Some new approaches to modular and fuzzy metric and related best proximity results

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Fuzzy Sets and Systems ••• (••••) •••–•••

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www.elsevier.com/locate/fss

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Some new approaches to modular and fuzzy metric and related best proximity results

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M. Paknazar , M. De La Sen

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a Department of Mathematics Education, Farhangian University, Tehran, Iran b University of Basque Country, Spain

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Received 26 April 2018; received in revised form 25 November 2019; accepted 27 December 2019

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Abstract By using current methods of best proximity point theory we can not establish and prove best proximity point results in modular metric spaces. For this purpose, we introduce the notion of P D−modular metric spaces and then we establish some best proximity point theorems for certain new contraction mappings. As an application of the obtained results we establish some best proximity point results in fuzzy metric spaces. An examples is furnished to demonstrate the validity of the obtained results. © 2020 Elsevier B.V. All rights reserved.

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Keywords: P D−modular metric; Best proximity point

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1. Introduction and preliminaries

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Modular metric spaces are natural and interesting generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and others. The concept of Modular metric spaces was introduced in [2,3]. There exists an extensive literature about “Best proximity point Theorems in modular metric spaces and fuzzy metric spaces” (see, for instance, [1,12–15,20] and the papers referenced there). Here, we look at Modular metric space as the nonlinear version of the classical one introduced by Nakano [17] on the vector space and the modular function space introduced by Musielak [16] and Orlicz [18]. In this work we introduce the notion of P D−modular metric spaces and then we establish some best proximity point theorems for certain new contraction mappings. As an application of the obtained results we establish some best proximity point results in fuzzy metric spaces. An examples is furnished to demonstrate the validity of the obtained results. We are going to recall some necessary concepts, required definitions and primary results for coherence with the literature.

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E-mail addresses: [email protected] (M. Paknazar), [email protected] (M. De La Sen). https://doi.org/10.1016/j.fss.2019.12.012 0165-0114/© 2020 Elsevier B.V. All rights reserved.

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Definition 1.1. [2,3] A function ω : (0, +∞) × X × X → [0, +∞] is called a modular metric on X if the following axioms hold:

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(i) x = y if and only if ωλ (x, y) = 0 for all λ > 0; (ii) ωλ (x, y) = ωλ (y, x) for all λ > 0 and x, y ∈ X; (iii) ωλ+μ (x, y) ≤ ωλ (x, z) + ωμ (z, y) for all λ, μ > 0 and x, y, z ∈ X. In this Definition,if we utilize the condition

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instead of (i) then ω is said to be a pseudomodular metric on X. A modular metric ω on X is called regular if the following weaker version of (i) is satisfied

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x=y

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Xω = Xω (x0 ) = {x ∈ X : ωλ (x, x0 ) → 0 as

Xω and

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λ → +∞}

Xω∗

such that ωλ (x, x0 ) < +∞}.

are called modular spaces (around x0 ).

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δω (M) = sup{ω1 (x, y); x, y ∈ M} < ∞.

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Definition 1.4. (Schweizer and Sklar [5]) A binary operation  : [0, 1] × [0, 1] → [0, 1] is called a continuous t-norm if it satisfies the following assertions:

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(T1) (T2) (T3) (T4)

 is commutative and associative;  is continuous; a  1 = a for all a ∈ [0, 1]; a  b ≤ c  d when a ≤ c and b ≤ d, with a, b, c, d ∈ [0, 1].

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Definition 1.5. (George and Veeramani [5]) A fuzzy metric space is an ordered triple (X, M, ) where X = ∅,  is a continuous t-norm and M : X × X × (0, +∞) is a fuzzy set, satisfying the following assertions, for all x, y, z ∈ X and t, s > 0:

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(1) The sequence (xn )n∈N in Xω is said to be ω-convergent to x ∈ Xω if and only if ω1 (xn , x) → 0, as n → ∞. x will be called the ω-limit of (xn ). (2) The sequence (xn )n∈N in Xω is said to be ω-Cauchy if ω1 (xm , xn ) → 0, as m, n → ∞. (3) A subset M of Xω is said to be ω-closed if the ω-limit of a ω-convergent sequence of M always belongs to M. (4) A subset M of Xω is said to be ω-complete if any ω-Cauchy sequence in M is a ω-convergent sequence and its ω-limit is in M. (5) A subset M of Xω is said to be ω-bounded if we have

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Definition 1.3. Let Xω be a modular metric space.

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Xω∗ = Xω∗ (x0 ) = {x ∈ X : ∃λ = λ(x) > 0

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λ > 0.

and

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ωλ (x, y) = 0 for some

Definition 1.2. [2,3] Suppose that ω is a pseudomodular on X and x0 ∈ X is fixed. So the two sets

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if and only if

Remark 1.1. [3] Note that if ω is a pseudomodular metric on a set X, then the function λ → ωλ (x, y) is non-increasing on (0, +∞) for all x, y ∈ X.

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(i’) ωλ (x, x) = 0 for all λ > 0 and x ∈ X;

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(F1) M(x, y, 0) = 0,

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(F2) (F3) (F4) (F5)

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M(x, y, t) = 1 if and only if x = y, M(x, y, t) = M(y, x, t), M(x, y, t)  M(y, z, s) ≤ M(x, z, t + s), M(x, y, ·) : (0, +∞) → (0, 1] is left continuous.

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Definition 1.6. (George and Veeramani [5]) Let (X, M, ) be a fuzzy metric space. Then

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(i) a sequence {xn } converges to x ∈ X if and only if limn→+∞ M(xn , x, t) = 1 for all t > 0, (ii) a sequence {xn } in X is a Cauchy sequence if and only if for all  ∈ (0, 1) and t > 0, there exists n0 ∈ N where M(xn , xm , t) > 1 −  for all m, n ≥ n0 , (iii) the fuzzy metric space is complete if every Cauchy sequence converges to some x ∈ X.

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Definition 1.7. [4] Let (X, M, ∗) be a fuzzy metric space. The fuzzy metric M is called triangular whenever

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for all x, y, z ∈ X and all t > 0.

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Lemma 1.1 ([7]). Let (X, M, ∗) be a triangular fuzzy metric space. The function ω : (0, +∞) × X × X → [0, +∞] defined by ωλ (x, y) =

1 −1 M(x, y, λ)

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for all x, y ∈ X and all λ > 0 is a modular metric on X.

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Moreover, Kirk [10] explored some significant generalizations of the Banach contraction principle to the case of non-self mappings. Let A and B be nonempty subsets of a metric space (X, d). A mapping T : A → B is called a k-contraction if there exists k ∈ [0, 1) such that d(T x, T y) ≤ kd(x, y), for all x, y ∈ A. Evidently, k-contraction coincides with Banach contraction mapping if we take A = B. Furthermore, a non-self contractive mapping may not have a fixed point. In this case, we try to find an element x such that d(x, T x) is minimum, i.e., x and T x are in close proximity to each other. It is clear that d(x, T x) is at least d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}, we are interested in investigating the existence of an element x such that d(x, T x) = d(A, B). In this case x is a best proximity point of the non-self-mapping T . Evidently, a best proximity point reduces to a fixed point if T is a self-mapping. We denote by A0 and B0 the following sets: A0 = {x ∈ A : d(x, y) = d(A, B) for some y ∈ B}, B0 = {y ∈ B : d(x, y) = d(A, B) for some x ∈ A}.

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(1.2)

In 2003, Kirk et al. [11] established sufficient conditions for determining when the sets A0 and B0 are nonempty. Raj [19] introduced the following concept. Definition 1.8. Let (A, B) be a pair of nonempty subsets of a metric space (X, d) with A0 = ∅. Then the pair (A, B) is said to have the P -property if and only if for all x1, x2 ∈ A0 and y1 , y2 ∈ B0 ,  d(x1 , y1 ) = d(A, B) (1.3) ⇒ d(x1 , x2 ) = d(y1 , y2 ). d(x2 , y2 ) = d(A, B)

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Also Raj proved that any pair (A, B) of nonempty closed convex subsets of a real Hilbert space satisfies the P -property. Clearly for any nonempty subset A of (X, d), the pair (A, A) has the P -property. Recently, Zhang et al. [21] introduced the following notion and showed that it is weaker than P -property.

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1 1 1 −1≤ −1+ −1 M(x, y, t) M(x, z, t) M(z, y, t)

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Definition 1.9. Let (A, B) be a pair of nonempty subsets of a metric space (X, d) with A0 = ∅. Then the pair (A, B) is said to have the weak P -property if and only if for any x1, x2 ∈ A0 and y1 , y2 ∈ B0 ,

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d(x1 , y1 ) = d(A, B) and d(x2 , y2 ) = d(A, B) ⇒ d(x1 , x2 ) ≤ d(y1 , y2 ).

(1.4)

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Definition 1.10. [8] A non-self-mapping T is called α-proximal admissible if ⎧ ⎨ α(x1 , x2 ) ≥ 1, d(u1 , T x1 ) = d(A, B), =⇒ α(u1 , u2 ) ≥ 1 ⎩ d(u2 , T x2 ) = d(A, B),

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for all x1 , x2 , u1 , u2 ∈ A, where α : A × A → [0, ∞).

