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Article

Best Proximity Point Results in Non-Archimedean Modular Metric Space

1
Department of Mathematics, Farhangian University, Iran
2
IIDP, University of Basque Country, Bilbao-48940, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2017, 5(2), 23; https://doi.org/10.3390/math5020023
Submission received: 14 December 2016 / Revised: 15 March 2017 / Accepted: 20 March 2017 / Published: 5 April 2017

Abstract

:
In this paper, we introduce the new notion of Suzuki-type ( α , β , θ , γ ) -contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak P λ -property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature.
MSC:
2000 46N40; 47H10; 54H25; 46T99

1. Introduction and Preliminaries

Modular metric spaces are a 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 [1,2]. Here, we look at modular metric spaces as the nonlinear version of the classical one introduced by Nakano [3] on vector spaces and modular function spaces introduced by Musielak [4] and Orlicz [5].
Recently, many authors studied the behavior of the electrorheological fluids, sometimes referred to as “smart fluids” (e.g., lithium polymethacrylate). A perfect model for these fluids is obtained by using Lebesgue and Sobolev spaces, L p and W 1 , p , in the case that p is a function [6].
Let X be a nonempty set and ω : ( 0 , + ) × X × X [ 0 , + ] be a function; for simplicity, we will write:
ω λ ( x , y ) = ω ( λ , x , y ) ,
for all λ > 0 and x , y X .
Definition 1.
[1,2] A function ω : ( 0 , + ) × X × X [ 0 , + ] is called a modular metric on X if the following axioms hold:
(i) 
x = y if and only if ω λ ( x , y ) = 0 for all λ > 0 ;
(ii) 
ω λ ( x , y ) = ω λ ( y , x ) for all λ > 0 and x , y X ;
(iii) 
ω λ + μ ( x , y ) ω λ ( x , z ) + ω μ ( z , y ) for all λ , μ > 0 and x , y , z X .
If in the above definition, we utilize the condition:
(i’)
ω λ ( x , x ) = 0 for all λ > 0 and x X ;
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:
x = y if and only if ω λ ( x , y ) = 0 for some λ > 0 .
Again, ω is called convex if for λ , μ > 0 and x , y , z X , the inequality holds:
ω λ + μ ( x , y ) λ λ + μ ω λ ( x , z ) + μ λ + μ ω μ ( z , y ) .
Remark 1.
Note that if ω is a pseudomodular metric on a set X, then the function λ ω λ ( x , y ) is decreasing on ( 0 , + ) for all x , y X . That is, if 0 < μ < λ , then:
ω λ ( x , y ) ω λ μ ( x , x ) + ω μ ( x , y ) = ω μ ( x , y ) .
Definition 2.
References [1,2] suppose that ω be a pseudomodular on X and x 0 X and fixed. Therefore, the two sets:
X ω = X ω ( x 0 ) = { x X : ω λ ( x , x 0 ) 0 a s λ + }
and:
X ω * = X ω * ( x 0 ) = { x X : λ = λ ( x ) > 0 s u c h   t h a t ω λ ( x , x 0 ) < + } .
X ω and X ω * are called modular spaces (around x 0 ).
It is evident that X ω X ω * , but this inclusion may be proper in general. Assume that ω is a modular on X; from [1,2], we derive that the modular space X ω can be equipped with a (nontrivial) metric, induced by ω and given by:
d ω ( x , y ) = inf { λ > 0 : ω λ ( x , y ) λ } for all x , y X ω .
Note that if ω is a convex modular on X, then according to [1,2], the two modular spaces coincide, i.e., X ω * = X ω , and this common set can be endowed with the metric d ω * given by:
d ω * ( x , y ) = inf { λ > 0 : ω λ ( x , y ) 1 } for all x , y X ω .
Such distances are called Luxemburg distances.
Example 2.1 presented by Abdou and Khamsi [7] is an important motivation for developing the modular metric spaces theory. Other examples may be found in [1,2].
Definition 3.
Reference [8] assume X ω to be a modular metric space, M a subset of X ω and ( x n ) n N be a sequence in X ω . Therefore:
(1) 
( x n ) n N is called ω-convergent to x X ω if and only if ω λ ( x n , x ) 0 , as n + for all λ > 0 . x will be called the ω-limit of ( x n ) .
(2) 
( x n ) n N is called ω-Cauchy if ω λ ( x m , x n ) 0 , as m , n + for all λ > 0 .
(3) 
M is called ω-closed if the ω-limit of a ω-convergent sequence of M always belong to M.
(4) 
M is called ω-complete if any ω-Cauchy sequence in M is ω-convergent to a point of M .
(5) 
M is called ω-bounded if for all λ > 0 , we have δ ω ( M ) = sup { ω λ ( x , y ) ; x , y M } < + .
Recently Paknazar et al. [9] introduced the following concept.
Definition 4.
If in Definition 1, we replace (iii) by:
( i v ) ω max { λ , μ } ( x , y ) ω λ ( x , z ) + ω μ ( z , y )
for all λ , μ > 0 and x , y , z X
Then, X ω is called the non-Archimedean modular metric space. Since (iv) implies (iii), every non-Archimedean modular metric space is a modular metric space.
One of the most important generalizations of Banach contraction mappings was given by Geraghty [10] in the following form.
Theorem 1
(Geraghty [10]). Suppose that ( X , d ) is a complete metric space and T : X X is self-mapping. Suppose that there exists β : [ 0 , + ) [ 0 , 1 ) satisfying the condition:
β ( t n ) 1 implies t n 0 ,   a s   n + .
If T satisfies the following inequality:
d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) , for all x , y X ,
hence T has a unique fixed point.
Moreover, Kirk [11] 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 ) k d ( 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 T as a self-mapping.
The reader can refer to [12,13,14,15,16]. Note that best proximity point theorems furnish an approximate solution to the equation T x = x , when there are not any fixed points for T.
Here, we collect some notions and concepts that will be utilized throughout the rest of this work. We denote by A 0 and B 0 the following sets:
A 0 = { x A : d ( x , y ) = d ( A , B ) for some y B } , B 0 = { y B : d ( x , y ) = d ( A , B ) for some x A } .
In 2003, Kirk et al. [12] established sufficient conditions for determining when the sets A 0 and B 0 are nonempty.
Furthermore, in [14], the authors 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. [16] introduced the following notion and showed that it is weaker than the P-property.
Definition 5.
Let ( A , B ) be a pair of nonempty subsets of a metric space ( X , d ) with A 0 . Then, the pair ( A , B ) is said to have the weak P-property if and only if for any x 1 , x 2 A 0 and y 1 , y 2 B 0 :
d ( x 1 , y 1 ) = d ( A , B ) and d ( x 2 , y 2 ) = d ( A , B ) d ( x 1 , x 2 ) d ( y 1 , y 2 ) .
Finally, we recall the following result of Caballero et al. [17].
Theorem 2.
Assume that ( A , B ) is a pair of nonempty closed subsets of a complete metric space ( X , d ) , such that A 0 is nonempty. Let T : A B be a Geraghty-contraction satisfying T ( A 0 ) B 0 . Assume that the pair ( A , B ) has the P-property. Then, there exists a unique x * A such that d ( x * , T x * ) = d ( A , B ) .
Recently, Kumam et al. [18] introduced the useful notion of triangular α -proximal admissible mapping as follows. See also [19]:
Definition 6
(Reference [18]). Let A and B be two nonempty subsets ofa metric space ( X , d ) and α : A × A [ 0 , + ) be a function. We say that a non-self-mapping T : A B is triangular α-proximal admissible if, for all x , y , z , x 1 , x 2 , u 1 , u 2 A :
( T 1 ) α ( x 1 , x 2 ) 1 d ( u 1 , T x 1 ) = d ( A , B ) d ( u 2 , T x 2 ) = d ( A , B ) α ( u 1 , u 2 ) 1 , ( T 2 ) α ( x , z ) 1 α ( z , y ) 1 α ( x , y ) 1 .
Let Θ denote the set of all functions θ : R + 4 R + satisfying:
  • ( Θ 1 ) θ is continuous and increasing in all of its variables;
  • ( Θ 2 ) θ ( t 1 , t 2 , t 3 , t 4 ) = 0 iff t 1 . t 2 . t 3 . t 4 = 0 .
For more details on Θ , see [20].
Let F denote the set of all functions β : [ 0 , + ) [ 0 , 1 ) satisfying the condition:
β ( t n ) 1 implies t n 0 , as n + .

