Best proximity point results in topological spaces and extension of Banach contraction principle
aa r X i v : . [ m a t h . GN ] J u l BEST PROXIMITY POINT RESULTS IN TOPOLOGICALSPACES AND EXTENSION OF BANACH CONTRACTIONPRINCIPLE
SUMIT SOM , SUPRITI LAHA , LAKSHMI KANTA DEY Abstract.
In this paper, we introduce the notion of topologically Banachcontraction mapping defined on an arbitrary topological space X with thehelp of a continuous function g : X × X → R and investigate the existenceof fixed points of such mapping. Moreover, we introduce two types of map-pings defined on a non-empty subset of X and produce sufficient conditionswhich will ensure the existence of best proximity points for these mappings.Our best proximity point results also extend some existing results from met-ric spaces or Banach spaces to topological spaces. More precisely, our newlyintroduced mappings are more general than that of the corresponding notionsintroduced by Bunlue and Suantai [Arch. Math. (Brno), 54(2018), 165-176].We present several examples to validate our results and justify its motivation.To study best proximity point results, we introduce the notions of g -closed, g -sequentially compact subsets of X and produce examples to show that thereexists a non-empty subset of X which is not closed, sequentially compact underusual topology but is g -closed and g -sequentially compact. Introduction
Metric fixed point theory is an essential part of mathematics. It gives neces-sary and sufficient conditions that will ensure the existence of solutions of theequation
U x = x where U is a self-mapping defined on a metric space M . Sucha solution is called a fixed point of the mapping U. Banach contraction principlefor standard metric spaces is a pioneer result in this connection. It has a lot ofapplications in the area of differential equations, integral equations. Over theyears, many Mathematicians have weakened the metric structure and prove theBanach contraction principle for such spaces, but till now, no result is found,which extends the Banach contraction principle from metric spaces to arbitrarytopological spaces. In this paper, we take an arbitrary topological space X anda real-valued continuous function g defined on the Cartesian product X × X. Then we introduce the notion of topologically Banach contraction mapping de-fined on a topological space X with respect to g and investigate the existence offixed points of such mapping. As a consequence, we extend the famous Banachcontraction principle from standard metric spaces to topological spaces and wecan retrieve the Banach contraction principle for metric spaces as a particularcase of our theorem. On the other hand, if U : A → B is mapping where A, B
Mathematics Subject Classification. H
10, 54 H
25, 46 A Key words and phrases.
Best proximity point, topological space, Banach contractionprinciple. are non-empty subsets of the metric space (
M, ρ ) , A = B and U ( A ) ∩ A = ∅ then the mapping U has no fixed points. So, in case of a non-self map, oneseek for an element in the domain space whose distance from its image is mini-mum i.e., the interesting problem is to minimize ρ ( x, U x ) such that x ∈ A. Since ρ ( x, U x ) ≥ D ( A, B ) = inf { ρ ( x, y ) : x ∈ A, y ∈ B } , so one can search for anelement x ∈ A such that ρ ( x, U x ) = D ( A, B ) . Best proximity point problemsdeal with this situation. In the year 2011, Basha [1] investigated the existenceof best proximity points of proximal contractions. In the years 2013 and 2015,Gabeleh [3,4] introduced the notion of proximal nonexpansive mappings, Berindeweak proximal contractions in the context of metric spaces and investigated theexistence of best proximity points of those classes of mappings. For more re-sults, see [5] and the references therein. In the year 2018, Bunlue and Suantai [2]introduced the notion of proximal weak contraction and proximal Berinde nonex-pansive mappings and discussed the existence of best proximity points for thoseclasses of mappings. All these results are formulated in the framework of metricspaces or Banach spaces where the standard metric or norm plays an importantrole. Recently in the year 2020, Raj and Piramatchi [6] presented a way in whichwe can extend the best proximity point results from standard metric spaces totopological spaces, and it is exciting. In this paper, we have introduced the no-tions of topologically proximal weak contraction, topologically proximal Berindenon-expansive on topological spaces and discuss the existence of best proximitypoints for these mappings. We have presented ample examples to validate ourresults. Moreover, we have introduced these notions w.r.t a continuous func-tion and present examples which show that there exist two continuous functionssuch that the mappings are topologically proximal weak contraction or topolog-ically proximal Berinde non-expansive with respect to one continuous functionbut not with respect to another continuous function. Our best proximity pointresult about topologically proximal weak contractions also extends the Banachcontraction principle for non-self mappings.On the other hand, in 1970, Takahashi [7] first introduced the notion of con-vexity in metric spaces, and with the help of this notion, in this paper, we havedefined the concept of topologically convex structure on an arbitrary topologicalspace. Our best proximity results improve and extend the results in [1–4] fromstandard metric spaces, Banach spaces to topological spaces. If the underlyingspace is metrizable with respect to the metric d , then by taking the continuousfunction g = d we will recover those results. To build the theory of this paper, weintroduce the notion of g -closed, g -sequentially compact subset of the topologicalspace X with the help of the continuous function g defined on X × X and presentexamples to show that there exists a non-empty subset of X which is not closedand sequentially compact with respect to the usual topology but is g -closed and g -sequentially compact for some continuous function g. Main results
We introduce some definitions which will be necessary for the development ofour results.
EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 3
Definition 2.1.
Let X be a topological space and g : X × X → R be a continuousfunction. Let { x n } be a sequence in X and x ∈ X. Then { x n } is said to be g -convergent to x if | g ( x n , x ) | → as n → ∞ i.e., for a given ε > k ∈ N such that | g ( x n , x ) | < ε ∀ n ≥ k. Definition 2.2.
Let X be a topological space and g : X × X → R be a continuousfunction. Let { x n } be a sequence in X. Then { x n } is said to be g -Cauchy if | g ( x n , x m ) | → n, m → ∞ i.e., for a given ε > k ∈ N such that | g ( x n , x m ) | < ε ∀ n, m ≥ k. Definition 2.3.
Let X be a topological space and g : X × X → R be a continuousfunction. X is said to be g -complete if every g -Cauchy sequence { x n } is g -convergent to a point x ∈ X. Lemma 2.4.
Let X be a topological space and g : X × X → R be a con-tinuous function such that g ( x, y ) = 0 ⇒ x = y and | g ( x, z ) | ≤ | g ( y, x ) | + | g ( y, z ) | ∀ x, y, z ∈ X. Then the limit of a g -convergent sequence is unique.Proof. The proof is straightforward, so omitted. (cid:3)
We show by an example that if the conditions of the Lemma 2.4 are violated,then the g -limit may not be unique. Example 2.5.
