Next Article in Journal
Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras
Next Article in Special Issue
On Fixed Point Results for Modified JS-Contractions with Applications
Previous Article in Journal
Conditions of Functional Null Controllability for Some Types of Singularly Perturbed Nonlinear Systems with Delays
Previous Article in Special Issue
Recent Advances on the Results for Nonunique Fixed in Various Spaces
Open AccessArticle

Best Proximity Point Results for Geraghty Type Ƶ-Proximal Contractions with an Application

1
Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
3
Institut Supérieur d’Informatique et des Techniques de Communication, Université de Sousse, H. Sousse 4000, Tunisia
4
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
5
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
*
Author to whom correspondence should be addressed.
Axioms 2019, 8(3), 81; https://doi.org/10.3390/axioms8030081
Received: 18 June 2019 / Revised: 10 July 2019 / Accepted: 11 July 2019 / Published: 18 July 2019
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

Abstract

In this study, we establish the existence and uniqueness theorems of the best proximity points for Geraghty type Ƶ-proximal contractions defined on a complete metric space. The presented results improve and generalize some recent results in the literature. An example, as well as an application to a variational inequality problem are also given in order to illustrate the effectiveness of our generalizations.
Keywords: best proximity point; Ƶ-contraction; geraghty type contraction; simulation function; admissible mapping; variational inequality best proximity point; Ƶ-contraction; geraghty type contraction; simulation function; admissible mapping; variational inequality

1. Introduction

Numerous problems in science and engineering defined by nonlinear functional equations can be solved by reducing them to an equivalent fixed-point problem. In fact, an operator equation
G x = 0
may be expressed as a fixed-point equation T x = x . Accordingly, the Equation (1) has a solution if the self-mapping T has a fixed point. However, for a non-self mapping T : P Q , the equation T x = x does not necessarily admit a solution. Here, it is quite natural to find an approximate solution x * such that the distance d ( x * , T x * ) is minimum, in which case x * and T x * are in close proximity to each other. Herein, the optimal approximate solution x * , for which d ( x * , T x * ) = d ( P , Q ) , is called a best proximity point of T . The main aim of the best proximity point theory is to give sufficient conditions for finding the existence of a solution to the nonlinear programming problem,
min ξ P d ( ξ , T ξ ) .
Moreover, a best proximity point generates to a fixed point if the mapping under consideration is a self-mapping. For more details on this research subject, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].
In 2015, Khojasteh et al. [16] presented the notion of Z -contraction involving a new class of mappings—namely, simulation functions, and proved new fixed-point theorems via different methods to others in the literature. For more details, see [17,18,19,20].
Definition 1
([16]). A simulation function is a mapping ζ : [ 0 , ) × [ 0 , ) R so that:
( ζ 1 )
ζ ( 0 , 0 ) = 0 ;
( ζ 2 )
ζ ( μ , η ) < η μ for all μ , η > 0 ;
( ζ 3 )
If ( μ n ) , ( η n ) are sequences in ( 0 , ) so that lim n μ n = lim n η n > 0 , then
lim sup n ζ ( μ n , η n ) < 0 .
Theorem 1
([16]). Let ( M , d ) be a complete metric space and T : M M be a Z -contraction with respect to ζ Z —that is,
ζ ( d ( T ξ , T ω ) , d ( ξ , ω ) ) 0 , f o r a l l ξ , ω M .
Then, T admits a unique fixed point (say τ X ) and, for each ξ 0 M , the Picard sequence { T n ξ 0 } is convergent to τ.
In this study, we will consider simulation functions satisfying only the condition ( ζ 2 ) . For the sake of convenience, we identify the set of all simulation functions satisfying only the condition ( ζ 2 ) by Z .
The main concern of the paper is to establish theorems on the existence and uniqueness of best proximity points for Geraghty type Z -proximal contractions in complete metric spaces. The obtained results complement and extend some known results from the literature. An example, as well as an application to a variational inequality problem, is also given in order to illustrate the effectiveness of our generalizations.