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2. On P D−modular and fuzzy metric spaces

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Definition 2.1. Let Xω be a modular metric space and A, B ∈ X. We say that the mapping T : A → B is ω1 −continuous if and only if for every sequence (xn )n∈N with ω1 (xn , x) → 0 as n → ∞ we have, ω1 (T xn , T x) → 0 as n → ∞. Definition 2.2. We say that the modular metric space Xω is P D−modular metric space if there exist μ1 > 0 and μ2 ≥ 0 such that μ1 + μ2 = 1 and ωμ1 λ+μ2 γ (x, y) ≤ ωλ (x, z) + ωγ (z, y)

ωλ (x, y) = e

λ − λ+1

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d(z, y)

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That is, ω is a P D−modular metric.

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Similarly we have the following examples. Example 2.2. Let (X, d) be a metric space. Define ω : (0, +∞) × X × X → [0, +∞] by ωλ (x, y) = e

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− 1.

Then, ω is a P D−modular metric.

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λ − λ+1

= ωλ (x, z) + ωγ (z, y).

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λ

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λ+γ − λ+γ +1

ωμ1 λ+μ2 μ (x, y) = ωλ (x, y) = e− λ+1 d(x, y)

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Now assume that μ1 = 1 and μ2 = 0. Therefore, for all λ ≥ γ > 0 and x, y, z ∈ X we can write

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= ωλ (x, z) + ωγ (z, y).

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λ+γ − λ+γ +1

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Proof. Clearly, ω is a modular metric. Indeed, for all λ, γ > 0 and x, y, z ∈ X we have

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d(x, y).

Then, ω is a P D−modular metric.

ωλ+γ (x, y) = e

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Example 2.1. Let (X, d) be a metric space. Define ω : (0, +∞) × X × X → [0, +∞] by

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holds for all λ ≥ γ > 0 and x, y, z ∈ X.

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Example 2.3. Let (X, d) be a metric space. Define ω : (0, +∞) × X × X → [0, +∞] by

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Also we introduce fuzzy version of Definition 2.2 as follows.

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Definition 2.3. Let (X, M, ∗) be a fuzzy metric space. We say M is a P D−fuzzy metric if there exist μ1 > 0 and μ2 ≥ 0 such that μ1 + μ2 = 1 and 1 1 1 −1≤ −1+ −1 M(x, y, μ1 t + μ2 s) M(x, z, t) M(z, y, s) holds for all x, y, z ∈ X and all t ≥ s > 0. Example 2.4. Let (X, d) be a metric space. Define M : X × X × (0, +∞) → (0, 1] by M(x, y, t) =

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Proof. Let t, s > 0 and x, y ∈ X. Since, d(x, y) ≤ d(x, z) + d(z, y), then we get t +s (t + s)2 M(x, y, t + s) = = 2 t + s + d(x, y) (t + s) + (t + s)d(x, y) ≥

+ s)2

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(t + s)2 ≥ (t + s)2 + (t + s)d(x, z) + (t + s)d(z, y) + d(x, z)d(z, y) (t + s)2 = (t + s + d(x, z))(t + s + d(z, y)) t +s t +s = × t + s + d(x, z) t + s + d(z, y) s t × ≥ t + d(x, z) s + d(z, y) = M(x, z, t)  M(z, y, s).

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So, (M, X, ) is a fuzzy metric space. Assume that μ1 = 1 and μ2 = 0. Then, for all t ≥ s > 0 and x, y, z ∈ X we get, 1 1 1 −1= − 1 = d(x, y) M(x, y, μ1 t + μ2 s) M(x, y, t) t 1 1 ≤ d(x, z) + d(z, y) t t 1 1 ≤ d(x, z) + d(z, y) t s 1 1 −1+ −1 = M(x, y, t) M(x, y, s) Therefore, (M, X, ) is P D−fuzzy metric space.

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(t (t + s)2 + (t + s)d(x, z) + (t + s)d(z, y)

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Denote a  b = ab for all a, b ∈ [0, 1]. Then, (M, X, ) is a P D−fuzzy metric space.

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Example 2.5. Let (X, d) be a metric space. Define M : X × X × (0, +∞) → (0, 1] by

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Denote a  b = ab for all a, b ∈ [0, 1]. Then, (M, X, ) is a P D−fuzzy metric space.

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The result that follows emphasizes that a P D−modular metric can be associated to each P D−fuzzy metric. Lemma 2.1. Let (X, M, ∗) be a P D−fuzzy metric space. Put

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ωλ (x, y) =

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1 1 1 1 −1= −1≤ −1+ −1 M(x, y, t) M(x, y, μ1 t + μ2 t) M(x, z, t) M(z, y, t) for all x, y, z ∈ X. This says us (X, M, ∗) is a triangular fuzzy metric space. So, from Lemma 1.1, ωλ is a modular metric on X. For rest of the proof, we can write,

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ωμ1 λ+μ2 μ (x, y) = M(x,y,μ11 λ+μ2 μ) − 1 1 1 − 1 + M(z,y,μ) −1 ≤ M(x,z,λ) = ωλ (x, z) + ωμ (z, y)

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and so ωλ is a P D−modular metric on X.

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Definition 2.4. Let (X, M, ∗) be a P D−fuzzy metric space.

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(1) (2) (3) (4)

The sequence (xn )n∈N in X is said to be M1 -convergent to x ∈ X if and only if M(xn , x, 1) → 1, as n → ∞. The sequence (xn )n∈N in X is said to be M1 -Cauchy if M(xm , xn , 1) → 1, as m, n → ∞. A subset A of X is said to be M1 -closed if the M1 -limit of a M1 -convergent sequence of A always belongs to A. A subset A of X is said to be M1 -complete if any M1 -Cauchy sequence in A is a M1 -convergent sequence and its M1 -limit is in A.

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Let A and B be two subsets of P D−modular metric space Xω . We denote by A0 (1) and B0 (1) the following sets: A0 (1) = {x ∈ A : ω1 (x, y) = ω1 (A, B) for some y ∈ B}, B0 (1) = {y ∈ B : ω1 (x, y) = ω1 (A, B) for some x ∈ A}, where ω1 (A, B) = inf{ω1 (x, y) : x ∈ A, y ∈ B}. Also, let A and B be two subsets of P D−fuzzy metric space (X, M, ∗). We denote by A0 (1) and B0 (1) the following sets: A0 (1) = {x ∈ A : B0 (1) = {y ∈ B :

1 M(x,y,1) 1 M(x,y,1)

− 1 = M(A, B, 1) for some y ∈ B}, − 1 = M(A, B, 1) for some x ∈ A},

1 where M(A, B, t) = inf{ M(x,y,t) − 1 : x ∈ A, y ∈ B}.

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Proof. First of all from Definition 2.3 we have,

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for all x, y ∈ X and all λ > 0. Then ωλ is a P D−modular metric on X.

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Definition 2.5. Let (A, B) be a pair of nonempty subsets of a P D−modular metric space Xω with A0 (1) = ∅. Then the pair (A, B) is said to have the weak P1 -property if and only if for any x1 , x2 ∈ A0 (1) and y1 , y2 ∈ B0 (1), ω1 (x1 , y1 ) = ω1 (A, B) and ω1 (x2 , y2 ) = ω1 (A, B) ⇒ ω1 (x1 , x2 ) ≤ ω1 (y1 , y2 ).

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Example 2.6. Let X = R2 endowed with the P D−modular metric ωλ : X × X × [0, +∞) given by

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d(x,y) λ

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d(x, y) = d((x1 , x2 ), (y1 , y2 )) = |x1 − y1 | + |x2 − y2 |,

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for all x = (x1 , x2 ), y = (y1 , y2 ) ∈ X. From Example (2.2) we know that Xω is an P D−modular metric space. Define the sets     A = (a, 0); a ∈ [1, 2] and B = (b, 0); b ∈ [−1, 0]

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so that d(A, B) = 1 and ω1 (A, B) = e − 1. Then A0 (1) = {(1, 0)} and B0 (1) = {(0, 0)}. Now, it is easy to check that the pair (A, B) have the weak P1 -property.

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Definition 2.7. Let (X, M, ∗) be a P D−fuzzy metric space and A, B ∈ X. We say the mapping T : A → B is M1 −continuous if and only if for every sequence (xn )n∈N with M(xn , x, 1) → 1 as n → ∞ we have, M(T xn , T x, 1) → 1 as n → ∞. The proofs of the following Lemmas are straightforward and are therefore omitted.

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Lemma 2.2. Let Xω be a P D−modular. Assume (xn )n∈N and (yn )n∈N be two sequences in Xω such that limn→∞ ω1 (xn , x) = limn→∞ ω1 (yn , y) = 0. Then, limn→∞ ω1 (xn , yn ) = ω1 (x, y).

