Global optimization problem and probabilistic distance

Abstract

In this article, the probabilistic metric distance between two disjoint sets is utilised to define the essential criteria for the existence and uniqueness of the best proximity point, which takes into account the global optimization problem. In order to solve this problem, we pretend that we are trying to obtain the optimal approximation to the solution of a fixed point equation. Here, we introduce two types of probabilistic proximal contraction mappings and use a geometric property called Ω-property in the context of probabilistic metric spaces. We also obtain some consequences for self-mappings, which give the fixed point results. Some examples are provided to validate the findings. As an application, we obtain the solution to a second-order boundary value problem using a minimum t-norm in the context of probabilistic metric spaces.

AMS Subject Classification 2010: 47H10; 54H25.

1 Introduction

This manuscript considers a global optimization problem in the context of probabilistic metric space (PM space). PM space is the probabilistic generalization of metric space in which the distribution function plays the role of metric. Our objective is to identify the minimum distance between two subsets of a PM space. We use two different types of non-self contraction mappings to solve proximity point problems and propose the concept of probabilistic distance between any two subsets of a PM space.

The proximity point problem originated in the work of Eldred and Veeramani [4] in 2006 and has in subsequent times, developed vastly through many works. The following is the description of this problem. If P and Q are non-empty subsets of a metric space (Y, δ), a pair (p₁, q₁) ∈ P × Q is called a best proximity pair if δ (p₁, q₁) = δ (P, Q) = inf {δ (p₁, q₁) : p₁ ∈ P and q₁ ∈ Q}. Further, if $\mathfrak{B}$ is a mapping from P to Q, a point w ∈ P is called a best proximity point (with respect to $\mathfrak{B}$ ) if at the point w the function $\delta (w,\mathfrak{B}w)$ attains its global minimum with the value δ (P, Q); that is, $\delta (w,\mathfrak{B}w)=\delta (P,Q)$ . Thus the problem is a problem of global minimization. In another approach to this problem, it can be viewed as an approximate fixed point problem [1, 12]. For a non-self mapping $\mathfrak{B}:P\to Q$ , the idea of a fixed point is a point for which $u=\mathfrak{B}u$ is not pertinent when P and Q are disjoint. Even in the cases where P∩ Q ≠ ∅, a fixed point of the function $\mathfrak{B}$ , only exists under special conditions. But it may be possible to find some approximate fixed point of $\mathfrak{B}$ by minimizing the function $\delta (u,\mathfrak{B}u)$ . If the minimized value is δ (P, Q), we obtain a proximity point at which the proximity pair is realized. Thus, the proximity point problem is to find an optimal approximate solution of the fixed point equation $\mathfrak{B}u=u$ .

Several exciting results regarding the global optimality approach for single-valued and multi-valued contraction in the setting of various spaces appeared in the literature, for examples, [5, 14–16] and references therein.

In this article, a novel approach is taken to tackle the challenge of defining essential criteria for the existence and uniqueness of the best proximity point within the framework of probabilistic metric spaces. To validate the proposed findings, the article furnishes a set of examples that not only illustrate the concepts but also underscore their practical applicability. These examples provide concrete evidence of the validity and effectiveness of the presented approach. As an application, we obtain the solution to a second-order boundary value problem in the context of probabilistic metric spaces by utilizing a minimum t-norm. This application highlights the framework’s adaptability and ability to handle various real-world problems.

2 Mathematical preliminaries

Some technical accounts of several concepts associated with the structure of the probabilistic metric space are given below, which are required in our main discussion.

Definition 2.1. [6, 18] A distribution function is a mapping $ϝ:ℝ$ $\to {ℝ}^{+}$ that is non-decreasing and left continuous with $\underset{l\in ℝ}{inf}ϝ\left(l\right)=0$ and $\underset{l\in ℝ}{sup}ϝ\left(l\right)=1$ . Heavyside function is a standard example of the distribution function.

Every pair of elements in PM spaces is assigned a distribution function, which makes them probabilistic generalizations of metric spaces. Schweizer and Sklar [18] comprehensively considered several aspects of such spaces.

Definition 2.2. A probabilistic metric space (PM-space) (see [6, 18]) is an ordered pair (Y, ϝ), where Y is a non-empty set and ϝ is a mapping from Y × Y into the set of all distribution functions. The distribution function ϝ (u, v) is denoted by ϝ_u,v and satisfies the following conditions for all u, v, w ∈ Y,

(i)    ϝ_u,v (0) =0,

(ii)  ϝ_u,v (l) =1 for all l > 0 if and only if u = v,

(iii) ϝ_u,v (l) = ϝ _v,u (l) for all l > 0,

(iv)  if ϝ_u,v (l₁) =1 and ϝ_v,w (l₂) =1 then ϝ_u,w (l₁ + l₂) =1, for l₁, l₂ > 0.

Example 2.1. Consider Y = [1, 2] and ${ϝ}_{u,v}\left(l\right)=\frac{l}{l+|u-v|}$ , for all u, v ∈ Y. Then (Y, ϝ) is a PM space.

A Menger space is a particular type of PM-space in which the triangle inequality is hypothesised using a t-norm.

Shi et al. [21] introduced the following definition of n-th order t-norm.

Definition 2.3. A mapping $\Delta :{\Pi }_{i=1}^n[0,1]\to [0,1]$ is called a n-th order t-norm [21] if the following conditions are satisfied:

(i) Δ (0, 0, …, 0) =0, Δ (p₁, 1, 1, …, 1) = p₁ for all p₁ ∈ [0, 1],

(ii) Δ (p₁, p_2,, p₃, …, p _n ) = Δ (p₂, p₁, p₃, …, p _n ) = Δ (p₂, p₃, p₁, …, p _n )

= … = Δ (p₂, p₃, p₄, …, p _n , p₁),

(iii) p _i  ≥ q _i ,   i=1,2,3,…,n    implies    Δ (p₁, p₂, p₃, …, p _n ) ≥ Δ (q₁, q₂, q₃, …, q _n ),

(iv) Δ (Δ (p₁, p₂, p₃, …, p _n ) , q₂, q₃, …, q _n )

= Δ (p₁, Δ (p₂, p₃, …, p _n , q₂) , q₃, …, q _n )

= Δ (p₁, p₂, Δ (p₃, p₄, …, p _n , q₂, q₃) , q₄, …, q _n )

⋮

= Δ (p₁, p₂, …, p_n-1, Δ (p _n , q₂, q₃, …, q _n )). When n = 2, we have a binary t-norm, which is commonly known as t-norm.

Some examples of different types of t-norms are:

(i) The Lukasiewicz t-norm, Δ = T _L , defined by $T_L(p_1,p_2)=max\{p_1+p_2-1,0\}.$ (ii) The minimum t-norm, Δ = T _m , defined by $T_m(p_1,p_2)=min\{p_1,p_2\}.$ (iii) The product t-norm, Δ = T _p , defined by $T_p(p_1,p_2)=p_1\times p_2.$

Definition 2.4. A Menger space, also known as a Menger PM-space (see [6, 18]) is a triplet (Y, ϝ , Δ), where Y is a non-empty set, ϝ is a function defined on Y × Y to the set of distribution functions and Δ is a t-norm, satisfying the following conditions:

ϝ_u,v (0) =0forallu, v ∈ Y,

ϝ_u,v (z) =1forallz > 0and u, v ∈ Yifandonlyifu = v,

ϝ_u,v (z) = ϝ _v,u (z) forallu, v ∈ Y, z > 0 and

ϝ_u,v (κ + μ) ≥ Δ (F_u,w (κ) , F_w,v (μ)) forallκ, μ ≥ 0andu, v, w ∈ Y .

