Random Best Proximity Points for α-Admissible Mappings via Simulation Functions

Abstract

In this paper, we introduce a new concept of random $\alpha$-proximal admissible and random $\alpha$-$\mathcal{Z}$-contraction. Then we establish random best proximity point theorems for such mapping in complete separable metric spaces.

1. Introduction

Some well known random fixed point theorems are generalizations of classical fixed point theorems. Random fixed point theorems for contraction mapping in a Polish space, i.e., a separable complete metric space, were proved by Špaček [1], Hanš [2,3]. In 1966, Mukhejea [4] proved the random fixed point theorem of Schauder’s type in an atomic probability measure space. In 1976, Bharuch-Reid [5] introduced the random fixed point theorems that have been used to establish the uniqueness, existence, and measurability of solutions of random operator equations. In 1977. Itoh [6] extended some random fixed point theorems of Špaček and Hanš for a multivalued contraction mapping in separable complete metric spaces and solved some random differential equations with random fixed point theorems in Banach spaces. In 1984, Sehgal and Waters [7] proved the random fixed point theorem of the classical Rothe’s fixed point theorem. After that, many authors have extended, generalized and improved random fixed point theorems in several ways [8,9,10,11,12,13,14,15,16].

In 2012, Samet et al. [17] introduced a new class of $\alpha$-$\psi$-contractive type mapping and establish fixed point theorems for such mapping in complete metric spaces. Afterwards, Jleli and Samet [18] introduced a new class of $\alpha$-$\psi$-contractive type mapping to the case of non-selfmapping and establish best proximity point theorems for such mapping in complete metric spaces. Recently, several authors have investigated the existence and applications of fixed point and best proximity point theorems for $\alpha$-$\psi$-contractive mapping; see [19,20,21,22,23] and the references therein.

In 2015, Khojasteh et al. [24] introduced the notion of simulation function and proved some fixed point theorem in metric space. Later, Samet [25] and Tchier et al. [26] introduced the best proximity point theorems involving simulation functions. In 2016, Karapinar [27] introduced the notion of $\alpha$-admissible, $\mathcal{Z}$-contraction and proved fixed point theprems in complete metric space. In 2017, Karapinar and Khojasted [28] proved the existence of best proximity point theorems of certain mapping via simulation function of complete metric space.

In 2017, Anh [29] introduced the concept of random best proximity point of a random operator. Thereafter, many authors have focused on various existence theorems of random best proximity point; for detail, see [30,31,32].

Recently, Tchier and Vetro [33] introduced the concepts of random $\alpha$-admissible and random $\alpha$-$\psi$-contractive mappings and established random fixed point theorems.

The purpose of this paper is to present some random best proximity point theorems for certain mapping via simulation functions in separable metric space.

2. Preliminaries

Throughout this paper, let $(M,d)$ be a Polish space, and $(\Omega ,\Sigma )$ be a measurable space, where $\Sigma$ is a $\sigma$-algebra of subsets of $\Omega$. Let U and V are two nonempty subsets of $M$. The following notations will be used herein:

$$d_{\omega }(U,V):=inf\{d(u\left(\omega \right),v\left(\omega \right)):u:\Omega \to U,v:\Omega \to V,\omega \in \Omega \};$$

$$U_0:=\{u:\Omega \to U:d(u\left(\omega \right),v\left(\omega \right))=d_{\omega }(U,V)\mathrm{for}\mathrm{some}\mathrm{mapping}v:\Omega \to V\};$$

$$V_0:=\{v:\Omega \to V:d(u\left(\omega \right),v\left(\omega \right))=d_{\omega }(U,V)\mathrm{for}\mathrm{some}\mathrm{mapping}u:\Omega \to U\}.$$

Definition 1.

A mapping $T:\Omega \to M$ is called Σ-measurable if for any open subset N of $M$, the set

$$T^{-1}\left(N\right)=\{\omega \in \Omega :T\left(\omega \right)\in N\}\in \Sigma .$$

Notice that when we say that a set A is measurable we mean that A is $\Sigma$-measurable.

Definition 2.

A mapping $T:\Omega \to M$ is called a random operator if $T(·,x)$ is a measurable for any $x\in X.$

Definition 3.

A measurable mapping $\xi :\Omega \to M$ is called a random fixed point of $T:\Omega \to M$ if

$$\xi \left(\omega \right)=T(\omega ,\xi (\omega \left)\right),$$

for all $\omega \in \Omega .$

Definition 4.

Let $U$, V be two closed subsets of a Polish space M and $T:\Omega \times U\to V$ a random operator. A measurable mapping $\xi :\Omega \to U$ is called a random best proximity point of T if

$$d(\xi \left(\omega \right),T(\omega ,\xi \left(\omega \right)))=d_{\omega }(U,V),$$

for any $\omega \in \Omega .$

Clearly, the random best proximity point of a random fixed point of T if $U\cap V\ne \varnothing .$ This means that the concept of a random best proximity point is an extension of the concept of random fixed point.

