On the existence and convergence of best proximity points in Menger probabilistic metric spaces

In this article we will study the existence of best proximity point of some special mappings like cyclic mappings in a Menger probabilistic metric space.


Introduction
If A and B are nonempty closed subsets of a complete metric space (X, d) such that A ̸ = B, a non-self mapping T : A → B does not necessarily have a fixed point.Finding an optimal approximate solution to the equation T x = x for a mapping T which has no fixed point is one of the important branches of the nonlinear analysis which KY Fan [9] in 1969 was the first one that began this subject by introducing approximation point and after him many authors such as Prolla [17], Reich [18], Sehgal and Singh [21,22] and etc. extended this subject.Basha and Veeramani [4] in 1997 introduced another useful compliment which is called a best proximity point.This point is the most optimal solution to the problem of minimizing the real valued function x → d(x, hx) over the domain A of a nonself mapping h : A → B. If the mapping is a self-mapping, the best proximity point reduces to a fixed point .On the other hand, the concept of a probabilistic metric space is generalization of a metric space which has been introduced by Karl Menger [15] in 1942.Fixed point theory in a probabilistic metric space is an important branch of probabilistic analysis and many results on the existence of fixed points or solutions of nonlinear equations under various types of conditions in Menger Probabilistic metric spaces have been extensively studied by many scholars.In this article we will study the existence of best proximity point of some special mappings like cyclic mappings in a Menger probabilistic metric space.

A special distribution function which is important for us is denoted by H
Remember that we can say F x,y (t) = α means that P(d(x, y) ≤ t) = α, that is, Definition 2.4.Let (X, F, T ) be a Menger probabilistic metric space.
(1) A sequence {x n } in X is said to be converge to x ∈ X if for any given ε > 0 and λ > 0, there exists a positive integer N = N(ε, λ ) such that F x n ,x (ε) > 1 − λ , whenever n ≥ N.
(2) A sequence {x n } in X is called a Cauchy sequence, if for any ε > 0 and λ > 0, there exists a positive integer (3) (X, F, T ) is said to be complete, if each Cauchy sequence in X converges to some point in X.
If (X, F, T ) is a Menger probabilistic metric space and A and B are nonempty subsets of X, we define, Definition 2.5.Let (X, F, T ) be a Menger probabilistic metric space and A and B be two nonempty subsets of X.The mapping h : Definition 2.6.Let (X, F, T ) be a Menger probabilistic metric space, A and B be two nonempty subsets of X and h : A → B be a mapping.We say that a * is a best proximity point of the mapping h if for all t > 0, Definition 2.7.Let (X, F, T ) be a Menger probabilistic metric space, A and B be two nonempty subsets of X and h : A ∪ B → A ∪ B be a cyclic mapping.We say that a * is a best proximity point of the cyclic mapping h if for all t > 0, International Scientific Publications and Consulting Services Definition 2.8.Let (X, F, T ) be a Menger probabilistic metric space, A and B be two nonempty subsets of X and H, S : A → B be two mappings.We say that a * is a common best proximity point of f and g if for all t > 0, Lemma 2.1.[16]Let (X, F, T ) be a Menger probabilistic metric space and define E λ ,F : for each λ ∈ (0, 1) and x, y ∈ X.Then we have, 1.For each µ ∈ (0, 1), there exists λ ∈ (0, 1) such that, for any x 1 , x 2 , ..., x n ∈ X.

Main section
For the beginning of our main results, let us start with reminding the definition of A 0 and B 0 .For nonempty subsets A and B of a Menger probabilistic metric space (X, F, T ), A 0 and B 0 are the following sets, We define the class Θ of all functions θ : [0, +∞) → [0, +∞) such that θ is onto and strictly increasing which ∑ +∞ n=1 θ n (t) < ∞ for all t > 0, where θ n (t) denotes the nth iterative mapping of θ (t).Also, we define the class Γ of all functions γ : [0, +∞) → [0, +∞) such that γ is onto and nondecreasing with γ(t) ≥ t.Theorem 3.1.Let A and B be nonempty subsets of a complete Menger probabilistic metric space (X, F, T ) such that A ̸ = B and A 0 and B 0 are nonempty and closed.Also, assume that {h n } is a sequence of cyclic mappings h n : A ∪ B → A ∪ B such that for each i, j ∈ N and t > 0, where m ∈ N, x, y ∈ X, γ ∈ Γ, and θ i, j : [0, +∞) → [0, +∞) is a function such that there exists θ ∈ Θ which θ i, j (t) ≤ θ (t).

If there exists x
then, there exist a sequence {x 2n } = {h 2m (x 2n−2 )} ⊆ A and a * ∈ A 0 such that lim n→+∞ x 2n = a * and a * is best proximity point of {h n }.