2. Preliminaries

Let P and Q be two non-empty subsets of a metric space, ( M , d ) . Consider:
d ( P , Q ) : = inf d ( ρ , ν ) : ρ P , ν Q ; P 0 : = ρ P : d ( ρ , ν ) = d ( P , Q ) for some ν Q ; Q 0 : = ν Q : d ( ρ , ν ) = d ( P , Q ) for some ρ P .
Denote by
B e s t ( T ) = u P : d ( u , T u ) = d ( P , Q ) ,
the set of all best proximity points of a non-self-mapping T : P Q . In the study [5], Caballero et al. familiarized the notion of Geraghty contraction for non-self-mappings as follows:
Definition 2
([5]). Let P , Q be two non-empty subsets of a metric space, ( M , d ) . A mapping T : P Q is called a Geraghty contraction if there is β Σ , so that for all ξ , ω P
d ( T ξ , T ω ) β ( d ( ξ , ω ) ) · d ( ξ , ω ) ,
where the class Σ is the set of functions β : [ 0 , ) [ 0 , 1 ) , satisfying
β ( t n ) 1 t n 0 .
In the paper [10], Jleli and Samet initiated the concepts of α - ψ -proximal contractive and α -proximal admissible mappings. They provided related best-proximity-point results. Subsequently, Hussain et al. [7] modified the aforesaid notions and substantiated certain best-proximity-point theorems.
Definition 3
([10]). Let T : P Q and α : P × P 0 , be given mappings. Then, T is called α-proximal admissible if
α ( u 1 , u 2 ) 1 d ( p 1 , T u 1 ) = d ( P , Q ) d ( p 2 , T u 2 ) = d ( P , Q ) α ( p 1 , p 2 ) 1 ,
for all u 1 , u 2 , p 1 , p 2 P .
Definition 4
([7]). Let T : P Q and α , η : P × P 0 , be given mappings. Such T is said to be ( α , η ) -proximal admissible if
α ( u 1 , u 2 ) η ( u 1 , u 2 ) d ( p 1 , T u 1 ) = d ( P , Q ) d ( p 2 , T u 2 ) = d ( P , Q ) α ( p 1 , p 2 ) η ( p 1 , p 2 ) ,
for all u 1 , u 2 , p 1 , p 2 P .
Note that if η ( u , v ) = 1 for all u , v P , then Definition 4 corresponds to Definition 3.
Very recently, Tchier et al. in [14] initiated the concept of Z -proximal contractions.
Definition 5
([14]). Let P and Q be two non-empty subsets of a metric space, ( M , d ) . A non-self-mapping T : P Q is called a Z -proximal contraction if there is a simulation function ζ so that
d ( ρ , T u ) = d ( P , Q ) d ( ν , T v ) = d ( P , Q ) ζ ( d ( ρ , ν ) , d ( u , v ) ) 0 ,
for all ρ , ν , u , v P .
Now, we introduce a new concept which will be efficiently used in our results.
Definition 6.
Let T : P Q and α , η : P × P 0 , be given mappings. Then, T is said to be triangular ( α , η ) -proximal admissible, if
(1)
T is ( α , η ) -proximal admissible;
(2)
α ( u , v ) η ( u , v ) a n d α ( v , z ) η ( v , z ) i m p l i e s t h a t α ( u , z ) η ( u , z ) , for all u , v , z P .
Now, we describe a new class of contractions for non-self-mappings which generalize the concept of Geraghty-contractions.
Definition 7.
Let P and Q be two non-empty subsets of a metric space ( M , d ) , ζ Z and α , η : P × P 0 , and β Σ . A non-self-mapping T : P Q is said to be a Geraghty type Z -proximal contraction, if for all u , v , ρ , ν P , the following implication holds:
α ( u , v ) η ( u , v ) d ( ρ , T u ) = d ( P , Q ) d ( ν , T v ) = d ( P , Q ) ζ ( d ( ρ , ν ) , β ( d ( u , v ) ) d ( u , v ) ) 0 .
Remark 1.
If T : P Q is a Geraghty type Z -proximal contraction, then by ( ζ 2 ) and Definition 7, the following implication holds for all u , v , ρ , ν P with u v :
α ( u , v ) η ( u , v ) d ( ρ , T u ) = d ( P , Q ) d ( ν , T v ) = d ( P , Q ) d ( ρ , ν ) < β ( d ( u , v ) ) d ( u , v ) .