Menger spaces are the particular types of PM-spaces. A metric space becomes a Menger space if we write ϝ_u,v (l) = H (l - δ (u, v)), where H is a Heavyside function. Sehgal and Bharucha-Reid [19] initiated the fixed point result in PM-spaces. Following that, many results arose in the literature, and Hadzic and Pap [6] provided a comprehensive survey. For details on convergence, Cauchy sequence and completeness in Menger spaces (see [6, 18]).

Definition 2.5. [20] Let P and Q be two nonempty subsets of PM-space (Y, ϝ , Δ). We define P₀ and Q₀ as follows:

P₀ = {u ∈ P : ∃  v ∈ Q such that  ϝ _u,v (l) = ϝ _P,Q (l) for all l > 0},

Q₀ = {v ∈ Q : ∃  u ∈ P such that  ϝ _u,v (l) = ϝ _P,Q (l) for all l > 0}.

The distance of a point u ∈ Y from a nonempty set P for l > 0 is defined as ${ϝ}_{u,P}\left(l\right)=\underset{p_1\in P}{sup}{ϝ}_{u,p_1}\left(l\right)$

and

the distance between two nonempty sets P and Q for l > 0 is defined as ${ϝ}_{P,Q}\left(l\right)=sup\left\{{ϝ}_{p_1,q_1}\right(l):p_1\in P,q_1\in Q\}.$

An element u^* ∈ P is said to be the best proximity point of the mapping $\mathfrak{B}:P\to Q$ if it satisfies the condition that for all l > 0

${ϝ}_{u^{*},\mathfrak{B}{u}^{*}}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ ,

where $\mathfrak{B}{u}^{*}$ denotes the $\mathfrak{B}$ image of u^*.

Here, we use a geometric property called Ω-property in the context of PM-spaces.

Definition 2.6. [20] Let (P, Q) be a pair of nonempty subsets of a PM-space (Y, ϝ , Δ). Then the pair (P, Q) is said to have the Ω-property if and only if

ϝ_u1,v1 (l) = ϝ _P,Q (l)

and

ϝ_u2,v2 (l) = ϝ _P,Q (l),

which imply that ϝ_u1,u2 (l) = ϝ _v1,v2 (l) where for all l > 0 and u₁, u₂ ∈ P, v₁, v₂ ∈ Q.

The following result of Saha et al. [13] will be used in the sequel. The authors showed that the distance between two non-empty disjoint subsets of probabilistic metric spaces is a distribution function.

Lemma 2.1. Let P and Q be two non-empty subsets of a probabilistic metric space (Y, ϝ , Δ), then ϝ_P,Q (l), l > 0 is a distribution function.

The main features of the paper are:

1. We establish two proximity point theorems for two non-self contraction mappings.

2. Some consequences are obtained for self-mapping.

3. Some examples are given here to validate our results.

4. An application to second-order differential equation is provided.

3 Main Result

In this section we establish some generalized best proximity point results on PM spaces.

Before the main results, we define two probabilistic proximal contraction mappings.

Definition 3.1. Let (P, Q) be a pair of nonempty subsets of a Menger space (Y, ϝ , Δ), where Δ is a continuous t-norm which satisfies Δ (p, p) = p, p ∈ [0, 1]. A mapping $\mathfrak{B}:P\to Q$ is called

generalized type-I proximal contraction if ${ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left\{{ϝ}_{u,v}\right(\frac{l_1}{p}),{ϝ}_{u,\mathfrak{B}u}(\frac{l_2}{q}),{ϝ}_{v,\mathfrak{B}v}(\frac{l_3}{r}\left)\right\},$(3.1) where l, l₁, l₂, l₃ > 0 with l = l₁ + l₂ + l₃ and 0 < p + q + r < 1

and

generalized type-II proximal contraction if $\begin{array}{cc}\hfill & {ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left\{{ϝ}_{u,v}\right(\frac{l_1}{p}),{ϝ}_{u,\mathfrak{B}u}(\frac{l_2}{q}),\hfill \\ \hfill & {ϝ}_{v,\mathfrak{B}v}\left(\frac{l_3}{r}\right),{ϝ}_{u,\mathfrak{B}v}\left(\frac{l_4}{s}\right),{ϝ}_{v,\mathfrak{B}u}\left(\frac{l_5}{t}\right)\},\hfill \end{array}$(3.2) where l = l₁ + l₂ + l₃ + l₄ + l₅ and 0 < p + q + r + s + t < 1.

Theorem 3.1. Let (P, Q) be a pair of nonempty subsets of a complete Menger space (Y, ϝ , Δ), where Δ is a continuous t-norm which satisfies Δ (p, p) = p, p ∈ [0, 1]. Suppose that P is closed and $\mathfrak{B}:P\to Q$ is a generalized type-II proximal contractive mapping satisfying the following conditions: (i) $\mathfrak{B}\left(P_0\right)\subseteq Q_0$ and (P, Q) satisfies the Ω-property,

(ii) there exist u₀, u₁ ∈ P₀ such that ${ϝ}_{u_1,\mathfrak{B}{u}_0}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ for all l > 0,

(iii) for each sequence {u _n } in Y tends to u^*, if $\left\{\mathfrak{B}{u}_n\right\}$ is convergent, then it converges to $\mathfrak{B}{u}^{*}$ .

Then there exists an element u^* ∈ P₀ such that ${ϝ}_{u^{*},\mathfrak{B}{u}^{*}}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ , that is, $\mathfrak{B}$ has a best proximity point.

Proof. By an assumption of the theorem there exist u₀, u₁ ∈ P₀ such that ${ϝ}_{u_1,\mathfrak{B}{u}_0}\left(l\right)={ϝ}_{P,Q}\left(l\right),$ for all l > 0. Since $\mathfrak{B}\left(P_0\right)\subseteq Q_0$ , there exists u₂ ∈ P₀ such that ${ϝ}_{u_2,\mathfrak{B}{u}_1}\left(l\right)={ϝ}_{P,Q}\left(l\right).$ So, for all l > 0, ${ϝ}_{u_1,\mathfrak{B}{u}_0}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ and ${ϝ}_{u_2,\mathfrak{B}{u}_1}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ .

Again, as $\mathfrak{B}\left(P_0\right)\subseteq Q_0$ , there exists u₃ ∈ P₀ such that ${ϝ}_{u_3,\mathfrak{B}{u}_2}\left(l\right)={ϝ}_{P,Q}\left(l\right).$ So, for all l > 0, ${ϝ}_{u_2,\mathfrak{B}{u}_1}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ and ${ϝ}_{u_3,\mathfrak{B}{u}_2}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ .

Using induction, for all l > 0, we have

${ϝ}_{u_n,\mathfrak{B}{u}_{n-1}}\left(l\right)={ϝ}_{P,Q}\left(l\right)\mathrm{for}\mathrm{all}n>0$(3.3)

and

${ϝ}_{u_{n+1},\mathfrak{B}{u}_n}\left(l\right)={ϝ}_{P,Q}\left(l\right)\mathrm{for}\mathrm{all}n>0.$(3.4)

Using (3.3) and (3.4) and Ω-property of the pair (P, Q), for all l > 0, n ≥ 1, we get

${ϝ}_{u_n,u_{n+1}}\left(l\right)={ϝ}_{\mathfrak{B}{u}_{n-1},\mathfrak{B}{u}_n}\left(l\right).$(3.5)

Using (3.2), (3.3), (3.4) and definition of ϝ_P,Q (l), for all l > 0, we have $\begin{array}{cc}\hfill {ϝ}_{u_{n+1},u_n}\left(l\right)& ={ϝ}_{\mathfrak{B}{u}_n,\mathfrak{B}{u}_{n-1}}\left(l\right)\hfill \\ \hfill & \ge min\left\{{ϝ}_{u_n,u_{n-1}}\right(\frac{l_1}{p}),{ϝ}_{u_n,\mathfrak{B}{u}_n}(\frac{l_2}{q}),{ϝ}_{u_{n-1},\mathfrak{B}{u}_{n-1}}(\frac{l_3}{r}),\hfill \\ \hfill & {ϝ}_{u_n,\mathfrak{B}{u}_{n-1}}\left(\frac{l_4}{s}\right),{ϝ}_{u_{n-1},\mathfrak{B}{u}_n}\left(\frac{l_5}{t}\right)\}\hfill \\ \hfill & \ge min\left\{{ϝ}_{u_n,u_{n-1}}\right(\frac{l_1}{p}),\Delta ({ϝ}_{u_n,u_{n+1}}\left(\frac{l_2^{\prime }}{q}\right),{ϝ}_{u_{n+1},\mathfrak{B}{u}_n}\left(\frac{l_2^{″}}{q}\right)),\hfill \\ \hfill & \Delta \left({ϝ}_{u_{n-1},u_n}\right(\frac{l_3^{\prime }}{r}),{ϝ}_{u_n,\mathfrak{B}{u}_{n-1}}(\frac{l_3^{″}}{r}\left)\right),{ϝ}_{P,Q}\left(\frac{l_4}{s}\right),\hfill \\ \hfill & \Delta \left({ϝ}_{u_{n+1},u_{n-1}}\right(\frac{l_5^{\prime }}{t}),{ϝ}_{u_{n+1},\mathfrak{B}{u}_n}(\frac{l_5^{\prime }}{t}\left)\right)\}\hfill \\ \hfill & =min\left\{{ϝ}_{u_n,u_{n-1}}\right(\frac{l_1}{p}),\Delta ({ϝ}_{u_n,u_{n+1}}\left(\frac{l_2^{\prime }}{q}\right),{ϝ}_{P,Q}\left(\frac{l_2^{″}}{q}\right)),\hfill \\ \hfill & \Delta \left({ϝ}_{u_{n-1},u_n}\right(\frac{l_3^{\prime }}{r}),{ϝ}_{P,Q}(\frac{l_3^{″}}{r}\left)\right),{ϝ}_{P,Q}\left(\frac{l_4}{s}\right),\Delta \left({ϝ}_{u_{n+1},u_{n-1}}\right(\frac{l_5^{\prime }}{t}),{ϝ}_{P,Q}(\frac{l_5^{\prime }}{t}\left)\right)\}\hfill \\ \hfill & =min\left\{{ϝ}_{u_n,u_{n-1}}\right(\frac{l_1}{p}),{ϝ}_{u_n,u_{n+1}}(\frac{l_2^{\prime }}{q}),{ϝ}_{u_{n-1},u_n}(\frac{l_3^{\prime }}{r}),{ϝ}_{u_{n+1},u_{n-1}}(\frac{l5\prime }{t}\left)\right\}\hfill \\ \hfill & \ge min\left\{{ϝ}_{u_n,u_{n-1}}\right(\frac{l_1}{p}),{ϝ}_{u_n,u_{n+1}}(\frac{l_2^{\prime }}{q}),{ϝ}_{u_{n-1},u_n}(\frac{l3\prime }{r}),\hfill \\ \hfill & \Delta \left({ϝ}_{u_n,u_{n+1}}\right(\frac{l_5^{\mathit{iii}}}{t}),{ϝ}_{u_n,u_{n-1}}(\frac{l_5^{\mathit{iv}}}{t}\left)\right)\},\hfill \end{array}$(3.6) where $l2\prime +l2″=l_2$ , $l3\prime +l3″=l_3$ , $l{5}^{\prime }+l{5}^{″}=l_5$ and .

Now, here we get two possible cases.

Case-I: If $min\left({ϝ}_{u_n,u_{n+1}}\right(\frac{l_5^{\mathit{iii}}}{t}),{ϝ}_{u_n,u_{n-1}}(\frac{l_5^{\mathit{iv}}}{t}\left)\right)={ϝ}_{u_n,u_{n+1}}\left(\frac{l_5^{\mathit{iii}}}{t}\right),$ then using the property of t-norm, we get $\begin{array}{cc}\hfill & \Delta \left({ϝ}_{u_n,u_{n+1}}\right(\frac{l_5^{\mathit{iii}}}{t}),{ϝ}_{u_n,u_{n-1}}(\frac{l_5^{\mathit{iv}}}{t}\left)\right)\ge \hfill \\ \hfill & \Delta \left({ϝ}_{u_n,u_{n+1}}\right(\frac{l_5^{\mathit{iii}}}{t}),{ϝ}_{u_n,u_{n+1}}(\frac{l_5^{\mathit{iii}}}{t}\left)\right)={ϝ}_{u_n,u_{n+1}}\left(\frac{l_5^{\mathit{iii}}}{t}\right).\hfill \end{array}$ Therefore, from (3.6), we have

$\begin{array}{cc}\hfill & {ϝ}_{u_{n+1},u_n}\left(l\right)\ge min\left\{{ϝ}_{u_n,u_{n-1}}\right(\frac{l_1}{p}),{ϝ}_{u_n,u_{n+1}}(\frac{l_2^{\prime }}{q}),\hfill \\ \hfill & {ϝ}_{u_{n-1},u_n}\left(\frac{l3\prime }{r}\right),{ϝ}_{u_n,u_{n+1}}\left(\frac{l_5^{\mathit{iii}}}{t}\right)\}.\hfill \end{array}$(3.7) Choose $l_1=\frac{\mathit{pl}}{p+q+r+t}$ , , , $l_5^{\mathit{iii}}=\frac{\mathit{tl}}{p+q+r+t}$ , that is, $\frac{l_1}{p}=\frac{l}{\lambda }$ , , , $\frac{l_5^{\mathit{iii}}}{t}=\frac{l}{\lambda }$ , where λ = p + q + r + t, then obviously 0 < λ < 1. From (3.7), we get $\begin{array}{cc}\hfill {ϝ}_{u_{n+1},u_n}\left(l\right)& \ge min\left\{{ϝ}_{u_n,u_{n-1}}\right(\frac{l}{\lambda }),{ϝ}_{u_n,u_{n+1}}(\frac{l}{\lambda }),\hfill \\ \hfill & {ϝ}_{u_{n-1},u_n}\left(\frac{l}{\lambda }\right),{ϝ}_{u_n,u_{n+1}}\left(\frac{l}{\lambda }\right)\}\hfill \\ \hfill & =min\left\{{ϝ}_{u_n,u_{n+1}}\right(\frac{l}{\lambda }),{ϝ}_{u_{n-1},u_n}(\frac{l}{\lambda }\left)\right\}.\hfill \end{array}$ We now claim that for l > 0,

