Best proximity point results for Prešić type nonself operators in $b$-metric spaces

1.   Introduction

In 1922, Banach proved his famous result known as the Banach contraction principle, which is a simple and powerful result with a wide range of applications ^[12]. Many generalizations of Banach contraction principle can be seen in the literature, see e.g., ^{[5,9,16,17,19,21,22,23,24]}.

Consider the $k$th order nonlinear difference equation

with initial values $x_0, \ldots, x_{k-1}\in X,$ where $k\geq 1$ is a positive integer and $f: X^k\rightarrow X.$ This difference equation can be discussed with the perspective of fixed point theory by considering the fact that $x^*$ is a fixed point of $f$ if and only if it is the solution of (1.1) exist, that is,

The first step in this direction is taken by Pre$\hat{\rm{s}}$i$\acute{c}$ in 1965 by establishing a generalization of the Banach contraction principle in the following manner:

Theorem 1.1. ^[28] Let $(X, d)$ be a complete metric space. Given $k\geq 1$ a positive integer and $f: X^k\rightarrow X.$ Assume also that

where $\eta_1, \eta_2, \ldots, \eta_k$ are positive constants such that $\sum\limits_{i = 1}^{k} \eta_i \in (0, 1).$ Then there exists a unique $x^*\in X$ such that $x^* = f(x^*, x^*, \ldots x^*),$ that is, $f$ has a unique fixed point $x^*\in X.$ Moreover, for any initial values $x_0, \ldots, x_{k-1}\in X$ the iterative sequence given by $(6.1)$ converges to $x^*.$

Note that for $k = 1,$ the map $f:X\rightarrow X$ becomes a self map and hence the above Theorem is the generalization of Banach contraction principle (for contractions defined on $X^k$). In ^[14], Theorem 1.1 is generalized by $\acute{\rm{C}}$iri$\acute{\rm{c}}$ and Pre$\hat{\rm{s}}$i$\acute{c}$ in the following way:

Theorem 1.2. ^[14]Let $(X, d)$ be a complete metric space. Given $k\geq 1$ a positive integer and $f: X^k\rightarrow X.$ Suppose that

where $\mu \in(0, 1)$ is a constant. Then there exists a unique $x^*\in X$ such that $x^* = f(x^*, x^*, \ldots x^*),$ that is, $f$ has a unique fixed point $x^*\in X.$ Moreover, for any initial values $x_0, \ldots, x_{k-1}\in X$, the iterative sequence given by $(6.1)$ converges to $x^*.$

Note that a fixed point of the operator $f: X^k\rightarrow X$ can be considered as the equilibrium point of the $k$th order nonlinear difference Eq (6.1). Therefore, the above theorems can be taken as a tool to ensure the existence and uniqueness of the $k$th order nonlinear difference equation. Some other generalizations are obtained by P$\hat{\rm{a}}$curar in ^[13,27]. Recently, Ali et al. ^[3] studied the existence of an approximate solution of the equation $x = f(x, x, \ldots, x),$ where $f: H^k\rightarrow K.$ This equation has a solution if $H$ and $K$ have some common element, but has no solution otherwise. Hence in that case we can only get the approximate solution of the equation. The approximate solution of $x = f(x, x, \ldots, x),$ with the error term $d(H, K)$ is called a best proximity point of $f: H^k\rightarrow K.$ The classical result of approximation theory given by Fan ^[18] is a great source of inspiration for various researchers in study of approximate solutions of $x = f(x).$ This result is given as follows:

Theorem 1.3. Let $H$ be a nonempty compact convex subset of a normed linear space $X$ and $f:H\rightarrow X$ be a continuous function. Then there exists $x\in X$ such that

Recently Altun et al. ^[7,8] investigated certain best proximity points results on KW-type nonlinear contractions and fractals. Furthermore Ali et al. ^[3] used the metric space $(X, d)$ endowed with a graph and proved some best proximity results. These results are the generalizations of already existing results which are stated earlier.

Czerwik ^[15] gave a generalization of the famous Banach fixed point theorem in so-called $b$-metric spaces. For some important results on $b$-metric spaces, we refer the reader to ^{[2,4,10,11,25,26]}.

The purpose of present research is to extend the results of Ali et al.^[3], in the setting of $b$-metric spaces equipped with an order. Hence, many results in literature become special cases of results presented in this article. Our paper also contains some examples for the validation of presented results and an application for further authentication.

2.   Preliminaries

We include the following definitions before giving the main results.