Definition 5.

Let $(\Omega ,\Sigma )$ be a measurable space, X and Y be two metric spaces. A mapping $h:\Omega \times X\to Y$ is called Carathéodory if, for all $x\in X,$ the mapping $\omega \to h(\omega ,x)$ is Σ-measurable and for all $\omega \in \Omega ,$ the mapping $x\to h(\omega ,x)$ is continuous.

Definition 6

([24]). A simulation function is a mapping $\zeta :[0,\infty )\times [0,\infty )\to \mathbb{R}$ satisfying the following conditions:

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

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

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

$\zeta (t,s)<s-t$ for all $t,s>0$;

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

if $\left\{t_n\right\},$ $\left\{s_n\right\}$ are sequences in $(0,\infty )$ such that ${lim}_{n\to \infty }{t}_n={lim}_{n\to \infty }{s}_n>0,$ then

$$\underset{n\to \infty }{lim\; sup}\zeta (t_n,s_n)<0.$$

Denote with $\mathcal{Z}$ the family of all simulation functions $\zeta :[0,\infty )\times [0,\infty )\to \mathbb{R}.$ Due to the axiom $\left({\zeta }_2\right)$, we have

$$\zeta (t,t)<0,\mathrm{for}\mathrm{all}t>0.$$

Denote with $\mathrm{\Psi }$ the family of non-decreasing functions $\psi :[0,\infty )\to [0,\infty )$ satisfying the following conditions:

$\left(i_1\right)$

$\psi \left(t\right)<t,$ for any $t\in {\mathbb{R}}^{+}$;

$\left(i_2\right)$

$\psi$ is continuous at 0.

Lemma 1

([34]). Let $(X,d)$ be a metric space and let $\left\{y_n\right\}$ be a sequence in X such that $d(y_{n+1},y_n)$ is nonincreasing and that

$$\underset{n\to \infty }{lim}d(y_{n+1},y_n)=0.$$

If $\left\{y_{2n}\right\}$ is not a Cauchy sequence, then there exist an $ϵ>0$ and two sequences $\left\{m_k\right\}$ and $\left\{n_k\right\}$ of positive integers such that the following four sequences tend to ϵ when $k\to \infty$:

$$d(y_{2m_k},y_{2n_k}),d(y_{2m_k},y_{2n_{k+1}}),d(y_{2m_k-1},y_{2n_k}),d(y_{2m_k-1},y_{2n_k+1}).$$

3. Main Results

We start with the following definition.

Definition 7.

Let $T:\Omega \times M\to M$ and $\alpha :\Omega \times M\times M\to [0,\infty ).$ We say that T is a random triangular weak-α-admissible if

$$\alpha (\omega ,x(\omega ),y(\omega \left)\right)\ge 1and\alpha (\omega ,y(\omega ),z(\omega \left)\right)\ge 1\Rightarrow \alpha (\omega ,x(\omega ),z(\omega \left)\right)\ge 1,$$

for all $x,y,z\in M$ and $\omega \in \Omega .$

Definition 8.

Let $(\Omega ,\Sigma )$ be a measurable space, $(M,d)$ be a separable metric space, U and V are two nonempty subsets of M, $T:\Omega \times U\to V$ and $\alpha :\Omega \times U\times U\to [0,\infty ).$ We say that T is a random α-proximal admissible if

$$\left.\begin{array}{c}\alpha (\omega ,x(\omega ),y(\omega \left)\right)\ge 1\hfill \\ d(u\left(\omega \right),T(\omega ,x\left(\omega \right)))=d_{\omega }(U,V)\hfill \\ d(v\left(\omega \right),T(\omega ,y\left(\omega \right)))=d_{\omega }(U,V)\hfill \end{array}\right\}\Rightarrow \alpha (\omega ,u\left(\omega \right),v\left(\omega \right))\ge 1,$$

for all $x,y,u,v\in M$ and $\omega \in \Omega .$

Definition 9.

Let $(\Omega ,\Sigma )$ be a measurable space, $(M,d)$ be a separable metric space, U and V are two nonempty subsets of M, $\psi \in \mathrm{\Psi }$, and $\alpha :\Omega \times M\times M\to [0,\infty ).$ We say that $T:\Omega \times U\to V$ is a random α-ψ-$\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z}$ if T is a random α-proximal admissible and

$$\left.\begin{array}{c}\alpha (\omega ,x(\omega ),y(\omega \left)\right)\ge 1\hfill \\ d(u\left(\omega \right),T(\omega ,x\left(\omega \right)))=d_{\omega }(U,V)\hfill \\ d(v\left(\omega \right),T(\omega ,y\left(\omega \right)))=d_{\omega }(U,V)\hfill \end{array}\right\}\Rightarrow \zeta (d(u\left(\omega \right),v\left(\omega \right)),\psi \left(d(x\left(\omega \right),y\left(\omega \right))\right))\ge 0,$$

for all $x,y,u,v\in M$ and $\omega \in \Omega .$

Definition 10.