2. Best Proximity Point Results

At first, we introduce the following concept, which will be suitable for our main Theorem.
Definition 7.
Suppose that ( A , B ) is a pair of nonempty subsets of a modular metric space X ω with A 0 λ for all λ > 0 . We say the pair ( A , B ) has the weak P λ -property if and only if for any x 1 , x 2 A 0 , y 1 , y 2 B 0 and λ > 0 :
ω λ ( x 1 , y 1 ) = ω λ ( A , B ) and ω λ ( x 2 , y 2 ) = d ( A , B ) ω λ ( x 1 , x 2 ) ω λ ( y 1 , y 2 ) ,
where:
ω λ ( A , B ) = : inf { ω λ ( x , y ) | x A and y B } ,
A 0 λ = : { x A : ω λ ( x , y ) = ω λ ( A , B ) for some y B } .
Now, let us introduce the concept of Suzuki-type ( α , β , θ , γ ) -contractive mapping.
Definition 8.
Let A and B be two nonempty subsets of a modular metric space X ω where A 0 λ for all λ > 0 and α : X ω × X ω [ 0 , ) is a function. A mapping T : A B is said to be a Suzuki-type ( α , β , θ , γ ) contractive mapping if there exists β F and θ Θ , such that for all x , y A and λ > 0 with 1 2 ω λ * ( x , T x ) ω λ ( x , y ) and α ( x , y ) 1 , one has:
ω λ ( T x , T y ) β M ( x , y ) M ( x , y ) + γ N ( x , y , θ ) N ( x , y , θ )
where γ : [ 0 , ) [ 0 , ) is a bounded function, ω λ * ( x , y ) = ω λ ( x , y ) ω λ ( A , B ) ,
M ( x , y ) = max { ω λ ( x , y ) , ω λ ( x , T x ) + ω λ ( y , T y ) 2 ω λ ( A , B ) , ω λ ( x , T y ) + ω λ ( y , T x ) 2 ω λ ( A , B ) }
and:
N ( x , y , θ ) = θ ( ω λ ( x , T x ) ω λ ( A , B ) , ω λ ( y , T y ) ω λ ( A , B ) , ω λ ( x , T y ) ω λ ( A , B ) , ω λ ( y , T x ) ω λ ( A , B ) ) .
Now, we are ready to prove our main result.
Theorem 3.
Let A and B be two nonempty subsets of a non-Archimedean modular metric space X ω with ω regular, such that A is ω complete and A 0 λ is nonempty for all λ > 0 . Assume that T is a Suzuki-type ( α , β , θ , γ ) -contractive mapping satisfying the following assertions:
(i) 
T ( A 0 λ ) B 0 λ for all λ > 0 , and the pair ( A , B ) satisfies the weak P λ -property,
(ii) 
T is a triangular α-proximal admissible mapping,
(iii) 
there exist elements x 0 and x 1 in A 0 λ for all λ > 0 , such that:
ω λ ( x 1 , T x 0 ) = ω λ ( A , B ) and α ( x 0 , x 1 ) 1
(iv) 
if { x n } is a sequence in A, such that α ( x n , x n + 1 ) 1 for all n N { 0 } with x n x A as n , then α ( x n , x ) 1 for all n N .
Then, there exists an x * in A, such that ω λ ( x * , T x * ) = ω λ ( A , B ) for all λ > 0 . Further, the best proximity point is unique if, for every x , y A , such that ω λ ( x , T x ) = ω λ ( A , B ) = ω λ ( y , T y ) , we have α ( x , y ) 1 .
Proof. 
By (iii), there exist elements x 0 and x 1 in A 0 λ for all λ > 0 , such that:
ω λ ( x 1 , T x 0 ) = ω λ ( A , B ) and α ( x 0 , x 1 ) 1 .
On the other hand, T ( A 0 λ ) B 0 λ for all λ > 0 . Therefore, there exists x 2 A 0 , such that:
ω λ ( x 2 , T x 1 ) = ω λ ( A , B ) .
Now, since T is triangular α -proximal admissible, we have α ( x 1 , x 2 ) 1 . That is:
ω λ ( x 2 , T x 1 ) = ω λ ( A , B ) and α ( x 1 , x 2 ) 1 .
Again, since T ( A 0 λ ) B 0 λ for all λ > 0 , there exists x 3 A 0 λ , such that:
ω λ ( x 3 , T x 2 ) = ω λ ( A , B ) .
Thus, we have:
ω λ ( x 2 , T x 1 ) = ω λ ( A , B ) and ω λ ( x 3 , T x 2 ) = ω λ ( A , B ) and α ( x 1 , x 2 ) 1 .
Again, since T is triangular α -proximal admissible, α ( x 2 , x 3 ) 1 . Hence:
ω λ ( x 3 , T x 2 ) = ω λ ( A , B ) and α ( x 2 , x 3 ) 1 .
Continuing this process, we get:
ω λ ( x n + 1 , T x n ) = ω λ ( A , B ) and α ( x n , x n + 1 ) 1 for all n N { 0 } .
Since ( A , B ) has the weak P λ -property, we derive that:
ω λ ( x n , x n + 1 ) ω λ ( T x n 1 , T x n ) for any n N .
Now, by (6), we get:
ω λ ( x n 1 , T x n 1 ) ω λ ( x n 1 , x n ) + ω λ ( x n , T x n 1 ) = ω λ ( x n 1 , x n ) + ω λ ( A , B ) .
Clearly, if there exists n 0 N , such that ω λ ( x n 0 , x n 0 + 1 ) = 0 , then we have nothing to prove. In fact:
0 = ω λ ( x n 0 , x n 0 + 1 ) = ω λ ( T x n 0 1 , T x n 0 ) .
Since ω is regular, we get, T x n 0 1 = T x n 0 . Thus, we conclude that:
ω λ ( A , B ) = ω λ ( x n 0 , T x n 0 1 ) = ω λ ( x n 0 , T x n 0 ) .
For the rest of the proof, we suppose that ω λ ( x n , x n + 1 ) > 0 for any n N . Now, from (8), we deduce that:
1 2 ω λ * ( x n 1 , T x n 1 ) ω λ * ( x n 1 , T x n 1 ) ω λ ( x n , x n 1 ) .
Applying (6) and (7), we obtain:
M ( x n 1 , x n ) = max { ω λ ( x n 1 , x n ) , ω λ ( x n 1 , T x n 1 ) + ω λ ( x n , T x n ) 2 ω λ ( A , B ) , ω λ ( x n 1 , T x n ) + ω λ ( x n , T x n 1 ) 2 ω λ ( A , B ) } max ω λ ( x n 1 , x n ) , ω λ ( x n 1 , x n ) + ω λ ( x n , T x n 1 ) + ω λ ( x n , x n + 1 ) + ω λ ( x n + 1 , T x n ) 2 ω λ ( A , B ) , ω λ ( x n 1 , x n + 1 ) + ω λ ( x n + 1 , T x n ) + ω λ ( x n , T x n 1 ) 2 ω λ ( A , B ) } = max ω λ ( x n 1 , x n ) , ω λ ( x n 1 , x n ) + ω λ ( A , B ) + ω λ ( x n , x n + 1 ) + ω λ ( A , B ) 2 ω λ ( A , B ) , ω λ ( x n 1 , x n + 1 ) + ω λ ( A , B ) + ω λ ( A , B ) 2 ω λ ( A , B ) } = max ω λ ( x n 1 , x n ) , ω λ ( x n 1 , x n ) + ω λ ( x n , x n + 1 ) 2 , ω λ ( x n 1 , x n + 1 ) 2 max { ω λ ( x n 1 , x n ) , ω λ ( x n 1 , x n ) + ω λ ( x n , x n + 1 ) 2 } max { ω λ ( x n 1 , x n ) , ω λ ( x n , x n + 1 ) } .