Definition 2.1. ^[6] Consider a metric space $(X, d)$. Suppose $H$ and $K$ are two non-empty subsets of $X$. An element $x \in H$ is said to be a best proximity point of the mapping $T : H \rightarrow K$ if

Remark 2.1. From the above definition, it is obvious that a best proximity point reduces to a fixed point for self-mappings.

Basha and Shahzad ^[29] have presented the following definition:

Definition 2.2. Consider a complete metric space $(X, d)$. Suppose that $H, K$ are non empty subsets of $X$. If each sequence $\lbrace k_{n} \rbrace$ in $K$ with $d(h, k_{n})\rightarrow d(h, K)$, for some $h \in H$, has a convergent subsequence. Then, $K$ is called approximately compact with respect to $H$.

Ali et al. ^[3] introduced path admissible mappings as follows:

Definition 2.3. Suppose that $H, K$ are nonempty subsets of a metric space $(X, d)$ endowed with a binary relation $\mathcal{R}$. Then $T : H \times H \rightarrow K$ is said to be path admissible, if

where $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$.

Here, by $w_1\mathcal{R}w_2$ we mean that $w_1$ and $w_2$ are related with each other under the binary relation $\mathcal{R}$ and $h_{1}Ph_{3}$, we mean that for above mentioned $h_{1}, h_{2}, h_{3} \in X\; \mathrm{we \; have}\; h_{1}\mathcal{R} h_{2}$ and $h_{2}\mathcal{R}h_{3}$.

Definition 2.4. Suppose $H, K$ are non empty subsets of a metric space $(X, d)$. An element $h_{*} \in H$ is said to be a best proximity point of $T : H \times H \rightarrow K$ if

where

3.   Main results

First, we recall some definitions which are used in the sequel. Let $(X, d_b)$ be a $b$-metric space with coefficient $b \geq 1$. Suppose that $H$ and $K$ are two nonempty subsets of $X$, then define the following sets:

Definition 3.1. Consider a $b$-metric space $(X, d_b)$ with coefficient $b \geq 1$. Suppose that $H$ and $K$ are nonempty subsets of $X$. The element $h_{*} \in H$ is said to be a best proximity point of the mapping $T: H \rightarrow K$ if

Definition 3.2. Consider a $b$-metric space $(X, d_b)$ with coefficient $b \geq 1$ and let $H$ and $K$ be two nonempty subsets of $X$. Then $K$ is said to be approximately compact with respect to $H$, if each sequence $\lbrace k_{n} \rbrace \subseteq K$ with $d_b (h, k_{n})\rightarrow d_b (h, K)$ for some $h \in H$, has a convergent subsequence.

Definition 3.3. Let $(X, d_b)$ be a $b$-metric space with coefficient $b \geq 1$ and $\mathcal{R}$ is the binary relation on $X$. Suppose $H$, $K$ are nonempty subsets of $X$. A mapping $T : H \times H \rightarrow K$ is called path admissible, whenever $\forall\; h_1, h_2, h_3, w_1, w_2 \in H$ we have

here, by $w_{1}\mathcal{R} w_{2}$ mean that $w_1$ and $w_2$ are related with each other under the binary relation $\mathcal{R}$ and $h_{1}Ph_{3}$ we mean that for above mentioned $h_{1}, h_{2}, h_{3} \in H$, we have $h_{1}\mathcal{R} h_{2}$ and $h_{2}\mathcal{R}h_{3}.$

Theorem 3.1. Suppose that $(X, d_b)$ is a complete $b$-metric space with coefficient $b \geq 1$ endowed with a binary relation $\mathcal{R}$, where $d_ b$ is a continuous functional. Assume that $H$ and $K$ are nonempty closed subsets of $X$. Consider a mapping $T: H \times H \rightarrow K$ such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$ with $h_{1}Ph_{3}$ that is $h_1 \mathcal{R}h_2, \; h_2\mathcal{R} h_3$ and $d_b(w_{1}, T(h_{1}, h_{2})) = d_b(H, K) = d_b(w_{2}, T(h_{2}, h_{3}))$, we have:

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$. Furthermore, suppose that the subsequent conditions are true:

(1) $T$ is path admissible;

(2) $\exists \; h_{0}, h_{1}, h_{2} \in H$ which satisfy $d_b(h_{2}, T(h_{0}, h_{1})) = d_b(H, K)$ and $h_{0}Ph_{2}$;

(3) $T(H \times H _{0})\subseteq K _{0}$;

(4) $K$ is approximately compact with respect to $H$;

(5) If $\lbrace h_{j}\rbrace\subseteq X$ such that $h_{j}Ph_{j+2}$ for each $j \in \mathbb{N}$ and $h_{j} \rightarrow x_{\ast}$ as $j \rightarrow \infty$, then $h_{j}\mathcal{R} x_{\ast}$ for all $j \in \mathbb{N}$ and $x_{\ast}\mathcal{R} x_{\ast}.$

Then $T$ has a best proximity point.