3. Main Results

Our first result is as follows.
Theorem 2.
Let ( P , Q ) be a pair of non-empty subsets of a complete metric space ( M , d ) so that P 0 is non-empty, T : P Q and α , η : P × P 0 , be given mappings. Suppose that:
(i)
P is closed and T ( P 0 ) Q 0 ;
(ii)
T is triangular ( α , η ) -proximal admissible;
(iii)
There are u 0 , u 1 P 0 so that d ( u 1 , T u 0 ) = d ( P , Q ) and α u 0 , u 1 η u 0 , u 1 ;
(iv)
T is a continuous Geraghty type Z -proximal contraction.
Then, T has a best proximity point in P . If α ( u , v ) η ( u , v ) for all u , v B e s t ( T ) , then T has a unique best proximity point u * P . Moreover, for every u P , lim n T n u = u * .
Proof. 
From the condition ( i i i ) , there are u 0 , u 1 P 0 so that
d ( u 1 , T u 0 ) = d ( P , Q ) and α u 0 , u 1 η u 0 , u 1 .
Since T ( P 0 ) Q 0 , there is u 2 P 0 so that
d ( u 2 , T u 1 ) = d ( P , Q ) .
Thus, we get
α ( u 0 , u 1 ) η ( u 0 , u 1 ) , d ( u 1 , T u 0 ) = d ( P , Q ) , d ( u 2 , T u 1 ) = d ( P , Q ) .
Since T is ( α , η ) -proximal admissible, we get α u 1 , u 2 η u 1 , u 2 . Now, we have
d ( u 2 , T u 1 ) = d ( P , Q ) and α u 1 , u 2 η u 1 , u 2 .
Again, since T ( P 0 ) Q 0 , there exists u 3 P 0 such that
d ( u 3 , T u 2 ) = d ( P , Q ) ,
and thus,
α ( u 1 , u 2 ) η ( u 1 , u 2 ) , d ( u 2 , T u 1 ) = d ( P , Q ) , d ( u 3 , T u 2 ) = d ( P , Q ) .
Since T is ( α , η ) -proximal admissible, this implies that α u 2 , u 3 η u 2 , u 3 . Thus, we have
d ( u 3 , T u 2 ) = d ( P , Q ) and α u 2 , u 3 η u 2 , u 3 .
By repeating this process, we build a sequence u n in P 0 P so that
d ( u n + 1 , T u n ) = d ( P , Q ) and α u n , u n + 1 η u n , u n + 1 ,
for all n N 0 . If there is n 0 so that u n 0 = u n 0 + 1 , then
d ( u n 0 , T u n 0 ) = d ( u n 0 + 1 , T u n 0 ) = d ( P , Q ) .
That is, u n 0 is a best proximity point of T . We should suppose that u n u n + 1 , for all n .
From (8), for all n N , we get
α u n 1 , u n η u n 1 , u n , d ( u n , T u n 1 ) = d ( P , Q ) , d ( u n + 1 , T u n ) = d ( P , Q ) .
On the grounds that T is a Geraghty type Z -proximal contraction, by utilizing Remark 1, we deduce that
d ( u n , u n + 1 ) < β ( d ( u n 1 , u n ) ) d ( u n 1 , u n ) ,
which requires that d ( u n , u n + 1 ) < d ( u n 1 , u n ) , for all n . Therefore, the sequence d ( u n , u n + 1 ) is decreasing, and so there is λ 0 so that lim n d u n , u n + 1 = λ . Now, we shall show that λ = 0 . On the contrary, assume that λ > 0 . Then, taking into account (9), for any n N ,
d ( u n , u n + 1 ) < β ( d ( u n 1 , u n ) ) d ( u n 1 , u n ) < d ( u n 1 , u n ) .
This yields, for any n N ,
0 < d ( u n , u n + 1 ) d ( u n 1 , u n ) < β ( d ( u n 1 , u n ) ) < 1 .
Taking n , we find that
lim n β ( d ( u n 1 , u n ) ) = 1 ,
and since β Σ , lim n d ( u n 1 , u n ) = 0 . This contradicts our assumption lim n d ( u n 1 , u n ) = λ > 0 . Therefore, we get
lim n d ( u n 1 , u n ) = 0 , f o r a l l n N .
We shall prove that u n is Cauchy in P. By contradiction, suppose that u n is not a Cauchy sequence, so there is an ε > 0 for which we can find u m k and u n k of u n such that n k is the smallest index for which n k > m k > k and
d u m k , u n k ε and d u m k , u n k 1 < ε .
We have
ε d u m k , u n k d u m k , u n k 1 + d u n k 1 , u n k < ε + d u n k 1 , u n k .
Taking k , by (10), we get
lim k d u m k , u n k = ε .
By triangular inequality,
d u m k + 1 , u n k + 1 d u m k , u n k d u m k + 1 , u m k + d u n k , u n k + 1 ,
