Some Best Proximity Point Problems and their Applications

Abstract

Approximation theory is a subject with a long history and a huge importance in classical and contemporary research. Over the years, the theory has become so extensive that it intersects with every other branch of analysis. One of the problems in approximation theory is to detect a point that minimizes the distance between two subsets $\mathcal{E}_{1},\mathcal{E}_{2}$ of a metric space $(\mathcal{W},d)$. Once this is done, it motivates us to study the solution of minimization problem. For example, \begin{align*} \min_{\acute{k} \in \mathcal{E}_{1}}d(\acute{k},\mathcal{B}\acute{k}), ~\min_{\acute{m} \in \mathcal{E}_{2}}d(\acute{m},\mathcal{B}\acute{m}) ~\text{and}~~ \min_{(\acute{k},\acute{m})\in \mathcal{E}_{1}\times \mathcal{E}_{2}}d(\acute{k},\acute{m}), \end{align*} where $ \mathcal{B}$ is a mapping on $\mathcal{E}_{1} \cup \mathcal{E}_{2}$, such that $\mathcal{B}(\mathcal{E}_{1}) \subseteq \mathcal{E}_{1}~ \text{and} ~ \mathcal{B}(\mathcal{E}_{2}) \subseteq \mathcal{E}_{1}$. It is fascinating to arise a question whether is possible to find a pair $(\acute{k},\acute{m}) \in \mathcal{E}_{1} \times \mathcal{E}_{2}$ which is a solution of above problem that is, to find a pair $(\acute{k},\acute{m}) \in \mathcal{E}_{1} \times \mathcal{E}_{2}$ such that $\acute{k}=\mathcal{B}\acute{k}, \acute{m}=\mathcal{B}\acute{m} ~\text{and}~~ d(\acute{k},\acute{m}) = d(\mathcal{E}_{1},\mathcal{E}_{2})$. If such a pair exists, it is called the best proximity pair for a mapping $\mathcal{B}$. If we take $\mathcal{B}$ a non-self mapping we find an approximate solution $\acute{k}$ such that the error $d(\acute{k}, \mathcal{B}\acute{k})$ is minimum. The existence of an approximate solution $\acute{k}$, called best proximity point, that is, to find $\acute{k} \in \mathcal{E}_{1}$ such that \begin{align*} d(\acute{k}, \mathcal{B}\acute{k}) = d(\mathcal{E}_{1},\mathcal{E}_{2}) = \inf\left\lbrace d(\acute{k}, \acute{m}) : \acute{k} \in \mathcal{E}_{1}, \acute{m} \in \mathcal{E}_{2}\right\rbrace. \end{align*} Approximation theory can be used to solve many kinds of problem such as systems of nonlinear matrices, integral and differential equations, fractals, split feasibility problems, and variational inequalities. The study of approximation theory is appropriately inspired by the fact that particular instances of approximation frequently arise from problems connected with science and technology.

The first aim of this project is to construct algorithms for the existence and uniqueness of a best proximity point. Another aim of this project is to discuss some applications of best proximity points.

In the first chapter, we provide the supplementary material such as some definitions, preliminary results that are useful for upcoming chapters. It also includes the literature survey, thesis goals, as well as a synopsis of the information included in each of the thesis's chapters.

In the second chapter, we discuss the existence of best proximity points for non-self mappings satisfying some contractive conditions in the setting of metric spaces, relational metric spaces, quasi partial metric spaces, normed and binormed linear spaces. Furthermore, we discuss the existence of best proximity pair results using noncyclic contraction mapping at the end of this chapter.

Third chapter concerns with iterative schemes. In this chapter, we propose some algorithms that converge to a best proximity point and fixed point. Also, we introduce some algorithms which converge to solution of split fixed and split best proximity point problem.

The fourth chapter deals with applications of best proximity point problems. In this chapter, we present the solution for variational inequality problems in the framework of Hilbert spaces. We provide a solution for a system of differential equations in the context of metric spaces. We also solve the model that spreads a virus using a non-linear integral equation.