Existence results in Banach space for a nonlinear impulsive system

Abstract

We deal with three important aspects of a generalized impulsive fractional order differential equation (DE) involving a nonlinear p-Laplacian operator: the existence of a solution, the uniqueness and the Hyers-Ulam stability. Our problem involves Caputo's fractional derivative. For these goals, we establish an equivalent fractional integral form of the problem and use a topological degree approach for the existence and uniqueness of the solution (EUS). Next, we check the stability of the suggested problem and then demonstrate the results via an illustrative example. In the literature, we could not find articles on the Hyers-Ulam stability of the impulsive fractional order DEs with φ p operator.

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

“…In this paper, we mainly consider a kind of ψ-Hilfer fractional order differential equation. The addressed equation has time-varying delay terms and non-instantaneous impulsive effects, which are quite different from the related references discussed in the literature [18,19,21,22,[38][39][40][41][42]. The nonlinear fractional order differential system studied in the present paper is more generalized and more practical.…”

“…In [20], Ameen et al studied the Ulam stability and existence theorems for Caputo generalized fractional differential equations where the kernel of the fractional derivative was function dependent so that the result generalized many existing results in history. Further, for more details about some other properties of the solutions, we can see [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”

Some new local fractional inequalities associated with generalized $(s,m)$-convex functions and applications

“…Fractional calculus based on differential and difference equations is of considerable importance due to their connection with real-world problems that depend not only on the instant time but also on the previous time, in particular, modeling the phenomena by means of fractals, random walk processes, control theory, signal processing, acoustics, and so on (see [1][2][3][4][5][6][7][8][9][10][11][12]). It has been shown that fractional-order models are much more adequate than integer-order models.…”

Existence and data-dependence theorems for fractional impulsive integro-differential system

“…We suggest the readers for more detail about the fractional calculus and its application to the work in [1][2][3][4][5][6][7]. Among the fractional operators, the Caputo-Fabrizio [8][9][10] and the Atangana-Baleanu fractional differential operators with nonsingular kernel [4][5][6][7][11][12][13][14][15] are recently well studied operators.…”