional Order Generalized Hilfer-Type Fractional Derivative with Non-Instantaneous Impulses

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.

On $k$-Generalized $\psi$-Hilfer Boundary Value Problems with Retardation and Anticipation

“…By (20), the operator T is a contraction on F. Hence, by Banach's contraction principle, T has a unique xed point x ∈ F, which is a solution to our problem (1)-(4).…”

“…In recent years, fractional calculus has proven to be a very valuable method for addressing the complexity structures from dierent branches of science and engineering. It concerns the generalization of the integer order dierentiation and integration of a function to non-integer order, and its theory and application are solid and growing works [1,2,3,7,13,14,15,20,21,22,23,24]. The authors of [16,5,6,12] explored the existence, stability and uniqueness of solutions for various problems with fractional dierential equation and inclusions concerning retarded or advanced arguments.…”

On Ψ-Hilfer Fractional Integro-Differential Equations with Non-Instantaneous Impulsive Conditions

“…For some problems involving N-InI in psychology see [32]. For some recent works, on N-InI FDEs, see, e.g., [33][34][35][36][37] and references therein.…”

“…In [33], A. Salim et al established the BVP for implicit fractional order generalized Hilfer-type FD with N-InI of the form…”

On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions

“…Non-instantaneous ( N-InI) systems are types of systems which are more suitable to study the dynamics of evolution processes. For more details, one can refer [17,[25][26][27][28][29][30].…”