${ϝ}_{u_{n+1},u_n}\left(\frac{l}{\lambda }\right)>{ϝ}_{u_n,u_{n-1}}\left(\frac{l}{\lambda }\right).$(3.8) If possible, let for some z > 0, $\begin{array}{cc}\hfill {ϝ}_{u_{n+1},u_n}\left(\frac{z}{\lambda }\right)<{ϝ}_{u_n,u_{n-1}}\left(\frac{z}{\lambda }\right).& \hfill \end{array}$ Then, we have $\begin{array}{cc}\hfill {ϝ}_{u_{n+1},u_n}\left(z\right)& \ge min\left\{{ϝ}_{u_{n+1},u_n}\right(\frac{z}{\lambda }),{ϝ}_{u_{n-1},u_n}(\frac{z}{\lambda }\left)\right\}\hfill \\ \hfill & ={ϝ}_{u_{n+1},u_n}\left(\frac{z}{\lambda }\right).\hfill \end{array}$ For λ ∈ (0, 1), $\frac{z}{\lambda }>z$ and using the monotone property of ϝ, we get $\begin{array}{cc}\hfill {ϝ}_{u_{n+1},u_n}\left(\frac{z}{\lambda }\right)\ge {ϝ}_{u_{n+1},u_n}\left(z\right),& \hfill \end{array}$ which is a contradiction. Hence $\begin{array}{cc}\hfill {ϝ}_{u_{n+1},u_n}\left(l\right)& \ge {ϝ}_{u_n,u_{n-1}}\left(\frac{l}{\lambda }\right)\hfill \\ \hfill & \ge {ϝ}_{u_{n-1},u_{n-2}}\left(\frac{l}{{\lambda }^2}\right).\hfill \end{array}$ Taking n-times iteration successively, we have $\begin{array}{cc}\hfill {ϝ}_{u_{n+1},u_n}\left(l\right)\ge {ϝ}_{u_0,u_1}\left(\frac{l}{{\lambda }^n}\right).& \hfill \end{array}$ Now, taking limit on both sides and since λ ∈ (0, 1), we have $\underset{n\to \infty }{lim}{ϝ}_{u_{n+1},u_n}\left(l\right)\ge \underset{n\to \infty }{lim}{ϝ}_{u_0,u_1}\left(\frac{l}{{\lambda }^n}\right)=1.$(3.9) Case II: If $min\left({ϝ}_{u_n,u_{n+1}}\right(\frac{l_5^{\mathit{iii}}}{t}),{ϝ}_{u_n,u_{n-1}}(\frac{l_5^{\mathit{iv}}}{t}\left)\right)={ϝ}_{u_n,u_{n-1}}\left(\frac{l_5^{\mathit{iv}}}{t}\right)$ , then using the property of t-norm, we have $\begin{array}{cc}\hfill & \Delta \left({ϝ}_{u_n,u_{n+1}}\right(\frac{l_5^{\mathit{iii}}}{t}),{ϝ}_{u_n,u_{n-1}}(\frac{l_5^{\mathit{iv}}}{t}\left)\right)\ge \hfill \\ \hfill & \Delta \left({ϝ}_{u_n,u_{n-1}}\right(\frac{l_5^{\mathit{iv}}}{t}),{ϝ}_{u_n,u_{n-1}}(\frac{l_5^{\mathit{iv}}}{t}\left)\right)={ϝ}_{u_n,u_{n-1}}\left(\frac{l_5^{\mathit{iv}}}{t}\right).\hfill \end{array}$ Using a similar procedure as in Case-I, we get

$\underset{n\to \infty }{lim}{ϝ}_{u_{n+1},u_n}\left(l\right)\ge \underset{n\to \infty }{lim}{ϝ}_{u_0,u_1}\left(\frac{l}{{\lambda }^n}\right)=1.$(3.10) Combining Case-I and Case-II, we have

$\underset{n\to \infty }{lim}{ϝ}_{u_{n+1},u_n}\left(l\right)=1.$(3.11) Now, we have to prove that {u _n } is a Cauchy sequence. If possible, let {u _n } be not a Cauchy sequence. Then, there exist ϵ > 0 and 0 < θ < 1 for which we can find subsequences {u_m(k)} and {u_n(k)} of {u _n } with n (k) > m (k) > k such that ${ϝ}_{u_{m\left(k\right)},u_{n\left(k\right)}}\left(ϵ\right)<1-\theta$(3.12)

and

${ϝ}_{u_{m\left(k\right)},u_{n\left(k\right)-1}}\left(ϵ\right)\ge 1-\theta .$(3.13) Take ϵ₁ < ϵ. Using the monotone property of ϝ, for all k > 0, we have $\begin{array}{cc}\hfill {ϝ}_{u_{m\left(k\right)},u_{n\left(k\right)}}\left({ϵ}_1\right)\le {ϝ}_{u_{m\left(k\right)},u_{n\left(k\right)}}\left(ϵ\right).& \hfill \end{array}$ Now from (3.12), $\begin{array}{l}1-\theta >{Ϝ}_{u_{m(k)},u_{n(k)}}(ϵ)\\ \ge \Delta ({Ϝ}_{u_{m(k)},u_{n(k)-1}}({ϵ}^{\prime }),{Ϝ}_{u_{n(k)},u_{n(k)-1}}(ϵ-{ϵ}^{\prime })).\end{array}$(3.14) Further, we choose is chosen sufficiently small such that . Using the left continuity property of ϝ and by (3.13), we have $ϝum\left(k\right),un\left(k\right)-1\left({ϵ}^{\prime }\right)\ge 1-\theta .$(3.15) Again using (3.11), for sufficiently large k, we get $ϝun\left(k\right),un\left(k\right)-1(ϵ-{ϵ}^{\prime })\ge 1-\theta .$(3.16) Now, using (3.15), (3.16) in (3.14), we have $\begin{array}{l}1-\theta >{Ϝ}_{u_{m(k)},u_{n(k)}}(ϵ)\\ \ge \Delta ({Ϝ}_{u_{m(k)},u_{n(k)-1}}({ϵ}^{\prime }),{Ϝ}_{u_{n(k)},u_{n(k)-1}}(ϵ-{ϵ}^{\prime }))n\\ \ge \Delta (1-\theta ,1-\theta )\\ =1-\theta \end{array}$ which is a contradiction. Hence {u _n } is a Cauchy sequence in P. Since (Y, ϝ , Δ) is complete and P is closed, there exists u^* ∈ P such that $\underset{n\to \infty }{lim}{u}_n=u^{*}.$(3.17) Using (3.4), for each m > n > 0 and l > 0, we have $\begin{array}{cc}\hfill {ϝ}_{u_{m+1},\mathfrak{B}{u}_m}\left(l\right)={ϝ}_{P,Q}\left(l\right)& \hfill \end{array}$