Let $(\Omega ,\Sigma )$ be a measurable space, $(M,d)$ be a separable metric space, U and V are two nonempty subsets of M, and $\alpha :\Omega \times M\times M\to [0,\infty ).$ We say that $T:\Omega \times U\to V$ is a random α-$\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z}$ if T is a random α-proximal admissible and

$$\left.\begin{array}{c}\alpha (\omega ,x(\omega ),y(\omega \left)\right)\ge 1\hfill \\ d(u\left(\omega \right),T(\omega ,x\left(\omega \right)))=d_{\omega }(U,V)\hfill \\ d(v\left(\omega \right),T(\omega ,y\left(\omega \right)))=d_{\omega }(U,V)\hfill \end{array}\right\}\Rightarrow \zeta (d(u\left(\omega \right),v\left(\omega \right)),d(x\left(\omega \right),y\left(\omega \right)))\ge 0,$$

for all $x,y,u,v\in M$ and $\omega \in \Omega .$

Notice that Definition 9 dose not yield Definition 10. Indeed, for $\psi \left(t\right)=t,$ the implication can be happen but $\psi \left(t\right)=t\notin \mathrm{\Psi }.$

Definition 11.

Let $(\Omega ,\Sigma )$ be a measurable space, $(M,d)$ be a separable metric space, U and V are two nonempty subsets of M, and $\alpha :\Omega \times M\times M\to [0,\infty ).$ We say that $T:\Omega \times U\to V$ is a generalized random α-$\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z}$ if T is a random α-proximal admissible and

$$\left.\begin{array}{c}\alpha (\omega ,x(\omega ),y(\omega \left)\right)\ge 1\hfill \\ d(u\left(\omega \right),T(\omega ,x\left(\omega \right)))=d_{\omega }(U,V)\hfill \\ d(v\left(\omega \right),T(\omega ,y\left(\omega \right)))=d_{\omega }(U,V)\hfill \end{array}\right\}\Rightarrow \zeta (d(u\left(\omega \right),v\left(\omega \right)),r(x\left(\omega \right),y\left(\omega \right)))\ge 0,$$

for all $x,y,u,v\in M$ and $\omega \in \Omega ,$ with $x\left(\omega \right)\ne y\left(\omega \right),$ where

$$r(x\left(\omega \right),y\left(\omega \right))=max\left\{d(x\left(\omega \right),y\left(\omega \right)),\frac{d\left(x\right(\omega ),u(\omega \left)\right)d\left(y\right(\omega ),v(\omega \left)\right)}{d\left(x\right(\omega ),y(\omega \left)\right)}\right\}.$$

We can now state the main result of this paper.

Theorem 1.

Let $(\Omega ,\Sigma )$ be a measurable space, let $(M,d)$ be a Polish space, U and V are two nonempty subsets of M and $\alpha :\Omega \times M\times M\to [0,\infty ).$ Suppose that $T:\Omega \times U\to V$ is a random α-ψ-$\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z}$ and ζ is non-decreasing with respect to second component. The hypotheses are the following:

$\left(A_1\right)$

T is a random triangular weak-α-admissible,

$\left(A_2\right)$

U is closed with respect to the topology induced by $d,$

$\left(A_3\right)$

$T(\Omega \times U_0)\subset V_0,$

$\left(A_4\right)$

there exist measurable mappings $x_0,x_1:\Omega \to U$ such that, for all $\omega \in \Omega ,$ $d(x_1\left(\omega \right),T(\omega ,x_0\left(\omega \right)))=d(U,V)$ and $\alpha (\omega ,x_0\left(\omega \right),x_1\left(\omega \right))\ge 1,$

$\left(A_5\right)$

T is a Carathéodory mapping.

Then T has a random best proximity point, that is, there exists $\xi :\Omega \to U$ which is a measurable such that $d(\xi \left(\omega \right),T(\omega ,\xi \left(\omega \right)))=d_{\omega }(U,V)$ for all $\omega \in \Omega .$

Proof of Theorem 1.

By hypothese $\left(A_4\right),$ we have there exists measurable mapping $x_0,x_1:\Omega \to U$ such that $\alpha (\omega ,x_0\left(\omega \right),x_1\left(\omega \right))\ge 1$ and

$$d(x_1\left(\omega \right),T(\omega ,x_0\left(\omega \right)))=d_{\omega }(U,V),$$

for all $\omega \in \Omega .$ The hypthese $\left(A_3\right)$ implies that $T(\omega ,x_1\left(\omega \right))\in V_0,$ which yields there exists measurable mapping $x_2:\Omega \to U_0$ such that

$$d(x_2\left(\omega \right),T(\omega ,x_1\left(\omega \right)))=d_{\omega }(U,V),$$

for all $\omega \in \Omega .$ Since $\alpha (\omega ,x_0\left(\omega \right),x_1\left(\omega \right))\ge 1$ and T is a random $\alpha$-proximal admissible, we have that $\alpha (\omega ,x_1\left(\omega \right),x_2\left(\omega \right))\ge 1.$ Iteratively, a sequence $\left\{x_n\left(\omega \right)\right\}\subset U_0$ can be constructed as follows:

$$d(x_{n+1}\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d(U,V),\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,\omega \in \Omega ,$$