Proof. Using condition $(ii)$, we have $h_{0}, h_{1}, h_{2} \in H$ satisfying

that is, $h_{0}\mathcal{R} h_{1}, h_{1}\mathcal{R}h_{2}$. From condition $(iv)$, $T(h_{1}, h_{2}) \in K _{0}$, and by the definition of $K _{0}$, we have $h_{3} \in H$ which satisfies

Due to condition $(i)$, we have $h_{2}\mathcal{R}h_{3}$. Hence, $h_{1}Ph_{3}$. By continuing same process, we build a sequence $\lbrace h_j\rbrace_{\geq 2} \subseteq H$ which satisfies

and $h_{j-1}Ph_{j+1}$. That is, $h_{j-1}\mathcal{R}h_{j}, h_{j}\mathcal{R}h_{j+1} \; \; \forall \; \; j \in \mathbb{N}.$ From $(3.2)$, we have

For convenience, we take $c _{j} = d_b(h_{j}, h_{j+1})$ for each $j \in \mathbb{N} \cup \lbrace 0\rbrace$. Then we can rewrite $(3.4)$ as

By using induction, we can get $c_{n-1} \leq Z\psi^n\; \mathrm{where}\; \; \psi = \Gamma^{1/2}.$ It is obviously true for $j = 0, 1$ by considering

since $Z$ is $\max \lbrace c _{0}/ \psi, c _{1}/ \psi^{2}\rbrace$, one writes

We obtain

Therefore, we have

Hence,

By using triangle inequality, we get

Thus, $\lbrace h_{j}\rbrace$ is a Cauchy sequence in $H$, so there is an element $h_{\ast} \in H$ such that $h_{j}\rightarrow h_{\ast}$ and $h_{j} \in H _{0}$ which satisfies

that is, $h_{j-1}\mathcal{R}h_{\ast}$.

Furthermore, we have to prove that $d_b(h_{\ast}, T(h_{j-1}, h_{j}))\rightarrow d_b(h_{\ast}, K)\; \; \; as\; \; \; j\rightarrow \infty$. Consider,

Therefore,

Since $T$ is approximately compact with respect to $H$, the sequence $\{ T(h_{j-1}, h_{j})\}$ has a subsequence $\{ T(h_{j_{m-1}}, h_{j_{m}})\}$, which converges to a point $k_{\ast} \; \in \; K$. That is,

Hence, $h_{\ast} \in H _{0}$. As we know $T(h_{j}, h_{\ast}) \in K _{0}$, we have $g \in H$ satisfying

By assumption $(vi)$, we have $h_{j}\mathcal{R}h_{\ast}$ for all $j \in \mathbb{N}$. Thus, we have

Hence, we get $h_{j-1}Ph_{\ast}$. Also, $h_{j-1}\mathcal{R}h_{j},$ and $h_{j}\mathcal{R}h_{\ast}$ for all $j \in \mathbb{N}.$ Hence, from $(6.2)$,

Taking $j \rightarrow \infty$, we obtain $d_b(h_{\ast}, g) = 0, \; \mathrm{that\; is}\; \; g = h_{\ast}$. Putting $g = h_{\ast}$ in (3.7), we have

That is, $h_{\ast}\mathcal{R} h_{\ast}$. Furthermore, we know that $T(h_{\ast}, h_{\ast})\in K _{0}$, and we have an element $t \in H$ which satisfies

Condition $(vi)$ implies that $h_{\ast}\mathcal{R}h_{\ast}$. Hence,

Therefore, $h_{j}Ph_{\ast}\; \mathrm{for\; each}\; j \in \mathbb{N}$, that is, $h_{j}\mathcal{R}h_{\ast}, h_{\ast}\mathcal{R}h_{\ast}$ for each $j \in \mathbb{N}.$ Thus, from $(3.2)$,

Taking limit as $j \rightarrow \infty,$ we have $d_b(h_{\ast}, t) = 0, \; \; \mathrm{that\; is}\; \; t = h_{\ast}.$ Putting $t = h_{\ast}$ in (3.8), we have