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Lemma 2.3. Let (X, M, ∗) be a P D−fuzzy metric space. Assume (xn )n∈N and (yn )n∈N be two sequences in X such that limn→∞ M(xn , x, 1) = limn→∞ M(yn , y, 1) = 0. Then, limn→∞ M(xn , yn , 1) = M(x, y, 1).

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Lemma 2.4. Let (X, M, ∗) be a P D−fuzzy metric space and (xn )n∈N be a sequence. Further, assume that ωλ (x, y) = 1 M(x,y,λ) − 1.

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(m1) (m2) (m3) (m4) (m5)

The sequence (xn )n∈N is M1 -convergent to x ∈ X if and only if it is ω-convergent to x ∈ X, The sequence (xn )n∈N in X is M1 -Cauchy if and only it is ω-Cauchy, A subset A of X is M1 -closed if and only if it is ω-closed, A subset A of X is M1 -complete if and only if it is ω-complete. Let A, B ⊂ X. Let T : A → B be a mapping. Then T is M1 −continuous if and only if it is an ω1 −continuous mapping.

3. On hybrid (α, ) − ω-contraction

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We are going to introduce the notion of hybrid (α, ) − ω-contraction in modular metric spaces and then state and prove our main results in the setting of P D−modular metric spaces. Consistent with Jleli and Samet [9], we denote by the set of all functions : [0, +∞) → [1, +∞) satisfying the following conditions:

46 47 48 49 50

50 51

20

44

45 46

19

43

43 44

18

36

36 37

17

33

33 34

16

30

30 31

13

27

27 28

12

25

25 26

10

21

21

23

9

15

Definition 2.6. Let (A, B) be a pair of nonempty subsets of a P D−fuzzy metric space (X, M, ∗) with A0 (1) = ∅. Then the pair (A, B) is said to have the weak PM1 -property if and only if for any x1 , x2 ∈ A0 and y1 , y2 ∈ B0 ,

20

8

14

Also we introduce fuzzy version of Definition 2.5 as follows.

19

22

1 2

− 1,

where d : X × X → [0, +∞) is the metric

6 7

7

( 1 ) is increasing; ( 2 ) for all sequence {αn } ⊆ (0, +∞), lim αn = 0 if and only if lim (αn ) = 1; n→+∞

n→+∞

51 52

JID:FSS AID:7787 /FLA

1

[m3SC+; v1.304; Prn:6/01/2020; 15:47] P.8 (1-22)

M. Paknazar, M. De La Sen / Fuzzy Sets and Systems ••• (••••) •••–•••

8

( 3 ) there exist 0 < k < 1 and  ∈ (0, +∞] such that lim

t→0+

2 3 4 5 6 7 8 9 10 11

16 17 18 19 20 21

24 25

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

3 4

we have

11





r

L [ ω1 (u, v) ]α(x, y) ≤ M (x, y, u, v) N (x, y, u, v) where 0 ≤ r < 1, L ≥ 0, ∈ and      ω1 (x, v) + ω1 (y, u) M (x, y, u, v) = max ω1 (x, y)), ω1 (x, u) , ω1 (y, v)), 2 and

6 7 8 9 10

12 13 14 15 16 17 18



      N (x, y, u, v) = min ω1 (x, y)), ω1 (x, u) , ω1 (y, v) , ω1 (x, v) , ω1 (y, u) .

19 20 21 22

Theorem 3.1. Let A and B are two ω−closed subsets of an ω−complete P D−modular metric space with ω regular Xω . Let T : A → B be a non-self-mapping such that T (A0 (1)) ⊆ B0 (1) and A0 (1) = ∅. Assume that there exist two functions α : A × A → [0, +∞) and ∈ such that the following assertions hold:

23 24 25 26

26 27

2

5

22 23

1

Definition 3.1. Let (X, ω) be a modular metric space and T : A → B be a non-self-mapping. Also suppose that α : X × X → [0, +∞) is a function. We say that T is a hybrid (α, ) − ω-contraction if for all x, y, u, v ∈ A with ⎧ ⎨ ω1 (u, v) > 0 ω1 (u, T x) = ω1 (A, B) ⎩ ω1 (v, T y) = ω1 (A, B)

13

15

= .

We now introduce a concept of hybrid (α, ) − ω-contraction:

12

14

(t)−1 tk

(i) T is an α-proximal admissible mapping, (ii) T is a hybrid (α, ) − ω-contraction and ω−continuous mapping, (iii) there exist elements x0 and x1 in A0 (1) such that, ω1 (x1 , T x0 ) = ω1 (A, B) and α(x0 , x1 ) ≥ 1. Then there exists

x∗

∈ Xω such that ω1

(x ∗ , T x ∗ ) = ω

1 (A, B).

Proof. By (iii) there exist elements x0 and x1 in A0 (1) such that, ω1 (x1 , T x0 ) = ω1 (A, B) and α(x0 , x1 ) ≥ 1. On the other hand T (A0 (1)) ⊆ B0 (1), there exists x2 ∈ A0 (1) such that, ω1 (x2 , T x1 ) = ω1 (A, B). Now, since, T is an α-proximal admissible mapping, so we have α(x1 , x2 ) ≥ 1. That is, ω1 (x2 , T x1 ) = ω1 (A, B), α(x1 , x2 ) ≥ 1. Since, T (A0 (1)) ⊆ B0 (1) then there exists x3 ∈ A0 (1) such that, ω1 (x3 , T x2 ) = ω1 (A, B). Thus we have, ω1 (x2 , T x1 ) = ω1 (A, B), ω1 (x3 , T x2 ) = ω1 (A, B), α(x1 , x2 ) ≥ 1. Again since, T is an α-proximal admissible mapping, so α(x2 , x3 , t) ≥ 1. Hence, ω1 (x3 , T x2 ) = ω1 (A, B), α(x2 , x3 ) ≥ 1.

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

JID:FSS AID:7787 /FLA

[m3SC+; v1.304; Prn:6/01/2020; 15:47] P.9 (1-22)

M. Paknazar, M. De La Sen / Fuzzy Sets and Systems ••• (••••) •••–•••

1

Continuing this process, we get,

2

9 10 11 12 13 14

18 19 20 21 22 23 24 25 26 27 28

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

and

7 8 9 10 11 12 13 14 15

M (xn−1 , xn , xn , xn+1 )    = max ω1 (xn−1 , xn )), ω1 (xn−1 , xn ) , ω1 (xn , xn+1 )),   ω1 (xn−1 , xn+1 ) + ω1 (xn , xn ) 2    ω1 (xn−1 , xn+1 ) = max ω1 (xn−1 , xn )), ω1 (xn , xn+1 )), ( ) 2    ωμ1 +μ2 (xn−1 , xn+1 ) = max ω1 (xn−1 , xn )), ω1 (xn , xn+1 )), ( ) 2

   ω1 (xn−1 , xn ) + ω1 (xn , xn+1 ) ≤ max ω1 (xn−1 , xn )), ω1 (xn , xn+1 )), 2   = max{ ω1 (xn−1 , xn )), ω1 (xn , xn+1 ) }

17

30

6

where,

16

29

5

That is, xn0 is best proximity point of T . Hence, we assume xn = xn+1 for all n ∈ N. Now, the regularity of ω implies ω1 (xn , xn+1 ) > 0 for all n ∈ N. Since T is a hybrid (α, ) − ω-contraction, we derive   ω1 (xn xn+1 ) ≤ [ ω1 (xn xn+1 ) ]α(xn−1 ,xn ) (3.2)

r

L ≤ M (xn−1 , xn , xn , xn+1 ) N (xn−1 , xn , xn , xn+1 )

15

3 4

ω1 (xn0 , T xn0 ) = ω1 (xn0 +1 , T xn0 ) = ω1 (A, B).

6

8

2

(3.1)

If there exists n0 ∈ N ∪ {0} such that xn0 = xn0 +1 , then,

5

7

1

ω1 (xn+1 , T xn ) = ω1 (A, B), α(xn , xn+1 ) ≥ 1 for all n ∈ N ∪ {0}.

3 4

9

16 17 18 19 20 21 22 23 24 25 26 27 28 29



30

   N (xn−1 , xn , xn , xn+1 ) = min ω1 (xn−1 , xn )), ω1 (xn−1 , xn ) , ω1 (xn , xn+1 )),    ω1 (xn−1 , xn+1 ) , ω1 (xn , xn )    = min ω1 (xn−1 , xn )), ω1 (xn−1 , xn ) , ω1 (xn , xn+1 )),   ω1 (xn−1 , xn+1 ) , 1 = 1

31 32 33 34 35 36 37 38 39

and so from (3.2) we can write,    ω1 (xn , xn+1 ) ≤ [max{ ω1 (xn−1 , xn )), ω1 (xn , xn+1 ) }]r [1]L = [max{ ω1 (xn−1 , xn )), ω1 (xn , xn+1 ) }]r

40

Now assume that,    max{ ω1 (xn−1 , xn )), ω1 (xn , xn+1 ) } = ω1 (xn , xn+1 )

44

then we have,    ω1 (xn , xn+1 ) ≤ [ ω1 (xn , xn+1 ) ]r < ω1 (xn , xn+1 )

47

which is a contradiction. Hence,   ω1 (xn , xn+1 ) ≤ [ ω1 (xn−1 , xn ) ]r

41 42 43

45 46

48 49 50 51

(3.3)

52

JID:FSS AID:7787 /FLA

[m3SC+; v1.304; Prn:6/01/2020; 15:47] P.10 (1-22)

M. Paknazar, M. De La Sen / Fuzzy Sets and Systems ••• (••••) •••–•••

10

1 2 3 4 5 6 7 8 9 10

for all n ∈ N. Therefore,   r 1 < ω1 (xn , xn+1 ) ≤ ω1 (xn−1 , xn )  r 2 n ≤ ω1 (xn−2 , xn−1 ) ≤ · · · ≤ (ω1 (x0 , x1 ))r .