Consider R with usual topology. Define g : R × R → R be defined by g (( x, y ) , ( u, v )) = xu. Then g is a continuous function. Let A =[ − , × [ − , . For the function g we have, g (cid:16) (1 , , (0 , (cid:17) = 0 but (1 , = (0 , . Also g (cid:16) (1 , , (1 , (cid:17) = 1 = 0 . So here, g (cid:16) ( x, y ) , ( u, v ) (cid:17) = 0 < ( x, y ) = ( u, v ) . Also, if we take x = (1 , , y = (0 , , z = (4 ,
0) then | g ( x, z ) | > | g ( y, x ) | + | g ( y, z ) | . Now consider the sequence { x n } ⊂ A defined by x n = ( n , . Then it can be seenthat the sequence { x n } is g -convergent to (0 ,
1) and also to ( , . So the limit isnot unique. In fact, this sequence has infinitely many g -limits.Now we introduce the notion of topologically Banach contraction mapping ina topological space X with respect to a continuous function as follows: Definition 2.6.
Let X be a topological space and g : X × X → R be a continuousfunction. Let T : A → B be a mapping where A, B ⊆ X and A, B = φ. Themapping T is said to be topologically Banach contraction w.r.t g if there exists α ∈ (0 ,
1) such that (cid:12)(cid:12)(cid:12) g (cid:16) T ( x ) , T ( y ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ α (cid:12)(cid:12)(cid:12) g ( x, y ) (cid:12)(cid:12)(cid:12) for all x, y ∈ A. S. SOM, S. LAHA, L.K. DEY
We present an example of a topologically Banach contraction mapping f withrespect to a real valued continuous function g, which is not a contraction map-ping with respect to the metric d with respect to which the space is metrizable.Also, in this example we show that, though, f is a topologically Banach con-traction mapping with respect to a real valued continuous function g, may notbe a topologically Banach contraction mapping with respect another real valuedcontinuous function h. Example 2.7.
Consider R with the usual topology. Let A = [ , × [0 ,
1] and B = [1 , × [0 , . Consider g : R × R → R by g (cid:16) ( x , y ) , ( u , v ) (cid:17) = min { y , v } . Define T : A → B by T ( x, y ) = (2 x, y ) , ( x, y ) ∈ A. Let x = ( t , p ) and x = ( t , p ) ∈ A. Now (cid:12)(cid:12)(cid:12) g (cid:16) T ( x ) , T ( x ) (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) g (cid:16) (2 t , p , (2 t , p (cid:17)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12) g ( x , x ) (cid:12)(cid:12)(cid:12) . So, the mapping T is topologically Banach contraction w.r.t g. It can be seenthat this mapping is not a contraction with respect to the usual metric on R . Now define h : R × R → R by h (cid:16) ( x , y ) , ( u , v ) (cid:17) = x u . Then h is acontinuous function. Now we will show that the mapping T is not topologicallyBanach contraction w.r.t h. Let α ∈ (0 , , x = ( , , y = (1 , . Now (cid:12)(cid:12)(cid:12) h ( T ( x ) , T ( y )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) h ((1 , , (2 , (cid:12)(cid:12)(cid:12) = 2 > α | h ( x, y ) | = α . So the mapping T is not topologically Banach contraction w.r.t h. Now, we give an example of a topologically Banach contraction mapping on anon-metrizable topological space.
Example 2.8.
Consider R ω , the space of all real valued sequences with the boxtopology. Then the space R ω is not metrizable. Define g : R ω × R ω → R by g (cid:16) ( x n ) , ( y n ) (cid:17) = x y ; ( x n ) , ( y n ) ∈ R ω . Then g is a continuous function on R ω × R ω with respect to the box topology.Define f : R ω → R ω by f ( x , x , . . . ) = (0 , x , x , . . . ); ( x , x , . . . ) ∈ R ω . Now let x = ( x n ) , y = ( y n ) ∈ R ω and α ∈ (0 , . Then0 = (cid:12)(cid:12)(cid:12) g (cid:16) f ( x ) , f ( y ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ α (cid:12)(cid:12)(cid:12) g (cid:16) x, y (cid:17)(cid:12)(cid:12)(cid:12) . So, f is a topologically Banach contraction mapping w.r.t g. We present our first desired result ‘Banach contraction principle’ in a topolog-ical space.
Theorem 2.9.
Let X be a g -complete topological space where g : X × X → R is a continuous function such that g ( x, y ) = 0 = ⇒ x = y, | g ( x, y ) | = | g ( y, x ) | , | g ( x, z ) | ≤ | g ( x, y ) | + | g ( y, z ) | for all x, y, z ∈ X. Let U : X → X be a topologicallyBanach contraction mapping w.r.t g . Then U has a unique fixed point and for EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 5 any p ∈ X, the sequence p n +1 = U ( p n ) for all n ≥ will converge to the uniquefixed point of U. Proof. As U : X → X is topologically Banach contraction map w.r.t g so thereexists β ∈ (0 ,
1) such that | g ( U ( x ) , U ( y )) | ≤ β | g ( x, y ) | for all x, y ∈ X. Let p ∈ X and define a sequence { p n } ⊂ X by p n +1 = U ( p n ) for all n ≥ , n ∈ N . Now | g ( p n +1 , p n ) | = | g ( U ( p n ) , U ( p n − ) |≤ α | g ( p n , p n − ) | = α | g ( U ( p n − ) , U ( p n − ) | ... ≤ α n | g ( p , p ) | . Suppose that m > n and n ∈ N . Let m = n + r where r ≥ . Now, | g ( p n , p n + r ) | ≤ | g ( p n , p n +1 ) | + | g ( p n +1 , p n +2 ) | + · · · + | g ( p n + r − , p n + r ) | = ⇒ | g ( p n , p n + r ) | ≤ (cid:16) β n + β n +1 + · · · + β n + r − (cid:17) | g ( p , p ) | = ⇒ | g ( p n , p n + r ) | ≤ β n (cid:16) − β r − β (cid:17) | g ( p , p ) | → n, r → ∞ . This shows that the sequence { p n } n ≥ is a g -Cauchy sequence. Since X is g -complete, so the sequence { p n } n ≥ is g -convergent to p ∗ ∈ X (say) . Now since U is topologically Banach contraction map so, | g ( U ( p n ) , U ( p ∗ )) | ≤ β | g ( p n , p ∗ ) | → n → ∞ . This shows that the sequence { U ( p n ) } n ≥ is g -convergent to U ( p ∗ ) . But p n +1 = U ( p n ) so, { U ( p n ) } n ≥ is g -convergent to p ∗ . As the continuous function g satisfiesthe conditions of Lemma 2.4, so the limit is unique. So, U ( p ∗ ) = p ∗ . So themapping U has a fixed point. Now suppose U has two fixed points p ∗ and p ∗∗ , p ∗ = p ∗∗ . Now, | g ( U ( p ∗ ) , U ( p ∗∗ )) | ≤ β | g ( p ∗ , p ∗∗ ) | = ⇒ | g ( p ∗ , p ∗∗ ) | ≤ β | g ( p ∗ , p ∗∗ ) | . This is a contradiction since | g ( p ∗ , p ∗∗ ) | > . So the mapping U has unique fixedpoint. (cid:3) Note 2.10.
