Best Proximity Point Results for Geraghty Type Ƶ-Proximal Contractions with an Application

Abstract

In this study, we establish the existence and uniqueness theorems of the best proximity points for Geraghty type 𝒵-proximal contractions defined on a complete metric space. The presented results improve and generalize some recent results in the literature. An example, as well as an application to a variational inequality problem are also given in order to illustrate the effectiveness of our generalizations.

1. Introduction

Numerous problems in science and engineering defined by nonlinear functional equations can be solved by reducing them to an equivalent fixed-point problem. In fact, an operator equation

$$Gx=0$$

(1)

may be expressed as a fixed-point equation $\mathcal{T}x=x.$ Accordingly, the Equation (1) has a solution if the self-mapping $\mathcal{T}$ has a fixed point. However, for a non-self mapping $\mathcal{T}:P\to Q,$ the equation $\mathcal{T}x=x$ does not necessarily admit a solution. Here, it is quite natural to find an approximate solution $x^{*}$ such that the distance $d(x^{*},\mathcal{T}{x}^{*})$ is minimum, in which case $x^{*}$ and $\mathcal{T}{x}^{*}$ are in close proximity to each other. Herein, the optimal approximate solution $x^{*},$ for which $d(x^{*},\mathcal{T}{x}^{*})=d(P,Q)$, is called a best proximity point of $\mathcal{T}.$ The main aim of the best proximity point theory is to give sufficient conditions for finding the existence of a solution to the nonlinear programming problem,

$$\underset{\xi \in P}{min}d(\xi ,\mathcal{T}\xi ).$$

(2)

Moreover, a best proximity point generates to a fixed point if the mapping under consideration is a self-mapping. For more details on this research subject, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].

In 2015, Khojasteh et al. [16] presented the notion of $\mathcal{Z}$-contraction involving a new class of mappings—namely, simulation functions, and proved new fixed-point theorems via different methods to others in the literature. For more details, see [17,18,19,20].

Definition 1

([16]). A simulation function is a mapping $\zeta :[0,\infty )\times [0,\infty )\to \mathbb{R}$ so that:

$\left({\zeta }_1\right)$

$\zeta (0,0)=0$;

$\left({\zeta }_2\right)$

$\zeta (\mu ,\eta )<\eta -\mu$ for all $\mu ,\eta >0$;

$\left({\zeta }_3\right)$

If $\left({\mu }_n\right),\left({\eta }_n\right)$ are sequences in $(0,\infty )$ so that ${\displaystyle \underset{n\to \infty }{lim}{\mu }_n=\underset{n\to \infty }{lim}{\eta }_n>0,}$ then

$$\underset{n\to \infty }{lim\; sup}\zeta ({\mu }_n,{\eta }_n)<0.$$

(3)

Theorem 1

([16]). Let $(M,d)$ be a complete metric space and $\mathcal{T}:M\to M$ be a $\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z}$—that is,

$$\zeta \left(d\right(\mathcal{T}\xi ,\mathcal{T}\omega ),d(\xi ,\omega \left)\right)\ge 0,forall\xi ,\omega \in M.$$

Then, $\mathcal{T}$ admits a unique fixed point (say $\tau \in X$) and, for each ${\xi }_0\in M$, the Picard sequence $\left\{{\mathcal{T}}^n{\xi }_0\right\}$ is convergent to τ.

In this study, we will consider simulation functions satisfying only the condition $\left({\zeta }_2\right).$ For the sake of convenience, we identify the set of all simulation functions satisfying only the condition $\left({\zeta }_2\right)$ by $\mathcal{Z}$.

The main concern of the paper is to establish theorems on the existence and uniqueness of best proximity points for Geraghty type $\mathcal{Z}$-proximal contractions in complete metric spaces. The obtained results complement and extend some known results from the literature. An example, as well as an application to a variational inequality problem, is also given in order to illustrate the effectiveness of our generalizations.

2. Preliminaries

Let P and Q be two non-empty subsets of a metric space, $(M,d).$ Consider:

$$\begin{array}{cc}\hfill d(P,Q)& :=inf\left\{d(\rho ,\nu ):\rho \in P,\nu \in Q\right\};\hfill \\ \hfill P_0& :=\left\{\rho \in P:d(\rho ,\nu )=d(P,Q)\mathrm{for}\mathrm{some}\nu \in Q\right\};\hfill \\ \hfill Q_0& :=\left\{\nu \in Q:d(\rho ,\nu )=d(P,Q)\mathrm{for}\mathrm{some}\rho \in P\right\}.\hfill \end{array}$$

Denote by

$$B_{est}\left(\mathcal{T}\right)=\left\{u\in P:d(u,\mathcal{T}u)=d(P,Q)\right\},$$

the set of all best proximity points of a non-self-mapping $\mathcal{T}:P\to Q$. In the study [5], Caballero et al. familiarized the notion of Geraghty contraction for non-self-mappings as follows:

Definition 2

([5]). Let $P,Q$ be two non-empty subsets of a metric space, $(M,d).$ A mapping $\mathcal{T}:P\to Q$ is called a Geraghty contraction if there is $\beta \in \mathsf{\Sigma }$, so that for all $\xi ,\omega \in P$

$$d(\mathcal{T}\xi ,\mathcal{T}\omega )\le \beta \left(d\right(\xi ,\omega \left)\right)·d(\xi ,\omega ),$$

(4)

where the class Σ is the set of functions $\beta :[0,\infty )\to [0,1)$, satisfying

$$\begin{array}{c}\hfill \beta \left(t_n\right)\to 1⟹t_n\to 0.\end{array}$$

In the paper [10], Jleli and Samet initiated the concepts of $\alpha$-$\psi$-proximal contractive and $\alpha$-proximal admissible mappings. They provided related best-proximity-point results. Subsequently, Hussain et al. [7] modified the aforesaid notions and substantiated certain best-proximity-point theorems.