15 16 17 18 19 20 21 22

8 9 10

(3.5)

13 14

(3.6)

17

Now, let ∈ (0, ). From the definition of limit, there exists n1 ∈ N such that  ω1 (xn , xn+1 ) − 1 ≥ C −1 for all n ≥ n1 [ω1 (xn , xn+1 )]k

18 19 20 21

and so

22



n[ω1 (xn , xn+1 )] ≤ nC[ ω1 (xn , xn+1 ) − 1] k

23

for all n ≥ n1 .

24

From (3.4), we deduce

25

rn

n[ω1 (xn , xn+1 )]k ≤ nC[ (ω1 (x0 , x1 )) − 1]

for all

26

n ≥ n1 .

27

Taking the limit as n → +∞ in the above inequality, we have

28 29

lim n[ω1 (xn , xn+1 )]k = 0

30

n[ω1 (xn , xn+1 )]k ≤ 1

34

39 40 41

for all

32 33

n ≥ N1 .

34 35

Thus

36

38

30 31

From (3.7), it follows that there exists N1 ∈ N such that

33

37

(3.7)

n→+∞

31

35

15 16

C −1

29

32

11 12

Thus there exist 0 < k < 1 and 0 <  ≤ +∞ such that, from ( 3) :  ω1 (xn , xn+1 ) − 1 lim = . n→+∞ [ω1 (xn , xn+1 )]k

27 28

7

lim ω1 (xn , xn+1 ) = 0.

26

4

6

n→+∞

24

3

5

and, since ∈ , we obtain

23

25

(3.4)

n→+∞

12

14

2

Taking the limit as n → +∞ in (3.4), we get  lim ω1 (xn , xn+1 ) = 1

11

13

1

ω1 (xn , xn+1 ) ≤

1

36

for all

n1/k Now, for m > n, by (3.8), we get

n ≥ N1 .

(3.8)

37 38 39 40

ω1 (xn , xm ) = ωμ1 +μ2 (xn , xm )

41

42

≤ ω1 (xn , xn+1 ) + ω1 (xn+1 , xm )

42

43

= ω1 (xn , xn+1 ) + ωμ1 +μ2 (xn+1 , xm )

43

44 45 46 47 48 49 50 51 52

≤ ω1 (xn , xn+1 ) + ω1 (xn+1 , xn+2 ) + ω1 (xn+2 , xm ) ≤ ω1 (xn , xn+1 ) + ω1 (xn+1 , xn+2 ) + ω1 (xn+2 , xn+3 ) + ω1 (xn+3 , xm ) .. . m−1 m−1   1 = ω1 (xi , xi+1 ) ≤ . i 1/k i=n

Since 0 < k < 1, then

i=n

44 45 46 47 48 49 50 51 52

JID:FSS AID:7787 /FLA

[m3SC+; v1.304; Prn:6/01/2020; 15:47] P.11 (1-22)

M. Paknazar, M. De La Sen / Fuzzy Sets and Systems ••• (••••) •••–•••

1 2 3 4 5 6 7 8

∞  1 =0 1/k n→+∞ i

1

lim

2

i=n

and hence ω1 (xn , xm ) → 0 as m, n → +∞. Thus, we have proved that {xn } is a ω−Cauchy sequence. The hypothesis of ω−completeness of X ensures that there exists x ∗ ∈ X such that ω1 (xn+1 , x ∗ ) → 0 as n → +∞. Now, since T is an ω−continuous mapping, then ω1 (T xn , T x ∗ ) → 0 as n → +∞. Thus, ∗

∗

∗

∗

∗

11

≤ ω1 (x , xn+1 ) + ω1 (xn+1 , T xn ) + ω1 (T xn , T x )

12

= ω1 (x ∗ , xn+1 ) + ω1 (A, B) + ω1 (T xn , T x ∗ )

13

taking limit as n → +∞, we get ω1

(x ∗ , T x ∗ ) = ω

1 (A, B).

13

Hence, T has a best proximity point. 2

14 15

15 16 17 18

Theorem 3.2. Let A and B are two ω−closed subsets of an ω−complete P D−modular metric space with ω regular Xω . Let T : A → B be a non- self-mapping such that T (A0 (1)) ⊆ B0 (1) and A0 (1) = ∅. Assume that there exist two functions α : A × A → [0, +∞) and ∈ such that the following assertions hold:

21 22

27 28 29

22 23

32 33 34 35 36 37 38 39 40 41 42 43 44 45

24

(iv) if {xn } is a sequence in X for all n ∈ N ∪ {0} such that α(xn , xn+1 ) ≥ 1 with ω1 (xn , x) → 0 as n → +∞, then α(xn , x) ≥ 1 for all n ∈ N ∪ {0}, (v) for any sequence {yn } in B0 (1) and x ∈ A satisfying ω1 (x, yn ) → ω1 (A, B) as n → +∞, then x ∈ A0 (1). Then there exists

30 31

x∗

∈ Xω such that ω1

(x ∗ , T x ∗ ) = ω

48 49 50 51 52

25 26 27 28 29

1 (A, B).

30 31

Proof. By (iii) there exist elements x0 and x1 in A0 (1) such that,

32

ω1 (x1 , T x0 ) = ω1 (A, B) and α(x0 , x1 ) ≥ 1.

33

As in the proof of Theorem 3.1, we can deduce that a sequence {xn } starting at x0 is ω−Cauchy and so it converges to a point x ∗ ∈ X where, ω1 (xn+1 , T xn ) = ω1 (A, B), α(xn , xn+1 ) ≥ 1 for all n ∈ N ∪ {0}.

(3.9)

34 35 36 37 38

Also, from (iv) we have α(xn , x ∗ ) ≥ 1 for all n ∈ N ∪ {0}. Then,

39

ω1 (A, B) = ω1 (xn+1 , T xn ) = ωμ1 +μ2 (xn+1 , T xn ) ≤ ω1 (xn+1 , x ∗ ) + ω1 (x ∗ , T xn ) = ω1 (xn+1 , x ∗ ) + ωμ1 +μ2 (x ∗ , T xn ) ≤ ω1 (xn+1 , x ∗ ) + ω1 (x ∗ , xn+1 ) + ω1 (xn+1 , T xn ) = ω1 (xn+1 , x ∗ ) + ω1 (x ∗ , xn+1 ) + ω1 (A, B).

40 41 42 43 44 45 46

46 47

18

21

ω1 (x1 , T x0 ) = ω1 (A, B) and α(x0 , x1 ) ≥ 1,

24

26

17

20

(i) T is an α-proximal admissible mapping, (ii) T is a weak (α, ) − ω-contraction where continuous, (iii) there exist elements x0 and x1 in A0 (1) such that,

23

25

16

19

19 20

6

10

∗

12

5

9

= ωμ1 +μ2 (x ∗ , T xn ) + ω1 (T xn , T x ∗ )

11

4

8

≤ ω1 (x ∗ , T xn ) + ω1 (T xn , T x ∗ )

10

3

7

ω1 (A, B) ≤ ω1 (x , T x ) = ωμ1 +μ2 (x , T x )

9

14

11

That is,

47

∗

∗

∗

∗

ω1 (A, B) ≤ ω1 (xn+1 , x ) + ω1 (x , T xn ) ≤ ω1 (xn+1 , x ) + ω1 (x , xn+1 ) + ω1 (A, B). By letting limit as n → ∞ in the above inequality, we derive, ∗

lim ω1 (x , T xn ) = ω1 (A, B)

n→∞

48 49 50 51 52

JID:FSS AID:7787 /FLA

[m3SC+; v1.304; Prn:6/01/2020; 15:47] P.12 (1-22)

M. Paknazar, M. De La Sen / Fuzzy Sets and Systems ••• (••••) •••–•••

12

1 2 3 4

and then by condition (iv), x ∗ ∈ A0 (1). Since, T (A0 (1)) ⊂ B0 (1), then there exists z ∈ A0 (1) such that, ω1 (z, T x ∗ ) = ω1 (A, B). First assume that, for each n ∈ N, there exists kn ∈ N such that ω1 (xkn +1 , z) = 0 and kn > kn−1 where k0 = 1. Note that,

5

7 8 9 10 11 12 13

(x ∗ , z) = 0.