The preceding theorem is an extension of Banach contraction princi-ple from metric space to general topological space X with a continuous real-valuedfunction g defined on X × X. If the space X is metrizable with respect to a metric d then by taking g = d, we will get the Banach contraction principle for standardmetric spaces. S. SOM, S. LAHA, L.K. DEY
Example 2.11.
Consider X = [0 ,
1] with the usual standard subspace topologyinherit from R . Define g : X × X → R by g ( x, y ) = x − y . Then g is a continuous function on R × R . Define T : X → X by T ( x ) = x , x ∈ X. It can be seen that [0 ,
1] is g -complete. Here the continuous function g satisfiesall the conditions of Theorem 2.9. But g is not a metric, since g can take negetivevalues. Let x, y ∈ [0 , . Now (cid:12)(cid:12)(cid:12) g ( T ( x ) , T ( y )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) g ( x , y (cid:12)(cid:12)(cid:12) = 14 | ( x − y | ) = 14 (cid:12)(cid:12)(cid:12) g ( x, y ) (cid:12)(cid:12)(cid:12) . So T is topologically Banach contraction map w.r.t g . By previous Theorem 2.9, T has unique fixed point. Here p ∗ = 0 is the fixed point of T. Now in the upcoming example, we will show that if any one of the conditionsof the continuous function g defined in Theorem 2.9 is violated, then there mayexist infinitely many fixed points of the mapping T. Example 2.12.
Consider the space R with the usual topology. Define g : R × R → R by g (cid:16) ( x, y ) , ( u, v ) (cid:17) = y − v, ( x, y ) , ( u, v ) ∈ R . Then g is a continuous function. Here g (cid:16) (1 , , (4 , (cid:17) = 0 but (1 , = (4 , . Sothe function g does not satisfied all the conditions of Theorem 2.9. Now define T : R → R by T (( x, y )) = ( x, y , ( x, y ) ∈ R . Now we will show that T is topologically Banach contraction map w.r.t g. Let( x , y ) , ( x , y ) ∈ R . Now (cid:12)(cid:12)(cid:12) g (cid:16) T ( x , y ) , T ( x , y ) (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) g (cid:16) ( x , y , ( x , y (cid:17)(cid:12)(cid:12)(cid:12) = 12 | y − y | = 12 (cid:12)(cid:12)(cid:12) g (cid:16) ( x , y ) , ( x , y ) (cid:17)(cid:12)(cid:12)(cid:12) . So, T is a topologically Banach contraction map w.r.t g. Let { ( x n , y n ) } be a g -Cauchy sequence in R . So (cid:12)(cid:12)(cid:12) g (cid:16) ( x n , y n ) , ( x m , y m ) (cid:17)(cid:12)(cid:12)(cid:12) → n, m → ∞ = ⇒ | y n − y m | → n, m → ∞ . So the sequence { y n } is a Cauchy sequence of real numbers. Let y n → y ∈ R as n → ∞ . Fix n ∈ N . Now (cid:12)(cid:12)(cid:12) g (cid:16) ( x n , y n ) , ( x n , y ) (cid:17)(cid:12)(cid:12)(cid:12) = | y n − y | → n → ∞ . This shows that the sequence { ( x n , y n ) } is g -convergent to ( x n , y ) . So R is g -complete. It can be seen that for any x ∈ R , p ∗ = ( x,
0) is a fixed point of T. Sothere are infinitely many fixed points of T. EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 7
Corollary 2.13.
Let X be a g -complete topological space where g : X × X → R is a continuous function such that g ( x, y ) = 0 = ⇒ x = y, | g ( x, y ) | = | g ( y, x ) | , | g ( x, z ) | ≤ | g ( x, y ) | + | g ( y, z ) | for all x, y, z ∈ X. Let U : X → X be a mappingsuch that for some n ∈ N , U n : X → X is a topologically Banach contractionmapping w.r.t g . Then U has a unique fixed point.Proof. The proof follows from Theorem 2.9, so omitted. (cid:3)
Now we recall the following definition from [6].
Definition 2.14. [6] Let A , B be non-empty subsets of a topological space X .Let g : X × X → R be a continuous function. Define D g ( A, B ) = inf {| g ( x, y ) | : x ∈ A, y ∈ B } . In this paper, we will use the following definitions. A g = { x ∈ A : | g ( x, y ) | = D g ( A, B ) for some y ∈ B } and B g = { y ∈ B : | g ( x, y ) | = D g ( A, B ) for some x ∈ A } . We like to introduce the definition of topologically proximal weak contractionin a topological space X as follows: Definition 2.15.
Let (
A, B ) be a pair of non-empty subsets of a topological space X. A mapping f : A → B is said to be topologically proximal weak contractionwith respect to a continuous function g : X × X → R if there exists β ∈ (0 , N ≥ | g ( u , f ( x )) | = D g ( A, B ) | g ( u , f ( x )) | = D g ( A, B ) (cid:27) = ⇒ | g ( u , u ) | ≤ β | g ( x , x ) | + N | g ( x , u ) | for all x , x , u , u ∈ A. Note 2.16.
In Definition 2.15, if we take A = B and N = 0 we may not getthe Definition 2.6 because the continuous function g does not necessarily satisfy g ( x, y ) = 0 < x = y as we already see in Example 2.5. Note 2.17.
If the topological space X is metrizable with respect to a metric d ,then by taking g = d we will get the notion of proximal weak contraction forstandard metric spaces introduced by Bunlue and Suantai in [2]. In particular,if we take g = d and N = 0 then we will get the notion of proximal contractionintroduced by Basha in [1]. In our last definition, we mention that the mapping f is a topologically proximalweak contraction with respect to the continuous mapping g , and it is important.In our upcoming example we will show that there exist two subsets A and B in a topological space X and a mapping f : A → B such that f is topologi-cally proximal weak contraction with respect to a continuous function g but isnot topologically proximal weak contraction with respect to another continuousfunction h. S. SOM, S. LAHA, L.K. DEY
Example 2.18.