Thus:
M ( x n 1 , x n ) max { ω λ ( x n 1 , x n ) , ω λ ( x n , x n + 1 ) } .
Furthermore:
N ( x n 1 , x n , θ ) = θ ( ω λ ( x n 1 , T x n 1 ) ω λ ( A , B ) , ω λ ( x n , T x n ) ω λ ( A , B ) ,   ω λ ( x n 1 , T x n ) ω λ ( A , B ) , ω λ ( x n , T x n 1 ) ω λ ( A , B ) ) = θ ( ω λ ( x n 1 , T x n 1 ) ω λ ( A , B ) , ω λ ( x n , T x n ) ω λ ( A , B ) ,   ω λ ( x n 1 , T x n ) ω λ ( A , B ) , 0 ) = 0 .
Since T is a Suzuki-type ( α , β , θ , γ ) -contractive mapping, we have:
ω λ ( x n , x n + 1 ) ω λ ( T x n 1 , T x n ) β ( M ( x n 1 , x n ) ) M ( x n 1 , x n ) + γ ( N ( x n 1 , x n , θ ) ) N ( x n 1 , x n , θ ) < M ( x n 1 , x n ) + γ ( N ( x n 1 , x n , θ ) ) N ( x n 1 , x n , θ ) .
From (10) to (12), we deduce:
ω λ ( x n , x n + 1 ) < max { ω λ ( x n 1 , x n ) , ω λ ( x n , x n + 1 ) } .
Now if, max { ω λ ( x n 1 , x n ) , ω λ ( x n , x n + 1 ) } = ω λ ( x n , x n + 1 ) then,
ω λ ( x n , x n + 1 ) < ω λ ( x n , x n + 1 ) ,
which is a contradiction. Hence:
ω λ ( x n 1 , x n ) M ( x n 1 , x n ) max { ω λ ( x n 1 , x n ) , ω λ ( x n , x n + 1 ) } = ω λ ( x n 1 , x n ) ,
and so:
M ( x n 1 , x n ) = ω λ ( x n 1 , x n ) ,
for all n N . Now, by (12), we get:
ω λ ( x n , x n + 1 ) = ω λ ( T x n 1 , T x n ) β ( ω λ ( x n 1 , x n ) ) ω λ ( x n 1 , x n ) < ω λ ( x n 1 , x n ) ,
for all n N . Consequently, { ω λ ( x n , x n + 1 ) } is a non-increasing sequence, which is bounded from below, and so, lim n ω λ ( x n , x n + 1 ) : = L exists. Let L > 0 . Then, from (14), we have:
ω λ ( x n , x n + 1 ) ω λ ( x n 1 , x n ) β ( ω λ ( x n 1 , x n ) ) 1 ,
for each n 1 , which implies:
lim n β ( ω λ ( x n , x n + 1 ) ) = 1 .
On the other hand, since β F , we conclude:
L = lim n ω λ ( x n , x n + 1 ) = 0 .
Since, ω λ ( x n , T x n 1 ) = ω λ ( A , B ) holds for all n N and ( A , B ) satisfies the weak P λ -property, so for all m , n N with n < m , we obtain, ω λ ( x m , x n ) ω λ ( T x m 1 , T x n 1 ) . Note that:
M ( x m , x n ) = max { ω λ ( x m , x n ) , ω λ ( x m , T x m ) + ω λ ( x n , T x n ) 2 ω λ ( A , B ) , ω λ ( x m , T x n ) + ω λ ( x n , T x m ) 2 ω λ ( A , B ) } max { ω λ ( x m , x n ) , ω λ ( x m , x m + 1 ) + ω λ ( x m + 1 , T x m ) + ω λ ( x n , x n + 1 ) + ω λ ( x n + 1 , T x n ) 2 ω λ ( A , B ) , ω λ ( x m , x n + 1 ) + ω λ ( x n + 1 , T x n ) + ω λ ( x n , x m + 1 ) + ω λ ( x m + 1 , T x m ) 2 ω λ ( A , B ) } = max ω λ ( x m , x n ) , ω λ ( x m , x m + 1 ) + ω λ ( x n , x n + 1 ) 2 , ω λ ( x m , x n + 1 ) max { ω λ ( x m , x n ) , ω λ ( x m , x m + 1 ) + ω λ ( x n , x n + 1 ) 2 , ω λ ( x m , x n ) + ω λ ( x n , x n + 1 ) } .
As lim n ω λ ( x n , x n + 1 ) = 0 , we have:
lim m , n ω λ ( x m , x n ) lim m , n M ( x m , x n ) lim m , n ω λ ( x m , x n ) ,
that is:
lim m , n M ( x m , x n ) = lim m , n ω λ ( x m , x n ) .
Furthermore:
0 N ( x m , x n , θ ) = θ ( ω λ ( x m , T x m ) ω λ ( A , B ) , ω λ ( x n , T x n ) ω λ ( A , B ) ,      ω λ ( x m , T x n ) ω λ ( A , B ) , ω λ ( x n , T x m ) ω λ ( A , B ) ) θ ( ω λ ( x m , x m + 1 ) + ω λ ( A , B ) ω λ ( A , B ) , ω λ ( x n , T x n ) ω λ ( A , B ) ,      ω λ ( x m , T x n ) ω λ ( A , B ) , ω λ ( x n , T x m ) ω λ ( A , B ) ) θ ( ω λ ( x m , x m + 1 ) , ω λ ( x n , T x n ) ω λ ( A , B ) , ω λ ( x m , T x n ) ω λ ( A , B ) ,      ω λ ( x n , T x m ) ω λ ( A , B ) ) .
Again, by lim n ω λ ( x n , x n + 1 ) = 0 , we have:
0 lim m , n N ( x m , x n , θ ) lim m , n θ ( ω λ ( x m , x m + 1 ) , ω λ ( x n , T x n ) ω λ ( A , B ) , ω λ ( x m , T x n ) ω λ ( A , B ) , ω λ ( x n , T x m ) ω λ ( A , B ) ) lim m , n θ ( 0 , ω λ ( x n , T x n ) ω λ ( A , B ) , ω λ ( x m , T x n ) ω λ ( A , B ) , ω λ ( x n , T x m ) ω λ ( A , B ) ) = 0 .
That is:
lim m , n N ( x m , x n , θ ) = 0 .
Now, we show that { x n } is a Cauchy sequence. On the contrary, assume that:
ε = lim sup  m , n ω λ ( x n , x m ) > 0 .
Now, since lim n + ω λ ( x n , x n + 1 ) = 0 , then:
ω λ ( A , B ) lim m + ω λ ( x m , T x m ) lim m + [ ω λ ( x m , x m + 1 ) + ω λ ( x m + 1 , T x m ) ] = lim m + [ ω λ ( x m , x m + 1 ) + ω λ ( A , B ) ] = ω λ ( A , B ) ,
which implies that lim m + ω λ ( x m , T x m ) = ω λ ( A , B ) , that is:
lim m + 1 2 ω λ * ( x m , T x m ) = lim m + 1 2 [ ω λ ( x m , T x m ) ω λ ( A , B ) ] = 0 .
On the other hand, from (18), it is follows that there exists N N , such that, for all m , n N , we have:
1 2 ω λ * ( x m , T x m ) ω λ ( x n , x m ) .
Furthermore, we can show that:
α ( x m , x n ) 1 , where n > m .
Indeed, since T is a triangular α -proximal admissible mapping and:
α ( x m , x m + 1 ) 1 α ( x m + 1 , x m + 2 ) 1 ,
from Condition (T2) of Definition 6, we have:
α ( x m , x m + 2 ) 1 .
Again, since T is a triangular α -proximal admissible mapping and:
α ( x m , x m + 2 ) 1 α ( x m + 2 , x m + 3 ) 1 ,
