Periodic boundary value problems for fractional semilinear integro-differential equations with non-instantaneous impulses

Abstract

In this paper, we study periodic boundary value problems of fractional semilinear integro-differential equations with non-instantaneous impulses in Banach spaces. By the measure of noncompactness, the theory of β-resolvent family, and the fixed point theorem, we obtain several sufficient conditions on the existence of mild solutions for such problems. Finally, an example is given to show the main results of this paper.

Stability and solvability analysis for a class of optimal control problems described by fractional differential equations with non-instantaneous impulses

“…According to the instantaneity and continuity of the effects, impulses are divided into instantaneous and non-instantaneous ones. Most of the mathematical models extracted from impulsive phenomena are characterized by impulsive differential equations, which can be classified under two categories in accordance with the types of impulses: non-instantaneous impulsive differential equations [1][2][3][4][5][6][7][8]and instantaneous ones [9][10][11][12][13] .…”

“…In [10], Zhu and Liu studied the following periodic boundary value problem of nonlinear evolution equations of fractional order with non-instantaneous impulses…”

Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

“…Recently, we found that many researchers have set out to study a new type of impulsive (called non-instantaneous impulsive) fractional differential equations, where the impulsive action starts at an arbitrary fixed point and remains active on a finite time interval, which is very different from the classical instantaneous impulsive case that changes are relatively short compared to the overall duration of the process. For an extensive collection of non-instantaneous impulsive results, we refer the reader to the related literature, such as the monograph [5] and the papers [6][7][8][9][10][11][12][13][14][15][16][17].…”

On a delayed epidemic model with non-instantaneous impulses

“…For example, the introduction of a drug or a vaccine and the absorption for the body is a gradual process, then one is forced to consider the vaccine as a non-instantaneous impulse since it starts abruptly but remains active on a finite interval of time. Some recent results on non-instantaneous impulsive differential equations can be found in [1,3,4,11,21,13,23,20,24,6,14] and in monograph [2]. Further technical details are given in Section 2.…”