Consider R with the usual topology. Let A = { } × [ − , B = { } × [ − , . Let T : A → B be defined by T (0 , y ) = (1 , y ) . Let g : R × R → R be defined by g (cid:16) ( x, y ) , ( u, v ) (cid:17) = y − v . Then g is a continuousfunction. Now we will show that T is a topologically proximal weak contractionwith respect to g. It is clear that D g ( A, B ) = 0 . Let x = (0 , p ) , x = (0 , p ) , u =(0 , y ) , u = (0 , y ) ∈ A and | g ( x , T ( u )) | = 0 and | g ( x , T ( u )) | = 0 . So (cid:12)(cid:12)(cid:12) g ((0 , p ) , (1 , y (cid:12)(cid:12)(cid:12) = 0= ⇒ p − y
16 = 0 . Similarly, from the second equation, we get, p − y
16 = 0 . Now, | g ( x , x ) | = p − p = ( y − y ) = | g ( u , u ) | . This shows that T is atopologically proximal weak contraction with respect to g with β = and N = 0 . Now let h : R × R → R be defined by h (cid:16) ( x, y ) , ( u, v ) (cid:17) = min { y, v } . It can beseen that D h ( A, B ) = 0 . Let x = (0 , ) , x = (0 , ) , u = (0 , , u = (0 , ∈ A and | h ( x , T ( u )) | = 0 and | h ( x , T ( u )) | = 0 . Now if β ∈ (0 ,
1) and N ≥ | h ( x , x ) | > β | h ( u , u ) | + N | h ( u , x ) | = 0 . This shows that T is not topologically proximal weak contraction with respect to h. In our next example, we show that the notion of topologically proximal weakcontraction with respect to a continuous function is indeed more general thanthe notion of proximal weak contraction introduced by Bunlue and Suantai in[2]. We show that, there exists a topological space X with a continuous realvalued function g , two non-empty disjoint subsets A, B of X and a function f : A → B such that f is topologically proximal weak contraction w.r.t g butif the topological space is metrizable with respect to a metric d then f is notproximal weak contraction w.r.t the metric d. Example 2.19.
Consider R with the usual topology. Let g : R × R → R bedefined by g ( x, y ) = x − y , x, y ∈ R . Then g is a continuous function. Let A = { , , , , } and B = {− , − , − , } . Let f : A → B be defined by f (0) = f (3) = f (5) = 4 , f (1) = − , f (2) = − . Then it can be seen that D g ( A, B ) = 0 . Let β = and N = 1 . Now (cid:12)(cid:12)(cid:12) g (cid:16) , f (1) (cid:17)(cid:12)(cid:12)(cid:12) = D g ( A, B )and (cid:12)(cid:12)(cid:12) g (cid:16) , f (2) (cid:17)(cid:12)(cid:12)(cid:12) = D g ( A, B ) . EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 9
Now 3 = (cid:12)(cid:12)(cid:12) g (cid:16) , (cid:17)(cid:12)(cid:12)(cid:12) ≤ . (cid:12)(cid:12)(cid:12) g (cid:16) , (cid:17)(cid:12)(cid:12)(cid:12) + 1 . (cid:12)(cid:12)(cid:12) g (cid:16) , (cid:17)(cid:12)(cid:12)(cid:12) . This shows that f is topologicallyproximal weak contraction w.r.t g with β = and N = 1 . Let d denote the usualmetric on R and D ( A, B ) = inf { d ( x, y ) : x ∈ A, y ∈ B } = 1 . Now d (cid:16) , f (0) (cid:17) = D ( A, B )and g (cid:16) , f (1) (cid:17) = D ( A, B ) . But 5 = d (5 , > .d (0 ,
1) + 1 .d (1 , . So f is not proximal weak contraction with respect to the usual metric on R with β = and N = 1 . In the following, we present a sufficient condition for topologically proximalweak contraction mappings to have a unique best proximity point in arbitrarytopological space X. Before that, we introduce the definition of a g -closed set and g -sequentially compact set as follows: Definition 2.20.
Let X be a topological space and g : X × X → R be acontinuous function. A non-empty subset A of X is said to be g -closed if every g -convergent sequence { x n } ⊂ A , converges to a point in A. Definition 2.21.
Let X be a topological space and g : X × X → R be acontinuous function. A non-empty subset A of X is said to be g -sequentiallycompact if every sequence { x n } n ∈ N in A has a g -convergent subsequence { x n k } which converges to a point in A. In the upcoming example we show that there exists a non-empty subset A oftopological space X such that A is g -closed but not closed in X with respect tothe usual topology. We also find a non-empty set which is g -sequentially compactbut not sequentially compact with respect to the usual topology. Example 2.22.
Consider R with the usual topology and let g : R × R → R bedefined by g ( x, y ) = x − y + . Let A = (0 , ∞ ) . Then A is not closed with respectto the usual topology in R . Let { x n } be a sequence in A which is g -convergent to x ∈ R . So (cid:12)(cid:12)(cid:12) g ( x n , x ) (cid:12)(cid:12)(cid:12) → n → ∞ = ⇒ (cid:12)(cid:12)(cid:12) x n − x + 12 (cid:12)(cid:12)(cid:12) → n → ∞ = ⇒ x n → ( x −
12 ) as n → ∞ . But since { x n } is a sequence in (0 , ∞ ) so, we have x − ≥ . This shows that x ≥ and A is g -closed.Now consider R with the usual topology and g : R × R → R be defined by g (cid:16) ( x, y ) , ( u, v ) (cid:17) = y − v, ( x, y ) , ( u, v ) ∈ R . Then g is a continuous function. Let B = { n : n ∈ N } × { } ∪ { n : n ∈ N } . Then B is not sequentially compact in R with respect to the usual topology. If { x n } is a sequence in B with finite range and { x n } is either constant or ultimatelyconstant sequence then it is clear that there exists p ∈ B such that (cid:12)(cid:12)(cid:12) g ( x n , p ) (cid:12)(cid:12)(cid:12) → n → ∞ . So, in this case, we take the subsequence as the sequence itself and { x n } is g -convergent to p ∈ B. On the other hand, let { x n } is a sequence in B withfinite range and ( p, q ) ∈ B is a cluster point of the sequence. In this case, wetake the subsequence as, x n k = ( p, q ) for all k ∈ N . So, in this case { x n k } is g -convergent to ( p, q ) ∈ B. Now let { ( p n , t n ) } be a sequence in B with infiniterange and t n → n → ∞ . In this case the sequence { ( p n , t n ) } is g -convergentto (1 , ∈ B since (cid:12)(cid:12)(cid:12) g (cid:16) ( p n , t n ) , (1 , (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) t n (cid:12)(cid:12)(cid:12) → n → ∞ . Similarly if { ( p n , t n ) } is a sequence in B with infinite range and p n → n → ∞ then similarly we can show that there exists a subsequence of { ( p n , t n ) } which is g -convergent to some element of B. So, B is g -sequentially compact in R butnot sequentially compact.In our next example we show that a non-empty subset, which is g -closed, where g is a real valued continuous function on X × X , may not be h -closed with respectto another real valued continuous function h on X × X. Example 2.23.
