On positive solutions of a system of equations generated by Hadamard fractional operators

Hussein A. H. Salem,Amira M. Abdalla,Kinga Cichoń

doi:10.1186/s13662-020-02702-0

Abstract

This paper is devoted to studying some systems of quadratic differential and integral equations with Hadamard-type fractional order integral operators. We concentrate on general growth conditions for functions generating right-hand side of considered systems, which leads to the study of Hadamard-type fractional operators on Orlicz spaces. Thus we need to prove some properties of such type of operators. In contrast to the case of Caputo or Riemann–Liouville type of fractional operators, it is not a convolution-type operator, so we need to study some of their new properties. Some more general problems than systems of quadratic integral equations are also studied, and the results are new even in the context of a single integral equation with the Hadamard fractional operator. The paper concludes with illustrative examples.

Highlights

The study of quadratic differential and integral equations has a long history
Over the last years, the theory of quadratic integral equations with nonsingular kernels has received a lot of attention
We wish to consider the most general case, i.e., when Hadamard fractional integral operators are acting on Orlicz spaces, and we apply the obtained results for the study of systems of quadratic equations of fractional order

Summary

Introduction

The study of quadratic differential and integral equations has a long history. They have been investigated by many authors. We wish to consider the most general case, i.e., when Hadamard fractional integral operators are acting on Orlicz spaces (instead of Lebesgue spaces), and we apply the obtained results for the study of systems of quadratic equations of fractional order. The obtained results are new in the context of singularity of the system of quadratic equations and of the Hadamard fractional operators with general growth conditions on Orlicz spaces. For an approach allowing to generalize growth conditions with the use of Orlicz spaces, see [14, Theorem 4.16] or [15, Remark 2], or [13, Theorem 5.1], but only for the case of quadratic integral equations with Riemann–Liouville integrals, or [16] for basics about Hadamard integral operators on Orlicz spaces

Preliminaries

System of Hadamard-type fractional integral equations