and

$\begin{array}{cc}\hfill {ϝ}_{u_{n+1},\mathfrak{B}{u}_n}\left(l\right)={ϝ}_{P,Q}\left(l\right).& \hfill \end{array}$ Using Ω-property, we get $\begin{array}{cc}\hfill {ϝ}_{u_{m+1},u_{n+1}}\left(l\right)={ϝ}_{\mathfrak{B}{u}_m,\mathfrak{B}{u}_n}\left(l\right)\mathrm{for}\mathrm{all}l>0.& \hfill \end{array}$ As the sequence, {u _n } is a Cauchy sequence, the above equation implies that $\left\{\mathfrak{B}{u}_n\right\}$ is a Cauchy sequence and hence it is convergent, that is, $\underset{n\to \infty }{lim}\mathfrak{B}{u}_n=\mathfrak{B}{u}^{*}.$(3.18) For any arbitrary ρ > 0 and using (3.4), we get $\begin{array}{cc}\hfill & {ϝ}_{u^{*},\mathfrak{B}{u}^{*}}\left(l\right)\hfill \\ \hfill & \ge \Delta \left({ϝ}_{u^{*},u_{n+1}}\right(\rho ),{ϝ}_{u_{n+1},\mathfrak{B}{u}^{*}}(l-\rho \left)\right)\hfill \\ \hfill & \ge \Delta \left({ϝ}_{u^{*},u_{n+1}}\right(\rho ),\Delta ({ϝ}_{u_{n+1},\mathfrak{B}{u}_n}(l-2\rho ),{ϝ}_{\mathfrak{B}{u}_n,\mathfrak{B}{u}^{*}}\left(\rho \right)\left)\right)\hfill \\ \hfill & =\Delta \left({ϝ}_{u^{*},u_{n+1}}\right(\rho ),\Delta ({ϝ}_{P,Q}(l-2\rho ),{ϝ}_{\mathfrak{B}{u}_n,\mathfrak{B}{u}^{*}}\left(\rho \right)\left)\right).\hfill \end{array}$ Taking limit n→ ∞ in the above inequality, we have $\begin{array}{cc}\hfill & {ϝ}_{u^{*},\mathfrak{B}{u}^{*}}\left(l\right)\ge \hfill \\ \hfill & \Delta (1,\Delta ({ϝ}_{P,Q}(l-2\rho ),1\left)\right)={ϝ}_{P,Q}(l-2\rho ).\hfill \end{array}$ As ρ is arbitrary positive number and by Lemma 2.1, ϝ_P,Q (l) is left continuous, from above we have $\begin{array}{cc}\hfill {ϝ}_{u^{*},\mathfrak{B}{u}^{*}}\left(l\right)\ge {ϝ}_{P,Q}\left(l\right).& \hfill \end{array}$ It implies $\begin{array}{cc}\hfill {ϝ}_{u^{*},\mathfrak{B}{u}^{*}}\left(l\right)={ϝ}_{P,Q}\left(l\right).& \hfill \end{array}$ Hence, it completes the proof.

It is easy to see the following result.

Theorem 3.2. Let (P, Q) be a pair of nonempty subsets of a complete Menger space (Y, ϝ , Δ), where Δ is a continuous t-norm which satisfies Δ (p, p) = p, p ∈ [0, 1]. Suppose that P is closed and $\mathfrak{B}:P\to Q$ is a generalized type-I proximal contractive mapping satisfying the following conditions:

(i) $\mathfrak{B}\left(P_0\right)\subseteq Q_0$ and (P, Q) satisfies the Ω-property,

(ii) there exist u₀, u₁ ∈ P₀ such that ${ϝ}_{u_1,\mathfrak{B}{u}_0}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ for all l > 0,

(iii) for each sequence {u _n } in Y tends to u^*, if $\mathfrak{B}{u}_n$ is convergent, then it converges to $\mathfrak{B}{u}^{*}$ .

Then there exists an element u^* ∈ P₀ such that ${ϝ}_{u^{*},\mathfrak{B}{u}^{*}}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ , that is, $\mathfrak{B}$ has a proximity point. If we take self mapping, that is, P = Q, then we get a fixed point result as a consequence of Theorems 3.1 and 3.2. The following corollaries are deduced from the work of Babacev [2] and Shams and Jafari [17].

Corollary 3.1. Let P be a nonempty closed subset of a complete Menger space (Y, ϝ , Δ), where Δ is a continuous t-norm which satisfies Δ (p, p) = p, p ∈ [0, 1] and let $\mathfrak{B}:P\to P$ be a self mapping satisfying the following condition $\begin{array}{cc}\hfill & {ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left({ϝ}_{u,v}\right(\frac{l_1}{p}),{ϝ}_{u,\mathfrak{B}u}(\frac{l_2}{q}),{ϝ}_{v,\mathfrak{B}v}(\frac{l_3}{r}),\hfill \\ \hfill & {ϝ}_{u,\mathfrak{B}v}\left(\frac{l_4}{s}\right),{ϝ}_{v,\mathfrak{B}u}\left(\frac{l_5}{t}\right))\hfill \end{array}$(3.19) where u, v ∈ P, l = l₁ + l₂ + l₃ + l₄ + l₅ and 0 < p + q + r + s + t < 1. Then $\mathfrak{B}$ has a unique fixed point in P.

Now, taking l = l₁ = l₂ = l₃ = l₄ = l₅ and p = q = r = s = t = α, we get another corollary.

Corollary 3.2. Let P be a nonempty closed subset of a complete Menger space (Y, ϝ , Δ), where Δ is a continuous t-norm which satisfies Δ (p, p) = p, p ∈ [0, 1] and let $\mathfrak{B}:P\to P$ be a self mapping satisfying the following condition for all l > 0, 0 < α < 1, u, v ∈ P, $\begin{array}{cc}\hfill & {ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left({ϝ}_{u,v}\right(\frac{l}{\alpha }),{ϝ}_{u,\mathfrak{B}u}(\frac{l}{\alpha }),{ϝ}_{v,\mathfrak{B}v}(\frac{l}{\alpha }),\hfill \\ \hfill & {ϝ}_{u,\mathfrak{B}v}\left(\frac{l}{\alpha }\right),{ϝ}_{v,\mathfrak{B}u}\left(\frac{l}{\alpha }\right)).\hfill \end{array}$(3.20) Then $\mathfrak{B}$ has a unique fixed point in P.

Corollary 3.3. Let P be a nonempty closed subset of a complete Menger space (Y, ϝ , Δ), where Δ is a continuous t-norm which satisfies Δ (p, p) = p, p ∈ [0, 1] and let $\mathfrak{B}:P\to P$ be a self mapping satisfying the following condition ${ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left({ϝ}_{u,v}\right(\frac{l_1}{\omega }),{ϝ}_{u,\mathfrak{B}u}(\frac{l_2}{\phi }),{ϝ}_{v,\mathfrak{B}v}(\frac{l_3}{\psi }\left)\right)$(3.21) where u, v ∈ P, l, l₁, l₂, l₃ > 0 with l = l₁ + l₂ + l₃ and 0 < ω + φ + ψ < 1. Then $\mathfrak{B}$ has a unique fixed point in P. Taking l = l₁ = l₂ = l₃ and ω = φ = ψ = c, we get another result for self mapping.

Corollary 3.4. Let P be a nonempty closed subset of a complete Menger space (Y, ϝ , Δ), where Δ is a continuous t-norm which satisfies Δ (p, p) = p, p ∈ [0, 1] and let $\mathfrak{B}:P\to P$ be a self mapping satisfying the following condition for all u, v ∈ P, l > 0, and 0 < c < 1,

${ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left({ϝ}_{u,v}\right(\frac{l}{c}),{ϝ}_{u,\mathfrak{B}u}(\frac{l}{c}),{ϝ}_{v,\mathfrak{B}v}(\frac{l}{c}\left)\right).$(3.22) Then $\mathfrak{B}$ has a unique fixed point in P.

Remark 3.1.

It is interesting to see the above results for different types of t-norms.

We know that contraction mapping may be self or non-self. In the case of self-mapping, we can find the solution for the equation $\mathfrak{B}y=y$ , known as a fixed point. But in our present work, we consider the non-self mapping, which means two sets are disjoint. We aim to find the optimal distance, the smallest distance between y and $\mathfrak{B}y$ .

Best proximity points may be considered a generalization of the best approximation. From the best approximation results, we get an idea about the existence of an approximate solution. The best proximity point theorems are used to solve the problems to find an approximate optimal solution. That is the best proximity point of $\mathfrak{B}$ . If we consider self-mapping, the best proximity theorem reduces to a fixed point result.