Consider R with the usual topology and let g : R × R → R be defined by g ( x, y ) = x − y + . Let A = (0 , ∞ ) . Then from example 2.22, A is g -closed but A is not closed with respect to usual topology on R . Now let h : R × R → R be defined by h ( x, y ) = xy. Now we will show that the set A is not h -closed. Consider the sequence { n } in A. Now the sequence { n } is h -convergentto − since (cid:12)(cid:12)(cid:12) h ( 1 n , −
12 ) (cid:12)(cid:12)(cid:12) = − n → n → ∞ but − / ∈ A. So A is not h -closed. Theorem 2.24.
Let X be a g -complete topological space where g : X × X → R is a continuous function such that g ( x, y ) = 0 ⇔ x = y, | g ( x, y ) | = | g ( y, x ) | and | g ( x, z ) | ≤ | g ( x, y ) | + | g ( y, z ) | for all x, y, z ∈ X. Let ( A, B ) be a pair of non-empty subsets of X such that A g is non-empty and g -closed. Let T : A → B betopologically proximal weak contraction mapping w.r.t g with β ∈ (0 , , N ≥ such that T ( A g ) ⊆ B g . Then (1) there exists a best proximity point p ∗ ∈ A g of T and the sequence { p n } n ≥ defined by p ∈ A g and | g ( p n +1 , T ( p n )) | = D g ( A, B ) converges to the bestproximity point of T ;(2) moreover if (1 − β − N ) > , then the best proximity point p ∗ is unique.Proof. Let p ∈ A g . Since T ( A g ) ⊆ B g , we have T ( p ) ∈ B g . So, there exists p ∈ A g such that | g ( p , T ( p )) | = D g ( A, B ) . Similarly, as T ( p ) ∈ B g , so there EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 11 exists p ∈ A g such that | g ( p , T ( p )) | = D g ( A, B ) . Continuing this process, weget a sequence { p n } n ≥ ⊂ A g such that | g ( p n +1 , T ( p n )) | = D g ( A, B ) ∀ n ≥ . Now we will show that the sequence { p n } n ≥ is a g -Cauchy sequence. From theconstruction of the sequence we have, | g ( p n , T ( p n − )) | = D g ( A, B )and | g ( p n +1 , T ( p n )) | = D g ( A, B ) . As T is a topologically proximal weak contraction mapping w.r.t g , so we have, | g ( p n , p n +1 ) | ≤ β | g ( p n − , p n ) | + N | g ( p n , p n ) |⇒ | g ( p n , p n +1 ) | ≤ β | g ( p n − , p n ) | . So, we get | g ( p n , p n +1 ) | ≤ β n | g ( p , p ) | . Suppose that m > n and n ∈ N . Let m = n + r where r ≥ . Now, | g ( p n , p n + r ) | ≤ | g ( p n , p n +1 ) | + | g ( p n +1 , p n +2 ) | + · · · + | g ( p n + r − , p n + r ) | = ⇒ | g ( p n , p n + r ) | ≤ (cid:16) β n + β n +1 + · · · + β n + r − (cid:17) | g ( p , p ) | = ⇒ | g ( p n , p n + r ) | ≤ β n (cid:16) − β r − β (cid:17) | g ( p , p ) | → n, r → ∞ . This shows that the sequence { p n } n ≥ is a g -Cauchy sequence. Since X is g -complete, so the sequence { p n } n ≥ is g -convergent to a point p ∗ ∈ X. Since A g is g -closed so p ∗ ∈ A g . Since T ( p ∗ ) ∈ B g so there exists x ∈ A g such that | g ( x, T ( p ∗ ) | = D g ( A, B ) . Also, | g ( p n +1 , T ( p n )) | = D g ( A, B ) . Thus we have | g ( p n +1 , x ) | ≤ β | g ( p n , p ∗ ) | + N | g ( p ∗ , p n +1 ) | = ⇒ | g ( p n +1 , x ) | → n → ∞ . This shows that the sequence { p n } n ≥ is also g -convergent to x ∈ A g . But sincethe limit is unique as we see from Lemma 2.4, so, x = p ∗ . We have | g ( p ∗ , T ( p ∗ ) | = D g ( A, B ) that is, p ∗ is a best proximity point of T. Now, suppose the mapping T has two best proximity points p ∗ and p ∗∗ . So we have | g ( p ∗ , T ( p ∗ ) | = D g ( A, B )and | g ( p ∗∗ , T ( p ∗∗ ) | = D g ( A, B ) . As T is topologically proximal weak contraction, so we have, | g ( p ∗ , p ∗∗ ) | ≤ β | g ( p ∗ , p ∗∗ ) | + N | g ( p ∗ , p ∗∗ ) | = ⇒ (1 − β − N ) | g ( p ∗ , p ∗∗ ) | ≤ ⇒ | g ( p ∗ , p ∗∗ ) | = 0 [since (1 − β − N ) > ⇒ p ∗ = p ∗∗ [since g ( x, y ) = 0 ⇒ x = y ] . So the best proximity point is unique. (cid:3)
Example 2.25.