which yields that
lim k d x m k + 1 , x n k + 1 = ε .
Since T is triangular ( α , η ) -proximal admissible, by using (8), we infer
α ( u m , u n ) η ( u m , u n ) , for all n , m N with m < n .
Combining (8) and (14), for all k N , we have
α ( u m k , u n k ) η ( u m k , u n k ) , d ( u m k + 1 , T u m k ) = d ( P , Q ) , d ( u n k + 1 , T u n k ) = d ( P , Q ) .
Regarding the fact that T is a Geraghty type Z -proximal contraction, from Remark 1, we deduce that
d ( u m k + 1 , u n k + 1 ) < β ( d ( u m k , u n k ) ) d ( u m k , u n k ) < d ( u m k , u n k ) .
Taking the limit as k tends to on both sides of the last inequality, and using the Equations (12) and (13), we get
ε lim k β ( d ( u m k , u n k ) ) ε ε ,
which implies that lim k β ( d ( u m k , u n k ) ) = 1 , and so lim k d ( u m k , u n k ) = 0 which contradicts ε > 0 . Hence, { u n } is a Cauchy sequence in P. Since P is a closed subset of the complete metric space ( M , d ) , there is p P so that
lim n d ( u n , p ) = 0 .
Since T is continuous, we have
lim n d ( T u n , T p ) = 0 .
Combining (8), (15), and (16), we get
d ( P , Q ) = lim n d ( u n + 1 , T u n ) = d ( p , T p ) .
Therefore, u P is a best proximity point of T . Finally, we shall show that the set B e s t ( T ) is a singleton. Suppose that r is another best proximity point of T , that is, d ( r , T r ) = d ( P , Q ) . Then, by the hypothesis, we have α ( p , r ) η ( p , r ) —that is,
α ( p , r ) η ( p , r ) , d ( p , T p ) = d ( P , Q ) , d ( r , T r ) = d ( P , Q ) .
Then, from Remark 1, we deduce
d ( p , r ) < β ( d ( p , r ) ) d ( p , r ) < d ( p , r ) ,
which is a contradiction. Hence, we have a unique best proximity point of T . □
Let us consider the following assertion in order to remove the continuity on the operator T in the next theorem.
(C)
If a sequence u n in P is convergent to u P so that α u n , u n + 1 η u n , u n + 1 , then α u n , u η u n , u for all n N .
Theorem 3.
Let ( P , Q ) be a pair of non-empty subsets of a complete metric space ( M , d ) so that P 0 is non-empty, T : P Q and α , η : P × P 0 , be given mappings. Suppose that:
(i)
P is closed and T ( P 0 ) Q 0 ;
(ii)
T is triangular ( α , η ) -proximal admissible;
(iii)
there are u 0 , u 1 P 0 so that d ( u 1 , T u 0 ) = d ( P , Q ) and α u 0 , u 1 η u 0 , u 1 ;
(iv)
the condition ( C ) holds and T is a Geraghty type Z -proximal contraction.
Then, T has a best proximity point in P . If α ( u , v ) η ( u , v ) for all u , v B e s t ( T ) , then T has a unique best proximity point u * P . Moreover, for each u P , we have lim n T n u = u * .
Proof. 
Following the proof of Theorem 2, there exists a Cauchy sequence u n P 0 satisfying (8) and u n p . On account of (i), P 0 is closed, and so p P 0 . Also, since T ( P 0 ) Q 0 , there is z P 0 so that
d ( z , T p ) = d ( P , Q ) .
Taking ( C ) and (8) into account, we infer
α u n , p η u n , p , for all n N .
Since T is ( α , η ) -proximal admissible and
α ( u n , p ) η ( u n , p ) , d ( u n + 1 , T u n ) = d ( P , Q ) , d ( z , T p ) = d ( P , Q ) ,
so, we conclude that
α ( u n + 1 , z ) η ( u n + 1 , z ) , for all n N .
Considering (18), (19) and Remark 1, we have
d ( u n + 1 , z ) < β ( d ( u n , p ) ) d ( u n , p ) < d ( u n , p ) ,
which implies that lim n d ( u n + 1 , z ) = 0 . By the uniqueness of the limit, we obtain z = p . Thus, by (17), we deduce that d ( p , T p ) = d ( P , Q ) . Uniqueness of the best proximity point follows from the proof of Theorem 2. □