It may be noted that the best approximation theorem and best proximity point theorems are relevant with respect to their topological structures. A best approximate solution is not optimal where the best proximity solution is optimal.

4 Illustrations

In this section, we provide some examples that support our results.

By the property of t-norm, $\begin{array}{cc}\hfill & \Delta (p_1,q_1)\ge \Delta (q_1,q_1)=q_1,\mathrm{if}{p}_1>q_1\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & \Delta (p_1,q_1)\ge \Delta (p_1,p_1)=p_1,\mathrm{if}{q}_1>p_1.\hfill \end{array}$ If Δ is replaced by minimum norm, then Δ (p₁, q₁) = min {p₁, q₁} = q₁, if p₁ > q₁ and Δ (p₁, q₁) = min {p₁, q₁} = p₁, if q₁ > p₁. So, we choose the minimum t-norm in our examples.

Example 4.1. Suppose that $Y=ℝ$ and ϝ is a distribution function defined on Y × Y, defined as ${ϝ}_{u,v}\left(l\right)=\frac{l}{l+|u-v|}$ , u, v ∈ Y, l > 0. Define a t-norm as Δ (p₁, q₁) = min {p₁, q₁}. It is easy to see that (Y, ϝ , Δ) is a PM space. Choose P = [0, 1] and $Q=[\frac{5}{2},3]$ as closed subsets of Y. For only point 1 ∈ P and only point $\frac{5}{2}\in Q$ , ${ϝ}_{1,\frac{5}{2}}\left(l\right)=\frac{2l}{2l+3}={ϝ}_{P,Q}\left(l\right)$ for all l ≥ 0. So trivially the pair (P, Q) satisfies Ω-property. Define a mapping $\mathfrak{B}:P\to Q$ by $\mathfrak{B}u=-\frac{u}{2}+3$ , u ∈ P. Choose P₀ = {1} and $Q_0=\left\{\frac{5}{2}\right\}$ such that $\mathfrak{B}\left(P_0\right)\subseteq Q_0$ . Now, for all u, v ∈ P and l > 0, we have to show that ${ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left({ϝ}_{u,v}\right(\frac{l_1}{p}),{ϝ}_{u,\mathfrak{B}u}(\frac{l_2}{q}),{ϝ}_{v,\mathfrak{B}v}(\frac{l_3}{r}\left)\right),$(4.23) where l, l₁, l₂, l₃ > 0 with l = l₁ + l₂ + l₃ and 0 < p + q + r < 1. Here, ${ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)=\frac{l}{l+|\mathfrak{B}u-\mathfrak{B}v|}=\frac{2l}{2l+|u-v|}$ .

Further, take $p=q=r=\frac{1}{4}>0$ such that 0 < p + q + r < 1. From (4.23), we have $\frac{2l}{2l+|u-v|}\ge min$ $\left\{\frac{4l_1}{4l_1+|u-v|},\frac{4l_2}{4l_2+|u-\mathfrak{B}u|},\frac{4l_3}{4l_3+|v-\mathfrak{B}v|}\right\}.$ Now, we get the following three possible cases: Case I: Take $\frac{2l}{2l+|u-v|}\ge \frac{4l_1}{4l_1+|u-v|}$ . Then after rearranging, we get $\frac{|u-v|(l-2l_1)}{(4l+2|u-v\left|\right)(4l_1+|u-v\left|\right)}\ge 0$ , l - 2l₁ ≥ 0, that is, l ≥ 2l₁ > l₁ > 0.

Case II: Take $\frac{2l}{2l+|u-v|}\ge \frac{4l_2}{4l_2+|u-\mathfrak{B}u|}$ . Then after rearranging, we get $\frac{l|u-\mathfrak{B}u|-2l_2|u-v|}{(4l+2|u-v\left|\right)(4l_2+|u-\mathfrak{B}u\left|\right)}\ge 0$ . For u, v ∈ P = [0, 1], it is obviously $|u-\mathfrak{B}u|>|u-v|$ and l > 2l₂ > l₂ > 0.

Case III: Take $\frac{2l}{2l+|u-v|}\ge \frac{4l_3}{4l_3+|v-\mathfrak{B}v|}$ . Then after rearranging, we get $\frac{l|v-\mathfrak{B}v|-2l_3|u-v|}{(4l+2|u-v\left|\right)(4l_3+|v-\mathfrak{B}v\left|\right)}\ge 0$ . For u, v ∈ P = [0, 1], it is obviously $|v-\mathfrak{B}v|>|u-v|$ and l > 2l₃ > l₃ > 0.

Hence all the conditions of Theorem 3.2 hold for all u, v ∈ P, l > 2 max {l₁, l₂, l₃} and $\mathfrak{B}$ has a best proximity point 1 ∈ P.

Example 4.2. Suppose that $Y={ℝ}^2$ and and ϝ is a distribution function defined on Y × Y, defined as , (u, v) , (u′, v′) ∈ Y, l > 0. Define a minimum t-norm as

Δ (p₁, q₁) = min {p₁, q₁}. It is easy to see that (Y, ϝ , Δ) is a PM space.

Consider the closed subsets P and Q of Y as P = {(u, 1) :0 ≤ u ≤ 1} and Q = {(u, - 1) :0 ≤ u ≤ 1} . Let $\mathfrak{B}:P\to Q$ be a mapping defined by $\mathfrak{B}(u,1)=(\frac{u}{2},-1)$ , u ∈ P.

Here, $P_0=\left\{\right(\frac{u}{2},1):0\le u\le 1\}$ , Q₀ = Q such that $\mathfrak{B}\left(P_0\right)\subseteq Q_0$ . Also, ${ϝ}_{P,Q}\left(l\right)=\frac{l}{l+2}$ for all l > 0.

Now, we have to show that the pair (P, Q) satisfies the Ω-property.

Let κ₁ = (u₁, 1) , κ₂ = (u₂, 1) ∈ P and μ₁ = (v₁, - 1) , μ₂ = (v₂, - 1) ∈ Q with ${ϝ}_{{\kappa }_1,{\mu }_1}\left(l\right)={ϝ}_{P,Q}\left(l\right)(\mathrm{for}\mathrm{all}l>0)$(4.24)

and

${ϝ}_{{\kappa }_2,{\mu }_2}\left(l\right)={ϝ}_{P,Q}\left(l\right)(\mathrm{for}\mathrm{all}l>0).$(4.25) From (4.24), we get for all l > 0 $\begin{array}{cc}\hfill \frac{l}{l+|u_1-v_1|+2}=\frac{l}{l+2},& \hfill \end{array}$ which implies that u₁ = v₁. Similarly from (4.25), we get for all l > 0, u₂ = v₂.

Now for all l > 0, we have $\begin{array}{cc}\hfill {ϝ}_{{\kappa }_1,{\kappa }_2}\left(l\right)& =\frac{l}{l+|u_1-u_2|}\hfill \\ \hfill & =\frac{l}{l+|v_1-v_2|}\hfill \\ \hfill & ={ϝ}_{{\mu }_1,{\mu }_2}\left(l\right).\hfill \end{array}$ So, the pair (P, Q) satisfies the Ω-property.