(1)

and

$$\alpha (\omega ,x_n\left(\omega \right),x_{n+1}\left(\omega \right))\ge 1,\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,\omega \in \Omega .$$

(2)

If $x_n\left(\omega \right)=x_{n+1}\left(\omega \right)$ for some $n\in {\mathbb{N}}_0,$ $\omega \in \Omega ,$ then

$$d(x_n\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d(x_{n+1}\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d_{\omega }(U,V),$$

that is $x_n\left(\omega \right)$ is a random best proximity point. Assume that

$$x_n\left(\omega \right)\ne x_{n+1}\left(\omega \right)\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,\mathrm{for}\mathrm{one}\omega \in \Omega .$$

(3)

By combining (1)–(3), we get that

$$d(x_n\left(\omega \right),T(\omega ,x_{n-1}\left(\omega \right)))=d(x_{n+1}\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d_{\omega }(U,V),$$

for all $n\in \mathbb{N},$ $\omega \in \Omega$ and

$$\zeta (d(x_n\left(\omega \right),x_{n+1}\left(\omega \right)),\psi \left(d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))\right))\ge 0,\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,\omega \in \Omega .$$

(4)

Since T is a random $\alpha$-$\psi$-$\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z}.$ Regarding (3) and $\left({\zeta }_2\right),$ the inequality (4) yields that

$$\begin{array}{ccc}\hfill d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))& \le & \psi \left(d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))\right)\hfill \\ & <& d(x_{n-1}\left(\omega \right),x_n\left(\omega \right)),\hfill \end{array}$$

for all $n\in \mathbb{N},$ $\omega \in \Omega .$ It follows that $\left\{d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))\right\}$ is a non-increasing sequence bounded below. Then, there exists $r\ge 0$ such that $\left\{d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))\right\}\to r.$ We claim that $r=0.$ Assume on the contrary that $r>0.$ Obviously,

$$\underset{n\to \infty }{lim}d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))=\underset{n\to \infty }{lim}d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))=r.$$

(5)

From (5) and the property $\left({\zeta }_3\right)$ of simulation function and $\left(i_1\right)$ and $\zeta$ is non-decreasing with respect to second component, we get

$$\begin{array}{ccc}\hfill 0& \le & \underset{n\to \infty }{lim\; sup}\zeta (d(x_n\left(\omega \right),x_{n+1}\left(\omega \right)),\psi \left(d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))\right))\hfill \\ & \le & \underset{n\to \infty }{lim\; sup}\zeta (d(x_n\left(\omega \right),x_{n+1}\left(\omega \right)),d(x_{n-1}\left(\omega \right),x_n\left(\omega \right)))\hfill \\ & <& 0,\hfill \end{array}$$

which is a contradiction, that is

$$\underset{n\to \infty }{lim}d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))=0.$$

Next, to prove that $\left\{x_n\left(\omega \right)\right\}$ is a Cauchy sequence. Suppose, on the contrary, that $\left\{x_n\left(\omega \right)\right\}$ is not Cauchy sequence. Consequently, there exists $ϵ>0$ and subsequences $\left\{x_{m_k}\left(\omega \right)\right\}$ and $\left\{x_{n_k}\left(\omega \right)\right\}$ of $\left\{x_n\left(\omega \right)\right\},$ so that for $n_k>m_k>k,$ we have

$$d(x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right))\ge ϵ,$$

and

$$d(x_{m_k}\left(\omega \right),x_{n_k-1}\left(\omega \right))<ϵ.$$

By Lemma (1), we have

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

(6)

Since T is a random triangular weak-$\alpha$-admissible, from (2) we have

$$\alpha (\omega ,x_n\left(\omega \right),x_m\left(\omega \right))\ge 1,\mathrm{for}\mathrm{all}n,m\in {\mathbb{N}}_0,\mathrm{with}n>m,\omega \in \Omega .$$

Thus, we have

$$\alpha (\omega ,x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right))\ge 1,$$

and

$$\begin{array}{ccc}\hfill d(x_{m_k}\left(\omega \right),T(\omega ,x_{m_k-1}\left(\omega \right)))& =& d(x_{n_k}\left(\omega \right),T(\omega ,x_{n_k-1}\left(\omega \right)))\hfill \\ & =& d(U,V)\mathrm{for}\mathrm{all}k\in \mathbb{N}.\hfill \end{array}$$

(7)

Since T is a random $\alpha$-$\psi$-$\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z},$ the obtained expression (7) yields the following inequality:

$$0\le \zeta (d(x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right)),\psi \left(d(x_{m_k-1}\left(\omega \right),T(\omega ,x_{n_k-1}\left(\omega \right)))\right)),\mathrm{for}\mathrm{all}k\in \mathbb{N}.$$