Consider R with the usual topology and X = { } × [ − ,
1] withsubspace topology. Let g : X × X → R be defined by g (cid:16) ( x, y ) , ( u, v ) (cid:17) = y − v, ( x, y ) , ( u, v ) ∈ X. Then g is a continuous function on X × X. Now we will show that X is g -complete.Let { (1 , x n ) } be a g -Cauchy sequence in X. So (cid:12)(cid:12)(cid:12) g (cid:16) (1 , x n ) , (1 , x m ) (cid:17)(cid:12)(cid:12)(cid:12) → n, m → ∞ = ⇒ | x n − x m | → n, m → ∞ . So the sequence { x n } is a Cauchy sequence in [0 , . Since [ − ,
1] is complete, solet x n → p ∈ [ − ,
1] as n → ∞ . Now, (cid:12)(cid:12)(cid:12) g (cid:16) (1 , x n ) , (1 , p ) (cid:17)(cid:12)(cid:12)(cid:12) = | x n − p | → n → ∞ . So the sequence { (1 , x n ) } is g -convergent to (1 , p ) ∈ X. So X is g -complete. Nowlet A = { }× [ − ,
0] and B = { }× [0 , . Then D g ( A, B ) = 0 . Now let (1 , x ) ∈ A g . Then there exists (1 , y ) ∈ B such that | g ((1 , x ) , (1 , y )) | = 0 . So | x − y | = 0 . Thisis satisfied only by x = 0 . This shows that A g = { (1 , } . Also, B g = { (1 , } . So, A g is non-empty and g -closed. Now it can be seen that the function g is satisfiedall the conditions of Theorem 2.24.Now define f : A → B by f (1 , x ) = (1 , − x , (1 , x ) ∈ A. So, f (1 ,
0) = (1 ,
0) = ⇒ f ( A g ) ⊆ B g . Let (1 , p ) , (1 , p ) , (1 , u ) , (1 , u ) ∈ A suchthat (cid:12)(cid:12)(cid:12) g (cid:16) (1 , p ) , f (1 , u ) (cid:17)(cid:12)(cid:12)(cid:12) = 0and (cid:12)(cid:12)(cid:12) g (cid:16) (1 , p ) , f (1 , u ) (cid:17)(cid:12)(cid:12)(cid:12) = 0 . These two equations imply that p + u = 0 and p + u = 0 . Now (cid:12)(cid:12)(cid:12) g (cid:16) (1 , p ) , (1 , p ) (cid:17)(cid:12)(cid:12)(cid:12) = | p − p | = 12 | u − u | = 12 (cid:12)(cid:12)(cid:12) g (cid:16) (1 , u ) , (1 , u ) (cid:17)(cid:12)(cid:12)(cid:12) . This shows that the mapping f is topologically proximal weak contraction map-ping with respect to g. So all the conditions of Theorem 2.24 are satisfied. So,by the Theorem 2.24 the mapping f has a best proximity point in A g . Here p ∗ = (1 , ∈ A g is a best proximity point of f. In this example the best proxim-ity point is unique because here (1 − β − N ) = > . So the Theorem 2.24 holdsgood.
Note 2.26.
Theorem 2.24 is an extension and improvement of Theorem . of [2]from metric space to topological space X with a continuous function defined on X × X. If the topological space X is metrizable with respect to metric d thenby taking g = d we will get Theorem 3.1 of [2]. Also in Theorem 2.24 if wetake A = B, g = d and N = 0 then we get the Banach contraction principlefor topological spaces. So Theorem 2.24 is an extension of Banach contractionprinciple from standard metric spaces to topological spaces. EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 13
In a topological space X on which there is no linear space structure, it is hard todefine the notion of convex sets in X. In order to define the notion of convex setsin an arbitrary topological space X with a continuous function g : X × X → R ,we first introduce the notion of topologically convex structure on X as follows: Definition 2.27.
Let X be a topological space and g : X × X → R be acontinuous function. A continuous function H : X × X × [0 , → X is calledtopologically convex structure w.r.t g if the two conditions are satisfied:(1) | g ( x , H ( x, y, λ )) | ≤ λ | g ( x , x ) | +(1 − λ ) | g ( x , y ) | for all x , x, y ∈ X and λ ∈ [0 , | g ( H ( x, y, λ ) , H ( x , y , λ )) | ≤ λ | g ( x, x ) | +(1 − λ ) | g ( y, y ) | for all x , x, y, y ∈ X and λ ∈ [0 , . A non empty subset A of X is said to be convex if H ( x, y, λ ) ∈ A for all x, y ∈ A and λ ∈ [0 , . Now by using the notion of topologically convex structure, wewill define the concept of topologically r -starshaped subset of X as follows: Definition 2.28.
Let X be a topological space and g : X × X → R be acontinuous function. Let H : X × X × [0 , → X be a topologically convexstructure on X w.r.t g. A non empty subset A of X is called topologically r -starshaped if there exists a point r ∈ A such that H ( r, x, λ ) ∈ A for all x ∈ A and λ ∈ [0 , . Now we present a lemma that will be necessary for our upcoming theoremabout best proximity points.
Lemma 2.29.
Let X be a topological space and g : X × X → R be a continuousfunction. Let H : X × X × [0 , → X be a topologically convex structure on X w.r.t g. Let
A, B ( = φ ) ⊂ X such that A is topologically r -starshaped and B is topologically s -starshaped and | g ( r, s ) | = D g ( A, B ) . Then A g is topologically r -starshaped and B g is topologically s -starshaped.Proof. Since | g ( r, s ) | = D g ( A, B ) so r ∈ A g and A g = φ. Let x ∈ A g and λ ∈ [0 , . Since x ∈ A and A is topologically r -starshaped, so H ( r, x, λ ) ∈ A. Since x ∈ A g so there exists y ∈ B g such that | g ( x, y ) | = D g ( A, B ) . Since y ∈ B and B istopologically s -starshaped, so H ( s, y, λ ) ∈ B. Now D g ( A, B ) ≤ (cid:12)(cid:12)(cid:12) g ( H ( r, x, λ ) , H ( s, y, λ )) (cid:12)(cid:12)(cid:12) ≤ λ | g ( r, s ) | + (1 − λ ) | g ( x, y ) | = D g ( A, B ) . This shows that H ( r, x, λ ) ∈ A g and A g is topologically r -starshaped. Similarlywe can show that B g is topologically s -starshaped. (cid:3) Now we introduce the concept of topologically semi-sharp proximinal pair inan arbitrary topological space X as follows: Definition 2.30.
Let (
A, B ) be a pair of non-empty subsets of a topologicalspace X . The pair ( A, B ) is said to be a topologically semi-sharp proximinal pairw.r.t a continuous function g : X × X → R if for each x ∈ A there exists at mostone x ∗ in B such that | g ( x, x ∗ ) | = D g ( A, B ) . Lemma 2.31.
Let X be topological space and g : X × X → R be a continuousfunction. Let ( A, B )( = φ ) ⊂ X be a topologically semi-sharp proximinal pair w.r.t g such that A g , B g = φ. Then ( A g , B g ) is a topologically semi-sharp proximinalpair w.r.t g. Proof.
The proof is straightforward, so omitted. (cid:3)
Now we like to introduce the notion of topologically proximal Berinde non-expansive mapping in a topological space X as follows: Definition 2.32.
Let (
A, B ) be a pair of non-empty subsets of a topologicalspace X. Let g : X × X → R be a continuous function. A mapping f : A → B is said to be topologically proximal Berinde non-expansive w.r.t g , if there exists N ≥ | g ( u , f ( x )) | = D g ( A, B ) | g ( u , f ( x )) | = D g ( A, B ) (cid:27) = ⇒ | g ( u , u ) | ≤ | g ( x , x ) | + N | g ( x , u ) | for all x , x , u , u ∈ A. Note 2.33.