Example 1.
Let M = R 2 be endowed with the Euclidian metric, P = ( 0 , u ) : u 0 and Q = ( 1 , u ) : u 0 . Note that d ( P , Q ) = 1 , P 0 = P and Q 0 = Q . Let
β ( t ) = 1 1 + t , i f   t > 0 β ( t ) = 1 2 , o t h e r w i s e .
Then, β Σ . Define T : P Q and α : P × P 0 , by
T ( 0 , u ) = ( 1 , u 9 ) , i f   0 u 1 , ( 1 , u 2 ) , i f   u > 1 ,
and
α ( ( 0 , u ) , ( 0 , v ) ) = 2 η ( ( 0 , u ) , ( 0 , v ) ) , i f u , v [ 0 , 1 ] , o r u = v 0 , o t h e r w i s e .
Choose ζ ( t , s ) = 2 3 s t for all t , s [ 0 , ) . Let u , v , p , q 0 be such that
α ( ( 0 , u ) , ( 0 , v ) ) η ( ( 0 , u ) , ( 0 , v ) ) d ( ( 0 , p ) , T ( 0 , u ) ) = d ( P , Q ) = 1 d ( ( 0 , q ) , T ( 0 , v ) ) = d ( P , Q ) = 1 .
Then, u , v [ 0 , 1 ] or u = v .
u , v [ 0 , 1 ] . Here, T ( 0 , u ) = ( 1 , u 9 ) and T ( 0 , v ) = ( 1 , v 9 ) . Also,
1 + ( p u 9 ) 2 = 1 + ( q v 9 ) 2 = 1 ,
that is, p = u 9 and q = v 9 . So, α ( ( 0 , p ) , ( 0 , q ) ) d ( ( 0 , p ) , ( 0 , q ) ) . Moreover,
ζ ( d ( ( 0 , p ) , ( 0 , q ) ) , β ( d ( ( 0 , u ) , ( 0 , v ) ) ) d ( ( 0 , u ) , ( 0 , v ) ) ) = 2 3 β ( d ( ( 0 , u ) , ( 0 , v ) ) ) d ( ( 0 , u ) , ( 0 , v ) ) d ( ( 0 , u 9 ) , ( 0 , v 9 ) ) = 2 3 β ( | u v | ) | u v | | u v | 9 .
If u = v , then β ( | u v | ) = 1 2 and the right-hand side of the above inequality is equal to 0.
If u v , we have
ζ ( d ( ( 0 , p ) , ( 0 , q ) ) , β ( d ( ( 0 , u ) , ( 0 , v ) ) ) d ( ( 0 , u ) , ( 0 , v ) ) ) = 2 3 | u v | 1 + | u v | | u v | 9 0 .
u = v > 1 . Here, T ( 0 , u ) = ( 1 , u 2 ) and T ( 0 , v ) = ( 1 , v 2 ) . Similarly, we get that p = q = u 2 = v 2 . So, α ( ( 0 , p ) , ( 0 , q ) ) = 0 = η ( ( 0 , p ) , ( 0 , q ) ) .
Also, ζ ( d ( ( 0 , p ) , ( 0 , q ) ) , β ( d ( ( 0 , u ) , ( 0 , v ) ) ) d ( ( 0 , u ) , ( 0 , v ) ) ) 0 .
In each case, we get that T is an ( α , η ) -proximal admissible. It is also easy to see that T is triangular ( α , η ) -proximal admissible. Also, T is a Geraghty type Z -proximal contraction. Also, if u n = ( 0 , p n ) is a sequence in P such that α u n , u n + 1 η u n , u n + 1 for all n and u n = ( 0 , p n ) u = ( 0 , p ) as n , then p n p . We have p n , p n + 1 [ 0 , 1 ] or p n = p n + 1 . We get that p [ 0 , 1 ] or p n = p . This implies that α u n , u η u n , u for all n .
Moreover, there is ( u 0 , u 1 ) = ( ( 0 , 1 ) , ( 0 , 1 9 ) ) P 0 × P 0 so that
d ( u 1 , T u 0 ) = 1 = d ( P , Q ) and α u 0 , u 1 d u 0 , u 1 .
Consequently, all conditions of Theorem 3 are satisfied. Therefore, T has a unique best proximity point in P, which is ( 0 , 0 ) . On the other side, we indicate that (4) is not satisfied. In fact, for u = ( 0 , 2 ) , v = ( 0 , 3 ) , we have
d ( T u , T v ) = d ( T ( 0 , 2 ) , T ( 0 , 3 ) ) = d ( ( 0 , 4 ) , ( 0 , 9 ) ) = 5 > 1 2 = β ( d ( ( 0 , 2 ) , ( 0 , 3 ) ) ) d ( ( 0 , 2 ) , ( 0 , 3 ) ) = β ( d ( u , v ) ) d ( u , v ) .
Corollary 1.
Let ( P , Q ) be a pair of non-empty subsets of a complete metric space ( M , d ) , such that P 0 is non-empty. Suppose that T : P Q is a Geraghty-proximal contraction—that is, the following implication holds for all u , v , ρ , ν P :
d ( ρ , T u ) = d ( P , Q ) d ( ν , T v ) = d ( P , Q ) ζ ( d ( ρ , ν ) , β ( d ( u , v ) ) d ( u , v ) ) 0 .
Also, assume that P is closed and T ( P 0 ) Q 0 . Then, T has a unique best proximity point u * P . Moreover, for each u P , we have lim n T n u = u * .
Proof. 
We take α ( σ , ς ) = η ( σ , ς ) = 1 in the proof of Theorem 2 (resp. Theorem 3). □