Theorem 3.2. Let $H$ and $K$ be nonempty subsets of a complete $b$-metric space $(X, d_b)$ endowed with binary relation $\mathcal{R}$, where $b$-metric is a continuous functional. Consider a mapping $T: H \times H \rightarrow K$ such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$ with $h_{1}Ph_{3}$, that is, $h_{1}\mathcal{R}h_{2}, \; h_{2}\mathcal{R}h_{3}$ and

$d_b(w_{1}, T(h_{1}, h_{2})) = d_b(H, K) = d_b(w_{2}, T(h_{2}, h_{3}))$, we have

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$.

Furthermore, suppose that the subsequent conditions are true

(1) $T$ is path admissible;

(2) $\exists \; h_{0}, h_{1}, h_{2} \in H$ which satisfy $d_b(h_{2}, T(h_{0}, h_{1})) = d_b(H, K)$ and $h_{0}Ph_{2}$;

(3) $T(H \times H _{0})\subseteq K _{0}$;

(4) $K$ is approximately compact with respect to $H$;

(5) When $\lbrace h_{j}\rbrace\subseteq X$ such that $h_{j}Ph_{j+2}$ for each $j \in \mathbb{N}$ and $h_{j} \rightarrow x_{\ast}$ as $n \rightarrow \infty$, then $h_{j}\mathcal{R}x_{\ast}$ for all $j \in \mathbb{N}$ and $x_{\ast}\mathcal{R}x_{\ast}.$

Then there exists a point $h_{\ast} \in \; H$ which satisfies

that is, $T$ has a best proximity point.

Proof. Proceeding as in Theorem 3.1, we obtain a sequence $\lbrace h_j :j \in \mathbb{N}-{1} \rbrace$ in $H_0$ satisfying

and $h_{j-1}Ph_{j+1}$, that is $h_{j-1}\mathcal{R}h_{j}, h_{j}\mathcal{R}h_{j+1}\; \; \forall \; \; j \in \mathbb{N}.$

From $(3.9)$, we have

Following the proof of Theorem 3.1 and above inequality, $\lbrace h_{j} \rbrace$ is a Cauchy sequence in $H$ such that $h_{j} \rightarrow h_{\ast}$ and $h_{\ast} \in H _{0}$. As $T(h_{j}, h_{\ast}) \in K _{0}$, we have $w \in H$ satisfying

From assumption $(vi)$, we get $h_{j} \mathcal{R}h_{\ast}$ for all $j\in \mathbb{N}$. We already have

Thus, we get $h_{j-1}Ph_{\ast}$, that is $h_{j-1}\mathcal{R} h_{j}$ and $h_{j} \mathcal{R} h_{\ast}$ for all $j\in \mathbb{N}$. Hence, from $(3.9)$, we get

Taking limit as $j \rightarrow \infty$ in above inequality, we get $d_b(h_{\ast}, w) = 0,$ that is, $h_{\ast} = w.$ Using $w = h_{\ast}$ in $(3.10)$,

Further, note that $T(h_{\ast}, h_{\ast}) \in K _{0},$ and there is $q \in H$ which satisfies

Hypothesis $(vi)$ implies $h_{\ast} \mathcal{R} h_{\ast}$. Hence, we have

Thus, from $(3.9)$,

Letting $j\rightarrow \infty$, we have $q = h_{\ast}.$ Thus, we have

Example 3.1. Consider $X = \mathbb{R}^{2}$ endowed with the $b$-metric given by

Define a binary relation $\mathcal{R}$ on $\mathbb{R}^{2}$ as $s \mathcal{R} c$ if and only if $s_{1} \leq c_1 \mathrm\; {and}\; s_{2}\leq c_{2}.$ Take

Define

Let $\overline{h_{1}} = (0, h_{1}), \; \; \overline{h_{2}} = (0, h_{2}), \; \; \overline{h_{3}} = (0, h_{3})\in [-2, 2]$. To find $w_{1}$ and $w_{2}$, we have

For this, consider

That is,

Then

Using $(3.12)$ and $(3.13)$ in $(3.11)$, we obtain

That is,

Similarly,

From $(3.11)$, we obtain

Thus, $\overline{h_{1}}, \overline{h_{2}}, \overline{h_{3}}, \overline{w_{1}}, \overline{w_{2}} \in H,$ with $\overline{h_{1}}P\overline{h_{3}}$.