Letting $k\to \infty$ and keeping (6) and $\left({\zeta }_3\right)$ in mind, and regarding $\left({\zeta }_3\right),$ $\left(i_1\right)$ and $\zeta$ is non-decreasing with respect to second component, we get

$$\begin{array}{ccc}\hfill 0& \le & \underset{n\to \infty }{lim\; sup}\zeta (d(x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right)),\psi \left(d(x_{m_k-1}\left(\omega \right),T(\omega ,x_{n_k-1}\left(\omega \right)))\right))\hfill \\ & \le & \underset{n\to \infty }{lim\; sup}\zeta (d(x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right)),d(x_{m_k-1}\left(\omega \right),T(\omega ,x_{n_k-1}\left(\omega \right))))\hfill \\ & <& 0,\hfill \end{array}$$

which is a contradiction. Thus, we conclude that the sequence $\left\{x_n\left(\omega \right)\right\}$ is a Cauchy sequence. Since $(M,d)$ is a complete and U is closed subset of $(M,d)$ and T is a Carathéodory mapping, there exists $\xi :\Omega \to U$ such that

$$\left\{x_n\left(\omega \right)\right\}\to \xi \left(\omega \right)\mathrm{as}n\to +\infty \mathrm{for}\mathrm{all}\omega \in \Omega ,$$

(8)

it follows that $x_n$ is measurable for all $n\in \mathbb{N}$ and

$$x_{n+1}\left(\omega \right)=T(\omega ,x_n\left(\omega \right))\to T(\omega ,\xi \left(\omega \right))\mathrm{as}n\to +\infty \mathrm{for}\mathrm{all}\omega \in \Omega .$$

(9)

From (1), (8) and (9) we have

$$d_{\omega }(U,V)=\underset{n\to \infty }{lim}d(x_{n+1}\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d(\xi \left(\omega \right),T(\omega ,\xi \left(\omega \right))).$$

Therefore $\xi$ is a random best proximity point. ☐

Theorem 2.

Let $(\Omega ,\Sigma )$ be a measurable space, let $(M,d)$ be a Polish space, U and V are two nonempty subsets of M and $\alpha :\Omega \times M\times M\to [0,\infty ).$ Suppose that $T:\Omega \times U\to V$ is a random α-ψ-$\mathcal{Z}$-contraction mapping with respect to $\zeta \in \mathcal{Z}$ and ζ is non-decreasing with respect to second component. The hypotheses are the following:

$\left(B_1\right)$

T is a random triangular weak-α-admissible,

$\left(B_2\right)$

U is closed with respect to the topology induced by $d,$

$\left(B_3\right)$

$T(\Omega \times U_0)\subset V_0,$

$\left(B_4\right)$

there exist measurable mappings $x_0,x_1:\Omega \to U$ such that, for all $\omega \in \Omega ,$ $d(x_1\left(\omega \right),T(\omega ,x_0\left(\omega \right)))=d_{\omega }(U,V)$ and $\alpha (\omega ,x_0\left(\omega \right),x_1\left(\omega \right))\ge 1,$

$\left(B_5\right)$

T is a sup-measurable,

$\left(B_6\right)$

if $\left\{u_n\left(\omega \right)\right\}$ is a sequence in U such that $\alpha (\omega ,u_n\left(\omega \right),u_{n+1}\left(\omega \right))\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\},$ $\omega \in \Omega$ and $u_n\left(\omega \right)\to u_n\left(\omega \right)$ as $n\to +\infty ,$ then there is a subsequence $\left\{u_{n_k}\left(\omega \right)\right\}$ of $\left\{u_n\left(\omega \right)\right\}$ with $\alpha (\omega ,u_{n_k}\left(\omega \right),u\left(\omega \right))\ge 1$ for all $k,$ $\omega \in \omega .$

Then T has a random best proximity point, that is, there exists $\xi :\Omega \to U$ is a measurable such that $d\left(\xi \right(\omega ),T(\omega ,\xi \left(\omega \right)\left)\right)=d(U,V)$ for all $\omega \in \Omega .$

Proof of Theorem 2.

A similar reasoning as in the proof of Theorem 1 gives us that the sequence $\left\{u_n\left(\omega \right)\right\}$ is a Cauchy sequence. This means that there exists $\xi :\Omega \to U$ such that $u_n\left(\omega \right)\to \xi \left(\omega \right)$ as $n\to +\infty$ for all $\omega \in \Omega .$ Due to $\left(B_2\right),$ $U_0$ is closed. Regarding $\left(B_3\right),$ we note that $T(\omega ,\xi \left(\omega \right))\in V_0$ and hence

$$d(u_1\left(\omega \right),T(\omega ,\xi \left(\omega \right)))=d_{\omega }(U,V)\mathrm{for}\mathrm{some}{u}_1\in U_0,\omega \in \Omega .$$

