Conditions to guarantee the existence of solutions for a nonlinear and implicit integro‐differential equation with variable coefficients

Abstract

Using Babenko's approach, multivariate Mittag–Leffler (MM–L) function, and Krasnoselskii's fixed point theorem, we first investigate the existence of solutions to a Liouville–Caputo nonlinear integro‐differential equation with variable coefficients and initial conditions in a Banach space. Then the existence of a positive solution for a variant equation is studied. Finally, we provide examples to illustrate the applications of the main results obtained.

Uniqueness and Hyers–Ulam’s stability for a fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition

“…There are intensive studies on fractional PDEs using various approaches, such as integral transforms [8], analytical and numerical solutions [10], homotopy analysis technique [2,11], variational iteration method [12] and so on. Very recently, Li et al [7] investigated the uniqueness of solutions for the following fractional PDE with nonlocal initial value conditions for 2 < α ≤ 3, 0 < α 1 ≤ 1 and α 2 > 0 based on BCP, BA and the multivariate Mittag-Leffler function for a constant η:…”

Remarks on a fractional nonlinear partial integro-differential equation via the new generalized multivariate Mittag-Leffler function *

“…Babenko's approach [5] is a highly effective method that can be employed to solve various integral and differential equations [2,6]. To show this approach, we will consider the following equation in the space C[0, 1] (the space of all continuous functions over [0, 1]) for constants a and b:…”

“…Partial differential equations have played an important role in various scientific areas, such as physics and engineering [7,8,9]. There are many interesting studies on uniqueness and existence of solutions, based on the theory of fixed points, for fractional nonlinear PDEs with initial value or boundary condition problems, as well as for integral equations [6,10]. Ouyang and Zhu et al [11,12,13] studied the time fractional PDEs given below: [11] investigated the existence of the local solutions using Leray-Schauder's fixed point theorem.…”

Application of a matrix Mittag-Leffler function to the fractional partial integro-differential equation in R n

“…On the other hand, Babenko's approach [2], a technique introduced in 1986, is an effective means for dealing with both integral and differential equations that have initial conditions or boundary value problems as demonstrated in [4,7], respectively. To demonstrate this method, we will solve the following equation with initial conditions in the space C[0, 1]:…”