Definition 3

([10]). Let $\mathcal{T}:P\to Q$ and $\alpha :P\times P\to \left[0,\infty \right)$ be given mappings. Then, $\mathcal{T}$ is called α-proximal admissible if

$$\left.\begin{array}{c}\alpha (u_1,u_2)\ge 1\hfill \\ d(p_1,\mathcal{T}{u}_1)=d(P,Q)\hfill \\ d(p_2,\mathcal{T}{u}_2)=d(P,Q)\hfill \end{array}\right\}⟹\alpha (p_1,p_2)\ge 1,$$

for all $u_1,u_2,p_1,p_2\in P.$

Definition 4

([7]). Let $\mathcal{T}:P\to Q$ and $\alpha ,\eta :P\times P\to \left[0,\infty \right)$ be given mappings. Such $\mathcal{T}$ is said to be $(\alpha ,\eta )$-proximal admissible if

$$\left.\begin{array}{c}\alpha (u_1,u_2)\ge \eta (u_1,u_2)\hfill \\ d(p_1,\mathcal{T}{u}_1)=d(P,Q)\hfill \\ d(p_2,\mathcal{T}{u}_2)=d(P,Q)\hfill \end{array}\right\}⟹\alpha (p_1,p_2)\ge \eta (p_1,p_2),$$

for all $u_1,u_2,p_1,p_2\in P.$

Note that if $\eta (u,v)=1$ for all $u,v\in P,$ then Definition 4 corresponds to Definition 3.

Very recently, Tchier et al. in [14] initiated the concept of $\mathcal{Z}$-proximal contractions.

Definition 5

([14]). Let P and Q be two non-empty subsets of a metric space, $(M,d).$ A non-self-mapping $\mathcal{T}:P\to Q$ is called a $\mathcal{Z}$-proximal contraction if there is a simulation function ζ so that

$$\left.\begin{array}{c}d(\rho ,\mathcal{T}u)=d(P,Q)\hfill \\ d(\nu ,\mathcal{T}v)=d(P,Q)\hfill \end{array}\right\}⟹\zeta (d(\rho ,\nu ),d(u,v))\ge 0,$$

(5)

for all $\rho ,\nu ,u,v\in P.$

Now, we introduce a new concept which will be efficiently used in our results.

Definition 6.

Let $\mathcal{T}:P\to Q$ and $\alpha ,\eta :P\times P\to \left[0,\infty \right)$ be given mappings. Then, $\mathcal{T}$ is said to be triangular $(\alpha ,\eta )$-proximal admissible, if

(1)

$\mathcal{T}$ is $(\alpha ,\eta )$-proximal admissible;

(2)

$\alpha (u,v)\ge \eta (u,v)and\alpha (v,z)\ge \eta (v,z)impliesthat\alpha (u,z)\ge \eta (u,z),$ for all $u,v,z\in P.$

Now, we describe a new class of contractions for non-self-mappings which generalize the concept of Geraghty-contractions.

Definition 7.

Let P and Q be two non-empty subsets of a metric space $(M,d),$$\zeta \in \mathcal{Z}$ and $\alpha ,\eta :P\times P\to \left[0,\infty \right)$ and $\beta \in \mathsf{\Sigma }.$ A non-self-mapping $\mathcal{T}:P\to Q$ is said to be a Geraghty type $\mathcal{Z}$-proximal contraction, if for all $u,v,\rho ,\nu \in P$, the following implication holds:

$$\left.\begin{array}{c}\alpha (u,v)\ge \eta (u,v)\hfill \\ d(\rho ,\mathcal{T}u)=d(P,Q)\hfill \\ d(\nu ,\mathcal{T}v)=d(P,Q)\hfill \end{array}\right\}⟹\zeta (d(\rho ,\nu ),\beta \left(d(u,v)\right)d(u,v))\ge 0.$$

(6)

Remark 1.

If $\mathcal{T}:P\to Q$ is a Geraghty type $\mathcal{Z}$-proximal contraction, then by $\left({\zeta }_2\right)$ and Definition 7, the following implication holds for all $u,v,\rho ,\nu \in P$ with $u\ne v$:

$$\left.\begin{array}{c}\alpha (u,v)\ge \eta (u,v)\hfill \\ d(\rho ,\mathcal{T}u)=d(P,Q)\hfill \\ d(\nu ,\mathcal{T}v)=d(P,Q)\hfill \end{array}\right\}⟹d(\rho ,\nu )<\beta \left(d(u,v)\right)d(u,v).$$

(7)

3. Main Results

Our first result is as follows.

Theorem 2.