4. Some Consequences

In this section we give new fixed-point results on a metric space endowed with a partial ordering/graph by using the results provided in the previous section. Define
α , η : M × M [ 0 , ) , α u , v = η ( u , v ) , if   u v , 0 , otherwise .
Definition 8.
Let ( M , , d ) be a partially ordered metric space, ( P , Q ) be a pair of non-empty subsets of M, and T : P Q be a given mapping. Such T is said to be ⪯-proximal increasing if
u 1 u 2 d ( p 1 , T u 1 ) = d ( P , Q ) d ( p 2 , T u 2 ) = d ( P , Q ) p 1 p 2 ,
for all u 1 , u 2 , p 1 , p 2 P .
Then, the following result is a direct consequence of Theorem 2 (resp. Theorem 3).
Theorem 4.
Let ( P , Q ) be a pair of non-empty subsets of a complete ordered metric space ( M , , d ) so that P 0 is non-empty and T : P Q be a given non-self-mapping. Suppose that:
(i)
P is closed and T ( P 0 ) Q 0 ;
(ii)
T is ⪯-proximal increasing;
(iii)
There are u 0 , u 1 P 0 so that d ( u 1 , T u 0 ) = d ( P , Q ) and u 0 u 1 ;
(iv)
T is continuous or, for every sequence u n in P is convergent to u P so that u n u n + 1 , we have u n u for all n N ;
(v)
There exist ζ Z and β Σ , such that for all u , v , ρ , ν P ,
u v d ( ρ , T u ) = d ( P , Q ) d ( ν , T v ) = d ( P , Q ) ζ ( d ( ρ , ν ) , β ( d ( u , v ) ) d ( u , v ) ) 0 .
Then, T has a best proximity point in P . If u v for all u , v B e s t ( T ) , then T has a unique best proximity point u * P . Moreover, for every u P , lim n T n u = u * .
Now, we present the existence of the best proximity point for non-self mappings from a metric space M, endowed with a graph, into the space of non-empty closed and bounded subsets of the metric space. Consider a graph G, such that the set V G of its vertices coincides with M and the set E G of its edges contains all loops; that is, E G Δ , where Δ = u , u : u M . We assume G has no parallel edges, so we can identify G with the pair V G , E G .
Define
α , η : M × M [ 0 , + ) , α u , v = η ( u , v ) , if   u , v E G , 0 , otherwise .
Definition 9.
Let M , d be a complete metric space endowed with a graph G and ( P , Q ) be a pair of non-empty subsets of M and T : P Q be a given mapping. Such T is said to be triangular G-proximal, if
(1)
for all u 1 , u 2 , p 1 , p 2 P ,
( u 1 , u 2 ) E ( G ) d ( p 1 , T u 1 ) = d ( P , Q ) d ( p 2 , T u 2 ) = d ( P , Q ) ( p 1 , p 2 ) E ( G ) ;
(2)
( u , v ) E ( G ) a n d ( v , z ) E ( G ) i m p l i e s t h a t ( u , z ) E ( G ) , for all u , v , z P .
for all u 1 , u 2 , p 1 , p 2 P .
The following result is a direct consequence of Theorem 2 (resp. Theorem 3).
Theorem 5.
Let M , d be a complete metric space endowed with a graph G and ( P , Q ) be a pair of non-empty subsets of M so that P 0 is non-empty and T : P Q be a given non-self mapping. Suppose that:
(i)
P is closed and T ( P 0 ) Q 0 ;
(ii)
T is triangular G-proximal;
(iii)
There are u 0 , u 1 P 0 so that d ( u 1 , T u 0 ) = d ( P , Q ) and ( u 0 , u 1 ) E ( G ) ;
(iv)
T is continuous or, for every sequence u n in P is convergent to u P so that ( u n , u n + 1 ) E ( G ) , we have ( u n , u ) E ( G ) for all n N ;
(v)
There exist ζ Z and β Σ such that for all u , v , ρ , ν P ,
( u , v ) E ( G ) d ( ρ , T u ) = d ( P , Q ) d ( ν , T v ) = d ( P , Q ) ζ ( d ( ρ , ν ) , β ( d ( u , v ) ) d ( u , v ) ) 0 .
Then, T has a best proximity point in P . If ( u , v ) E ( G ) for all u , v B e s t ( T ) , then T has a unique best proximity point u * P . Moreover, for every u P , lim n T n u = u * .

