Research paperAn existence and uniqueness theorem for a fractional boundary value problem via new fixed point results on quasi metric spaces
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
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 holds for all ξ, η ∈ Ω. Example 7 Consider the quasi metric space (Ω, ρ), where and for all ξ, η ∈ Ω. We know that and are Q-functions on (Ω, ρ). Then every mapping F: Ω → Ω satisfying 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 and an integrable function we will consider the fractional boundary value problem given as where α ∈ (1, 2], and is Riemann-Liouville derivative of order γ. We know that for positive integer n and
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.
References (19)
- et al.
Some generalizations of ekeland-type variational principle with applications to equilibrium problems and fixed point theory
Nonlinear Anal
(2008) Completeness in quasi-metric spaces and ekeland variational principle
Topol Appl
(2011)On generalizations of the ekeland-type variational principles
Nonlin Anal
(2000)- et al.
On half-completion and bicompletion of quasi-metric spaces
Comment Math Univ Carolin
(1996) - et al.
Classification of completeness of quasi metric space and some new fixed point results
Nonlinear Funct Anal Appl
(2017) - et al.
Nonlinear contractions involving simulation functions in a metric space with a partial order
J Nonlinear Sci Appl
(2015) Functional analysis in asymmetric normed spaces
(2013)- et al.
An ordering on green’s functions for a family of two point boundary value problems for fractional differential equations
Commun Appl Anal
(2015) - et al.
Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition
Discrete Continuous Dyn Syst
(2015)
Cited by (1)
ANALYSIS OF SOLUTIONS FOR THE BOUNDARY VALUE PROBLEMS OF NONLINEAR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS INVOLVING GRONWALL’S INEQUALITY IN BANACH SPACES
2022, Journal of Applied Mathematics and Informatics