24



(3.11)

27 28 29

32 33 34 35 36 37 38 39 40

10 11 12

15

17 18 19

and

20



   N (xn , x ∗ , xn+1 , z) = min ω1 (xn , x ∗ )), ω1 (xn , xn+1 ) , ω1 (x ∗ , z) ,    ∗ ω1 (xn , z) , ω1 (x , xn+1 ) .

(3.12)

21 22 23 24 25

As is continuous then by taking limit as n → ∞ in (3.10), (3.11) and (3.12) we have,     ω1 (x ∗ , z) ≤ [ ω1 (x ∗ , z)) ]r [1]L = [ ω1 (x ∗ , z) ]r < ω1 (x ∗ , z) which is a contradiction. Hence,

z = x∗.

That is, again

x∗

is a best proximity point of T .

26 27 28

2

29 30

30 31

9

16

25 26

8

14

   M (xn , x , xn+1 , z) = max ω1 (xn , x ∗ )), ω1 (xn , xn+1 ) , ω1 (x ∗ , z)),   ω1 (xn , z) + ω1 (x ∗ , xn+1 ) 2 ∗

18

23

x∗

13

17

22

6

x∗

where

16

21

4

7

15

20

3

and so we get, ω1 Now the regularity of ω ensures that, = z. That is, = z is a best proximity point of T . Next we assume, ω1 (xn+1 , z) > 0. Since T is weak (α, ) − ω-contraction then we can write,   ∗ ω1 (xn+1 , z) ≤ [ ω1 (xn+1 , z) ]α(xn ,x ) (3.10) 



r L ≤ M (xn , x ∗ , xn+1 , z) N (xn , x ∗ , xn+1 , z)

14

19

2

5

ω1 (x ∗ , z) = ωμ1 +μ2 (x ∗ , z) ≤ ω1 (x ∗ , xkn +1 ) + ω1 (xkn +1 , z)

6

1

Example 3.1. Let X

= R2

endowed with the P D−modular metric ωλ : X × X × [0, +∞) given by

31 32

λ

ωλ (x, y) = e− λ+1 d(x, y),

33

where d : X × X → [0, +∞) is the metric d(x, y) = d((x1 , x2 ), (y1 , y2 )) = |x1 − y1 | + |x2 − y2 |, for all x = (x1 , x2 ), y = (y1 , y2 ) ∈ X. Thus, Xω is an ω−complete P D−modular metric space. Define the sets     A = (0, x) ∈ R2 ; x ∈ R and B = (1, x) ∈ R2 ; x ∈ R − 12

34 35 36 37 38 39 40 41

47

so that d(A, B) = 1 and ω1 (A, B) = e . Clearly, A and B are nonempty ω−closed subsets of X. Define T : A → B by ⎧ ⎪ 2π ) if (x1 , x2 ) ∈ AV , ⎨(1,   1 T (x1 , x2 ) = 1, 2n if (x1 , x2 ) = 0, n1 , ∀n ∈ N, ⎪ ⎩ (1, 0) if (x1 , x2 ) = (0, 0),

48

where

48

41 42 43 44 45 46

49 50 51 52

V=

0,

1 n



 : n ∈ N ∪ {(0, 0)}.

Notice that A0 (1) = A, B0 (1) = B. Also define, α : A × A → [0, ∞) by

42 43 44 45 46 47

49 50 51 52

JID:FSS AID:7787 /FLA

[m3SC+; v1.304; Prn:6/01/2020; 15:47] P.13 (1-22)

M. Paknazar, M. De La Sen / Fuzzy Sets and Systems ••• (••••) •••–•••

 α (0, x), (0, y), t =

1 2 3

Let,

4 5 6 7 8

So,

9 10 11 12 13

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

1 2 3 4 5 6 7 8

⎧ ⎨ (x, y) ∈ V , ω1 (u, T x) = ω1 (A, B), ⎩ ω1 (v, T y) = ω1 (A, B).

9 10 11 12 13

Then

((u, x), (v, y)) ∈ ((0, 0), (0, 0)),

16 17

1 if (0, x), (0, y) ∈ V , 0 otherwise.

⎧ ⎨ α(x, y) ≥ 1, ω1 (u, T x) = ω1 (A, B), ⎩ ω1 (v, T y) = ω1 (A, B).

14 15



13



   1 1 0, , 0, :n∈N . 2n n

That is, α(u, v) ≥ 1 or T is an α-proximal admissible mapping. Next, we distinguish the following cases: 

 1  1  1 1 0, 2n , 0, n and (v, y) = 0, 2m , 0, m , for all n, m ∈ N, we have      λ λ [ω1 (u, v)] ω1 (u, v) = d(u, v) d(u, v) t +1 t +1       1  1 λ 1 1  λ    = − − λ + 1  2n 2m  λ + 1  2n 2m        1  1  1 λ 1 λ  −   − 1 = √    λ+1 n m λ + 1 n m 2 2  1 = [ √ ω1 (x, y)] ω1 (x, y) 2 2  1 ≤ [ ω1 (x, y)] ω1 (x, y). 2  1  1 , 0, m , for all m ∈ N, we have (ii) if (u, x) = ((0, 0), (0, 0)) and (v, y) = 0, 2m      λ λ ω1 (u, v) ω1 (u, v) = d(u, v) d(u, v) λ+1 λ+1 

  λ 1 1 t = [ ] [ ] λ + 1 2m t + 1 2m   

1 1 1 t λ = √ [ ] [ ] t +1 m 2 2 λ+1 m  1 = [ √ ω1 (x, y)] ω1 (x, y) 2 2  1 ≤ [ ω1 (x, y)] ω1 (x, y). 2 Therefore, (i) if (u, x) =

√ (ω1 (u, v)) = [ (ω1 (u, v))]α(x,y) = eω1 (u,v) ω1 (u,v) √ [ 12 ω1 (x,y)] ω1 (x,y)

≤e

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

JID:FSS AID:7787 /FLA

[m3SC+; v1.304; Prn:6/01/2020; 15:47] P.14 (1-22)

M. Paknazar, M. De La Sen / Fuzzy Sets and Systems ••• (••••) •••–•••

14

= [e[ω1 (x,y)]

1 2

√

ω1 (x,y)

= [ (ω1 (x, y))]

3

1

]2

4 5

√

6

9

3

1 

L ≤ M (x, y, u, v) 2 N (x, y, u, v) .

5

8

2



4

7

1

1 2

So we omit details. We conclude that all the hypotheses of Theorem 3.1 are satisfied, where (t) = et t , for all t ∈ [0, ∞), and so there exists x ∗ ∈ A such that ω1 (x ∗ , T x ∗ ) = ω1 (A, B). Here, x ∗ = (0, 0) is a best proximity point of T . In Theorem 3.2, if we take, (t) = et , we obtain the following best proximity result.

14 15

Corollary 3.1. Let A and B be two ω−closed subsets of an ω−complete P D−modular metric space with ω regular Xω . Let T : A → B be a non- self-mapping such that T (A0 (1)) ⊆ B0 (1) and A0 (1) = ∅. Assume that there exists a function α : A × A → [0, +∞) such that

18 19 20 21 22 23

(i) T is an α-proximal admissible mapping, (ii) if for all x, y, u, v ∈ A with ⎧ ⎨ ω1 (u, v) > 0, ω1 (u, T x) = ω1 (A, B), ⎩ ω1 (v, T y) = ω1 (A, B),

28

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

49 50 51 52

15

20 21 22

r

L [ω1 (u, v)]α(x, y) ≤ M(x, y, u, v) N (x, y, u, v) , where 0 ≤ r < 1, L ≥ 0 and  ω1 (x, v) + ω1 (y, u) M(x, y, u, v) = max ω1 (x, y), ω1 (x, u), ω1 (y, v), 2

23 24 25 26 27 28 29

and



 N (x, y, u, v) = min ω1 (x, y), ω1 (x, u), ω1 (y, v), ω1 (x, v), ω1 (y, u) ,

(iii) there exist elements x0 and x1 in A0 (1) such that, ω1 (x1 , T x0 ) = ω1 (A, B) and α(x0 , x1 ) ≥ 1, (iv) if {xn } is a sequence in X for all n ∈ N ∪ {0} such that α(xn , xn+1 ) ≥ 1 with ω1 (xn , x) → 0 as n → +∞, then α(xn , x) ≥ 1 for all n ∈ N ∪ {0}, (v) for any sequence {yn } in B0 (1) and x ∈ A satisfying ω1 (x, yn ) → ω1 (A, B) as n → +∞, then x ∈ A0 (1). Then, there exists

x∗

∈ Xω such that ω1

(x ∗ , T x ∗ ) = ω

1 (A, B).