Notice that from $\left(B_6\right),$ we have

$$\alpha (\omega ,x_{n_k}\left(\omega \right),\xi \left(\omega \right))\ge 1\mathrm{for}\mathrm{all}k\in \mathbb{N},\omega \in \Omega .$$

Since T is a random $\alpha$-proximal admissible, and

$$\begin{array}{ccc}\hfill d(u_1\left(\omega \right),T(\omega ,\xi \left(\omega \right)))& =& d(x_{n_k+1}\left(\omega \right),T(\omega ,x_{n_k}\left(\omega \right)))\hfill \\ & =& d_{\omega }(U,V),\hfill \end{array}$$

(10)

we get that $\alpha (\omega ,x_{n_k+1}\left(\omega \right),u_1\left(\omega \right))\ge 1$ for all $k\in \mathbb{N}$, $\omega \in \Omega .$ Therefore,

$$\zeta (d(u_1\left(\omega \right),x_{n_k+1}\left(\omega \right)),\psi \left(d(\xi \left(\omega \right),x_{n_k}\left(\omega \right))\right))\ge 0.$$

Then $\left({\zeta }_2\right)$ imples that

$$\begin{array}{ccc}\hfill d(u_1\left(\omega \right),x_{n_k+1}\left(\omega \right))& \le & \psi \left(d(\xi \left(\omega \right),x_{n_k}\left(\omega \right))\right)\hfill \\ & <& d(\xi \left(\omega \right),x_{n_k}\left(\omega \right)),\hfill \end{array}$$

and so

$$\underset{k\to \infty }{lim}d(u_1\left(\omega \right),x_{n_k+1}\left(\omega \right))\to 0.$$

Thus, $u_1\left(\omega \right)=\xi \left(\omega \right)$ for all $\omega \in \Omega$ and (10) we have

$$d(\xi \left(\omega \right),T(\omega ,\xi \left(\omega \right)))=d_{\omega }(U,V).$$

The hypothesis $\left(B_5\right)$ that T is sub-measurable implies that $u_n$ is measurable for all $n\in \mathbb{N}$ and hence $\xi$ is measurable. Then $\xi$ is a random best proximity point. ☐

Theorem 3.

Let $(\Omega ,\Sigma )$ be a measurable space, let $(M,d)$ be a Polish space, U and V are two nonempty subsets of M and $\alpha :\Omega \times M\times M\to [0,\infty ).$ Suppose that $T:\Omega \times U\to V$ is a generalized random α-$\mathcal{Z}$-contraction mapping with respect to $\zeta \in \mathcal{Z}$. The hypotheses are the following:

$\left(C_1\right)$

T is a random triangular weak-α-admissible,

$\left(C_2\right)$

U is closed with respect to the topology induced by $d,$

$\left(C_3\right)$

$T(\Omega \times U_0)\subset V_0,$

$\left(C_4\right)$

there exist measurable mappings $x_0,x_1:\Omega \to U$ such that, for all $\omega \in \Omega ,$ $d(x_1\left(\omega \right),T(\omega ,x_0\left(\omega \right)))=d(U,V)$ and $\alpha (\omega ,x_0\left(\omega \right),x_1\left(\omega \right))\ge 1,$

$\left(C_5\right)$

T is a Carathéodory mapping.

Then T has a random best proximity point, that is, there exists $\xi :\Omega \to U$ is a measurable such that $d(\xi \left(\omega \right),T(\omega ,\xi \left(\omega \right)))=d_{\omega }(U,V)$ for all $\omega \in \Omega .$

Proof of Theorem 3.

By hypothesis $\left(C_4\right)$ we have there exists measurable mapping $x_0,x_1:\Omega \to U_0$ such that $\alpha (\omega ,x_0\left(\omega \right),x_1\left(\omega \right))\ge 1$ and

$$d(x_1\left(\omega \right),T(\omega ,x_0\left(\omega \right)))=d_{\omega }(U,V),$$

for all $\omega \in \Omega .$ Hypothesis $\left(C_3\right)$ implies that $T(\omega ,x_1\left(\omega \right))\in V_0$ which yields there exists measurable mapping $x_2:\Omega \to U_0$ such that

$$d(x_2\left(\omega \right),T(\omega ,x_1\left(\omega \right)))=d_{\omega }(U,V),$$

for all $\omega \in \Omega .$ Since $\alpha (\omega ,x_0\left(\omega \right),x_1\left(\omega \right))\ge 1$ and T is a random $\alpha$-proximal admissible, we have that $\alpha (\omega ,x_1\left(\omega \right),x_2\left(\omega \right))\ge 1.$ Iteratively, a sequence $\left\{x_n\left(\omega \right)\right\}\subset U_0$ can be constructed as follows:

$$d(x_{n+1}\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d_{\omega }(U,V),\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,\omega \in \Omega ,$$

(11)

and

$$\alpha (\omega ,x_n\left(\omega \right),x_{n+1}\left(\omega \right))\ge 1,\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,\omega \in \Omega .$$