Let $(P,Q)$ be a pair of non-empty subsets of a complete metric space $(M,d)$ so that $P_0$ is non-empty, $\mathcal{T}:P\to Q$ and $\alpha ,\eta :P\times P\to \left[0,\infty \right)$ be given mappings. Suppose that:

(i)

P is closed and $\mathcal{T}\left(P_0\right)\subseteq Q_0$;

(ii)

$\mathcal{T}$ is triangular $(\alpha ,\eta )$-proximal admissible;

(iii)

There are $u_0,u_1\in P_0$ so that $d(u_1,\mathcal{T}{u}_0)=d(P,Q)$ and $\alpha \left(u_0,u_1\right)\ge \eta \left(u_0,u_1\right)$;

(iv)

$\mathcal{T}$ is a continuous Geraghty type $\mathcal{Z}$-proximal contraction.

Then, $\mathcal{T}$ has a best proximity point in $P.$ If $\alpha (u,v)\ge \eta (u,v)$ for all $u,v\in B_{est}\left(\mathcal{T}\right),$ then $\mathcal{T}$ has a unique best proximity point $u^{*}\in P.$ Moreover, for every $u\in P,$${lim}_{n\to \infty }{\mathcal{T}}^{n}u=u^{*}$.

Proof. 

From the condition $\left(iii\right),$ there are $u_0,u_1\in P_0$ so that

$$d(u_1,\mathcal{T}{u}_0)=d(P,Q)\mathrm{and}\alpha \left(u_0,u_1\right)\ge \eta \left(u_0,u_1\right).$$

Since $\mathcal{T}\left(P_0\right)\subseteq Q_0$, there is $u_2\in P_0$ so that

$$d(u_2,\mathcal{T}{u}_1)=d(P,Q).$$

Thus, we get

$$\begin{array}{c}\alpha (u_0,u_1)\ge \eta (u_0,u_1),\hfill \\ d(u_1,\mathcal{T}{u}_0)=d(P,Q),\hfill \\ d(u_2,\mathcal{T}{u}_1)=d(P,Q).\hfill \end{array}$$

Since $\mathcal{T}$ is $(\alpha ,\eta )$-proximal admissible, we get $\alpha \left(u_1,u_2\right)\ge \eta \left(u_1,u_2\right).$ Now, we have

$$d(u_2,\mathcal{T}{u}_1)=d(P,Q)\mathrm{and}\alpha \left(u_1,u_2\right)\ge \eta \left(u_1,u_2\right).$$

Again, since $\mathcal{T}\left(P_0\right)\subseteq Q_0$, there exists $u_3\in P_0$ such that

$$d(u_3,\mathcal{T}{u}_2)=d(P,Q),$$

and thus,

$$\begin{array}{c}\alpha (u_1,u_2)\ge \eta (u_1,u_2),\hfill \\ d(u_2,\mathcal{T}{u}_1)=d(P,Q),\hfill \\ d(u_3,\mathcal{T}{u}_2)=d(P,Q).\hfill \end{array}$$

Since $\mathcal{T}$ is $(\alpha ,\eta )$-proximal admissible, this implies that $\alpha \left(u_2,u_3\right)\ge \eta \left(u_2,u_3\right).$ Thus, we have

$$d(u_3,\mathcal{T}{u}_2)=d(P,Q)\mathrm{and}\alpha \left(u_2,u_3\right)\ge \eta \left(u_2,u_3\right).$$

By repeating this process, we build a sequence $\left\{u_n\right\}$ in $P_0\subseteq P$ so that

$$d(u_{n+1},\mathcal{T}{u}_n)=d(P,Q)\mathrm{and}\alpha \left(u_n,u_{n+1}\right)\ge \eta \left(u_n,u_{n+1}\right),$$

(8)

for all $n\in \mathbb{N}\cup \left\{0\right\}.$ If there is $n_0$ so that $u_{n_0}=u_{n_0+1},$ then

$$d(u_{n_0},\mathcal{T}{u}_{n_0})=d(u_{n_0+1},\mathcal{T}{u}_{n_0})=d(P,Q).$$

That is, $u_{n_0}$ is a best proximity point of $\mathcal{T}$. We should suppose that $u_n\ne u_{n+1},$ for all $n.$

From (8), for all $n\in \mathbb{N},$ we get

$$\begin{array}{c}\alpha \left(u_{n-1},u_n\right)\ge \eta \left(u_{n-1},u_n\right),\hfill \\ d(u_n,\mathcal{T}{u}_{n-1})=d(P,Q),\hfill \\ d(u_{n+1},\mathcal{T}{u}_n)=d(P,Q).\hfill \end{array}$$

On the grounds that $\mathcal{T}$ is a Geraghty type $\mathcal{Z}$-proximal contraction, by utilizing Remark 1, we deduce that

$$\begin{array}{c}\hfill d(u_n,u_{n+1})<\beta \left(d(u_{n-1},u_n)\right)d(u_{n-1},u_n),\end{array}$$

(9)

which requires that $d(u_n,u_{n+1})<d(u_{n-1},u_n),$ for all $n.$ Therefore, the sequence $\left\{d(u_n,u_{n+1})\right\}$ is decreasing, and so there is $\lambda \ge 0$ so that ${lim}_{n\to \infty }d\left(u_n,u_{n+1}\right)=\lambda .$ Now, we shall show that $\lambda =0.$ On the contrary, assume that $\lambda >0.$ Then, taking into account (9), for any $n\in \mathbb{N},$

$$\begin{array}{c}\hfill d(u_n,u_{n+1})<\beta \left(d(u_{n-1},u_n)\right)d(u_{n-1},u_n)<d(u_{n-1},u_n).\end{array}$$

