Positive solutions for Hadamard fractional differential equations on infinite domain

Phollakrit Thiramanus,Sotiris K Ntouyas,Jessada Tariboon

doi:10.1186/s13662-016-0813-7

Abstract

This paper investigates the existence of nonnegative multiple solutions for nonlinear fractional differential equations of Hadamard type, with nonlocal fractional integral boundary conditions on an unbounded domain by means of Leggett-Williams and Guo-Krasnoselskii’s fixed point theorems. Two examples are discussed for illustration of the main work.

Highlights

Fractional calculus has gained considerable attention from both theoretical and the applied points of view in recent years
Another kind of fractional derivatives found in the literature is the fractional derivative due to Hadamard introduced in [ ], which differs from the aforementioned derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains logarithmic function of arbitrary exponent
In [ ] we studied a new class of boundary value problems on fractional differential equations with m-point Erdélyi-Kober fractional integral boundary conditions on an infinite interval of the form

Summary

Introduction

Fractional calculus has gained considerable attention from both theoretical and the applied points of view in recent years. The monographs [ – ] are commonly cited for the theory of fractional derivatives and integrals and applications to differential equations of fractional order. It has been noticed that most of the work on the topic is concerned with Riemann-Liouville or Caputo type fractional differential equations. Besides these fractional derivatives, another kind of fractional derivatives found in the literature is the fractional derivative due to Hadamard introduced in [ ], which differs from the aforementioned derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains logarithmic function of arbitrary exponent. A detailed description of Hadamard fractional derivative and integral can be found in [ , – ]

Objectives

Conclusion