(12)

If $x_n\left(\omega \right)=x_{n+1}\left(\omega \right)$ for some $n\in {\mathbb{N}}_0,$ $\omega \in \Omega ,$ then

$$d(x_n\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d(x_{n+1}\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d_{\omega }(U,V),$$

that is $x_n\left(\omega \right)$ is a random best proximity point. Assume that

$$x_n\left(\omega \right)\ne x_{n+1}\left(\omega \right)\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,\mathrm{for}\mathrm{one}\omega \in \Omega .$$

(13)

By combining (11)–(13), we get that

$$d(x_n\left(\omega \right),T(\omega ,x_{n-1}\left(\omega \right)))=d(x_{n+1}\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d_{\omega }(U,V),$$

for all $n\in \mathbb{N},$ $\omega \in \Omega$ and

$$\zeta (d(x_n\left(\omega \right),x_{n+1}\left(\omega \right)),r(x_{n-1}\left(\omega \right),x_n\left(\omega \right)))\ge 0,\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,\omega \in \Omega .$$

(14)

We have

$$\begin{array}{ccc}\hfill r(x_{n-1}\left(\omega \right),x_n\left(\omega \right))& =& max\left\{\frac{d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))}{d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))},d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))\right\}\hfill \\ & =& max\{d(x_{n+1}\left(\omega \right),x_n\left(\omega \right)),d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))\}.\hfill \end{array}$$

Suppose that for some $n_0=1,2,3,\dots .$

$$max\{d(x_{n_0+1}\left(\omega \right),x_{n_0}\left(\omega \right)),d(x_{n_0-1}\left(\omega \right),x_{n_0}\left(\omega \right))\}=d(x_{n_0+1}\left(\omega \right),x_{n_0}\left(\omega \right)).$$

On the other hand, since $d(x_{n_0}\left(\omega \right),x_{n_0+1}\left(\omega \right))>0,$ using the property $\left({\zeta }_2\right)$ of a simulation function, we obtain

$$\zeta (d(x_{n_0+1}\left(\omega \right),x_{n_0}\left(\omega \right)),d(x_{n_0+1}\left(\omega \right),x_{n_0}\left(\omega \right)))<0,$$

which is a contradiction. As consequence,

$$r(x_{n-1}\left(\omega \right),x_n\left(\omega \right))=d(x_{n-1}\left(\omega \right),x_n\left(\omega \right)),$$

for all $n\in \mathbb{N},$ $\omega \in \Omega .$ It means that

$$\zeta (d(x_n\left(\omega \right),x_{n+1}\left(\omega \right)),d(x_{n-1}\left(\omega \right),x_n\left(\omega \right)))\ge 0,\mathrm{for}\mathrm{all}n\in \mathbb{N},\omega \in \Omega .$$

Regarding $\left({\zeta }_2\right),$ the inequality (14) yields that

$$d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))\le d(x_{n-1}\left(\omega \right),x_n\left(\omega \right))\mathrm{for}\mathrm{all}n\in \mathbb{N},\omega \in \Omega .$$

Hence, $\left\{d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))\right\}$ is a non-increasing sequence bounded below. Then, there exists a $r\ge 0$ such that $\left\{d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))\right\}\to r.$ We claim that $r=0.$ Assume on the contrary that $r>0.$ Taking lim sup of (14) as $n\to \infty$ and regarding $\left({\zeta }_3\right),$ we find

$$0\le \underset{n\to \infty }{lim\; sup}\zeta (d(x_n\left(\omega \right),x_{n+1}\left(\omega \right)),d(x_{n-1}\left(\omega \right),x_n\left(\omega \right)))<0,$$

which is a contradiction, that is

$$\underset{n\to \infty }{lim}d(x_n\left(\omega \right),x_{n+1}\left(\omega \right))=0.$$

Next, to prove that $\left\{x_n\left(\omega \right)\right\}$ is a Cauchy sequence. Suppose, on the contrary, that $\left\{x_n\left(\omega \right)\right\}$ is not Cauchy sequence. Consequently, there exists $ϵ>0$ and subsequences $\left\{x_{m_k}\left(\omega \right)\right\}$ and $\left\{x_{n_k}\left(\omega \right)\right\}$ of $\left\{x_n\left(\omega \right)\right\},$ so that for $n_k>m_k>k,$ we have

$$d(x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right))\ge ϵ,$$

and

$$d(x_{m_k}\left(\omega \right),x_{n_k-1}\left(\omega \right))<ϵ.$$

By Lemma (1), we have

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

(15)

Also, by Lemma (1), we have

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

Since T is a random triangular weak-$\alpha$-admissible, from (12) we have

$$\alpha (\omega ,x_n\left(\omega \right),x_m\left(\omega \right))\ge 1,\mathrm{for}\mathrm{all}n,m\in {\mathbb{N}}_0,\mathrm{with}n>m,\omega \in \Omega .$$