This yields, for any $n\in \mathbb{N},$

$$\begin{array}{c}\hfill 0<\frac{d(u_n,u_{n+1})}{d(u_{n-1},u_n)}<\beta \left(d(u_{n-1},u_n)\right)<1.\end{array}$$

Taking $n\to \infty$, we find that

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}\beta \left(d(u_{n-1},u_n)\right)=1,}\end{array}$$

and since $\beta \in \mathsf{\Sigma },$${lim}_{n\to \infty }d(u_{n-1},u_n)=0.$ This contradicts our assumption ${lim}_{n\to \infty }d(u_{n-1},u_n)=\lambda >0.$ Therefore, we get

$$\underset{n\to \infty }{lim}d(u_{n-1},u_n)=0,foralln\in \mathbb{N}.$$

(10)

We shall prove that $\left\{u_n\right\}$ is Cauchy in P. By contradiction, suppose that $\left\{u_n\right\}$ is not a Cauchy sequence, so there is an $\epsilon >0$ for which we can find $\left\{u_{m_k}\right\}$ and $\left\{u_{n_k}\right\}$ of $\left\{u_n\right\}$ such that $n_k$ is the smallest index for which $n_k>m_k>k$ and

$$d\left(u_{m_k},u_{n_k}\right)\ge \epsilon \mathrm{and}d\left(u_{m_k},u_{n_k-1}\right)<\epsilon .$$

(11)

We have

$$\begin{array}{cc}\hfill \epsilon \le d\left(u_{m_k},u_{n_k}\right)& \le d\left(u_{m_k},u_{n_k-1}\right)+d\left(u_{n_k-1},u_{n_k}\right)\hfill \\ & <\epsilon +d\left(u_{n_k-1},u_{n_k}\right).\hfill \end{array}$$

Taking $k\to \infty$, by (10), we get

$$\underset{k\to \infty }{lim}d\left(u_{m_k},u_{n_k}\right)=\epsilon .$$

(12)

By triangular inequality,

$$\left|d\left(u_{m_k+1},u_{n_k+1}\right)-d\left(u_{m_k},u_{n_k}\right)\right|\le d\left(u_{m_k+1},u_{m_k}\right)+d\left(u_{n_k},u_{n_k+1}\right),$$

which yields that

$$\underset{k\to \infty }{lim}d\left(x_{m_k+1},x_{n_k+1}\right)=\epsilon .$$

(13)

Since $\mathcal{T}$ is triangular $(\alpha ,\eta )$-proximal admissible, by using (8), we infer

$$\alpha (u_m,u_n)\ge \eta (u_m,u_n),\mathrm{for}\mathrm{all}n,m\in \mathbb{N}\mathrm{with}m<n.$$

(14)

Combining (8) and (14), for all $k\in \mathbb{N},$ we have

$$\begin{array}{c}\alpha (u_{m_k},u_{n_k})\ge \eta (u_{m_k},u_{n_k}),\hfill \\ d(u_{m_k+1},\mathcal{T}{u}_{m_k})=d(P,Q),\hfill \\ d(u_{n_k+1},\mathcal{T}{u}_{n_k})=d(P,Q).\hfill \end{array}$$

Regarding the fact that $\mathcal{T}$ is a Geraghty type $\mathcal{Z}$-proximal contraction, from Remark 1, we deduce that

$$\begin{array}{c}\hfill d(u_{m_k+1},u_{n_k+1})<\beta \left(d(u_{m_k},u_{n_k})\right)d(u_{m_k},u_{n_k})<d(u_{m_k},u_{n_k}).\end{array}$$

Taking the limit as k tends to ∞ on both sides of the last inequality, and using the Equations (12) and (13), we get

$$\begin{array}{c}\hfill \epsilon \le \underset{k\to \infty }{lim}\beta \left(d(u_{m_k},u_{n_k})\right)\epsilon \le \epsilon ,\end{array}$$

which implies that ${lim}_{k\to \infty }\beta \left(d(u_{m_k},u_{n_k})\right)=1,$ and so ${lim}_{k\to \infty }d(u_{m_k},u_{n_k})=0$ which contradicts $\epsilon >0.$ Hence, $\left\{u_n\right\}$ is a Cauchy sequence in P. Since P is a closed subset of the complete metric space $(M,d)$, there is $p\in P$ so that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}d(u_n,p)=0.\end{array}$$

(15)

Since $\mathcal{T}$ is continuous, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}d(\mathcal{T}{u}_n,\mathcal{T}p)=0.\end{array}$$

(16)

Combining (8), (15), and (16), we get

$$\begin{array}{c}\hfill d(P,Q)=\underset{n\to \infty }{lim}d(u_{n+1},\mathcal{T}{u}_n)=d(p,\mathcal{T}p).\end{array}$$

