Coupled Systems of Sequential Caputo and Hadamard Fractional Differential Equations with Coupled Separated Boundary Conditions

Suphawat Asawasamrit,Sotiris Ntouyas,Jessada Tariboon,Woraphak Nithiarayaphaks

doi:10.3390/sym10120701

Abstract

This paper studies the existence and uniqueness of solutions for a new coupled system of nonlinear sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions, which include as special cases the well-known symmetric boundary conditions. Banach’s contraction principle, Leray–Schauder’s alternative, and Krasnoselskii’s fixed-point theorem were used to derive the desired results, which are well-illustrated with examples.

Highlights

Fractional differential equations appear in the mathematical modeling of many real-world phenomena occurring in engineering and scientific disciplines, for instance, see References [1,2,3,4,5,6]
The different definitions of Caputo and Hadamard fractional derivatives that appeared in System (2) are proposed to study the existence theory of solutions of a fractional differential system using a variety of fixed-point theorems
Taking the Riemann–Liouville fractional integral of order p1, p1 ∈ (0, 1], to the first equation of Problem (5) and applying Problem (4), we obtain for t ∈ [ a, b]

Summary

Introduction

Fractional differential equations appear in the mathematical modeling of many real-world phenomena occurring in engineering and scientific disciplines, for instance, see References [1,2,3,4,5,6]. In Reference [19], the authors discussed existence and the uniqueness of solutions for sequential Caputo and Hadamard fractional differential equations subject to separated boundary conditions as. Α1 x ( a) + α2 ( H D q x )( a) = 0, β 1 x (b) + β 2 ( H D q x )(b) = 0, where C D p and H D q are the Caputo and Hadamard fractional derivatives of orders p and q, respectively,. We established the existence criteria for a coupled system of sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions as: C. The different definitions of Caputo and Hadamard fractional derivatives that appeared in System (2) are proposed to study the existence theory of solutions of a fractional differential system using a variety of fixed-point theorems.

Preliminaries

Main Results

Conclusions