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Proof.By 3.2, there exists . Now, by using 3.1 we have, We can continue this process and so, by induction, for all λ ∈ (0, 1), Now, By lemma 2.2 and 3.3, By 1 of lemma 2.1, for all µ ∈ (0, 1), there exists λ ∈ (0, 1) such that, Since X is complete and A 0 is closed, so there exists a * ∈ A 0 such that lim n→+∞ x 2n = a * and since h n (A) ⊆ B and a * ∈ A 0 , consequently, h n (a * ) ∈ B 0 .Therefore, a * is a best proximity point of {h n }.Corollary 3.1.Let A and B be two nonempty subsets of a complete Menger probabilistic metric space (X, F, T ) such that A ̸ = B and A 0 and B 0 are nonempty and closed.Also, assume that {h n } is a sequence of cyclic mappings h n : A ∪ B → A ∪ B such that for each i, j ∈ N and t > 0, where m ∈ N, x, y ∈ X, γ ∈ Γ, and θ ∈ Θ.If there exists x 0 ∈ A 0 with, then, there exist a sequence {x 2n } = {h 2m (x 2n−2 )} ⊆ A and a * ∈ A 0 such that lim n→+∞ x 2n = a * and a * is best proximity point of {h n }.
By defining x 2n+1 = h 2m 2n+1 (x 2n−1 ), with an argument similar to the proof of theorem 3.1, we can conclude the next corollary.

Corollary 3.2. Let A and B be two nonempty subsets of a complete Menger probabilistic metric space (X, F, T )
such that A ̸ = B and A 0 and B 0 are nonempty and closed.Also, assume that {h n } is a sequence of cyclic mappings h n : A ∪ B → A ∪ B such that for each i, j ∈ N and t > 0, where m ∈ N, x, y ∈ X, γ ∈ Γ, and θ i, j : [0, +∞) → [0, +∞) is a function such that there exists θ ∈ Θ which θ i, j (t) ≤ θ (t).If there exists x 1 ∈ B 0 with, International Scientific Publications and Consulting Services Corollary 3.3.Let A and B be two nonempty subsets of a complete Menger probabilistic metric space (X, F, T ) such that A ̸ = B and A 0 and B 0 are nonempty and closed.Also, assume that {h n } is a sequence of cyclic mappings h n : A ∪ B → A ∪ B such that for each i, j ∈ N and t > 0, where m ∈ N, x, y ∈ X, γ ∈ Γ, and θ ∈ Θ.If there exists x 1 ∈ B 0 with, 3. there exists x 0 ∈ A 0 with, then, there exist a * ∈ A 0 and a sequence {x n } with x n = H(x n−1 ) such that H(x 2n ) = S(x 2n−2 ), lim n→+∞ x 2n = lim n→+∞ x 2n−2 = a * and a * is common best proximity point of H and S.
Proof.By 3, there exists (by 1, we can define a sequence with this property).Now, consider the subsequence {x 2n } ⊆ {x n }.
It is obvious that {x 2n } ⊆ A 0 too.Now, by 2, and We can continue this process and so, by induction, for all λ ∈ (0, 1), With the same argument as we used in theorem 3.1, we conclude that {x 2n } is a Cauchy sequence.Since X is complete and A 0 is closed, so there exists a * ∈ A 0 such that lim n→+∞ x 2n = a * .But since H and S are continuous, so lim n→+∞ H(x 2n ) = H(a * ) and lim n→+∞ S(x 2n−2 ) = S(a * ).Since H is a cyclic mapping, so we have Consequently, H(a * ) ∈ B 0 and a * is best proximity point of H. Also, since by 1, S(A 0 ) ⊆ H(A 0 ) and H is a cyclic mapping, so S(a * ) ∈ B 0 .Consequently a * is best proximity point of S too.
By defining {x n } such that x n = H(x n−1 ) and H(x 2n+1 ) = S(x 2n−1 ) next corollary is true too.
Let Ω be the class of all real continuous functions ω : (R + ) 4 → R such that ω is nondecreasing in the first argument and for u, v ≥ 0, if ω(u, v, u, v) ≥ 0 or ω(u, v, v, u) ≥ 0 then u ≥ v.The next theorem and the next corollary are true.
Theorem 3.3.Let (X, F, T ) be a complete Menger probabilistic metric space and A and B be nonempty subsets of X such that A ̸ = B and A 0 and B 0 are nonempty and closed.Also, let G, M : A → B and H, N : B → A be continuous mappings and the following conditions be satisfied for some ω ∈ Ω and all t > 0, Then for every {u n } and {v n } in X with n = 0, 1, 2, 3, ..., such that {u 2n } ⊆ A and , there exist a * and b * such that lim n→+∞ v 2n = a * and lim n→+∞ v 2n+1 = b * and also, for all t > 0, Moreover, if, 6. NM(x) = HG(x) for all x ∈ A 0 and MN(x) = GH(x) for all x ∈ B 0 ; Proof.Since by 4, there exists u 0 ∈ A 0 such that E F (u 0 , HGu 0 ) = sup{E λ ,F (u 0 , HGu 0 ) : λ ∈ (0, 1)} < ∞, so define v 2 = HGu 0 , v 4 = HGu 2 ,..., v 2n = HGu 2n−2 .By 1, it is obvious that v 2n ∈ A 0 .Also, since by 5, there exists On the other hand, since Gu 2n = v 2n+1 and Mu 2n = v 2n+3 , so with Since ω ∈ Ω, so we can conclude that, With the same argument as we used above, By continuing this process and by using induction,