Therefore, $u\in P$ is a best proximity point of $\mathcal{T}.$ Finally, we shall show that the set $B_{est}\left(\mathcal{T}\right)$ is a singleton. Suppose that r is another best proximity point of $\mathcal{T},$ that is, $d(r,\mathcal{T}r)=d(P,Q).$ Then, by the hypothesis, we have $\alpha (p,r)\ge \eta (p,r)$—that is,

$$\begin{array}{c}\alpha (p,r)\ge \eta (p,r),\hfill \\ d(p,\mathcal{T}p)=d(P,Q),\hfill \\ d(r,\mathcal{T}r)=d(P,Q).\hfill \end{array}$$

Then, from Remark 1, we deduce

$$\begin{array}{c}\hfill d(p,r)<\beta \left(d\right(p,r\left)\right)d(p,r)<d(p,r),\end{array}$$

which is a contradiction. Hence, we have a unique best proximity point of $\mathcal{T}$. □

Let us consider the following assertion in order to remove the continuity on the operator $\mathcal{T}$ in the next theorem.

(C)

If a sequence $\left\{u_n\right\}$ in P is convergent to $u\in P$ so that $\alpha \left(u_n,u_{n+1}\right)\ge \eta \left(u_n,u_{n+1}\right)$, then $\alpha \left(u_n,u\right)\ge \eta \left(u_n,u\right)$ for all $n\in \mathbb{N}.$

Theorem 3.

Let $(P,Q)$ be a pair of non-empty subsets of a complete metric space $(M,d)$ so that $P_0$ is non-empty, $\mathcal{T}:P\to Q$ and $\alpha ,\eta :P\times P\to \left[0,\infty \right)$ be given mappings. Suppose that:

(i)

P is closed and $\mathcal{T}\left(P_0\right)\subseteq Q_0$;

(ii)

$\mathcal{T}$ is triangular $(\alpha ,\eta )$-proximal admissible;

(iii)

there are $u_0,u_1\in P_0$ so that $d(u_1,\mathcal{T}{u}_0)=d(P,Q)$ and $\alpha \left(u_0,u_1\right)\ge \eta \left(u_0,u_1\right)$;

(iv)

the condition $\left(C\right)$ holds and $\mathcal{T}$ is a Geraghty type $\mathcal{Z}$-proximal contraction.

Then, $\mathcal{T}$ has a best proximity point in $P.$ If $\alpha (u,v)\ge \eta (u,v)$ for all $u,v\in B_{est}\left(\mathcal{T}\right),$ then $\mathcal{T}$ has a unique best proximity point $u^{*}\in P.$ Moreover, for each $u\in P,$ we have ${lim}_{n\to \infty }{\mathcal{T}}^{n}u=u^{*}$.

Proof. 

Following the proof of Theorem 2, there exists a Cauchy sequence $\left\{u_n\right\}\subset P_0$ satisfying (8) and $u_n\to p.$ On account of (i), $P_0$ is closed, and so $p\in P_0.$ Also, since $\mathcal{T}\left(P_0\right)\subseteq Q_0,$ there is $z\in P_0$ so that

$$\begin{array}{c}\hfill d(z,\mathcal{T}p)=d(P,Q).\end{array}$$

(17)

Taking $\left(C\right)$ and (8) into account, we infer

$$\begin{array}{c}\hfill \alpha \left(u_n,p\right)\ge \eta \left(u_n,p\right),\mathrm{for}\mathrm{all}n\in \mathbb{N}.\end{array}$$

Since $\mathcal{T}$ is $(\alpha ,\eta )$-proximal admissible and

$$\begin{array}{c}\alpha (u_n,p)\ge \eta (u_n,p),\hfill \\ d(u_{n+1},\mathcal{T}{u}_n)=d(P,Q),\hfill \\ d(z,\mathcal{T}p)=d(P,Q),\hfill \end{array}$$

(18)

so, we conclude that

$$\begin{array}{c}\hfill \alpha (u_{n+1},z)\ge \eta (u_{n+1},z),\mathrm{for}\mathrm{all}n\in \mathbb{N}.\end{array}$$

(19)

Considering (18), (19) and Remark 1, we have

$$\begin{array}{c}\hfill d(u_{n+1},z)<\beta \left(d(u_n,p)\right)d(u_n,p)<d(u_n,p),\end{array}$$

which implies that ${lim}_{n\to \infty }d(u_{n+1},z)=0.$ By the uniqueness of the limit, we obtain $z=p.$ Thus, by (17), we deduce that $d(p,\mathcal{T}p)=d(P,Q).$ Uniqueness of the best proximity point follows from the proof of Theorem 2. □

Example 1.

Let $M={\mathbb{R}}^2$ be endowed with the Euclidian metric, $P=\left\{(0,u):u\ge 0\right\}$ and $Q=\left\{(1,u):u\ge 0\right\}$. Note that $d(P,Q)=1,$ $P_0=P$ and $Q_0=Q.$ Let

$$\left\{\begin{array}{cc}\beta \left(t\right)=\frac{1}{1+t},\hfill & if t>0\hfill \\ \beta \left(t\right)=\frac{1}{2},\hfill & otherwise.\hfill \end{array}\right.$$