Also, we have

where

Using above equation in $(3.14)$, we get

Here, we say $\psi = \Gamma^{\frac{1}{2}} = \frac{1}{2} \in [0, 1)$. Now, we will prove condition $(i)$ of Theorem 3.2. Consider

Since $\overline w_{1} = (0, w_{1}) = (0, h_{2})$ and $w_{2} = (0, w_{2}) = (0, h_{3})$, we now prove

Similarly,

This implies that $\overline {w_{1}}\; \mathcal{R}\; \overline {w_{2}}$. Thus, $T$ is path admissible. Now, we will prove condition $(ii)$:

We need to consider

such that

and $(0, 0) P (0, \frac{5}{8})$. Moreover, assumption $(v)$ holds, that is, $h_{j}Ph_{j+2}$ for all $j\in \mathbb{N},$ and $h_{j} \rightarrow a$ as $j \rightarrow \infty$, then $h_{j}\mathcal{R} a$ for each $j \in \mathbb{N}$ and $a \mathcal{R}a$. Therefore, all axioms are true. Hence, $T$ has a best proximity point.

Theorem 3.3. Let $H$ and $K$ be nonempty closed subsets of a complete $b$-metric space $(X, d_b)$ with coefficient $b \geq 1$ endowed with a binary relation $\mathcal{R}$, where the $b$-metric is a continuous functional. Consider a mapping $T: H \times H \rightarrow K$ such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$ with $h_{1}Ph_{3}$, that is, $h_{1}\mathcal{R}h_{2}, h_{2}\mathcal{R}h_{3}$, and

$d_b(w_{1}, T(h_{1}, h_{2})) = d_b(H, K) = d_b(w_{2}, T(h_{2}, h_{3}))$, we have

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$.

Furthermore, suppose that the subsequent conditions are true

(1) $T$ is path admissible;

(2) $\exists \; h_{0}, h_{1}, h_{2} \in H$ which satisfy $d_b(h_{2}, T(h_{0}, h_{1})) = d_b(H, K)$ and $h_{0}Ph_{2}$;

(3) $T(H \times H _{0})\subseteq K _{0}$;

(4) $K$ is approximately compact with respect to $H$;

(5) if $\lbrace h_{j} \rbrace$ and $\lbrace \overline{h_{j}} \rbrace$ are in $X$ such that $h_{j} \rightarrow h$ and $\overline{h_{j}} \rightarrow \overline{h}$, then $T(h_{j}, \overline{h_{j}}) \rightarrow T(h, \overline{h}).$

Then there exists a point $h_{\ast} \in \; H$ so that

that is, $T$ has a best proximity point.

Proof. By using a similar argument as in Theorem 3.1, we build a sequence $\lbrace h_{j\geq 2}\rbrace\subseteq H$ which satisfies

and $h_{j-1}Ph_{j+1}$, that is, $h_{j-1}\mathcal{R}h_{j}, h_{j}\mathcal{R}h_{j+1} \; \; \forall \; \; j \in \mathbb{N}.$

From $(3.15)$, we have

Inductively, we get

By using triangle inequality and above inequality for each $j\in \mathbb{N}$, we have

This proves that $T(h_{j-1}, h_{j})$ is a Cauchy sequence in the closed subset $K$. Since $X$ is complete, there exists $k_{\ast} \; \in \; K$ such that $T(h_{j-1}, h_{j}) \rightarrow k_{\ast}$.

Moreover,

Therefore, $d_b(k_{\ast}, h_{j+1})\; \; \rightarrow \; \; d_b(k_{\ast}, H)$ as $j \rightarrow \infty$.

Using hypothesis $(v)$, $\lbrace h_{j} \rbrace$ has a subsequence $\lbrace h_{j_{l}} \rbrace$ that converges to an element $h_{\ast} \subseteq H$ such that

Then

Theorem 3.4. Let $H$ and $K$ be closed nonempty subsets of a complete $b$-metric space $(X, d_b)$ endowed with a binary relation $\mathcal{R}$ where the $b$-metric is a continuous functional. Consider a mapping $T:H \times H \rightarrow K$ such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$ with $h_{1}Ph_{3}$, that is, $h_{1}\mathcal{R}h_{2}$ and $h_{2}\mathcal{R}h_{3}$, and

$d_b(w_{1}, T(h_{1}, h_{2})) = d_b(H, K) = d_b(w_{2}, T(h_{2}, h_{3}))$, we have

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$.

