Research paper
An existence and uniqueness theorem for a fractional boundary value problem via new fixed point results on quasi metric spaces

https://doi.org/10.1016/j.cnsns.2020.105462Get rights and content

Highlights

  • In this paper, we present an existence and uniqueness result for positive solution of a kind of fractional boundary value problem with Riemann-Liouville fractional derivative. Before this, we provide two new fixed point results for mappings satisfying a certain conrtractive condition that obtained by simulation and Q-functions on quasi metric spaces. This study is important because of the first time that a fixed point theorem obtained in quasi metric spaces is applied to such an equation.

Abstract

In this paper, we present two fixed point results on quasi metric spaces by taking into account a contractive condition that obtained via simulation and Q-functions. Then one of these fixed point results has been applied to guarantee the existence and uniqueness of positive solution of a kind of fractional boundary value problem with Riemann-Liouville fractional derivative.

Section snippets

Introduction and preliminaries

The theory of fractional calculus and its applications nowadays are large subjects of mathematics, which are found in physics, engineering and other fields connected with real world problems. Based on this fact, there have been many papers dealing with the solutions of boundary value problems for nonlinear fractional differential equations with the boundary conditions. For example, in the recent paper [8], the authors consider the fractional boundary value problem given as{D0+αυ(t)+a(t)f(υ(t))=0

Fixed point results

We begin this section by introducing two new definitions.

Definition 3

Let (Ω, ρ) be a quasi metric space, q be a Q-function on Ω and F: Ω → Ω be a mapping. Then F is said to has q-property, if the implication q(ξ,η)=0q(Fξ,Fη)=0 holds for all ξ, η ∈ Ω.

Example 7

Consider the quasi metric space (Ω, ρ), where Ω=[0,) and ρ(ξ,η)=max{ηξ,0} for all ξ, η ∈ Ω. We know that q1(ξ,η)=max{ξ,η} and q2(ξ,η)=η are Q-functions on (Ω, ρ). Then every mapping F: Ω → Ω satisfying F0=0, has q-property with respect to both q1 and q2.

Example 8

Let

Existence and uniqueness result

In this section we present a novel application where, with the help of Theorem 1, we will show the existence and uniqueness of solution of a fractional boundary value problem: Here for a continuous function f:[0,)[0,) and an integrable function a:[0,1][0,) we will consider the fractional boundary value problem given as{D0+αυ(t)+a(t)f(υ(t))=0,t(0,1)υ(0)=D0+βυ(1)=0, where α ∈ (1, 2], β[0,1] and D0+γ is Riemann-Liouville derivative of order γ. We know that for positive integer n and γ(n1,n

CRediT authorship contribution statement

Ishak Altun: Conceptualization, Validation, Methodology, Formal analysis, Investigation, Writing - original draft, Writing - review & editing. Murat Olgun: Methodology, Formal analysis, Investigation, Resources, Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors are grateful to the referees because their suggestions contributed to improve the paper.

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