Integral inequalities with ‘maxima’ and their applications to Hadamard type fractional differential equations

Phollakrit Thiramanus,Jessada Tariboon,Sotiris K Ntouyas

doi:10.1186/1029-242x-2014-398

Abstract

AbstractIn this paper, some new integral inequalities with ‘maxima’ are established involving Hadamard integral. Applications to Hadamard fractional differential equations with ‘maxima’ are also presented.MSC:26A33, 26D10, 26D15.

Highlights

1 Introduction It is well known that integral inequalities play a dominant role in the study of quantitative properties of solutions of differential and integral equations [ – ]
Fractional inequalities are important in studying the existence, uniqueness, and other properties of fractional differential equations
In [, ], the authors established some weakly singular integral inequalities of Gronwall-Bellman type and applied them in the qualitative analysis of solutions to certain fractional differential equations of the Caputo type. Another kind of fractional derivative that appears in the literature is the fractional derivative due to Hadamard, introduced in [ ], which differs from the RiemannLiouville and Caputo derivatives in the sense that the kernel of the integral contains a logarithmic function of an arbitrary exponent

Summary

Introduction

It is well known that integral inequalities play a dominant role in the study of quantitative properties of solutions of differential and integral equations [ – ]. In [ , ], the authors established some weakly singular integral inequalities of Gronwall-Bellman type and applied them in the qualitative analysis of solutions to certain fractional differential equations of the Caputo type. Another kind of fractional derivative that appears in the literature is the fractional derivative due to Hadamard, introduced in [ ], which differs from the RiemannLiouville and Caputo derivatives in the sense that the kernel of the integral contains a logarithmic function of an arbitrary exponent. Let us recall here the definitions of Hadamard’s fractional integral and derivative [ ].

For t

It is obvious that a

Then for