from Condition (T2) of Definition 6, we have:
α ( x m , x m + 3 ) 1 .
Continuing this process, we get (19).
Now, using the triangle inequality, we have:
ω λ ( x n , x m ) ω λ ( x n , x n + 1 ) + ω λ ( x n + 1 , x m + 1 ) + ω λ ( x m + 1 , x m ) .
From (5) and (20) we have:
ω λ ( x n , x m ) ω λ ( x n , x n + 1 ) + ω λ ( T x n , T x m ) + ω λ ( x m + 1 , x m ) ω λ ( x n , x n + 1 ) + β ( M ( x n , x m ) ) M ( x n , x m ) + γ ( N ( x n , x m , θ ) ) N ( x n , x m , θ ) + ω λ ( x m + 1 , x m ) .
Now, (16), (17), (21) and: lim n ω λ ( x n , x n + 1 ) = 0 , imply:
lim m , n ω λ ( x n , x m ) lim m , n β ( M ( x n , x m ) ) lim m , n M ( x m , x n ) + lim m , n γ ( N ( x n , x m , θ ) ) lim m , n N ( x m , x n , θ ) = lim m , n β ( M ( x n , x m ) ) lim m , n ω λ ( x m , x n ) .
By (18), we get:
1 lim m , n β ( M ( x n , x m ) ) .
Therefore, lim m , n β ( M ( x n , x m ) ) = 1 , so lim m , n M ( x n , x m ) = 0 . This implies:
lim m , n ω λ ( x n , x m ) = 0 ,
which is a contradiction. Therefore, { x n } is a Cauchy sequence. Since ( x n ) A and ( A , d ) is a complete metric space, we can find x * A , such that x n x * as n . From (iv), we know that, α ( x n , x ) 1 for all n N . Next, using (14), we have:
ω λ * ( x n , T x n ) = ω λ ( x n , T x n ) ω λ ( A , B ) ω λ ( x n , x n + 1 ) + ω λ ( x n + 1 , T x n ) ω λ ( A , B ) = ω λ ( x n , x n + 1 ) ,
and:
ω λ * ( x n + 1 , T x n + 1 ) = ω λ ( x n + 1 , T x n + 1 ) ω λ ( A , B ) ω λ ( T x n , T x n + 1 ) + ω λ ( x n + 1 , T x n ) ω λ ( A , B ) = ω λ ( T x n , T x n + 1 ) = ω λ ( x n + 1 , x n + 2 ) ω λ ( x n , x n + 1 ) .
Therefore, (22) and (23) imply that:
1 2 [ ω λ * ( x n , T x n ) + ω λ * ( x n + 1 , T x n + 1 ) ] ω λ ( x n , x n + 1 ) .
Now, suppose that:
1 2 ω λ * ( x n , T x n ) > ω λ ( x n , x * ) and 1 2 ω λ * ( x n + 1 , T x n + 1 ) > ω λ ( x n + 1 , x * ) ,
for some n N . Hence, using (24), we can write:
ω λ ( x n , x n + 1 ) ω λ ( x n , x * ) + ω λ ( x n + 1 , x * ) < 1 2 [ ω λ * ( x n , T x n ) + ω λ * ( x n + 1 , T x n + 1 ) ] ω λ ( x n , x n + 1 ) ,
which is a contradiction. Then, for any n N , either:
1 2 ω λ * ( x n , T x n ) ω λ ( x n , x * ) or 1 2 ω λ * ( x n + 1 , T x n + 1 ) ω λ ( x n + 1 , x * )
holds.
We shall show that ω λ ( x * , T x * ) = ω λ ( A , B ) . Suppose, to the contrary, that:
ω λ ( x * , T x * ) ω λ ( A , B ) .
From (5) with x = x n and y = x * , we get:
ω λ ( T x n , T x * ) β M ( x n , x * ) M ( x n , x * ) + γ N ( x n , x * , θ ) N ( x n , x * , θ ) .
On the other hand:
M ( x n , x * ) = max { ω λ ( x n , x * ) , ω λ ( x n , T x n ) + ω λ ( x * , T x * ) 2 ω λ ( A , B ) , ω λ ( x n , T x * ) + ω λ ( x * , T x n ) 2 ω λ ( A , B ) } max { ω λ ( x n , x * ) , ω λ ( x n , x n + 1 ) + ω λ ( x n + 1 , T x n ) + ω λ ( x * , T x * ) 2 ω λ ( A , B ) , ω λ ( x n , x * ) + ω λ ( x * , T x * ) + ω λ ( x * , x n + 1 ) + ω λ ( x n + 1 , T x n ) 2 ω λ ( A , B ) } = max { ω λ ( x n , x * ) , ω λ ( x n , x n + 1 ) + ω λ ( A , B ) + ω λ ( x * , T x * ) 2 ω λ ( A , B ) , ω λ ( x n , x * ) + ω λ ( x * , T x * ) + ω λ ( x * , x n + 1 ) + ω λ ( A , B ) 2 ω λ ( A , B ) } ,
and so:
lim k M ( x n , x * ) ω λ ( x * , T x * ) ω λ ( A , B ) 2 .
Furthermore, we have:
ω λ ( x * , T x * ) ω λ ( x * , T x n ) + ω λ ( T x n , T x * ) ω λ ( x * , x n + 1 ) + ω λ ( x n + 1 , T x n ) + ω λ ( T x n , T x * ) ω λ ( x * , x n + 1 ) + ω λ ( A , B ) + ω λ ( T x n , T x * ) .
Taking limit as n in the above inequality, we have:
ω λ ( x * , T x * ) ω λ ( A , B ) lim n ω λ ( T x n , T x * ) .
Further, we get:
ω λ ( x n , T x n ) ω λ ( x n , x n + 1 ) + ω λ ( x n + 1 , T x n ) = ω λ ( x n , x n + 1 ) + ω λ ( A , B ) .
Taking the limit as n in the above inequality, we get:
lim n ω λ ( x n , T x n ) ω λ ( A , B ) ,
and so, lim n ω λ ( x n , T x n ) = ω λ ( A , B ) . Now, we have:
lim n N ( x n , x * , θ ) = θ ( lim n ω λ ( x n , T x n ) ω λ ( A , B ) , ω λ ( x * , T x * ) ω λ ( A , B ) , lim n ω λ ( x n , T x * ) ω λ ( A , B ) , lim n ω λ ( x * , T x n ) ω λ ( A , B ) ) = θ ( 0 , ω λ ( x * , T x * ) ω λ ( A , B ) , lim n ω λ ( x n , T x * ) ω λ ( A , B ) , lim n ω λ ( x * , T x n ) ω λ ( A , B ) ) = 0 ,
that is:
lim n N ( x n , x * , θ ) = 0 .
From (25) to (28), we deduce that:
ω λ ( x * , T x * ) ω λ ( A , B ) lim n ω λ ( T x n , T x * ) lim n β ( M ( x n , x * ) ) lim n M ( x n , x * ) + lim n γ ( N ( x n , x * , θ ) ) lim n N ( x n , x * , θ ) = lim n β ( M ( x n , x * ) ) ( ω λ ( x * , T x * ) ω λ ( A , B ) 2 ) < ω λ ( x * , T x * ) ω λ ( A , B ) ,
which is a contradiction. Therefore, ω λ ( x * , T x * ) = ω λ ( A , B ) , and x * is a best proximity point of T. We now show the uniqueness of the best proximity point of T . Suppose that x * and y * are two distinct best proximity points of T. This implies:
ω λ ( x * , T x * ) = ω λ ( A , B ) = ω λ ( y * , T y * ) .
Using the weak P 1 -property, we have:
ω λ ( x * , y * ) ω λ ( T x * , T y * ) .
Since:
M ( x * , y * ) = max { ω λ ( x * , y * ) , ω λ ( x * , T x * ) + ω λ ( y * , T y * ) 2 ω λ ( A , B ) , ω λ ( x * , T y * ) + ω λ ( y * , T x * ) 2 ω λ ( A , B ) } = max { ω λ ( x * , y * ) , 0 , ω λ ( x * , T y * ) + ω λ ( y * , T x * ) 2 ω λ ( A , B ) } max { ω λ ( x * , y * ) , 0 , ω λ ( x * , T x * ) + ω λ ( T x * , T y * ) + ω λ ( y * , T y * ) + ω λ ( T y * , T x * ) 2 ω λ ( A , B ) } max { ω λ ( x * , y * ) , 0 , ω λ ( A , B ) + ω λ ( x * , y * ) + ω λ ( A , B ) + ω λ ( y * , x * ) 2 ω λ ( A , B ) } = ω λ ( x * , y * ) .
Furthermore:
N ( x * , y * , θ ) = θ ( ω λ ( x * , T x * ) ω λ ( A , B ) , ω λ ( y * , T y * ) ω λ ( A , B ) , ω λ ( x * , T y * ) ω λ ( A , B ) , ω λ ( y * , T x * ) ω λ ( A , B ) ) = θ ( ω λ ( A , B ) ω λ ( A , B ) , ω λ ( A , B ) ω λ ( A , B ) , ω λ ( x * , T y * ) ω λ ( A , B ) , ω λ ( y * , T x * ) ω λ ( A , B ) ) = θ 0 , 0 , ω λ ( x * , T y * ) ω λ ( A , B ) , ω λ ( y * , T x * ) ω λ ( A , B ) = 0 .
As T is a Suzuki-type ( α , β , θ , γ ) -contractive mapping and 1 2 ω λ * ( x * , T x * ) = 0 ω λ ( x * , y * ) and α ( x * , y * ) 1 , then, we obtain:
ω λ ( x * , y * ) ω λ ( T x * , T y * ) β ( M ( x * , y * ) ) M ( x * , y * ) + γ ( N ( x * , y * , θ ) ) N ( x * , y * , θ ) = β ( ω λ ( x * , y * ) ) ω λ ( x * , y * ) < ω λ ( x * , y * ) ,
which is a contradiction. This completes the proof of the theorem. ☐
If in Theorem 3, we take β ( t ) = r where r [ 0 , 1 ) and γ ( t ) = L where L 0 , then we obtain the following best proximity point result.
Corollary 1.
Let ( A , B ) be a pair of nonempty subsets of a non-Archimedean modular metric space X ω with ω regular, such that A is complete and A 0 λ is nonempty for all λ > 0 . Let T : A B be a non-self mapping, such that T ( A 0 λ ) B 0 λ for all λ > 0 and for all x , y A with 1 2 ω λ * ( x , T x ) ω λ ( x , y ) and α ( x , y ) 1 ; one has:
ω λ ( T x , T y ) r M ( x , y ) + L N ( x , y , θ )
where r [ 0 , 1 ) , L 0 and θ Θ . Suppose that the pair ( A , B ) has the weak P 1 -property and the following assertions hold:
(i) 
T is a triangular α-proximal admissible mapping,
(ii) 
there exist elements x 0 and x 1 in A 0 λ for all λ > 0 , such that:
ω λ ( x 1 , T x 0 ) = ω λ ( A , B ) and α ( x 0 , x 1 ) 1 .
(iii) 
if { x n } is a sequence in A, such that α ( x n , x ) 1 for all n N with x n x A as n , then α ( x n , x ) 1 for all n N .
Then, there exists an x * in A, such that ω λ ( x * , T x * ) = ω λ ( A , B ) for all λ > 0 . Further, the best proximity point is unique if, for every x , y A , such that ω λ ( x , T x ) = ω λ ( A , B ) = ω λ ( y , T y ) , we have: α ( x , y ) 1 .
If in Corollary 1 we take, θ ( t 1 , t 2 , t 3 , t 4 ) = min { t 1 , t 2 , t 3 , t 4 } , we obtain the following best proximity result.
Corollary 2.
Let ( A , B ) be a pair of nonempty subsets of a non-Archimedean modular metric space X ω with ω regular, such that A is complete and A 0 λ is nonempty for all λ > 0 . Let T : A B be a non-self mapping, such that T ( A 0 λ ) B 0 λ for all λ > 0 and for all x , y A with 1 2 ω λ * ( x , T x ) ω λ ( x , y ) and α ( x , y ) 1 ; we have:
ω λ ( T x , T y ) r M ( x , y ) + L N ( x , y )
where r [ 0 , 1 ) , L 0 ,
M ( x , y ) = max { ω λ ( x , y ) , ω λ ( x , T x ) + ω λ ( y , T y ) 2 ω λ ( A , B ) , ω λ ( x , T y ) + ω λ ( y , T x ) 2 ω λ ( A , B ) }
and:
N ( x , y ) = min ω λ ( x , T x ) , ω λ ( y , T y ) , ω λ ( x , T y ) , ω λ ( y , T x ) ω λ ( A , B ) .
Suppose that the pair ( A , B ) has the weak P λ -property and the following assertions hold:
(i) 
T is a triangular α-proximal admissible mapping,
(ii) 
there exist elements x 0 and x 1 in A 0 λ for all λ > 0 , such that:
ω λ ( x 1 , T x 0 ) = ω λ ( A , B ) and α ( x 0 , x 1 ) 1 .
(iii) 
if { x n } is a sequence in A, such that α ( x n , x n + 1 ) 1 for all n N with x n x A as n , then α ( x n , x ) 1 for all n N .
Then, there exists an x * in A, such that ω λ ( x * , T x * ) = ω λ ( A , B ) for all λ > 0 . Further, the best proximity point is unique if, for every x , y A , such that ω λ ( x , T x ) = ω λ ( A , B ) = ω λ ( y , T y ) , we have α ( x , y ) 1 .
The following example illustrates our results.
Example 1.
Consider the space X = R 2 endowed with the non-Archimedean modular metric ω : X × X ( 0 , + ) given by:
ω λ ( ( x 1 , x 2 ) , ( y 1 , y 2 ) ) = 1 λ ( | x 1 y 1 | + | x 2 y 2 | ) ,
for all ( x 1 , x 2 ) , ( y 1 , y 2 ) X . Define the sets:
A = { ( 1 , 0 ) , ( 4 , 5 ) , ( 5 , 4 ) } ( , 1 ] × ( , 1 ]
and:
B = { ( 0 , 0 ) , ( 0 , 4 ) , ( 4 , 0 ) } [ 10 , ) × [ 10 , )
so that ω λ ( A , B ) = 1 λ , A 0 λ = { ( 1 , 0 ) } , B 0 λ = { ( 0 , 0 ) } for all λ > 0 , and the pair ( A , B ) has the weak P λ -property. Furthermore, let T : A B be defined by:
T ( x 1 , x 2 ) = ( 10 x 1 2 , 15 x 2 4 ) if x 1 , x 2 ( , 1 ] , ( x 1 , 0 ) if x 1 , x 2 ( , 1 ] with x 1 x 2 , ( 0 , x 2 ) if x 1 , x 2 ( , 1 ] with x 1 > x 2 .
Notice that T ( A 0 λ ) B 0 λ for all λ > 0 .
Now, consider the function β : [ 0 , + ) [ 0 , 1 ) given by:
β ( t ) = 0 if t = 0 , ln ( 1 + t ) t if 0 < t 1 , 8 9 if 1 < t 10 , 10 11 if t > 10 ,
and note that β F . Furthermore, define α : X × X [ 0 , ) by:
α ( x , y ) = 2 , x , y { ( 1 , 0 ) , ( 4 , 5 ) , ( 5 , 4 ) } 1 4 , otherwise . .
Clearly, ω λ ( ( 1 , 0 ) , T ( 1 , 0 ) ) = ω λ ( A , B ) = 1 λ and α ( ( 1 , 0 ) , ( 1 , 0 ) ) 1 .
Assume that 1 2 ω λ * ( x , T x ) ω λ ( x , y ) and α ( x , y ) 1 , for some x , y A . Then:
x = ( 1 , 0 ) , y = ( 4 , 5 ) or x = ( 1 , 0 ) , y = ( 5 , 4 ) or y = ( 1 , 0 ) , x = ( 4 , 5 ) or y = ( 1 , 0 ) , x = ( 5 , 4 ) .
Since ω λ ( T x , T y ) = ω λ ( T y , T x ) and M ( x , y ) = M ( y , x ) for all x , y A , without any loss of generality, we can assume that:
( x , y ) = ( ( 1 , 0 ) , ( 4 , 5 ) )   or   ( x , y ) = ( ( 1 , 0 ) , ( 5 , 4 ) ) .
Now, we want to distinguish the following cases:
(i)
if ( x , y ) = ( ( 1 , 0 ) , ( 4 , 5 ) ) , then:
ω λ ( T ( 1 , 0 ) , T ( 4 , 5 ) ) = 4 λ 8 9 · 8 λ = β ( M ( ( 1 , 0 ) , ( 4 , 5 ) ) ) [ M ( ( 1 , 0 ) , ( 4 , 5 ) ) ] ;
(ii)
if ( x , y ) = ( ( 1 , 0 ) , ( 5 , 4 ) ) , then:
ω λ ( T ( 1 , 0 ) , T ( 5 , 4 ) ) = 4 8 9 · 8 λ = β ( M ( ( 1 , 0 ) , ( 5 , 4 ) ) ) [ M ( ( 1 , 0 ) , ( 5 , 4 ) ) ] .
Consequently, we have:
1 2 ω λ * ( x , T x ) ω λ ( x , y ) and α ( x , y ) 1 ω λ ( T x , T y ) β ( M ( x , y ) ) [ M ( x , y ) ]
and hence, T is a Suzuki-type ( α , β , θ , γ ) -contractive mapping with γ ( t ) = 0 . Let:
α ( x , y ) 1 ω λ ( u , T x ) = ω λ ( A , B ) = 1 λ ω λ ( v , T y ) = ω λ ( A , B ) = 1 λ ,
then:
x , y { ( 1 , 0 ) , ( 4 , 5 ) , ( 5 , 4 ) } ω λ ( u , T x ) = ω λ ( A , B ) = 1 λ ω λ ( v , T y ) = ω λ ( A , B ) = 1 λ ,
and so, u = v = ( 1 , 0 ) . i.e., α ( u , v ) 1 . Furthermore, assume that α ( x , y ) 1 and α ( y , z ) 1 . Then, x , y , z { ( 1 , 0 ) , ( 4 , 5 ) , ( 5 , 4 ) } , i.e., α ( x , z ) 1 . Therefore, T is a triangular α proximal admissible mapping. Moreover, if { x n } is a sequence, such that α ( x n , x n + 1 ) 1 for all n N { 0 } and x n x as n + , then { x n } { ( 1 , 0 ) , ( 4 , 5 ) , ( 5 , 4 ) } , and hence, x { ( 1 , 0 ) , ( 4 , 5 ) , ( 5 , 4 ) } . Consequently, α ( x n , x ) 1 for all n N { 0 } . Hence, as you see, all of the conditions of Theorem 3 hold true, and T has a unique best proximity point. Here, x = ( 1 , 0 ) is the unique best proximity point of T.
If in Theorem 3, we take α ( x , y ) = 1 for all x , y A , then we can deduce the following corollary.
Corollary 3.
Let ( A , B ) be a pair of nonempty subsets of a non-Archimedean modular metric space X ω with ω regular, such that A is complete and A 0 λ is nonempty for all λ > 0 . Let T : A B be a non-self mapping, such that T ( A 0 λ ) B 0 λ for all λ > 0 , and there exists β F and θ Θ , such that 1 2 ω λ * ( x , T x ) ω λ ( x , y ) implies:
ω λ ( T x , T y ) β M ( x , y ) M ( x , y ) + γ N ( x , y , θ ) N ( x , y , θ ) .
Suppose that the pair ( A , B ) has the weak P λ -property. Then, there exists a unique x * in A, such that ω λ ( x * , T x * ) = ω λ ( A , B ) for all λ > 0 .
We investigate the Suzuki-type result of Zhang et al. [16] in the setting of non-Archimedean modular metric space as follows:
Corollary 4.
Let ( A , B ) be a pair of nonempty and closed subsets of a complete non-Archimedean modular metric space X ω with ω regular, such that A 0 λ is nonempty for all λ > 0 . Let T : A B be a non-self mapping, such that T ( A 0 λ ) B 0 λ for all λ > 0 , and there exists r [ 0 , 1 ) , such that 1 2 ω λ * ( x , T x ) ω λ ( x , y ) implies:
ω λ ( T x , T y ) r ω λ ( x , y )
for all x , y A . Suppose that the pair ( A , B ) has the weak P λ -property. Then there exists a unique point x * in A, such that ω λ ( x * , T x * ) = ω λ ( A , B ) for all λ > 0 .
Corollary 5.
(Suzuki-type result of Suzuki [21]) Let ( A , B ) be a pair of nonempty and closed subsets of a complete non-Archimedean modular metric space X ω with ω regular, such that A 0 λ is nonempty for all λ > 0 . Let T : A B be a non-self mapping, such that T ( A 0 λ ) B 0 λ for all λ > 0 , and there exists r [ 0 , 1 ) , such that 1 2 ω λ * ( x , T x ) ω λ ( x , y ) implies:
ω λ ( T x , T y ) r ω λ ( x , T x ) + ω λ ( y , T y ) 2 ω λ ( A , B )
for all x , y A . Suppose that the pair ( A , B ) has the weak P λ -property. Therefore, there exists a unique point x * in A, such that ω λ ( x * , T x * ) = ω λ ( A , B ) for all λ > 0 .
Corollary 6.
Let ( A , B ) be a pair of nonempty subsets of a non-Archimedean modular metric space X ω with ω regular, such that A is complete and A 0 λ is nonempty for all λ > 0 . Let T : A B be a non-self mapping, such that T ( A 0 λ ) B 0 λ for all λ > 0 , and there exists r [ 0 , 1 ) , such that 1 2 ω λ * ( x , T x ) ω λ ( x , y ) implies:
ω λ ( T x , T y ) r ω λ ( x , T y ) + ω λ ( y , T x ) 2 ω λ ( A , B )
for all x , y A 0 . Suppose that the pair ( A , B ) has the weak P λ -property. Then, there exists a unique point x * in A, such that ω λ ( x * , T x * ) = ω λ ( A , B ) for all λ > 0 .