Then, $\beta \in \mathsf{\Sigma }.$ Define $\mathcal{T}:P\to Q$ and $\alpha :P\times P\to \left[0,\infty \right)$ by

$$\mathcal{T}(0,u)=\left\{\begin{array}{cc}(1,\frac{u}{9}),\hfill & if 0\le u\le 1,\hfill \\ (1,u^2),\hfill & if u>1,\hfill \end{array}\right.$$

and

$$\alpha ((0,u),(0,v))=\left\{\begin{array}{cc}2\eta \left(\right(0,u),(0,v\left)\right),\hfill & ifu,v\in [0,1],oru=v\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

Choose $\zeta (t,s)=\frac{2}{3}s-t$ for all $t,s\in [0,\infty )$. Let $u,v,p,q\ge 0$ be such that

$$\left\{\begin{array}{c}\alpha \left(\right(0,u),(0,v\left)\right)\ge \eta \left(\right(0,u),(0,v\left)\right)\hfill \\ d\left(\right(0,p),\mathcal{T}(0,u\left)\right)=d(P,Q)=1\hfill \\ d\left(\right(0,q),\mathcal{T}(0,v\left)\right)=d(P,Q)=1.\hfill \end{array}\right.$$

Then, $u,v\in [0,1]$ or $u=v$.

$u,v\in [0,1]$. Here, $\mathcal{T}(0,u)=(1,\frac{u}{9})$ and $\mathcal{T}(0,v)=(1,\frac{v}{9})$. Also,

$$\sqrt{1+{(p-\frac{u}{9})}^2}=\sqrt{1+{(q-\frac{v}{9})}^2}=1,$$

that is, $p=\frac{u}{9}$ and $q=\frac{v}{9}$. So, $\alpha \left(\right(0,p),(0,q\left)\right)\ge d\left(\right(0,p),(0,q\left)\right).$ Moreover,

$$\begin{array}{c}\zeta \left(d\right((0,p),(0,q)),\beta (d\left(\right(0,u),(0,v\left)\right)\left)d\right((0,u),(0,v)\left)\right)\hfill \\ =\frac{2}{3}\beta \left(d((0,u),(0,v))\right)d((0,u),(0,v))-d((0,\frac{u}{9}),(0,\frac{v}{9}))\hfill \\ =\frac{2}{3}\beta \left(\right|u-v\left|\right)|u-v|-\frac{|u-v|}{9}.\hfill \end{array}$$

If $u=v$, then $\beta \left(\right|u-v\left|\right)=\frac{1}{2}$ and the right-hand side of the above inequality is equal to 0.

If $u\ne v$, we have

$$\begin{array}{c}\zeta \left(d\right((0,p),(0,q)),\beta (d\left(\right(0,u),(0,v\left)\right)\left)d\right((0,u),(0,v)\left)\right)\hfill \\ =\frac{2}{3}\frac{|u-v|}{1+|u-v|}-\frac{|u-v|}{9}\ge 0.\hfill \end{array}$$

$u=v>1$. Here, $\mathcal{T}(0,u)=(1,u^2)$ and $\mathcal{T}(0,v)=(1,v^2)$. Similarly, we get that $p=q=u^2=v^2$. So, $\alpha \left(\right(0,p),(0,q\left)\right)=0=\eta \left(\right(0,p),(0,q\left)\right).$

Also, $\zeta \left(d\right((0,p),(0,q)),\beta (d\left(\right(0,u),(0,v\left)\right)\left)d\right((0,u),(0,v)\left)\right)\ge 0$.

In each case, we get that $\mathcal{T}$ is an $(\alpha ,\eta )$-proximal admissible. It is also easy to see that $\mathcal{T}$ is triangular $(\alpha ,\eta )$-proximal admissible. Also, $\mathcal{T}$ is a Geraghty type $\mathcal{Z}$-proximal contraction. Also, if $\left\{u_n=(0,p_n)\right\}$ is a sequence in P such that $\alpha \left(u_n,u_{n+1}\right)\ge \eta \left(u_n,u_{n+1}\right)$ for all n and $u_n=(0,p_n)\to u=(0,p)$ as $n\to \infty ,$ then $p_n\to p$. We have $p_n,p_{n+1}\in [0,1]$ or $p_n=p_{n+1}$. We get that $p\in [0,1]$ or $p_n=p$. This implies that $\alpha \left(u_n,u\right)\ge \eta \left(u_n,u\right)$ for all $n.$

Moreover, there is $(u_0,u_1)=((0,1),(0,\frac{1}{9}))\in P_0\times P_0$ so that

$$\begin{array}{c}\hfill d(u_1,\mathcal{T}{u}_0)=1=d(P,Q)\mathrm{and}\alpha \left(u_0,u_1\right)\ge d\left(u_0,u_1\right).\end{array}$$

Consequently, all conditions of Theorem 3 are satisfied. Therefore, $\mathcal{T}$ has a unique best proximity point in P, which is $(0,0).$ On the other side, we indicate that (4) is not satisfied. In fact, for $u=(0,2),v=(0,3),$ we have

$$\begin{array}{cc}\hfill d(\mathcal{T}u,\mathcal{T}v)& =d\left(\mathcal{T}\right(0,2),\mathcal{T}(0,3\left)\right)=d\left(\right(0,4),(0,9\left)\right)\hfill \\ & =5>\frac{1}{2}=\beta \left(d((0,2),(0,3))\right)d((0,2),(0,3))\hfill \\ & =\beta \left(d\right(u,v\left)\right)d(u,v).\hfill \end{array}$$

Corollary 1.