5. A Variational Inequality Problem

Let C be a non-empty, closed, and convex subset of a real Hilbert space H, with inner product · , · and a norm · . A variational inequality problem is given in the following:
Find   u C   so   that   S u , v u 0   for   all   v C ,
where S : H H is a given operator. The above problem can be seen in operations research, economics, and mathematical physics, especially in calculus of variations associated with the minimization of infinite-dimensional functionals. See [21] and the references therein. It appears in variant problems of nonlinear analysis, such as complementarity and equilibrium problems, optimization, and finding fixed points; see [21,22,23]. To solve problem (22), we define the metric projection operator P C : H C . Note that for every u H , there is a unique nearest point P C u C so that
u P C u u v , for   all   v C .
The two lemmas below correlate the solvability of a variational inequality problem to the solvability of a special fixed-point problem.
Lemma 1
([24]). Let z H . Then, u C is such that u z , y u 0 , for all y C iff u = P C z .
Lemma 2
([24]). Let S : H H . Then, u C is a solution of S u , v u 0 , for all v C , if u = P C ( u λ S u ) , with λ > 0 .
The main theorem of this section is:
Theorem 6.
Let C be a non-empty, closed, and convex subset of a real Hilbert space H. Assume that S : H H is such that P C ( I λ S ) : C C is a Geraghty-proximal contraction. Then, there is a unique element u * C , such that S u * , v u * 0 for all v C . Also, for any u 0 C , the sequence { u n } given as u n + 1 = P C ( u n λ S u n ) where λ > 0 and n N { 0 } , is convergent to u * .
Proof. 
We consider the operator T : C C defined by T x = P C ( x λ S x ) for all x C . By Lemma 2, u C is a solution of S u , v u 0 for all v C , if u = T u . Now, T verifies all the hypotheses of Corollary 1 with P = Q = C . Now, from Corollary 1, the fixed-point problem u = T u possesses a unique solution u * C .  □

Author Contributions

H.I. analyzed and prepared/edited the manuscript, H.A. analyzed and prepared/edited the manuscript, N.M. analyzed and prepared the manuscript, S.R. analyzed and prepared the manuscript.

Funding

This research received no external funding.

Acknowledgments

The third author would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

Conflicts of Interest

The authors declare that they have no competing interests regarding the publication of this paper.

