Existence of solutions of infinite system of nonlinear sequential fractional differential equations

Zahra Ahmadi,Giuseppe Caristi,Hamid Baghani,Rahmatollah Lashkaripour,Shapour Heidarkhani

doi:10.1186/s13662-020-02682-1

Abstract

In a recent paper (Filomat 32:4577–4586, 2018) the authors have investigated the existence and uniqueness of a solution for a nonlinear sequential fractional differential equation. To present an analytical improvement for Fazli–Nieto’s results with some conditions removed based on a new technique is the main objective of this paper. In addition, we introduce an infinite system of nonlinear sequential fractional differential equations and discuss the existence of a solution for them in the classical Banach sequence spaces c_{0} and ell_{p} by applying the Darbo fixed point theorem. Moreover, the proposed method is applied to several examples to show the clarity and effectiveness.

Highlights

1 Introduction and preliminaries As is well known, the fractional differential equations (FDEs) is a fundamental topic that considered as a powerful tool in many fields, for example, dynamic systems, rheology, blood flow phenomena, biophysics, electrical networks, modeled by different fractional order derivatives equations; see for details [2,3,4,5] and the references therein
On the other hand, during the last years, many studies have been done on the existence and uniqueness of solution of nonlinear initial fractional differential equations by the use of some fixed point theorems; see [9,10,11,12,13,14,15,16,17,18,19,20]
We address the following questions. (Q1) Is it possible to remove the non-decreasing conditions of the mappings f in Theorem 1.1 and Theorem 1.2? (Q2) Is it possible to remove assumption of the existence of a lower solution of the problems (1) and (4)? (Q3) Is it possible to define the problem (1) as an infinite system and discuss the existence results of the solution to it in spaces c0 and p? In the sequel, we prove that the non-decreasing condition of function f in Theorem 1.1 and Theorem 1.2 is not necessary

Summary

This problem is a special case of with α

It is easy to see that f x, u(x), Dαu(x) – f x, v(x), Dαv(x) ≤ ν2 v(x) – u(x) + ν1 Dαv(x) – Dαu(x). Applying Theorem 2.1 the linear initial value problem (6) possesses a unique solution in C1α–α[0, γ ]. It is simple to verify that Theorem 1.1 cannot be applied to our example. Because f is not increasing in all its arguments except for the first argument, that is, the condition (H2) of Theorem 1.1 is not satisfied

Let α

TJ p

Then we get

For every