Let $(P,Q)$ be a pair of non-empty subsets of a complete metric space $(M,d)$, such that $P_0$ is non-empty. Suppose that $\mathcal{T}:P\to Q$ is a Geraghty-proximal contraction—that is, the following implication holds for all $u,v,\rho ,\nu \in P$:

$$\left.\begin{array}{c}d(\rho ,\mathcal{T}u)=d(P,Q)\hfill \\ d(\nu ,\mathcal{T}v)=d(P,Q)\hfill \end{array}\right\}⟹\zeta (d(\rho ,\nu ),\beta \left(d(u,v)\right)d(u,v))\ge 0.$$

Also, assume that P is closed and $\mathcal{T}\left(P_0\right)\subseteq Q_0.$ Then, $\mathcal{T}$ has a unique best proximity point $u^{*}\in P.$ Moreover, for each $u\in P,$ we have ${lim}_{n\to \infty }{\mathcal{T}}^{n}u=u^{*}$.

Proof. 

We take $\alpha (\sigma ,\varsigma )=\eta (\sigma ,\varsigma )=1$ in the proof of Theorem 2 (resp. Theorem 3). □

4. Some Consequences

In this section we give new fixed-point results on a metric space endowed with a partial ordering/graph by using the results provided in the previous section. Define

$$\alpha ,\eta :M\times M\to [0,\infty ),\alpha \left(u,v\right)=\left\{\begin{array}{cc}\eta (u,v),\hfill & \mathrm{if} u⪯v,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Definition 8.

Let $(M,⪯,d)$ be a partially ordered metric space, $(P,Q)$ be a pair of non-empty subsets of M, and $\mathcal{T}:P\to Q$ be a given mapping. Such $\mathcal{T}$ is said to be ⪯-proximal increasing if

$$\left.\begin{array}{c}\hfill u_1⪯u_2\\ \hfill d(p_1,\mathcal{T}{u}_1)=d(P,Q)\\ \hfill d(p_2,\mathcal{T}{u}_2)=d(P,Q)\end{array}\right\}⟹p_1⪯p_2,$$

for all $u_1,u_2,p_1,p_2\in P.$

Then, the following result is a direct consequence of Theorem 2 (resp. Theorem 3).

Theorem 4.

Let $(P,Q)$ be a pair of non-empty subsets of a complete ordered metric space $(M,⪯,d)$ so that $P_0$ is non-empty and $\mathcal{T}:P\to Q$ be a given non-self-mapping. Suppose that:

(i)

P is closed and $\mathcal{T}\left(P_0\right)\subseteq Q_0$;

(ii)

$\mathcal{T}$ is ⪯-proximal increasing;

(iii)

There are $u_0,u_1\in P_0$ so that $d(u_1,\mathcal{T}{u}_0)=d(P,Q)$ and $u_0⪯u_1$;

(iv)

$\mathcal{T}$ is continuous or, for every sequence $\left\{u_n\right\}$ in P is convergent to $u\in P$ so that $u_n⪯u_{n+1}$, we have $u_n⪯u$ for all $n\in \mathbb{N};$

(v)

There exist $\zeta \in \mathcal{Z}$ and $\beta \in \mathsf{\Sigma }$, such that for all $u,v,\rho ,\nu \in P,$

$$\left.\begin{array}{c}\hfill u⪯v\\ \hfill d(\rho ,\mathcal{T}u)=d(P,Q)\\ \hfill d(\nu ,\mathcal{T}v)=d(P,Q)\end{array}\right\}⟹\zeta (d(\rho ,\nu ),\beta \left(d(u,v)\right)d(u,v))\ge 0.$$

(20)

Then, $\mathcal{T}$ has a best proximity point in $P.$ If $u⪯v$ for all $u,v\in B_{est}\left(\mathcal{T}\right),$ then $\mathcal{T}$ has a unique best proximity point $u^{*}\in P.$ Moreover, for every $u\in P,$${lim}_{n\to \infty }{\mathcal{T}}^{n}u=u^{*}$.

Now, we present the existence of the best proximity point for non-self mappings from a metric space M, endowed with a graph, into the space of non-empty closed and bounded subsets of the metric space. Consider a graph G, such that the set $V\left(G\right)$ of its vertices coincides with M and the set $E\left(G\right)$ of its edges contains all loops; that is, $E\left(G\right)\supseteq \Delta ,$ where $\Delta =\left\{\left(u,u\right):u\in M\right\}$. We assume G has no parallel edges, so we can identify G with the pair $\left(V\left(G\right),E\left(G\right)\right)$.

Define

$$\alpha ,\eta :M\times M\to [0,+\infty ),\alpha \left(u,v\right)=\left\{\begin{array}{cc}\eta (u,v),\hfill & \mathrm{if} \left(u,v\right)\in E\left(G\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Definition 9.