References

  1. Abkar, A.; Gabeleh, M. Best proximity points for cyclic mappings in ordered metric spaces. J. Optim. Theory Appl. 2011, 150, 188–193. [Google Scholar] [CrossRef]
  2. Al-Thagafi, M.A.; Shahzad, N. Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 2009, 70, 1209–1216. [Google Scholar] [CrossRef]
  3. Aydi, H.; Felhi, A. On best proximity points for various α-proximal contractions on metric-like spaces. J. Nonlinear Sci. Appl. 2016, 9, 5202–5218. [Google Scholar] [CrossRef]
  4. Aydi, H.; Felhi, A. Best proximity points for cyclic Kannan-Chatterjea-Ćirić type contractions on metric-like spaces. J. Nonlinear Sci. Appl. 2016, 9, 2458–2466. [Google Scholar] [CrossRef]
  5. Caballero, J.; Harjani, J.; Sadarangani, K. A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012, 2012, 231. [Google Scholar] [CrossRef]
  6. Eldred, A.A.; Veeramani, P. Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323, 1001–1006. [Google Scholar] [CrossRef]
  7. Hussain, N.; Kutbi, M.A.; Salimi, P. Best proximity point results for modified α-ψ-proximal rational contractions. Abstr. Appl. Anal. 2013, 2013, 927457. [Google Scholar] [CrossRef]
  8. Hussain, N.; Latif, A.; Salimi, P. New fixed point results for contractive maps involving dominating auxiliary functions. J. Nonlinear Sci. Appl. 2016, 9, 4114–4126. [Google Scholar] [CrossRef]
  9. Işık, H.; Sezen, M.S.; Vetro, C. φ-Best proximity point theorems and applications to variational inequality problems. J. Fixed Point Theory Appl. 2017, 19, 3177–3189. [Google Scholar] [CrossRef]
  10. Jleli, M.; Samet, B. Best proximity points for α-ψ-proximal contractive type mappings and application. Bull. Sci. Math. 2013, 137, 977–995. [Google Scholar] [CrossRef]
  11. Basha, S.S.; Veeramani, P. Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103, 119–129. [Google Scholar] [CrossRef]
  12. Sahmim, S.; Felhi, A.; Aydi, H. Convergence Best Proximity Points for Generalized Contraction Pairs. Mathematics 2019, 7, 176. [Google Scholar] [CrossRef]
  13. Souyah, N.; Aydi, H.; Abdeljawad, T.; Mlaiki, N. Best proximity point theorems on rectangular metric spaces endowed with a graph. Axioms 2019, 8, 17. [Google Scholar] [CrossRef]
  14. Tchier, F.; Vetro, C.; Vetro, F. Best approximation and variational inequality problems involving a simulation function. Fixed Point Theory Appl. 2016, 2016, 26. [Google Scholar] [CrossRef]
  15. Samet, B.; Vetro, C.; Vetro, P. Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal. 2012, 75, 2154–2165. [Google Scholar] [CrossRef]
  16. Khojasteh, F.; Shukla, S.; Radenović, S. A new approach to the study of fixed point theorems via simulation functions. Filomat 2015, 29, 1189–1194. [Google Scholar] [CrossRef]
  17. Argoubi, H.; Samet, B.; Vetro, C. Nonlinear contractions involving simulation functions in metric space with a partial order. J. Nonlinear Sci. Appl. 2015, 8, 1082–1094. [Google Scholar] [CrossRef]
  18. Işık, H.; Gungor, N.B.; Park, C.; Jang, S.Y. Fixed point theorems for almost 𝒵-contractions with an application. Mathematics 2018, 6, 37. [Google Scholar] [CrossRef]
  19. Nastasi, A.; Vetro, P. Fixed point results on metric and partial metric spaces via simulation functions. J. Nonlinear Sci. Appl. 2015, 8, 1059–1069. [Google Scholar] [CrossRef]
  20. Radenovic, S.; Vetro, F.; Vujakovic, J. An alternative and easy approach to fixed point results via simulation functions. Demonstr. Math. 2017, 50, 223–230. [Google Scholar] [CrossRef]
  21. Kinderlehrer, D.; Stampacchia, G. An Introduction to Variational Inequalities and Their Applications; Academic Press: New York, NY, USA, 1980. [Google Scholar]
  22. Fang, S.C.; Petersen, E.L. Generalized variational inequalities. J. Optim. Theory Appl. 1982, 38, 363–383. [Google Scholar] [CrossRef]
  23. Todd, M.J. The Computations of Fixed Points and Applications; Springer: Berlin/Heidelberg, Germany, 1976. [Google Scholar]
  24. Deutsch, F. Best Approximation in Inner Product Spaces; Springer: New York, NY, USA, 2001. [Google Scholar]
Back to TopTop