In Theorem 3.2, if we take α(x, y) = 1, we obtain the following best proximity result. Corollary 3.2. Let A and B are two ω−closed subsets of an ω−complete P D−modular metric space with ω regular Xω . Let T : A → B be a non- self-mapping such that T (A0 (1)) ⊆ B0 (1) and A0 (1) = ∅ and the following assertions hold

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

47 48

14

19

29 30

13

18



25

27

11

17

we have

24

26

9

16

16 17

8

12

12 13

7

10

10 11

6

(i) if for all x, y, u, v ∈ A with ⎧ ⎨ ω1 (u, v) > 0, ω1 (u, T x) = ω1 (A, B), ⎩ ω1 (v, T y) = ω1 (A, B),

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1 2 3 4 5 6 7 8 9

15

we have

1 2



r

L ω1 (u, v) ≤ M (x, y, u, v) N (x, y, u, v) ,

3 4

where 0 ≤ r < 1, L ≥ 0 and ∈ , (ii) there exist elements x0 and x1 in A0 (1) such that

5 6 7

ω1 (x1 , T x0 ) = ω1 (A, B),

8

(iii) for any sequence {yn } in B0 (1) and x ∈ A satisfying ω1 (x, yn ) → ω1 (A, B) as n → +∞, then x ∈ A0 (1).

9 10

10 11

Then there exists x ∗ ∈ Xω such that ω1 (x ∗ , T x ∗ ) = ω1 (A, B).

11 12

12 13

4. Suzuki type best proximity point results

13 14

14 15 16 17 18

In this section, we are going to introduce the notion of weak GS −contraction and GS −contraction in modular metric spaces and then we state and prove our main results in the setting of P D−modular metric spaces. Let A and B be two subsets of a P D−fuzzy metric space (X, M, ∗). We denote by A0 (1) and B0 (1) the following sets:

21

A0 (1) = {x ∈ A : B0 (1) = {y ∈ B :

1 M(x,y,1) 1 M(x,y,1)

− 1 = M(A, B, 1) for some y ∈ B}, − 1 = M(A, B, 1) for some x ∈ A},

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

43 44 45 46

22

1 where M(A, B, 1) = inf{ M(x,y,1) − 1 : x ∈ A, y ∈ B}. Consistent with Geraghty [6] we denote by F the set of all functions β : [0, +∞) → [0, 1) satisfying the condition

51 52

24

26

We denote by

the set of all functions : [0, +∞) → [1, +∞) satisfying the following conditions:

27 28

( 1 ) (t1 ) < (t2 ) ⇔ t1 < t2 , ( 2 ) is continuous, ( 3 ) (t) = 1 ⇔ t = 0.

29

Definition 4.1. Let (A, B) be a pair of nonempty subsets of a modular metric space Xω . A mapping T : A → B is said to be a weak GS −contraction(Geraghty-Suzuki −contraction) if there exist β ∈ F and ∈

such that

33

1 ∗ ω (x, T x) ≤ ω1 (x, y) ⇒ (ω1 (T x, T y)) ≤ [ (U1 (x, y) − ω1 (A, B))]β(U1 (x, y)) 2 1

30 31 32

(4.1)

ω1∗ (x, y) = ω1 (x, y) − ω1 (A, B)

for all x, y ∈ A, where and   U1 (x, y) = max ω1 (x, y), ω1 (x, T x), ω1 (y, T y) .

34 35 36 37 38 39 40 41

Also we say T is a GS −contraction if there exist β ∈ F and ∈

such that 1 ∗ ω (x, T x) ≤ ω1 (x, y) ⇒ (ω1 (T x, T y)) ≤ [ (ω1 (x, y))]β(ω1 (x, y)) 2 1 for all x, y ∈ A.

42 43 44 45 46 47

Thus, we state and prove the following result of existence and uniqueness.

48 49

49 50

23

25

β(tn ) → 1 implies tn → 0, as n → +∞.

47 48

18

21

41 42

17

20

22 23

16

19

19 20

15

Theorem 4.1. Let A and B be two ω−closed subsets of an ω−complete P D−modular metric space with ω regular Xω . Let T : A → B be a weak GS −contraction such that T (A0 (1)) ⊆ B0 (1) where A0 (1) = ∅. Suppose that the pair (A, B) has the weak P1 -property. Thus there exists a unique x ∗ in A such that ω1 (x ∗ , T x ∗ ) = ω1 (A, B).

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16

1 2 3 4 5 6

Proof. Let us select an element x0 ∈ A0 ; since T x0 ∈ T (A0 ) ⊆ B0 , we can find x1 ∈ A0 such that ω1 (x1 , T x0 ) = ω1 (A, B). Further, since T x1 ∈ T (A0 ) ⊆ B0 , it follows that there is an element x2 in A0 such that ω1 (x2 , T x1 ) = ω1 (A, B). Recursively, we obtain a sequence {xn } in A0 such that ω1 (xn+1 , T xn ) = ω1 (A, B), for any n ∈ N.

9 10 11

14 15 16 17 18

(4.3)

ω1 (xn−1 , T xn−1 ) ≤ ω1 (xn−1 , xn ) + ω1 (xn , T xn−1 ) = ω1 (xn−1 , xn ) + ω1 (A, B)

23 24 25 26 27 28 29 30 31 32

10

(4.4)

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

13 14

ω1 (xn , T xn ) ≤ ω1 (xn , T xn−1 ) + ω1 (T xn−1 , T xn ) = ω1 (xn , xn+1 ) + ω1 (A, B).

15

Therefore, we have

16 17

U1 (xn−1 , xn ) = max{ω1 (xn−1 , xn ), ω1 (xn−1 , T xn−1 ), ω1 (xn , T xn )} (4.5) ≤ max{ω1 (xn−1 , xn ), ω1 (xn , xn+1 )} + ω1 (A, B).

19

21 22 23

0 = ω1 (xn0 , xn0 +1 ) = ω1 (T xn0 −1 , T xn0 ),

24

and so from regularity of ω we get, T xn0 −1 = T xn0 . Thus, we conclude that ω1 (A, B) = ω1 (xn0 , T xn0 −1 ) = ω1 (xn0 , T xn0 ).

18

20

Clearly, if there exists n0 ∈ N such that ω1 (xn0 , xn0 +1 ) = 0, then we have nothing more to prove, the conclusion is immediate. In fact,

25 26

(4.6)

27

For the rest of the proof, we suppose that ω1 (xn , xn+1 ) > 0 for any n ∈ N. Now from (4.4), we deduce that

28

1 ∗ ω (xn−1 , T xn−1 ) ≤ ω1∗ (xn−1 , T xn−1 ) ≤ ω1 (xn , xn−1 ) 2 1 and by (4.1), we get

30

29

31 32 33

(ω1 (xn , xn+1 )) = (ω1 (T xn−1 , T xn )) ≤ [ (U1 (xn−1 , xn ) − ω1 (A, B))]β(U1 (xn−1 ,xn )) < (U1 (xn−1 , xn ) − ω1 (A, B)),

34

(4.7)

35 36 37

which implies,

38

ω1 (xn , xn+1 ) < U1 (xn−1 , xn ) − ω1 (A, B),

39 40

so by applying (4.5), we obtain

41

ω1 (xn , xn+1 ) < U1 (xn−1 , xn ) − ω1 (A, B) ≤ max{ω1 (xn−1 , xn ), ω1 (xn , xn+1 )}.

42 43

Now, if max{ω1 (xn−1 , xn ), ω1 (xn , xn+1 )} = ω1 (xn , xn+1 ), then

44 45

ω1 (xn , xn+1 ) < ω1 (xn , xn+1 ),

46

which is a contradiction and hence U1 (xn−1 , xn ) ≤ max{ω1 (xn−1 , xn ), ω1 (xn , xn+1 )} + ω1 (A, B) = ω1 (xn−1 , xn ) + ω1 (A, B).

47 48

(4.8)

49 50

Therefore, by (4.7) we get

51

51 52

11 12

and by (4.2) and (4.3) we obtain

33 34

8 9

Now, by (4.2), we get

20

22

5

7

ω1 (xn , xn+1 ) = ω1 (T xn−1 , T xn ), for any n ∈ N.

19

21

3

6

Since (A, B) has the weak P1 -property, we derive that

12 13

2

4

(4.2)

7 8

1

(ω1 (xn , xn+1 )) ≤ [ (ω1 (xn−1 , xn ))]β(U1 (xn−1 ,xn )) < (ω1 (xn−1 , xn )),

(4.9)

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1 2 3 4 5 6 7 8 9 10 11 12

17

for all n ∈ N. Evidently, from the properties of we can get, ω1 (xn , xn+1 ) < ω1 (xn−1 , xn ). This says us that {ω1 (xn , xn+1 )} is a decreasing sequence and bounded below and so

4

n→+∞

5

exists. Suppose L > 0. Then,

6

lim (ω1 (xn+1 , xn+2 )) = lim (ω1 (xn , xn+1 )) = (L) > 1.

7

n→∞

8

For this from (4.9), we have

9 10

(L) ≤ [ (L)]limn→∞ β(U1 (xn−1 ,xn )) ≤ (L).

11 12

So we can deduce,

13

13 14 15 16 17 18 19 20

lim β(U1 (xn , xn+1 )) = 1.