Furthermore, suppose that the subsequent conditions are true:

(1) $T$ is path admissible;

(2) $\exists \; h_{0}, h_{1}, h_{2} \in H$ which satisfy $d_b(h_{2}, T(h_{0}, h_{1})) = d_b(H, K)$ and $h_{0}Ph_{2}$;

(3) $T(H \times H_{0})\subseteq K _{0}$;

(4) $K$ is approximately compact with respect to $H$;

(5) if $\lbrace h_{j} \rbrace$, $\lbrace \overline{h_{j}} \rbrace$ in $X$ such that $h_{j} \rightarrow h$ and $\overline{h_{j}} \rightarrow \overline{h}$, then $T(h_{j}, \overline{h_{j}}) \rightarrow T(h, \overline{h}).$

Then there exists a point $h_{\ast} \in \; H$ which satisfies

that is, $T$ has a best proximity point.

Proof. Using the assumptions, we can build a sequence $\lbrace h_{j}\rbrace_{\geq 2}$ in $H_0$ which satisfies

and $h_{j-1}Ph_{j+1}$, that is, $h_{j-1}\mathcal{R}h_{j}$ and $h_{j}\mathcal{R}h_{j+1}$ for all $j \in \mathbb{N}.$ From $(3.16)$, we have

Therefore, we have

By using induction, we get

Hence,

By using triangle inequality,

By using $(3.18)$ in $(3.19)$, we get

Letting $j\rightarrow \infty$ in above inequality, we have

That is,

We get a Cauchy sequence $T(h_{j-1}, h_{j})$ in the closed subset $K$. Since $X$ is complete, consider $k_{\ast} \; \in \; K$ such that $T(h_{j-1}, h_{j}) \rightarrow k_{\ast}$. Moreover, Consider,

Therefore, $d_b(k_{\ast}, h_{j+1})\; \; \rightarrow \; \; d_b(k_{\ast}, H)$ as $j \rightarrow \infty$.

Condition (iv) implies, $\lbrace h_{j} \rbrace$ has a subsequence $\lbrace h_{j_{l}} \rbrace$ that converges to an element $h_{\ast} \in H$ such that

Hence,

Theorem 3.5. Consider a complete $b$-metric space $(X, d_b)$ with a coefficient $b \geq 1$ endowed with a binary relation $\mathcal{R}$, where $b$-metric is continuous. Suppose that $H$ and $K$ are non empty closed subsets of $\ X$. Consider a mapping $T:H \times H \rightarrow K$ such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$ with $h_{1}Ph_{3}$ such that $h_{1}\mathcal{R}h_{2},$ and $h_{2}\mathcal{R}h_{3}$, and $d_b(w_{1}, T(h_{1}, h_{2})) = d_b(H, K) = d_b(w_{2}, T(h_{2}, h_{3}))$, we have

where $\Gamma \in[0, 1)$ with $b \Gamma < 1$.

Furthermore, suppose that the subsequent conditions are true:

(1) $T$ is path admissible;

(2) There exist $h_{0}, h_{1}, h_{2} \in H$ which satisfy $d_b(h_{2}, T(h_{0}, h_{1})) = d_b(H, K)$ and $h_{0}Ph_{2}$;

(3) $T(H \times H_{0})\subseteq K _{0}$;

(4) $K$ is approximately compact with respect to $H$;

(5) When $\lbrace h_{j} \rbrace$, $\lbrace \overline{h_{j}} \rbrace \subseteq X$ such that $h_{j} \rightarrow h$ and $\overline{h_{j}} \rightarrow \overline{h}$, then $T(h_{j}, \overline{h_{j}}) \rightarrow T(h, \overline{h})$.

Then there exists a point $h_{\ast} \in \; H$ which satisfies

that is, $T$ has a best proximity point.

Proof. This theorem can be proved by using similar argument as in Theorem 3.4.

4.   On metric spaces endowed with a graph

In order to generalize the idea of partial ordering in metric spaces and partially ordered metric spaces, Jachymski ^[20] in $2008$ has introduced the idea of a metric space endowed with a graph. This section is about a consequence of our results in the setting of metric spaces endowed with a graph.

Theorem 4.1. Let $(X, d_b)$ be a complete $b$-metric space endowed with a graph $G$, where $d_b$ is a continuous functional. Suppose that $H$ and $K$ are non empty closed subsets of $X$. Consider a mapping $T: H \times H \rightarrow K$ such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$ with $h_{1}Ph_{3}$, that is, $h_1 \mathcal{R}h_2, \; h_2\mathcal{R} h_3$ and\\ $d_b(w_{1}, T(h_{1}, h_{2})) = d_b(H, K) = d_b(w_{2}, T(h_{2}, h_{3}))$, we have either

or

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$. Furthermore, assume that all the conditions of Theorem 3.1 are satisfied. Then $T$ has a best proximity point.