Now, for all u, v ∈ P and l > 0, we have to show that $\begin{array}{cc}\hfill & {ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left({ϝ}_{u,v}\right(\frac{l_1}{p}),{ϝ}_{u,\mathfrak{B}u}(\frac{l_2}{q}),{ϝ}_{v,\mathfrak{B}v}\hfill \\ \hfill & \left(\frac{l_3}{r}\right),{ϝ}_{u,\mathfrak{B}v}\left(\frac{l_4}{s}\right),{ϝ}_{v,\mathfrak{B}u}\left(\frac{l_5}{t}\right)),\hfill \end{array}$(4.26) where l, l₁, l₂, l₃, l₄, l₅ > 0 with l = l₁ + l₂ + l₃ + l₄ + l₅ and 0 < p + q + r + s + t < 1. Let κ = (u, 1) , μ = (v, 1) ∈ P.

Here, ${ϝ}_{\mathfrak{B}\kappa ,\mathfrak{B}\mu }\left(l\right)=\frac{l}{l+|\frac{u}{2}-\frac{v}{2}|}=\frac{2l}{2l+|u-v|}$ .

Further, take $p=q=r=s=t=\frac{1}{6}>0$ such that 0 < p + q + r + s + t < 1. From (4.26), we have $\begin{array}{cc}\hfill & \frac{2l}{2l+|u-v|}\ge min\{\frac{l_1}{l_1+6|u-v|},\hfill \\ \hfill & \frac{l_2}{l_2+12+6|u-\frac{u}{2}|},\frac{l_3}{l_3+12+6|v-\frac{v}{2}|},\hfill \\ \hfill & \frac{l_4}{l_4+12+6|u-\frac{v}{2}|},\frac{l_5}{l_5+12+6|v-\frac{u}{2}|}\}.\hfill \end{array}$

Now, we get the following five possible cases:

Case I: Take $\frac{2l}{2l+|u-v|}\ge \frac{l_1}{l_1+6|u-v|}$ . So after rearranging, we get $l\ge \frac{l_1}{12}>0$ . So Case-I is true.

Case II: Take $\frac{2l}{2l+|u-v|}\ge \frac{l_2}{l_2+12+6|u-\frac{u}{2}|}$ . Since 0 ≤ u, v ≤ 1, max |u - v|=1. Then, after rearranging, we get 24l + 6lu - l₂ ≥ 0. As l ≥ l₂, 24l + 6lu ≥ l₂. Hence Case-II is also true.

Case III: Take $\frac{2l}{2l+|u-v|}\ge \frac{l_3}{l_3+12+6|v-\frac{v}{2}|}$ . Since 0 ≤ u, v ≤ 1, max |u - v|=1. Then, after rearranging, we get 24l + 6lv - l₃ ≥ 0. As l ≥ l₃, 24l + 6lv ≥ l₃. Hence Case-III is also true.

Case IV: Take $\frac{2l}{2l+|u-v|}\ge \frac{l_4}{l_4+12+6|u-\frac{v}{2}|}$ . Since 0 ≤ u, v ≤ 1, max |u - v|=1. Then after rearranging, we get $24l+6l|u-\frac{v}{2}|-l_4\ge 0$ . As l ≥ l₄, $24l+6l|u-\frac{v}{2}|\ge l_4$ .

Hence Case-IV is also true.

Case V: Take $\frac{2l}{2l+|u-v|}\ge \frac{l_5}{l_5+12+6|v-\frac{u}{2}|}$ . Since 0 ≤ u, v ≤ 1, max |u - v|=1. Then after rearranging, we get $24l+6l|v-\frac{u}{2}|-l_5\ge 0$ . As l ≥ l₅, $24l+6l|v-\frac{u}{2}|\ge l_5$ . Therefore, Case V is also true.

Combining Cases I, II, III, IV and V, we say that all conditions of Theorem 3.1 hold.

Therefore (0, 1) ∈ P is the unique best proximity point of $\mathfrak{B}:P\to Q$ in P₀,

${ϝ}_{{\kappa }^{*},\mathfrak{B}{\kappa }^{*}}\left(l\right)={ϝ}_{P,Q}\left(l\right)$ , that is, $\frac{l}{l+|\frac{u}{2}-\frac{u}{4}|+2}=\frac{l}{l+2}$ .

The following example shows that we get a unique fixed point for self-mapping but a proximity point for non-self mapping under the same inequality.

Example 4.3. Let Y = {y₁, y₂, y₃, y₄}, here the t-norm Δ (p₁, q₁) = min(p₁, q₁) and ϝ_u,v (l) be defined as

${ϝ}_{y_1,y_2}\left(l\right)={ϝ}_{y_1,y_3}\left(l\right)={ϝ}_{y_1,y_4}\left(l\right)={ϝ}_{y_3,y_4}\left(l\right)=\{\begin{array}{cc}\hfill & 0,l\le 0\hfill \\ \hfill & 0.5,\mathit{if}0<l<7\hfill \\ \hfill & 1,\mathit{if}l\ge 7\hfill \end{array}$

and ${ϝ}_{y_2,y_4}\left(l\right)={ϝ}_{y_2,y_3}\left(l\right)=\{\begin{array}{cc}0,\hfill & \mathrm{if}l\le 0\hfill \\ 1,\hfill & \mathrm{if}l>0.\hfill \end{array}$ It is easy to see that (Y, ϝ , Δ) is a complete Menger space. Define a self mapping $\mathfrak{B}:Y\to Y$ as follows: $\mathfrak{B}{y}_1=y_2$ , $\mathfrak{B}{y}_2=y_2$ , $\mathfrak{B}{y}_3=y_2$ , $\mathfrak{B}{y}_4=y_3$ , then it satisfies the inequality (3.2). Also we get y₂ as fixed point, that is, $\mathfrak{B}{y}_2=y_2$ .

Now, if we consider the mapping $\mathfrak{B}:P\to Q$ where P = {y₁, y₂} and Q = {y₃, y₄}. If we define $\mathfrak{B}:P\to Q$ as follow $\mathfrak{B}{y}_1=y_3$ , $\mathfrak{B}{y}_2=y_4$ . Hence, the mapping is non-self. Here, we do not get any solution for the equation $\mathfrak{B}y=y$ , that is, there is no fixed point. But we can find out the proximity point.

Now, δ (P, Q) = inf {(p₁, q₁) : p₁ ∈ P, q₁ ∈ Q}.

So, δ (P, Q) = min {|y₁ - y₃|, |y₂ - y₄|}.

If $min\left\{\right|y_1-y_3|,|y_2-y_4\left|\right\}=|y_1-y_3|=|y_1-\mathfrak{B}{y}_1|$ , then y₁ is proximity point of $\mathfrak{B}$ in P.

Otherwise, y₂ will be a proximity point of $\mathfrak{B}$ in P.

5 Application

In this section, we give an application to find the solution of a second-order differential equation. We consider the following boundary value problem of second order differential equation:

$\begin{array}{cc}\hfill -\frac{d^{2}u}{{\mathit{dl}}^2}=g(l,u(l\left)\right),l\in [0,1]& \hfill \end{array}$ $\begin{array}{cc}\hfill u\left(0\right)=u\left(1\right)=0,& \hfill \end{array}$ where $g:[0,1]\times ℝ⟶ℝ$ is a continuous function.

It is easy to see that Green’s function G (l, z) for the associated boundary-values problem is $G(l,z)=\{\begin{array}{cc}\hfill & l(1-z),0\le l\le z\le 1\hfill \\ \hfill & -\mathit{zl}+z,0\le z\le l\le 1.\hfill \end{array}$ Let Y = C (I)  (I = [0, 1]) be the space of all continuous functions defined on I. It is well known that such a space with the metric given by $\begin{array}{cc}\hfill \delta (u,v)=\parallel u-v{\parallel }_{\infty }=\underset{l\in I}{max}\left|u\right(l)-v(l\left)\right|& \hfill \end{array}$ is a complete metric space. Choose a distribution function ${ϝ}_{u,v}\left(l\right)=\frac{l}{l+\delta (u,v)}$ . Define a t-norm as Δ (p₁, q₁) = min {p₁, q₁}. It is easy to see that (Y, ϝ , Δ) is a complete PM space.

