A fully Hadamard and Erdélyi–Kober-type integral boundary value problem of a coupled system of implicit differential equations

Fatima Zohra Berrabah,Benaouda Hedia,Johnny Henderson

doi:10.3906/mat-1812-46

Abstract

In this article, we give sufficient conditions for the existence of solutions for a new coupled system of second-order implicit differential equations with Hadamard and Erdelyi-Kober fractional integral boundary conditions and nonlocal conditions at the boundaries in Banach space. The main result is based on a Monch fixed point theorem combined with the measure of noncompactness of Kuratowski; an example is given to illustrate our approach.

Highlights

Fractional differential equations have large applications in a variety of fields such as electrical networks, signal and image processing, viscoelasticity, aerodynamics, economics, and so on, and has increased more attention from both theoretical and applied points of view in recent years.We note here that most of the work on the topic in the literature is based on Riemann–Liouville- and Caputo-type fractional differential equations; for this, we refer the readers to [1, 5, 6, 10, 11]
Another kind of fractional derivative that appears side by side to Riemann–Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 [13], which differs from the preceding ones in the sense that the kernel of the integral contains a logarithmic function of arbitrary exponent
We make use of the Mönch fixed point theorem combined with the Kuratowski measure of noncompactness to give our existence result

Summary

Introduction

Fractional differential equations have large applications in a variety of fields such as electrical networks, signal and image processing, viscoelasticity, aerodynamics, economics, and so on, and has increased more attention from both theoretical and applied points of view in recent years (for further details see [12, 16]).We note here that most of the work on the topic in the literature is based on Riemann–Liouville- and Caputo-type fractional differential equations; for this, we refer the readers to [1, 5, 6, 10, 11]. [3] The Hadamard fractional integral of order α ∈ R+ of a function f (t) , for all t > 0 is defined as [18] The Erdélyi–Kober fractional integral of order λ > 0 , with η > 0 and ε ∈ R, of a continuous function f : (0, ∞) → E is defined by

Results

Conclusion