Continuous Dependence of Solutions of Integer and Fractional Order Non-Instantaneous Impulsive Equations with Random Impulsive and Junction Points

Abstract

This paper gives continuous dependence results for solutions of integer and fractional order, non-instantaneous impulsive differential equations with random impulse and junction points. The notion of the continuous dependence of solutions of these equations on the initial point is introduced. We prove some sufficient conditions that ensure the solutions to perturbed problems have a continuous dependence. Finally, we use numerical examples to demonstrate the obtained theoretical results.

Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative

“…for each x ∈ [0, T] and some function G(x) > 0, where F is defined in (10). From Theorem 2, then there exists a solution y a : [0, T] → R of (1) such that…”

“…Fractional differential operators describe mechanical and physical processes with historical memory and spatial global correlation and for the basic theory-see [1][2][3]. Results on existence, stability and controllability for differential equations with Caputo, Riemann-Liouville and Hilfer type fractional derivatives can be found, for example, in [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Caputo and Fabrizio [20] introduced a new nonlocal derivative without a singular kernel and Atangana and Nieto [21] studied the numerical approximation of this new fractional derivative and established a modified resistance loop capacitance (RLC) circuit model.…”

Extremal Solutions for a Class of Tempered Fractional Turbulent Flow Equations in a Porous Medium

“…As a powerful tool of modeling the above phenomena, in recent years, the fractional calculus theory has been perfected gradually by many researchers, and various different types of fractional derivatives were studied, such as Riemann-Liouville derivatives [16,, Hadamardtype derivatives [63][64][65][66][67][68][69][70][71], Katugampola-Caputo derivatives [72], conformable derivatives [73][74][75][76], Caputo-Fabrizio derivatives [77,78], Hilfer derivatives [79][80][81][82], and tempered fractional derivatives [83]. ese works also enlarged and enriched the application of the fractional calculus in impulsive theories [84][85][86][87][88][89], chaotic system [90][91][92][93], and resonance phenomena [94][95][96]. Among them, by using the fixed point theorem of the mixed monotone operator, Zhang et al [9] established the result of uniqueness of the positive solution for the Riemann-Liouville-type turbulent flow in a porous medium:…”

Solutions for a Singular Hadamard-Type Fractional Differential Equation by the Spectral Construct Analysis

“…In fluid mechanics, when a fluid is subjected to a severe impact to form a fracture, singular points or singular domains also follow the fracture. Normally, at singular points and domains, the extreme behaviour such as blow-up phenomena [2,3], impulsive influence [4][5][6][7][8][9], and chaotic system [10][11][12][13], often leads to some difficulties for people in understanding and predicting the corresponding natural problems. Hence, the study of singularity for complex systems governed by differential equations [14][15][16][17][18][19][20][21][22][23][24][25][26][27] is important and interesting in deepening the understanding of the internal laws of dynamic system.…”