Proof. It follows by using the same procedure as in Theorems 3.1 and 3.2.

Theorem 4.2. Let $H$ and $K$ be closed nonempty subsets of a complete $b$-metric space $(X, d_b)$ endowed with a graph $G = (V(G), E)$ where the $b$-metric is a continuous functional. Consider a mapping $T:H \times H \rightarrow K$ such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$ with $h_{1}Ph_{3}$, that is, $(h_{1}, h_{2})\in E$ and $(h_{2}, h_{3}) \in E$, and $d_b(w_{1}, T(h_{1}, h_{2})) = d_b(H, K) = d_b(w_{2}, T(h_{2}, h_{3}))$, we have either

or

or

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$. Furthermore, assume that all the conditions of Theorem 3.3 are satisfied. Then $T$ has a best proximity point.

Proof. It follows by using the same arguments given in Theorem 3.3, Theorem 3.4 and ref{thm5a}.

5.   Application

Taking $A = B = X$ in Theorems 4.1 and 4.2, we obtain the following results, which guarantee the existence of a fixed point of the mapping $T:X\times X\rightarrow X.$

Theorem 5.1. Let $(X, d_b)$ be a complete $b$-metric space endowed with a graph $G$, where $d_b$ is a continuous functional. Let $T: X \times X \rightarrow X$ be a mapping such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in X$ with $h_{1}Ph_{3}$ that is $(h_1, h_2), \; (h_2, h_3) \in E$ satisfies one of the following inequalities

or

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$. Furthermore, assume that the following conditions are satisfied:

(1) $T$ is path admissible;

(2) There exist $a_0, a_1, a_3 \in X$ with $a_3 = T(a_0, a_1)$ and $a_0Pa_3$;

(3) If $\lbrace h_{j}\rbrace\subseteq X$ such that $h_{j}Ph_{j+2}$ for each $j \in \mathbb{N}$ and $h_{j} \rightarrow x_{\ast}$ as $j \rightarrow \infty$, then $(h_{j}, x_{\ast})\in E$ for all $j \in \mathbb{N}$ and $(x_{\ast}, x_{\ast})\in E.$

Then $T$ has a fixed point in $X.$

Theorem 5.2. Let $(X, d_b)$ be a complete $b$-metric space endowed with a graph $G$, where $d_b$ is a continuous functional. Let $T: X \times X \rightarrow X$ be a mapping such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in X$ with $h_{1}Ph_{3}$ that is $(h_1, h_2), \; (h_2, h_3) \in E$ satisfies one of the following inequalities:

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$. Furthermore, assume that the following conditions are satisfied:

(1) $T$ is path admissible;

(2) There exist $a_0, a_1, a_3 \in X$ with $a_3 = T(a_0, a_1)$ and $a_0Pa_3$;

(3) $T$ is continuous with respect to each coordinate.

Then, $T$ has a fixed point in $X.$

Suppose that $G = (V, E)$ where $V = X$ and $E = X \times X$, then Theorems 5.1 and 5.2 give rise to the following corollaries, respectively.

Corollary 5.1. Let $(X, d_b)$ be a complete $b$-metric space, where $d_b$ is a continuous functional and consider $T:X \times X\rightarrow X$ a mapping such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in X$, we have either

or

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$. Then $T$ has a fixed point.

Corollary 5.2. Let $(X, d_b)$ be a complete $b$-metric space, where $d_b$ is a continuous functional and consider $T:X \times X\rightarrow X$ as a mapping such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in X$ we have either

or

or

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$. Then $T$ has a fixed point.

6.   Doubled controlled metric space

In this section Theorem 3.1 has been proved in the setting of double controlled metric spaces introduced by Abdeljawad et al.^[1]. In ^[1] the notion of doubled controlled metric type space is given as follows.

Definition 6.1. Given non comparable functions $\alpha, \mu : X\times X \rightarrow [1, \infty).$ If $d_b:X \times X \rightarrow (0, \infty)$ satisfies

(1) $d_b(w_1, w_2) = 0 \Longleftrightarrow w_1 = w_2$

(2) $d_b(w_1, w_2) = d_b(w_2, w_1)$

(3) $d_b(w_1, w_3)\leq \alpha(w_1, w_2)d_b(w_1, w_2)+\mu(w_2, w_3)d_b(w_2, w_3)$ for all $w_1, w_2, w_3 \in X.$

Then $(X, d_b)$ is called a double controlled metric type by $\alpha$ and $\mu$.