3. Best Proximity Point Results in Metric Spaces Endowed with a Graph

Consistent with Jachymski [22], let X ω be a modular metric space, and Δ denotes the diagonal of the Cartesian product X ω × X ω . Assume that G is a directed graph, such that the set V ( G ) of its vertices coincides with X ω and the set E ( G ) of its edges contains all loops, i.e., E ( G ) Δ . We suppose that G has no parallel edges. We identify G with the pair ( V ( G ) , E ( G ) ) . Furthermore, we may handle G as a weighted graph (see [23], p. 309) by assigning to every edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length N ( N N ) is a sequence { x i } i = 0 N of N + 1 vertices, such that x 0 = x , x N = y and ( x i 1 , x i ) E ( G ) for i = 1 , , N . The foremost fixed point result in this area was given by Jachymski [22].
Definition 9
(Reference [22]). Let ( X , d ) be a modular metric space endowed with a graph G.We say that a self-mapping T : X X is a Banach G-contraction or simply a G-contraction if T preserves the edges of G, that is:
f o r   a l l   x , y X , ( x , y ) E ( G ) ( T x , T y ) E ( G )
and T decreases the weights of the edges of G in the following way:
α ( 0 , 1 )   s u c h   t h a t   f o r   a l l   x , y X , ( x , y ) E ( G ) d ( T x , T y ) α d ( x , y ) .
We define the following notion for modular metric spaces.
Definition 10.
Let X ω be a modular metric space endowed with a graph G. We say that a self-mapping T : X X is a Banach G-contraction or simply a G-contraction if T preserves the edges of G, that is:
f o r   a l l   x , y X , ( x , y ) E ( G ) ( T x , T y ) E ( G )
and T decreases the weights of the edges of G in the following way:
α ( 0 , 1 )   s u c h   t h a t   f o r   a l l   x , y X , ( x , y ) E ( G ) ω λ ( T x , T y ) α ω λ ( x , y ) .
Definition 11.
Let A and B be two nonempty subsets of a non-Archimedean modular metric space X ω endowed with a graph G and A 0 . A mapping T : A B is said to be a Suzuki-type G ( β , θ , γ ) -contractive mapping if there exists β F and θ Θ , such that for all x , y A with 1 2 ω λ * ( x , T x ) ω λ ( x , y ) and ( x , y ) E ( G ) , one has:
ω λ ( T x , T y ) β M ( x , y ) M ( x , y ) + γ N ( x , y , θ ) N ( x , y , θ )
and:
( x , y ) E ( G ) ω λ ( u , T x ) = ω λ ( A , B ) ω λ ( v , T y ) = ω λ ( A , B ) ( u , v ) E ( G ) .
Theorem 4.
Let A and B be two nonempty subsets of a non-Archimedean modular metric space X ω with ω regular endowed with a graph G, such that A is complete and A 0 λ is nonempty for all λ > 0 . Assume that T is a Suzuki-type G ( β , θ , γ ) -contractive mapping satisfying the following assertions:
(i) 
T ( A 0 λ ) B 0 λ for all λ > 0 , and the pair ( A , B ) satisfies the weak P-property,
(ii) 
( x , y ) E ( G ) and ( y , z ) E ( G ) implies ( x , z ) E ( G ) ,
(iii) 
there exist elements x 0 and x 1 in A 0 λ for all λ > 0 , such that:
ω λ ( x 1 , T x 0 ) = ω λ ( A , B ) and ( x 0 , x 1 ) E ( G ) .
(iv) 
if { x n } is a sequence in A, such that ( x n , x n + 1 ) E ( G ) for all n N { 0 } with x n x A as n , then ( x n , x ) E ( G ) for all n N .
Then, there exists an x * in A, such that ω λ ( x * , T x * ) = ω λ ( A , B ) for all λ > 0 .
Proof. 
Define α : X × X [ 0 , + ) with:
α ( x , y ) = 1 , if ( x , y ) E ( G ) 0 , otherwise .
At first, we show that T is a triangular α -proximal admissible mapping. For this goal, assume:
α ( x , y ) 1 ω λ ( u , T x ) = ω λ ( A , B ) ω λ ( v , T y ) = ω λ ( A , B ) .
Therefore, we have:
( x , y ) E ( G ) ω λ ( u , T x ) = ω λ ( A , B ) ω λ ( v , T y ) = ω λ ( A , B ) .
Since T is a Suzuki-type G ( β , θ , γ ) -contractive mapping, we get ( u , v ) E ( G ) , that is α ( u , v ) 1 . Furthermore, let α ( x , z ) 1 and α ( z , y ) 1 , then ( x , z ) E ( G ) and ( z , y ) E ( G ) . Consequently, from (iii), we deduce that ( x , y ) E ( G ) , that is, α ( x , y ) 1 . Thus, T is a triangular α -proximal admissible mapping with T ( A 0 ) B 0 . Now, assume that, 1 2 ω λ * ( x , T x ) ω λ ( x , y ) and α ( x , y ) 1 . Then, 1 2 ω λ * ( x , T x ) ω λ ( x , y ) and ( x , y ) E ( G ) . As T is a Suzuki-type G ( β , θ , γ ) -contraction, then we get:
ω λ ( T x , T y ) β M ( x , y ) M ( x , y ) + γ N ( x , y , θ ) N ( x , y , θ ) ,
and so, T is a Suzuki-type ( α , β , θ , γ ) -contractive mapping. From (iii), there exist x 0 , x 1 A 0 , such that ω λ ( x 1 , T x 0 ) = ω λ ( A , B ) and ( x 0 , x 1 ) E ( G ) , that is ω λ ( x 1 , T x 0 ) = ω λ ( A , B ) and α ( x 0 , x 1 ) 1 . Hence, all of the conditions of Theorem 3 are satisfied, and so, T has a best proximity point.  ☐