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Now, we claim that if Now, by 1 of lemma 2.1, for all µ ∈ (0, 1) there exists λ ∈ (0, 1) such that, as m, n → +∞.By 2 of lemma 2.1, {v 2n+1 } is a Cauchy sequence and since X is complete and B 0 is closed, so {v 2n+1 } is converges to some b * ∈ B 0 .Also, since Hu 2n−1 = v 2n and Nu 2n−1 = v 2n+2 , so with y 1 = u 2n+1 , y 2 = u 2n−3 , y 3 = u 2n−1 and y 4 = u 2n−5 and by 3, Since ω ∈ Ω, so with the same argument as we used above, we can conclude that {v 2n } is a convergent sequence too and there exists a * ∈ A 0 such that lim n→+∞ v 2n = a * .On the other hand, by the hypotheses we have v 2n = Hu 2n−1 and consequently, Gv 2n = SHu 2n−1 .Since by 1, GH(B 0 ) ⊆ B 0 , so Gv 2n ∈ B 0 and since B 0 is closed and G is continuous, by passing to limit as n → +∞, and hence, F a * ,Ga * (t) = F A,B (t).Again, since v 2n+1 = Gu 2n , consequently, Hv 2n+1 = HSu 2n .Since by 1, HG(A 0 ) ⊆ A 0 , so Hv 2n+1 ∈ A 0 and since A 0 is closed and G is continuous, by passing to limit as n → +∞, To prove the last part, consider that by 6, we can use the same sequence which was defined in the beginning of the proof and it is concluded that NM(A 0 ) ⊆ A 0 and MN(B 0 ) ⊆ B 0 .Also, we can say that v 2n = Nu 2n−3 .Consequently, Mv 2n = MNu 2n−3 and since MN(B 0 ) ⊆ B 0 , so Mv 2n ∈ B 0 .Also, since B 0 is closed and M is continuous, by passing to limit as n → +∞, Hence, F a * ,Ma * (t) = F A,B (t) too.Again, since v 2n+1 = Mu 2n−2 , consequently, Nv 2n+1 = NMu 2n−2 and since NM(A 0 ) ⊆ A 0 , so Nv 2n+1 ∈ A 0 .Since A 0 is closed and N is continuous, by passing to limit as n → +∞, Hence, F Nb * ,b * (t) = F A,B (t) and the proof is complete.
F a,b (t) = F A,B (t) f or some b ∈ B}, and B 0 := {b ∈ B : F a,b (t) = F A,B (t) f or some a ∈ A}.

Corollary 3 . 4 .
Let A and B be two nonempty subsets of a complete Menger probabilistic metric space (X, F, T ) such that A ̸ = B and A 0 and B 0 are nonempty and closed.If H, S : A ∪ B → A ∪ B are two continuous cyclic mappings with the following conditions, 1. S(B 0 ) ⊆ H(B 0 ); International Scientific Publications and Consulting Services

Proof.
It is enough to put x 1 = u 2n , x 2 = u 2n−2 , y 1 = u 2n−1 and y 2 = u 2n−3 .With an argument as we used in the theorem 3.3, we can prove this corollary too.Now, Let Ω′ be the class of all real continuous functions ω′ : (R + ) 4 → R such that ω ′ is nonincreasing in the first argument and for u, v ≥ 0, if ω ′ (u, v, u, v) ≤ 0 or ω ′ (u, v, v, u) ≤ 0 then u ≤ v. Theorem 3.3 and corollary 3.5 can be changed to the next corollaries.
4→ R such that ω(x, y, z,t) = x − y + z − t, it is obvious that for all x 1 , x 2 ∈ A and t > 0, ω(F