Suppose that $\mathfrak{B}:C\left(I\right)\to C\left(I\right)$ . Now, we have to show that the above-mentioned differential equation satisfies the following inequality ${ϝ}_{\mathfrak{B}u,\mathfrak{B}v}\left(l\right)\ge min\left\{{ϝ}_{u,v}\left(\frac{l}{\alpha }\right),{ϝ}_{u,\mathfrak{B}u}\left(\frac{l}{\alpha }\right),{ϝ}_{v,\mathfrak{B}v}\left(\frac{l}{\alpha }\right),{ϝ}_{u,\mathfrak{B}v}\left(\frac{l}{\alpha }\right),{ϝ}_{v,\mathfrak{B}u}\left(\frac{l}{\alpha }\right)\right\}.$(4.27) Therefore, $\frac{l}{l+\delta (\mathfrak{B}u,\mathfrak{B}v)}\ge min\left\{\frac{\frac{l}{\alpha }}{\frac{l}{\alpha }+\delta (u,v)},\frac{\frac{l}{\alpha }}{\frac{l}{\alpha }+\delta (u,\mathfrak{B}u)},\frac{\frac{l}{\alpha }}{\frac{l}{\alpha }+\delta (v,\mathfrak{B}v)},\frac{\frac{l}{\alpha }}{\frac{l}{\alpha }+\delta (u,\mathfrak{B}v)},\frac{\frac{l}{\alpha }}{\frac{l}{\alpha }+\delta (v,\mathfrak{B}u)}\right\},$

that is, $\frac{1}{l+\delta (\mathfrak{B}u,\mathfrak{B}v)}\ge min\left\{\frac{1}{l+\alpha \delta (u,v)},\frac{1}{l+\alpha \delta (u,\mathfrak{B}u)},\frac{1}{l+\alpha \delta (v,\mathfrak{B}v)},\frac{1}{l+\alpha \delta (u,\mathfrak{B}v)},\frac{1}{l+\alpha \delta (v,\mathfrak{B}u)}\right\}.$ So, $\delta (\mathfrak{B}u,\mathfrak{B}v)\le min\alpha \left\{\delta (u,v),\delta (u,\mathfrak{B}u),\delta (v,\mathfrak{B}v),\delta (u,\mathfrak{B}v),\delta (v,\mathfrak{B}u)\right\}.$ We have α > 0 such that for all u, v ∈ C (I); l, z ∈ I and $p_1,q_1\in ℝ$ , we get $\left|g\right(l,p_1)-g(l,q_1\left)\right|\le \alpha min\left\{\left|u\right(z)-v(z\left)\right|,\left|u\right(z)-\mathfrak{B}u(z\left)\right|,\left|v\right(z)-\mathfrak{B}v(z\left)\right|,\left|u\right(z)-\mathfrak{B}v(z\left)\right|,\left|v\right(z)-\mathfrak{B}u(z\left)\right|\right\}.$

Now, it is well known that u ∈ C² (I) is a solution of given differential equation is equivalent to that u ∈ C (I) is a solution of the integral equation $u\left(l\right)={\int }_0^{1}G(l,z)g(z,u(z\left)\right)\mathit{dz},\mathrm{for}\mathrm{all}l\in I.$(4.28) Define the operator $\mathfrak{B}:C\left(I\right)\to C\left(I\right)$ by $\mathfrak{B}\left(u\right(l\left)\right)={\int }_0^{1}G(l,z)g(z,u(z\left)\right)\mathit{dz},\mathrm{for}\mathrm{all}l\in I.$ For all u, v ∈ C (I); l, z ∈ I, we have $\begin{array}{cc}\hfill \left|\mathfrak{B}\right(u\left(l\right))-\mathfrak{B}(v\left(l\right)\left)\right|& =\left|{\int }_0^{1}G\right(l,z\left)\right[g(z,u(z\left)\right)-g(z,v(z\left)\right)\left]\mathit{dz}\right|\hfill \\ \hfill & \le {\int }_0^{1}G(l,z)\left|g\right(z,u\left(z\right))-g(z,v\left(z\right)\left)\right|\mathit{dz}\hfill \\ \hfill & \le {\int }_0^{1}G(l,z)\alpha min\left\{\delta \right(u,v),\delta (u,\mathfrak{B}u),\delta (v,\mathfrak{B}v),\delta (u,\mathfrak{B}v),\delta (v,\mathfrak{B}u\left)\right\}\mathit{dz}\hfill \\ \hfill & =\alpha min\left\{\delta \right(u,v),\delta (u,\mathfrak{B}u),\delta (v,\mathfrak{B}v),\delta (u,\mathfrak{B}v),\delta (v,\mathfrak{B}u\left)\right\}{\int }_0^{1}G(l,z)\mathit{dz}\hfill \\ \hfill & \le \alpha min\left\{\delta \right(u,v),\delta (u,\mathfrak{B}u),\delta (v,\mathfrak{B}v),\delta (u,\mathfrak{B}v),\delta (v,\mathfrak{B}u\left)\right\}\underset{l\in I}{sup}{\int }_0^{1}G(l,z)\mathit{dz}\hfill \\ \hfill & \le \alpha min\left\{\delta \right(u,v),\delta (u,\mathfrak{B}u),\delta (v,\mathfrak{B}v),\delta (u,\mathfrak{B}v),\delta (v,\mathfrak{B}u\left)\right\}\times \frac{1}{8}\hfill \\ \hfill & =0.\hfill \end{array}$

Also, $min\left\{\delta \right(u,v),\delta (u,\mathfrak{B}u),\delta (v,\mathfrak{B}v),\delta (u,\mathfrak{B}v),$ $\delta (v,\mathfrak{B}u)\}=min\{\delta (u,v),0,0,\delta (u,\mathfrak{B}v),\delta (v,\mathfrak{B}u)\}$ = 0 . It implies $\begin{array}{cc}\hfill & \delta (\mathfrak{B}u,\mathfrak{B}v)=min\left\{\delta \right(u,v),\delta (u,\mathfrak{B}u),\hfill \\ \hfill & \delta (v,\mathfrak{B}v),\delta (u,\mathfrak{B}v),\delta (v,\mathfrak{B}u)\},\hfill \end{array}$ for all u, v ∈ C (I). Therefore, all the conditions of Corollary 3.2 are satisfied, and we have a solution in C (I).

6 Conclusion

Probabilistic optimization is a vast area of applied mathematics having applications in different domains. This article considers the global optimization problem in probabilistic metric spaces to find the probabilistic distance between two sets. It can be explored as a continuation of the current work on how various types of probabilistic contractions can contribute to the proximity point theory in probabilistic metric space structure.

Declarations

Availability of data and materials

Data sharing does not apply to this article as no datasets were generated or analysed during the current study.

Ethical Approval

Not applicable.

Funding

Basque Government:Grant IT1555-22 Basque Government: Grant KK-2022/00090 MCIN/AEI 269.10.13039/501100011033 /FEDER,UE: Grant PID2021-1235430B-C21 MCIN/AEI 269.10.13039/501100011033 /FEDER,UE: Grant PID2021-1235430B-C22

Competing Interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contribute equally.