14

n→+∞

15

On the other hand, since β ∈ F , we get limn→+∞ U1 (xn , xn+1 ) = 0, that is,

16

(4.10)

17

Since, ω1 (xn , T xn−1 ) = ω1 (A, B) holds for all n ∈ N and the pair (A, B) satisfies the weak P1 -property, then for all m, n ∈ N, we can write ω1 (xm , xn ) = ω1 (T xm−1 , T xn−1 ). Using the fact that

19

L = lim ω1 (xn , xn+1 ) = 0. n→+∞

23 24 25 26 27 28 29 30 31

ω1 (xl , T xl ) ≤ ω1 (xl , xl+1 ) + ω1 (xl+1 , T xl ) = ω1 (xl , xl+1 ) + ω1 (A, B)

34 35 36 37 38 39 40 41 42 43

23

for all l ∈ N, we easily deduce

24

U1 (xm , xn ) = max{ω1 (xm , xn ), ω1 (xm , T xm ), ω1 (xn , T xn )} ≤ max{ω1 (xm , xn ), ω1 (xm , xm+1 ), ω1 (xn , xn+1 )} + ω1 (A, B).

25 26 27

Since lim ω1 (xn , xn+1 ) = 0, then we have n→+∞

limm,n→+∞ U1 (xm , xn ) ≤ limm,n→+∞ ω1 (xm , xn ) + ω1 (A, B).

28

(4.11)

We shall show that {xn } is a Cauchy sequence. If not, then we get m,n→+∞

ε=

ω1 (xn , xm ) > 0.

lim

m,n→+∞

ω1 (xn , xm ) > 0.

ω1 (xn , xm ) ≤ ω1 (xn , xn+1 ) + ω1 (xn+1 , xm+1 ) + ω1 (xm+1 , xm ). Now, since lim ω1 (xn , xn+1 ) = 0, then n→+∞

ω1 (A, B) ≤ lim ω1 (xm , T xm ) m→+∞

≤ lim [ω1 (xm , xm+1 ) + ω1 (xm+1 , T xm )] m→+∞

= lim [ω1 (xm , xm+1 ) + ω1 (A, B)] = ω1 (A, B), m→+∞

47

which implies limm→+∞ ω1 (xm , T xm ) = ω1 (A, B), that is

34 35

(4.12)

36 37 38 39

(4.13)

40 41 42 43 44 45 46 47 48 49

49

52

33

By using the triangular inequality, we have

46

51

30 31

Thus, without loss of generality, we can assume

45

50

29

32

lim

44

48

20

22

32 33

18

21

21 22

2 3

lim ω1 (xn , xn+1 ) := L

n→∞

1

1 ∗ 1 ω (xm , T xm ) = lim [ω1 (xm , T xm ) − ω1 (A, B)] = 0. m→+∞ 2 1 m→+∞ 2 On the other hand, from (4.12) it follows that there exists N ∈ N such that, for all m, n ≥ N , we have lim

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18

1 2 3

1 ∗ ω (xm , T xm ) ≤ ω1 (xn , xm ). 2 1 Now, from (4.1), (4.11), (4.13) and continuity of we have

4

6 7 8 9 10

16 17 18 19

ω1

8 9 10

12 13

(x ∗ , T x ∗ )

(x ∗ , T x

, T x∗)

≤ ω1 n ) + ω1 (T xn ≤ ω1 (x ∗ , xn+1 ) + ω1 (xn+1 , T xn ) + ω1 (T xn , T x ∗ ) ≤ ω1 (x ∗ , xn+1 ) + ω1 (A, B) + ω1 (T xn , T x ∗ ),

ω1

26

(x ∗ , T x ∗ ) − ω

1 (A, B) ≤ limn→+∞ ω1 (T xn

(4.14)

29 30

∗

n→+∞ ∗ ∗

36 37 38 39

limn→+∞ U1 (xn

, x∗) − ω

1 (A, B) = ω1

(x ∗ , T x ∗ ) − ω

1 (A, B).

(4.15)

ω1∗ (xn , T xn )

40 41

Next, we have

46

52

35

n→+∞

and hence

45

51

∗

= ω1 (x , T x )

44

50

34

∗

lim U1 (xn , x ) = max{ lim ω1 (x , xn ), lim ω1 (xn , T xn ), ω1 (x , T x )}

43

49

33

∗

n→+∞

41

48

32

that is limn→+∞ ω1 (xn , T xn ) = ω1 (A, B). Then, we get

40

47

26

28

n→+∞

37

42

25

31

36

39

19

27

lim ω1 (xn , T xn ) ≤ ω1 (A, B),

35

38

18

24

, T x ∗ ).

Taking limit as n → +∞ in the above inequality, we obtain

32

34

17

23

31

33

16

22

ω1 (xn , T xn ) ≤ ω1 (xn , xn+1 ) + ω1 (xn+1 , T xn ) = ω1 (xn , xn+1 ) + ω1 (A, B).

29

15

21

Also, we have

28

14

20

and taking limit as n → +∞, we get

25

30

7

that is limm,n→+∞ β(U1 (xn , xm )) = 1. Therefore, limm,n→+∞ U1 (xn , xm ) = 0 and consequently limm,n→+∞ ω1 (xn , xm ) = 0, which is a contradiction. Thus, {xn } is a Cauchy sequence. Since {xn } ⊂ A and A is an ω−closed subset of the ω−complete metric space (X, d), we can find x ∗ ∈ A such that xn → x ∗ , as n → +∞. We shall show that ω1 (x ∗ , T x ∗ ) = ω1 (A, B). Suppose to the contrary that ω1 (x ∗ , T x ∗ ) > ω1 (A, B). At first, we have

22

27

6

11

21

24

5

m,n→∞

20

23

4

1 ≤ lim β(U1 (xn , xm )),

13

15

3

and so,

12

14

2

(limm,n→+∞ ω1 (xn , xm )) ≤ (limm,n→+∞ [ω1 (xn , xn+1 ) + ω1 (T xn , T xm ) + ω1 (xm+1 , xm )]) = (limm,n→+∞ ω1 (T xn , T xm )) = limm,n→+∞

(ω1 (T xn , T xm ))  β(U1 (xn ,xm )) ≤ limm,n→+∞

(U1 (xn , xm ) − ω1 (A, B))]  ≤ limm,n→+∞ (ω1 (xn , xm ))]β(U1 (xn ,xm )) = [ (limm,n→+∞ ω1 (xn , xm ))]limm,n→+∞ β(U1 (xn ,xm )) ,

5

11

1

42

= ω1 (xn , T xn ) − ω1 (A, B) ≤ ω1 (xn , xn+1 ) + ω1 (xn+1 , T xn ) − ω1 (A, B) = ω1 (xn , xn+1 )

43

(4.16)

44 45 46

and

47

ω1∗ (xn+1 , T xn+1 )

= ω1 (xn+1 , T xn+1 ) − ω1 (A, B) ≤ ω1 (T xn , T xn+1 ) + ω1 (xn+1 , T xn ) − ω1 (A, B) = ω1 (T xn , T xn+1 ) = ω1 (xn+1 , xn+2 ) < ω1 (xn , xn+1 ),

48 49

(4.17)

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

and so (4.16) and (4.17) imply that

28

33

36 37 38 39 40 41 42 43 44 45 46 47 48

8 9 10 11 12 13 14 15 16 17 18 19

(4.19)

∗ ))

20 21 22 23

n→+∞

24

.

25 26

So we can conclude that,

27

1 ≤ lim β(U1 (xn , x ∗ )),

28

n→+∞

29

that is

30

∗

31

lim β(U1 (xn , x )) = 1,

n→+∞

32 33

which implies

34

lim U1 (xn , x ∗ ) = ω1 (x ∗ , T x ∗ ) = 0

35

n→+∞

and so ω1 (x ∗ , T x ∗ ) = 0 > ω1 (A, B), a contradiction. Therefore, ω1 (x ∗ , T x ∗ ) ≤ ω1 (A, B), that is ω1 (x ∗ , T x ∗ ) = ω1 (A, B). This means that x ∗ is a best proximity point of T and so the existence of a best proximity point is proved. We shall show the uniqueness of the best proximity point of T . Suppose that x ∗ and y ∗ are two distinct best proximity points of T , that is x ∗ = y ∗ . This implies that ∗

∗

∗

∗

ω1 (x , T x ) = ω1 (A, B) = ω1 (y , T y ).

(4.20)

Using the weak P1 -property, we have ω1 (x ∗ , y ∗ ) = ω1 (T x ∗ , T y ∗ ) U1 (x ∗ , y ∗ ) = max{ω1 (x ∗ , y ∗ ), ω1 (x ∗ , T x ∗ ), ω1 (y ∗ , T y ∗ )} = max{ω1 (x ∗ , y ∗ ), ω1 (A, B), ω1 (A, B)}

50

∗

∗

= ω1 (x , y ).