Thus, we have

$$\alpha (\omega ,x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right))\ge 1,$$

and

$$\begin{array}{ccc}\hfill d(x_{m_k}\left(\omega \right),T(\omega ,x_{m_k-1}\left(\omega \right)))& =& d(x_{n_k}\left(\omega \right),T(\omega ,x_{n_k-1}\left(\omega \right)))\hfill \\ & =& d_{\omega }(U,V)\mathrm{for}\mathrm{all}k\in \mathbb{N}.\hfill \end{array}$$

(16)

Since T is a generalized random $\alpha$-$\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z},$ the obtained expression (16) yields the following inequality:

$$0\le \zeta (d(x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right)),r(x_{m_k-1}\left(\omega \right),x_{n_k-1}\left(\omega \right))),\mathrm{for}\mathrm{all}k\in \mathbb{N}.$$

Since,

$$\begin{array}{c}r(x_{m_k-1}\left(\omega \right),x_{n_k-1}\left(\omega \right))\hfill \\ =max\left\{\frac{d(x_{m_k-1}\left(\omega \right),x_{m_k}\left(\omega \right))d(x_{n_k-1}\left(\omega \right),x_{n_k}\left(\omega \right))}{d(x_{m_k-1}\left(\omega \right),x_{n_k-1}\left(\omega \right))},d(x_{m_k-1}\left(\omega \right),x_{n_k-1}\left(\omega \right))\right\}.\hfill \end{array}$$

(17)

Taking limit from both sides of (17) concludes that

$$\underset{k\to \infty }{lim}r(x_{m_k-1}\left(\omega \right),x_{n_k-1}\left(\omega \right))=ϵ.$$

Letting $k\to \infty$ and keeping (15) and $\left({\zeta }_3\right)$ in mind, we get

$$0\le \underset{n\to \infty }{lim\; sup}\zeta (d(x_{m_k}\left(\omega \right),x_{n_k}\left(\omega \right)),r(x_{m_k-1}\left(\omega \right),x_{n_k-1}\left(\omega \right)))<0,$$

which is a contradiction. Thus, we conclude that the sequence $\left\{x_n\left(\omega \right)\right\}$ is a Cauchy sequence. Since $(M,d)$ is a complete and U is closed subset of $(M,d)$ and T is a Carathéodory mapping, there exists $\xi :\Omega \to U$ such that

$$\left\{x_n\left(\omega \right)\right\}\to \xi \left(\omega \right)\mathrm{as}n\to +\infty \mathrm{for}\mathrm{all}\omega \in \Omega ,$$

(18)

it follows that $x_n$ is measurable for all $n\in \mathbb{N}$ and

$$x_{n+1}\left(\omega \right)=T(\omega ,x_n\left(\omega \right))\to T(\omega ,\xi \left(\omega \right))\mathrm{as}n\to +\infty \mathrm{for}\mathrm{all}\omega \in \Omega .$$

(19)

From (11), (18) and (19) we have

$$d_{\omega }(U,V)=\underset{n\to \infty }{lim}d(x_{n+1}\left(\omega \right),T(\omega ,x_n\left(\omega \right)))=d(\xi \left(\omega \right),T(\omega ,\xi \left(\omega \right))).$$

Therefore $\xi$ is a random best proximity point. ☐

Corollary 1.

Let $(\Omega ,\Sigma )$ be a measurable space, let $(M,d)$ be a Polish space, U and V are two nonempty subsets of M and $\alpha :\Omega \times M\times M\to [0,\infty ).$ Suppose that $T:\Omega \times U\to V$ is a random α-$\mathcal{Z}$-contraction with respect to $\zeta \in \mathcal{Z}$. The hypotheses are the following:

$\left(D_1\right)$

T is a random triangular weak-α-admissible,

$\left(D_2\right)$

U is closed with respect to the topology induced by $d,$

$\left(D_3\right)$

$T(\Omega \times U_0)\subset V_0,$

$\left(D_4\right)$

there exist measurable mappings $x_0,x_1:\Omega \to U$ such that, for all $\omega \in \Omega ,$ $d(x_1\left(\omega \right),T(\omega ,x_0\left(\omega \right)))=d_{\omega }(U,V)$ and $\alpha (\omega ,x_0\left(\omega \right),x_1\left(\omega \right))\ge 1,$

$\left(D_5\right)$

T is a Carathéodory mapping.

Then T has a random best proximity point, that is, there exists $\xi :\Omega \to U$ is a measurable such that $d(\xi \left(\omega \right),T(\omega ,\xi \left(\omega \right)))=d_{\omega }(U,V)$ for all $\omega \in \Omega .$

4. Conclusions

We introduce the new concept of generalized $\alpha$-$\mathcal{Z}$-contraction, so-called a generalized random $\alpha$-$\mathcal{Z}$-contraction, in separable metric spaces and also proved its existence theorems in complete separable metric spaces. In particular, our results extend, generalize and improve the results given of Karapinar and Khojasted, in [28].