Integral boundary value problems for nonlinear non-instantaneous impulsive differential equations

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

“…Compared with some recent results in the literature, such as [6,[8][9][10][11][12][13] and some others, the chief contributions of our study contain at least the following four issues:…”

“…(1) In [6,[8][9][10], authors discussed several types of stability except the finite-time stability, and we first introduce the definition of finite-time stability into the ψ-Hilfer fractional differential equation with non-instantaneous impulses. (2) Compare with [6,[8][9][10][11][12][13], in system (1.4), we study the equation with time-varying delays, which is a significant breakthrough in dealing with a non-instantaneous impulsive ψ-Hilfer fractional differential system. 3The model we are concerned with is more generalized, and some ones in the references are the special cases of it.…”

“…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].…”

“…The study of the existence of solutions for fractional differential equations is one of the most interesting and valuable topics [8,11,[18][19][20][21][22]. In recent contributions [18,19] one studied equations of the following form: where c D α 0,t is the Caputo fractional derivative of the order α ∈ (n -1, n), n ∈ N, f : J × R → R, J = [0, T] and I k : R → R and 0 = τ 0 < τ 1 < • • • < τ m < τ m+1 = T, and we let x(τ + k ) = lim ε→0 + x(τ k + ε) and x(τk ) = lim ε→0 -x(τ k + ε) represent the right and left limits of x(t) at t = τ k , respectively.…”

“…We can see that Yang and Wang studied the following integral boundary value problems for fractional order nonlinear differential equations with non-instantaneous impulses in [11]:…”