If the topological space X is metrizable with respect to a metric d ,then by taking g = d we will get the notion of proximal Berinde nonexpansivemappings for standard metric spaces introduced by Bunlue and Suantai in [2]. Inparticular, if we take g = d and N = 0 then we will get the notion of proximalnonexpansive mappings introduced by Gabeleh in [3]. In our last definition, we mention that the mapping f is a topologically proxi-mal Berinde non-expansive w.r.t the continuous mapping g , and it is important.In our upcoming example, we will show that there exist two subsets A and B in atopological space X and a mapping f : A → B such that f is topologically prox-imal Berinde non-expansive w.r.t a continuous function g but is not topologicallyproximal Berinde non-expansive w.r.t another continuous function h. Example 2.34.
Consider X = [0 , × R with the usual subspace topology of R × R . Let A = { } × R and B = { } × R . Let T : A → B be defined by T (0 , y ) = (3 , y ) . Let g : R × R → R be defined by g (( x, y ) , ( u, v )) = y − v . Then g is a continuous function. Now we will show that T is a topologicallyproximal Berinde non-expansive w.r.t g. It is clear that D g ( A, B ) = 0 . Now let x = (0 , p ) , x = (0 , p ) , u = (0 , y ) , u = (0 , y ) ∈ A and | g ( x , T ( u )) | = 0 and | g ( x , T ( u )) | = 0 . So (cid:12)(cid:12)(cid:12) g ((0 , p ) , (3 , y )) (cid:12)(cid:12)(cid:12) = 0= ⇒ p − y = 0 . Similarly, from the second equation, we get, p − y = 0 . Now, | g ( x , x ) | = | p − p | = | y − y | = | g ( u , u ) | . This shows that T is atopologically proximal Berinde non-expansive w.r.t g with N = 0 . EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 15
Now let h : R × R → R be defined by h (( x, y ) , ( u, v )) = min { y, v } . It can beseen that D h ( A, B ) = 0 . Let x = (0 , , x = (0 , , u = (0 , , u = (0 , ∈ A and | h ( x , T ( u )) | = 0 and | h ( x , T ( u )) | = 0 . Now if N ≥ | h ( x , x ) | > | h ( u , u ) | + N | h ( u , x ) | = 0 . This shows that T is not topologically proximal Berinde non-expansive w.r.t h. In our next example, we show that the notion of topologically proximal Berindenon-expansive mapping with respect to a continuous function is indeed moregeneral than the notion of proximal Berinde non-expansive mapping introducedby Bunlue and Suantai in [2]. We show that, there exists a topological space X with a continuous real valued function g , two non-empty disjoint subsets A, B of X and a function f : A → B such that f is topologically proximal Berindenon-expansive w.r.t g but if the topological space is metrizable with respect to ametric d then f is not proximal Berinde non-expansive w.r.t the metric d. Example 2.35.
Consider R with the usual topology. Let g : R × R → R bedefined by g ( x, y ) = x − y , x, y ∈ R . Then g is a continuous function. Let A = { , , , , } and B = {− , − , − , } . Let f : A → B be defined by f (0) = f (3) = f (5) = 4 , f (1) = − , f (2) = − . Then it can be seen that D g ( A, B ) = 0 . Let N = 1 . Now (cid:12)(cid:12)(cid:12) g (cid:16) , f (1) (cid:17)(cid:12)(cid:12)(cid:12) = D g ( A, B )and (cid:12)(cid:12)(cid:12) g (cid:16) , f (2) (cid:17)(cid:12)(cid:12)(cid:12) = D g ( A, B ) . Now 3 = (cid:12)(cid:12)(cid:12) g (cid:16) , (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) g (cid:16) , (cid:17)(cid:12)(cid:12)(cid:12) + 1 . (cid:12)(cid:12)(cid:12) g (cid:16) , (cid:17)(cid:12)(cid:12)(cid:12) . This shows that f is topologicallyproximal Berinde non-expansive w.r.t g with N = 1 . Let d denote the usual metricon R and D ( A, B ) = inf { d ( x, y ) : x ∈ A, y ∈ B } = 1 . Now d (cid:16) , f (1) (cid:17) = D ( A, B )and g (cid:16) , f (0) (cid:17) = D ( A, B ) . But 3 = d (0 , > d (1 ,
0) + 1 .d (0 , . So f is not proximal Berinde non-expansive with respect to the usual metric on R with N = 1 . We will present a theorem regarding the existence of best proximity point of atopologically proximal Berinde non-expansive mapping in topological spaces.
Theorem 2.36.
Let X be a g -complete topological space where g : X × X → R is a continuous function such that g ( x, y ) = 0 ⇔ x = y, | g ( x, y ) | = | g ( y, x ) | , | g ( x, z ) | ≤ | g ( x, y ) | + | g ( y, z ) | for all x, y, z ∈ X and | g ( r, x ) | + | g ( y, s ) | =2 D g ( A g , B g )( A g , B g = φ ) for all x ∈ B g , y ∈ A g . Let H : X × X × [0 , → X be a topologically convex structure on X w.r.t g. Let ( A, B ) be a topologically semi-sharp proximinal pair of non-empty subsets of X such that A is r -starshaped, B is s -starshaped w.r.t g and | g ( r, s ) | = D g ( A, B ) . Assume A g is compact, g -sequentially compact and g -closed. Suppose that f : A → B satisfies the followingconditions: (1) f is a topologically proximal Berinde non-expansive mapping with respectto g ;(2) f ( A g ) ⊆ B g . Then there exists p ∗ ∈ A g such that | g ( p ∗ , f ( p ∗ )) | = D g ( A, B ) i.e. f has a bestproximity point in A g . Proof.