Remark 6.1. The class of double controlled metric is larger than $b$-metric. If $\alpha(w) = \mu (w) = b \geq 1$ for all $w\in X$ then, double controlled metric type is a $b$-metric with coefficient $b$.

The notion of convergence, Cauchyness and completeness can be extended naturally in the setting of double controlled metric type space as in ^[1].

Theorem 6.1. Suppose that $(X, d_b)$ be a complete double controlled metric type space by the functions $\alpha, \mu : X\times X \rightarrow [1, \infty)$ such that

and $\lim\limits_{n\rightarrow \infty}\alpha(u, u_n)$ and $\lim\limits_{n\rightarrow \infty}\mu(u, u_n)$ exist and are finite. Let $\mathcal{R}$ be a binary relation on $X$, where $d_ b$ is a continuous functional. Assume that $H$ and $K$ are nonempty closed subsets of $X$. Consider a mapping $T: H \times H \rightarrow K$ such that for each $h_{1}, h_{2}, h_{3}, w_{1}, w_{2} \in H$ with $h_{1}Ph_{3}$ that is $h_1 \mathcal{R}h_2, \; h_2\mathcal{R} h_3$ and

$d_b(w_{1}, T(h_{1}, h_{2})) = d_b(H, K) = d_b(w_{2}, T(h_{2}, h_{3}))$, we have:

where $\Gamma \in[0, 1)$ such that $b \Gamma < 1$. Furthermore, suppose that the subsequent conditions are true:

(1) $T$ is path admissible;

(2) $\exists \; h_{0}, h_{1}, h_{2} \in H$ which satisfy $d_b(h_{2}, T(h_{0}, h_{1})) = d_b(H, K)$ and $h_{0}Ph_{2}$;

(3) $T(H \times H _{0})\subseteq K _{0}$;

(4) $K$ is approximately compact with respect to $H$;

(5) If $\lbrace h_{j}\rbrace\subseteq X$ such that $h_{j}Ph_{j+2}$ for each $j \in \mathbb{N}$ and $h_{j} \rightarrow x_{\ast}$ as $j \rightarrow \infty$, then $h_{j}\mathcal{R} x_{\ast}$ for all $j \in \mathbb{N}$ and $x_{\ast}\mathcal{R} x_{\ast}.$

Then $T$ has a best proximity point.

Proof. Proceeding as in Theorem 3.1 till Eq (3.5) we obtain

$d_b(h_{j-1}, h_{j})\; \leq \; Z \psi^{j} \; \; \; \forall \; \; \; j \in \mathbb{N}.$ Now for $m > n$

Denoting $\mathbb{S}_q = \sum\limits_{i = 0}^{q}\left(\prod\limits_{j = 0}^{i}\mu(h_j, h_m) \right)\alpha(h_i, h_{i+1})\psi^{i},$ we have

The ratio test combined with (6.1) imply that the limit of the sequence $\{\mathbb{S}_n\}$ exists. Hence

implies that $\{h_n\}$ is a Cauchy sequence in $H.$ Since $H$ is complete, there exists some $h_{\ast}\in H$ such that $h_n\rightarrow h_{\ast}.$ Hence by $(vi)$ $h_n\mathcal{R}h_{\ast}\; \forall n\in \mathbb{N}.$ Furthermore, we have to prove that $d_b(h_{\ast}, T(h_{n-1}, h_{n}))\rightarrow d_b(h_{\ast}, K)\; \; \; as\; \; \; j\rightarrow \infty$. Consider,

Therefore $d_b(h_{\ast}, T(h_{n-1}, h_{n}))\rightarrow d_b(h_{\ast}, K)$ as $n\rightarrow \infty.$ The rest of the proof can be carried out in the same way as in Theorem 3.1 after Equation (3.6).

Remark 6.2. Note that Theorem 3.1 becomes a special case of Theorem 6.1 by taking $\alpha(w) = \mu (w) = b \geq 1$ for all $w\in X$.

Acknowledgments

The author Aiman Mukheimer and Suhad Subhi Aiadi would like to thank Prince Sultan University for paying APC and for the support through TAS research LAB.

Conflict of interest

The authors declare that they have no competing interests.