Non‐instantaneous impulsive fractional integro‐differential equations with proportional fractional derivatives with respect to another function

Abstract

This paper concerns the existence and uniqueness of solutions of non‐instantaneous impulsive fractional integro‐differential equations with proportional fractional derivatives with respect to another function. By the aid of the nonlinear alternative of Leray‐Schauder type and the Banach contraction mapping principle, the main results are demonstrated. Two examples are inserted to illustrate the applicability of the theoretical results.

Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives

“…From inequality (29) it follows that v (0) (t) ≤ v (1) (t), t ∈ (0, T]. Then, from condition (A1) and equality (17), we get 1) , k = 0, 1, 2, . .…”

“…Remark 1. Note that the generalized proportional fractional derivative of Riemann-Liouville fractional type leads to an appropriate definition of the impulsive conditions similar to the initial condition (see the last two equations in problem (1). Additionally, we consider the case when the lower limit of the fractional derivative is changed at any impulsive point.…”

“…Therefore, function v (1) ∈ PC 1−α,ρ ([0, T]) is a mild lower solution of PIVP (1). From the definition of functions v (i) (•), i = 1, 2, conditions (A1), (A2) with…”

“…) is a mild solution of the PIVP ( 1) on [0, T], i.e., the functions V(•) and W(•) satisfy the initial value problem in (1).…”

“…Fractional differential equations are effective in both theoretical and applied mathematics and arise in models of medicine, engineering, biochemistry, thermal and mechanical systems, acoustics and modeling of materials, etc. There are different forms of fractional derivatives and consequently numerous fractional derivatives have appeared (see, for example, [1][2][3][4][5][6] and the references cited therein). Jarad et al [7] introduced a new generalized proportional derivative which is well-behaved and has several advantages over classical derivatives and generalizes known derivatives in the literature.…”