Let $\left(M,d\right)$ be a complete metric space endowed with a graph G and $(P,Q)$ be a pair of non-empty subsets of M and $\mathcal{T}:P\to Q$ be a given mapping. Such $\mathcal{T}$ is said to be triangular G-proximal, if

(1)

for all $u_1,u_2,p_1,p_2\in P$,

$$\left.\begin{array}{c}\hfill (u_1,u_2)\in E\left(G\right)\\ \hfill d(p_1,\mathcal{T}{u}_1)=d(P,Q)\\ \hfill d(p_2,\mathcal{T}{u}_2)=d(P,Q)\end{array}\right\}⟹(p_1,p_2)\in E\left(G\right);$$

(2)

$(u,v)\in E\left(G\right)and(v,z)\in E\left(G\right)impliesthat(u,z)\in E\left(G\right),$ for all $u,v,z\in P.$

for all $u_1,u_2,p_1,p_2\in P.$

The following result is a direct consequence of Theorem 2 (resp. Theorem 3).

Theorem 5.

Let $\left(M,d\right)$ be a complete metric space endowed with a graph G and $(P,Q)$ be a pair of non-empty subsets of M so that $P_0$ is non-empty and $\mathcal{T}:P\to Q$ be a given non-self mapping. Suppose that:

(i)

P is closed and $\mathcal{T}\left(P_0\right)\subseteq Q_0$;

(ii)

$\mathcal{T}$ is triangular G-proximal;

(iii)

There are $u_0,u_1\in P_0$ so that $d(u_1,\mathcal{T}{u}_0)=d(P,Q)$ and $(u_0,u_1)\in E\left(G\right)$;

(iv)

$\mathcal{T}$ is continuous or, for every sequence $\left\{u_n\right\}$ in P is convergent to $u\in P$ so that $(u_n,u_{n+1})\in E\left(G\right)$, we have $(u_n,u)\in E\left(G\right)$ for all $n\in \mathbb{N};$

(v)

There exist $\zeta \in \mathcal{Z}$ and $\beta \in \mathsf{\Sigma }$ such that for all $u,v,\rho ,\nu \in P,$

$$\left.\begin{array}{c}\hfill (u,v)\in E\left(G\right)\\ \hfill d(\rho ,\mathcal{T}u)=d(P,Q)\\ \hfill d(\nu ,\mathcal{T}v)=d(P,Q)\end{array}\right\}⟹\zeta (d(\rho ,\nu ),\beta \left(d(u,v)\right)d(u,v))\ge 0.$$

(21)

Then, $\mathcal{T}$ has a best proximity point in $P.$ If $(u,v)\in E\left(G\right)$ for all $u,v\in B_{est}\left(\mathcal{T}\right),$ then $\mathcal{T}$ has a unique best proximity point $u^{*}\in P.$ Moreover, for every $u\in P,$${lim}_{n\to \infty }{\mathcal{T}}^{n}u=u^{*}$.

5. A Variational Inequality Problem

Let C be a non-empty, closed, and convex subset of a real Hilbert space H, with inner product $⟨·,·⟩$ and a norm $\parallel ·\parallel$. A variational inequality problem is given in the following:

$$\mathrm{Find} u\in C \mathrm{so} \mathrm{that} ⟨Su,v-u⟩\ge 0 \mathrm{for} \mathrm{all} v\in C,$$

(22)

where $S:H\to H$ is a given operator. The above problem can be seen in operations research, economics, and mathematical physics, especially in calculus of variations associated with the minimization of infinite-dimensional functionals. See [21] and the references therein. It appears in variant problems of nonlinear analysis, such as complementarity and equilibrium problems, optimization, and finding fixed points; see [21,22,23]. To solve problem (22), we define the metric projection operator $P_C:H\to C$. Note that for every $u\in H$, there is a unique nearest point $P_{C}u\in C$ so that

$$\parallel u-P_{C}u\parallel \le \parallel u-v\parallel ,\mathrm{for} \mathrm{all} v\in C.$$

The two lemmas below correlate the solvability of a variational inequality problem to the solvability of a special fixed-point problem.

Lemma 1

([24]). Let $z\in H$. Then, $u\in C$ is such that $⟨u-z,y-u⟩\ge 0$, for all $y\in C$ iff $u=P_{C}z$.

Lemma 2

([24]). Let $S:H\to H$. Then, $u\in C$ is a solution of $⟨Su,v-u⟩\ge 0$, for all $v\in C$, if $u=P_C(u-\lambda Su)$, with $\lambda >0$.

The main theorem of this section is:

Theorem 6.

Let C be a non-empty, closed, and convex subset of a real Hilbert space H. Assume that $S:H\to H$ is such that $P_C(I-\lambda S):C\to C$ is a Geraghty-proximal contraction. Then, there is a unique element $u^{*}\in C$, such that $⟨S{u}^{*},v-u^{*}⟩\ge 0$ for all $v\in C$. Also, for any $u_0\in C,$ the sequence $\left\{u_n\right\}$ given as $u_{n+1}=P_C(u_n-\lambda S{u}_n)$ where $\lambda >0$ and $n\in \mathbb{N}\cup \left\{0\right\}$, is convergent to $u^{*}$.

Proof. 

We consider the operator $\mathcal{T}:C\to C$ defined by $\mathcal{T}x=P_C(x-\lambda Sx)$ for all $x\in C$. By Lemma 2, $u\in C$ is a solution of $⟨Su,v-u⟩\ge 0$ for all $v\in C$, if $u=\mathcal{T}u$. Now, $\mathcal{T}$ verifies all the hypotheses of Corollary 1 with $P=Q=C.$ Now, from Corollary 1, the fixed-point problem $u=\mathcal{T}u$ possesses a unique solution $u^{*}\in C.$ □