4. Best Proximity Point Results in Partially-Ordered Metric Spaces

The existence of best proximity points in partially-ordered metric spaces has been investigated in recent years by many authors (see, [24] and the references therein). In this section, we introduce a new notion of Suzuki-type ordered ( β , θ , γ ) -contractive mapping and investigate the existence of the best proximity points for such mappings in partially-ordered non-Archimedean modular metric spaces by using the weak P λ -property.
Definition 12.
Let X ω be a partially-ordered modular metric space. We say that a non-self-mapping T : A B is proximally ordered-preserving if and only if, for all x 1 , x 2 , u 1 , u 2 A :
x 1 x 2 ω λ ( u 1 , T x 1 ) = ω λ ( A , B ) ω λ ( u 2 , T x 2 ) = ω λ ( A , B ) u 1 u 2 .
Definition 13.
Let A and B be two nonempty closed subsets of a partially-ordered modular metric space X ω and A 0 . A mapping T : A B is said to be a Suzuki-type ordered ( β , θ , γ ) -contractive mapping if there exists β F and θ Θ , such that for all x , y A with 1 2 ω λ * ( x , T x ) ω λ ( x , y ) and x y , we have:
ω λ ( T x , T y ) β M ( x , y ) M ( x , y ) + γ N ( x , y , θ ) N ( x , y , θ ) .
Theorem 5.
Let A and B be two nonempty closed subsets of a partially-ordered non-Archimedean modular metric space with ω regular, such that A is complete, A 0 λ is nonempty for all λ > 0 and the pair ( A , B ) has the weak P λ -property. Assume that T : A B satisfies the following conditions:
(i) 
T is proximally ordered-preserving, such that T ( A 0 λ ) B 0 λ for all λ > 0 ,
(ii) 
there exist elements x 0 , x 1 A 0 , such that:
ω λ ( x 1 , T x 0 ) = ω λ ( A , B ) and x 0 x 1 ,
(iii) 
T is a Suzuki-type ordered ( β , θ , γ ) -contractive mapping,
(iv) 
if { x n } is an increasing sequence in A converging to x A , then x n x for all n N .
Then, T has a best proximity point.

Author Contributions

The authors equally contribute in the preparation of this manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Chistyakov, V.V. Modular metric spaces, I: Basic concepts. Nonlinear Anal. 2010, 72, 1–14. [Google Scholar] [CrossRef]
  2. Chistyakov, V.V. Modular metric spaces, II: Application to superposition operators. Nonlinear Anal. 2010, 72, 15–30. [Google Scholar] [CrossRef]
  3. Nakano, H. Modulared Semi-Ordered Linear Spaces; Maruzen: Tokyo, Japan, 1950. [Google Scholar]
  4. Musielak, J. Orlicz Spaces and Modular Spaces, Lecture Notes in Math; Springer: Berlin, Germany, 1983; Volume 1034. [Google Scholar]
  5. Orlicz, W. Collected Papers, Part I, II; PWN Polish Scientific Publishers: Warsaw, Poland, 1988. [Google Scholar]
  6. Diening, L. Theoretical and Numerical Results for Electrorheological Fluids. Ph.D. Thesis, University of Freiburg, Freiburg, Germany, February 2002. [Google Scholar]
  7. Abdou, A.A.; Khamsi, M.A. On the fixed points of mappings in Modular Metric Spaces. Fixed Point Theory Appl. 2013, 2013, 229. [Google Scholar] [CrossRef]
  8. Mongkolkeha, C.; Sintunavarat, W.; Kumam, P. Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011, 1, 93. [Google Scholar] [CrossRef]
  9. Paknazar, M.; Kutbi, M.A.; Demma, M.; Salimi, P. On non-Archimedean Modular metric space and some nonlinear contraction mappings. J. Nonlinear Sci. Appl. 2017, in press. [Google Scholar]
  10. Geraghty, M. On contractive mappings. Proc. Am. Math. Soc. 1973, 40, 604–608. [Google Scholar] [CrossRef]
  11. Kirk, W.A. Contraction mappings and extensions. In Handbook of Metric Fixed Point Theory; Kirk, W.A., Sims, B., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2001; pp. 1–34. [Google Scholar]
  12. Kirk, W.A.; Reich, S.; Veeramani, P. Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24, 851–862. [Google Scholar] [CrossRef]
  13. Raj, V.S. A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal. 2011, 74, 4804–4808. [Google Scholar]
  14. Sadiq Basha, S. Extensions of Banach’s contraction principle. Numer. Funct. Anal. Optim. 2010, 31, 569–576. [Google Scholar] [CrossRef]
  15. Raj, V.S.; Veeramani, P. Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 2009, 10, 21–28. [Google Scholar]
  16. Zhang, J.; Su, Y.; Cheng, Q. A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl. 2013, 2013, 99. [Google Scholar] [CrossRef]
  17. Caballero, J.; Harjani, J.; Sadarangani, K. A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012, 2012, 231. [Google Scholar] [CrossRef]
  18. Kumam, P.; Salimi, P.; Vetro, C. Best proximity point results for modified α-proximal C-contraction mappings. Fixed Point Theory Appl. 2014, 2014, 99. [Google Scholar] [CrossRef]
  19. Alghamdi, M.A.; Shahzad, N.; Vetro, F. Best proximity points for some classes of proximal contractions. Abstr. Appl. Anal. 2013, 2013, 713252. [Google Scholar] [CrossRef]
  20. Kumam, P.; Roldán, A. On existence and uniqueness of g-best proximity points under (φ, θ, α, g)-contractivity conditions and consequences. Abstr. Appl. Anal. 2014, 2014, 234027. [Google Scholar] [CrossRef]
  21. Suzuki, T. The existence of best proximity points with the weak P-property. Fixed Point Theory Appl. 2013, 2013, 259. [Google Scholar] [CrossRef]
  22. Jachymski, J. The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 2008, 136, 1359–1373. [Google Scholar] [CrossRef]
  23. Johnsonbaugh, R. Discrete Mathematics; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 1997. [Google Scholar]
  24. Agarwal, R.P.; Hussain, N.; Taoudi, M.A. Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations. Abstr. Appl. Anal. 2012, 2012, 245872. [Google Scholar] [CrossRef]

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Paknazar, M.; Sen, M.D.l. Best Proximity Point Results in Non-Archimedean Modular Metric Space. Mathematics 2017, 5, 23. https://doi.org/10.3390/math5020023

AMA Style

Paknazar M, Sen MDl. Best Proximity Point Results in Non-Archimedean Modular Metric Space. Mathematics. 2017; 5(2):23. https://doi.org/10.3390/math5020023

Chicago/Turabian Style

Paknazar, Mohadeshe, and Manuel De la Sen. 2017. "Best Proximity Point Results in Non-Archimedean Modular Metric Space" Mathematics 5, no. 2: 23. https://doi.org/10.3390/math5020023

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