A new contribution to best proximity point theory on quasi metric spaces and an application to nonlinear integral equations. (English) Zbl 1535.54026

Aslantas, Mustafa

Optimization 72, No. 11, 2851-2864 (2023).

The main goal of this paper is to prove a best proximity result for \(BG\)-multivalued contractions defined on a quasi-metric space, by using the notion of a \(Q\)-function. Since in the case of quasi-metric spaces one encounters the lack of the symmetry, the author at the beginning of this article reconsiders definitions and notions related to the best proximity point theory in the quasi-metric setting. As a corollary from the best proximity result mentioned above, the author derives the following fixed point theorem.
Let \((\Gamma, \sigma)\) be a complete quasi-metric space, \(g\) be a \(Q\)-function and \(H:\Gamma\to \Gamma\). If there exists a Bianchini-Grandolfi gauge function \(\phi\) such that \[ g(H\zeta,H\nu)\leq \varphi(g(\zeta, \nu)), \] for all \(\zeta,\ \nu\in\Gamma\), then \(H\) has a unique fixed point \(\zeta^*\) in \(\Gamma\). Moreover, \(g(\zeta^*,\zeta^*)=0\).
As an application of the above fixed point theorem the author proves an existence and uniqueness result for positive continuous solutions to the following Volterra-Fredholm integral equation

•  Encyclopedia of Mathematics
•  Wikipedia

\[ \xi(\gamma)=\varphi(\gamma)+\alpha_1\int_0^1 K_1(\gamma,s,\xi(s))ds+\alpha_2\int_0^{\gamma} K_2(\gamma,s,\xi(s))ds, \] where \(\alpha_1\), \(\alpha_2\) are nonnegative constants such that \(\alpha_1+\alpha_2\leq 1\), \(0\leq\gamma\leq 1\), \(\phi:[0,1]\to [0,+\infty)\) and \(K_i:[0,1]^2\times [0,+\infty)\to [0,+\infty)\) are continuous functions for \(i=1,2\).

Reviewer: Dariusz Bugajewski (Poznań)

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
47H10	Fixed-point theorems

Keywords:

best proximity point theorem; fixed point theorem, nonlinear Volterra-Freldhom integral equation; quasi-metric space; Q-function

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