Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

Mohamed Jleli,Mokhtar Kirane,Bessem Samet

doi:10.1155/2017/5123240

Abstract

We consider a fractional boundary value problem involving a fractional derivative with respect to a certain function g. A Hartman-Wintner-type inequality is obtained for such problem. Next, several Lyapunov-type inequalities are deduced for different choices of the function g. Moreover, some applications to eigenvalue problems are presented.

Highlights

In this work, we are concerned with the following fractional boundary value problem:(Daα,gu) (t) + q (t) u (t) = 0, a < t < b, (1)u (a) = u (b) = 0, where (a, b) ∈ R2, a < b, α ∈ (1, 2), q : [a, b] → R is a continuous function, and Daα,g is the fractional derivative operator of order α with respect to a certain nondecreasing function g ∈ C1([a, b]; R) with g󸀠(x) > 0, for all x ∈ (a, b)
We are concerned with the following fractional boundary value problem: (Daα,gu) (t) + q (t) u (t) = 0, a < t < b, (1)
Several Lyapunov-type inequalities are deduced for different types of fractional derivatives

Summary

Introduction

We are concerned with the following fractional boundary value problem:. In [8], Hartman and Wintner proved that if boundary value problem (2) has a nontrivial solution, b. Due to the positive impact of fractional calculus on several applied sciences (see, for instance, [23]), several authors investigated Lyapunov-type inequalities for various classes of fractional boundary value problems. The main result obtained in [24] is the following fractional version of Theorem 1. If fractional boundary value problem (7) has a nontrivial solution, b. If fractional boundary value problem (9) has a nontrivial solution, e λ1−α In the same paper [37], the authors formulated the following question: How to get the Lyapunov inequality for the following Hadamard fractional boundary value problem:. Note that one of our obtained results is an answer to the above question

Preliminaries

Lyapunov-Type Inequalities for Different Choices of the Function g

Applications to Eigenvalue Problems