51

Also, we have

36 37 38 39 40 41 42 43 44

(4.21)

45 46

and so

49

52

7

(ω1 (x ∗ , T x ∗ ) − ω1 (A, B)) ≤ lim (ω1 (T xn , T x ∗ )) n→+∞

 ∗ β(U1 (xn ,x ∗ )) ≤ lim [ (U1 (xn , x ) − ω1 (A, B))]

34 35

6

1 ∗ 1 ∗ ω (xn , T xn ) ≤ ω1 (xn , z) or ω (xn+1 , T xn+1 ) ≤ ω1 (xn+1 , z) 2 1 2 1 holds. Therefore, by (4.1), (4.14) and (4.15) we deduce

31 32

5

which is a contradiction. Then, for any n ∈ N, either

= [ (ω1 (x ∗ , T x ∗ ) − ω1 (A, B))]limn→+∞ β(U1 (xn ,x

3 4

ω1 (xn , xn+1 ) ≤ ω1 (xn , z) + ω1 (xn+1 , z) 1 < [ω1∗ (xn , T xn ) + ω1∗ (xn+1 , T xn+1 )] 2 ≤ ω1 (xn , xn+1 ),

29 30

2

(4.18)

1 ∗ 1 ∗ ω (xn , T xn ) > ω1 (xn , z) and ω (xn+1 , T xn+1 ) > ω1 (xn+1 , z), 2 1 2 1 for some n ∈ N. Hence, by using (4.18), we can write

25

27

1

1 ∗ [ω (xn , T xn ) + ω1∗ (xn+1 , T xn+1 )] ≤ ω1 (xn , xn+1 ). 2 1 Now, we suppose that the following inequalities hold

24

26

19

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20

1 2 3 4 5

1 ∗ ∗ 1 ω1 (x , T x ∗ ) = [ω1 (x ∗ , T x ∗ ) − ω1 (A, B)] = 0 ≤ ω1 (x ∗ , y ∗ ). 2 2 Then, by (4.1) we have ∗

∗

∗

∗

(ω1 (x , y )) = (ω1 (T x , T y )) ≤ [ (U1 (x ∗ , y ∗ ) − ω1 (A, B))]β(U1 (x

7

= [ (ω1 (x ∗ , y ∗ ) − ω1 (A, B))]β(ω1 (x

9 10 11

∗

∗ ,y ∗ ))

∗ ,y ∗ ))

β(ω1 (x ∗ ,y ∗ ))

∗

≤ [ (ω1 (x , y ))] < (ω1 (x ∗ , y ∗ )) which is a contradiction. 2

16 17 18 19

22 23 24 25 26

In Theorem 4.2, if we take (t) = et we obtain the following best proximity result.

1 ∗ ω (x, T x) ≤ ω1 (x, y) ⇒ ω1 (T x, T y) ≤ β(ω1 (x, y))ω1 (x, y) 2 1 for all x, y ∈ A. Thus there exists a unique x ∗ in A such that ω1 (x ∗ , T x ∗ ) = ω1 (A, B).

31

34 35

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

10

13

15 16 17 18 19

21 22 23 24 25 26

28

30 31 32

Theorem 5.1. Let A and B be two M1 −closed subsets of an M1 −complete P D−fuzzy metric space with M1 regular (X, M, ∗). Let T : A → B be a non- self-mapping such that T (A0 (1)) ⊆ B0 (1) and A0 (1) = ∅. Assume that there exist two functions α : A × A → [0, +∞) and ∈ such that the following assertions hold:

33 34 35 36

36 37

9

29

By utilizing Lemma 2.1 and 2.4 and pervious best proximity point results in P D−modular metric spaces we can deduce the following best proximity results in the setting of P D−fuzzy metric space.

32 33

8

27

5. Some results in fuzzy metric spaces

29 30

7

20

Corollary 4.1. Let A and B are two ω−closed subsets of an ω−complete P D−modular metric space with ω regular Xω . Let T : A → B be a non-self mapping such that T (A0 (1)) ⊆ B0 (1) where A0 (1) = ∅. Also suppose that the pair (A, B) has the weak P1 -property. If there exist β ∈ F such that

27 28

6

14

Theorem 4.2. Let A and B are two ω−closed subsets of an ω−complete P D−modular metric space with ω regular Xω . Let T : A → B be a GS −contraction such that T (A0 (1)) ⊆ B0 (1) where A0 (1) = ∅. Also suppose that the pair (A, B) has the weak P1 -property. Thus there exists a unique x ∗ in A such that ω1 (x ∗ , T x ∗ ) = ω1 (A, B).

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Following arguments similar to those in the above Theorem we can prove the following Theorem.

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(i) T is an α-proximal admissible mapping and M1 −continuous, (ii) if there exists ∈ such that for all x, y, u, v ∈ A with ⎧ 1 −1>0 ⎪ ⎪ ⎨ M(u,v,1) 1 M(u,T x,1) − 1 = ω1 (A, B) ⎪ ⎪ ⎩ 1 − 1 = ω (A, B) M(v,T y,1)

1

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then we have  1

r

L [ M(u,v,1) − 1 ]α(x, y) ≤ M (x, y, u, v) N (x, y, u, v) where 0 ≤ r < 1, L ≥ 0 and

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  1 1 1 M (x, y, u, v) = max − 1), − 1 , −1 , M(x, y, 1) M(x, u, 1) M(y, v, 1)  1 1  M(x,v,1) − 1 + M(y,u,1) − 1 2

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and

 N (x, y, u, v) = min

  1 1 1 − 1), − 1 , −1 , M(x, y, 1) M(x, u, 1) M(y, v, 1)    1 1 − 1 , −1 . M(x, v, 1) M(y, u, 1)

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1 − 1 = M(A, B, 1) and α(x0 , x1 ) ≥ 1, M(x1 , T x0 , 1) (iv) if {xn } is a sequence in X for all n ∈ N ∪ {0} such that α(xn , xn+1 ) ≥ 1 with M(xn , x, 1) → 1 as n → +∞, then α(xn , x) ≥ 1 for all n ∈ N ∪ {0}, (v) for any sequence {yn } in B0 (1) and x ∈ A satisfying M(x, yn , 1) → M(A, B, 1) as n → +∞, then x ∈ A0 (1).

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Theorem 5.2. Let A and B be two M1 −closed subsets of an M1 −complete P D−fuzzy metric space with M1 regular (X, M, ∗). Let T : A → B be a non- self-mapping such that T (A0 (1)) ⊆ B0 (1) and A0 (1) = ∅. Assume that there exist two functions α : A × A → [0, +∞) and ∈ such that the following assertions hold:

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(i) T is an α-proximal admissible mapping, (ii) if there exists continuous ∈ such that for all x, y, u, v ∈ A with ⎧ 1 ⎪ ⎨ M(u,v,1) − 1 > 0 1 M(u,T x,1) − 1 = ω1 (A, B) ⎪ 1 ⎩ M(v,T y,1) − 1 = ω1 (A, B) then we have  1

r

L [ M(u,v,1) − 1 ]α(x, y) ≤ M (x, y, u, v) N (x, y, u, v)

Thus there exists

in A such that

− 1 = M(A, B, 1).

Theorem 5.3. Let A and B be two M1 −closed subsets of an M1 complete P D−fuzzy metric space with M1 regular (X, M, ∗). Let T (A0 (1)) ⊆ B0 (1) where A0 (1) = ∅. Also suppose that the pair (A, B) has the weak PM1 -property. If there exist β ∈ F and ∈

such that 1 ∗ 1 1 M (x, T x, 1) ≤ − 1 ⇒ ( − 1) ≤ [ (U1 (x, y) − M(A, B, 1))]β(U1 (x, y)) 2 M(x, y, 1) M(T x, T y, 1) for all x, y ∈ A, where

M ∗ (x, y, 1) =

1 M(x,y,1)

− 1 − M(A, B, 1) and



 1 1 1 U1 (x, y) = max − 1, − 1, −1 . M(x, y, 1) M(x, T x, 1) M(y, T y, 1) Thus, there exists a unique x ∗ in A such that

1 M(x ∗ ,T x ∗ ,1)

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(iv) if {xn } is a sequence in X for all n ∈ N ∪ {0} such that α(xn , xn+1 ) ≥ 1 with M(xn , x, 1) → 1 as n → +∞, then α(xn , x) ≥ 1 for all n ∈ N ∪ {0}, (v) for any sequence {yn } in B0 (1) and x ∈ A satisfying M(x, yn , 1) → M(A, B, 1) as n → +∞, then x ∈ A0 (1). 1 M(x ∗ ,T x ∗ ,1)

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1 − 1 = M(A, B, 1) and α(x0 , x1 ) ≥ 1, M(x1 , T x0 , 1)

x∗

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where 0 ≤ r < 1, L ≥ 0, (iii) there exist elements x0 and x1 in A0 (1) such that,

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15

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Then T has a best proximity point.

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(iii) there exist elements x0 and x1 in A0 (1) such that,

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− 1 = M(A, B, 1).

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Acknowledgement

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The second author is very grateful to the University of the Basque Country for Grant PGC17/33. References

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