Let x ∈ A g . Since f ( A g ) ⊆ B g , so f ( x ) ∈ B g . As (
A, B ) is a topologicallysemi-sharp proximinal pair of non-empty subsets of X such that A is r -starshaped, B is s -starshaped w.r.t g then by Lemma 2.29, we have A g is r -starshaped, B g is s -starshaped w.r.t g. Now define the sequence of functions f n : A g → B g by f n ( x ) = H ( s, f ( x ) , a n ) , x ∈ A g . Here the sequence ( a n ) ⊂ (0 ,
1) is such that a n → n → ∞ . Now we will showthat the sequence of functions { f n } is topologically proximal weak contractionfor each n ∈ N . Let p , p , q , q ∈ A g such that (cid:12)(cid:12)(cid:12) g ( p , f n ( q )) (cid:12)(cid:12)(cid:12) = D g ( A g , B g )and (cid:12)(cid:12)(cid:12) g ( p , f n ( q )) (cid:12)(cid:12)(cid:12) = D g ( A g , B g ) . As f ( q ) , f ( q ) ∈ B g so there exist r , r ∈ A g such that | g ( r , f ( q )) | = D g ( A, B )and | g ( r , f ( q )) | = D g ( A, B ) . Since f is topologically proximal Berinde non-expansive map so we have, | g ( r , r ) | ≤ | g ( q , q ) | + N | g ( q , r ) | , N ≥ . (2.1)From Lemma 2.29 and Lemma 2.31, we have the sets A g and B g are topo-logically r -starshaped and topologically s -starshaped respectively and ( A g , B g )is a topologically semi-sharp proximinal pair of the topological space X. So,
EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 17 H ( r, r , a n ) ∈ A g and H ( r, r , a n ) ∈ A g . Now D g ( A g , B g ) ≤ | g ( H ( r, r , a n ) , f n ( q )) | = | g ( f n ( q ) , H ( r, r , a n )) |≤ a n | g ( f n ( q ) , r ) | + (1 − a n ) | g ( f n ( q ) , r ) | = a n | g ( H ( s, f ( q ) , a n ) , r ) | + (1 − a n ) | g ( H ( s, f ( q ) , a n ) , r ) |≤ a n n a n | g ( r, s ) | + (1 − a n ) | g ( r, f ( q ) | o + (1 − a n ) n a n | g ( r , s ) | + (1 − a n ) | g ( r , f ( q ) | o = n a n + (1 − a n ) o D g ( A, B ) + a n (1 − a n ) n | g ( r, f ( q ) | + | g ( r , s ) | o ≤ D g ( A g , B g ) . = ⇒ | g ( H ( r, r , a n ) , f n ( q )) | = D g ( A g , B g ) . Similarly, we can show that | g ( H ( r, r , a n ) , f n ( q )) | = D g ( A g , B g ) . Since ( A g , B g ) is a topologically semi-sharp proximinal pair, so we have p = H ( r, r , a n ) , p = H ( r, r , a n ) . Since A g is a compact topological space so A g × A g is compact. Since g : X × X → R is continuous, so the mapping g restricted to A g × A g is continuous and hencebounded. So there exists M > | g ( x, y ) | ≤ M for all x, y ∈ A g . Now,from equation (2.1) we have, | g ( p , p ) | = | g ( H ( r, r , a n ) , H ( r, r , a n )) |≤ a n | g ( r, r ) | + (1 − a n ) | g ( r , r ) | = (1 − a n ) | g ( r , r ) |≤ (1 − a n ) | g ( q , q ) | + N (1 − a n ) | g ( q , r ) |≤ (1 − a n ) | g ( q , q ) | + N (1 − a n ) M | g ( q , p ) | . Since (1 − a n ) > N (1 − a n ) M ≥ { f n } are topologically proximal weak contraction w.r.t g for each n ∈ N and fromTheorem 2.24 we can say, the mapping f n : A g → B g has a best proximity point p ∗ n ∈ A g such that | g ( p ∗ n , f n ( p ∗ n )) | = D g ( A g , B g ) for all n ∈ N . Since A g is g -sequentially compact, so the sequence { p ∗ n } ⊂ A g has a subsequence { p ∗ n k } whichis g -convergent to p ∗ ∈ A g . Since f ( p ∗ n k ) ∈ B g so there exists y n k ∈ A g such that | g ( y n k , f ( p ∗ n k )) | = D g ( A, B ) . Also we have | g ( p ∗ n k , f n k ( p ∗ n k )) | = D g ( A g , B g ) . Now D g ( A g , B g ) ≤ (cid:12)(cid:12)(cid:12) g ( H ( r, y n k , a n k ) , f n k ( p ∗ n k )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) g ( f n k ( p ∗ n k ) , H ( r, y n k , a n k )) (cid:12)(cid:12)(cid:12) ≤ a n k (cid:12)(cid:12)(cid:12) g ( f n k ( p ∗ n k ) , r ) (cid:12)(cid:12)(cid:12) + (1 − a n k ) (cid:12)(cid:12)(cid:12) g ( f n k ( p ∗ n k ) , y n k ) (cid:12)(cid:12)(cid:12) = a n k (cid:12)(cid:12)(cid:12) g ( H ( s, f ( p ∗ n k ) , a n k ) , r ) (cid:12)(cid:12)(cid:12) + (1 − a n k ) (cid:12)(cid:12)(cid:12) g ( H ( s, f ( p ∗ n k ) , a n k ) , y n k ) (cid:12)(cid:12)(cid:12) ≤ a n k n a n k | g ( r, s ) | + (1 − a n k ) | g ( r, f ( p ∗ n k ) | o + (1 − a n k ) n a n k | g ( y n k , s ) | + (1 − a n k ) | g ( y n k , f ( p ∗ n k ) | o = n a n k + (1 − a n k ) o D g ( A, B ) + a n k (1 − a n k ) n | g ( r, f ( p ∗ n k ) | + | g ( y n k , s ) | o ≤ D g ( A g , B g ) . So we have, (cid:12)(cid:12)(cid:12) g ( H ( r, y n k , a n k ) , f n k ( p ∗ n k )) (cid:12)(cid:12)(cid:12) = D g ( A g , B g ) . So, p ∗ n k = H ( r, y n k , a n k ) . Now, | g ( p ∗ n k , y n k ) | = | g ( H ( r, y n k , a n k ) , y n k ) |≤ a n k | g ( y n k , r ) | + (1 − a n k ) | g ( y n k , y n k ) | = a n k | g ( y n k , r ) | → k → ∞ . Here we use the fact that the sequence a n → n → ∞ . This shows that thesequence { y n k } is g -convergent to p ∗ ∈ A g . As f ( p ∗ ) ∈ B g so there exists p ∗∗ ∈ A g such that | g ( p ∗∗ , f ( p ∗ )) | = D g ( A, B ) . Also we have | g ( y n k , f ( p ∗ n k )) | = D g ( A, B ) . Since f is topologically proximal Berinde non-expansive map w.r.t g so we have, | g ( y n k , p ∗∗ ) | ≤ | g ( p ∗ n k , p ∗ ) | + N | g ( y n k , p ∗ ) | = ⇒ | g ( y n k , p ∗∗ ) | → k → ∞ . So the sequence { y n k } is g -convergent to p ∗∗ ∈ A g . Since the limit is unique wehave p ∗ = p ∗∗ . So, | g ( p ∗ , f ( p ∗ )) | = D g ( A, B ) . Hence the mapping f has a bestproximity point in A g . (cid:3) Now we will provide an example to validate Theorem 2.36.
Example 2.37.
Consider R with the usual topology and X = { } × [ − ,
1] withsubspace topology. Let g : X × X → R be defined by g (cid:16) ( x, y ) , ( u, v ) (cid:17) = y − v, ( x, y ) , ( u, v ) ∈ X. Then g is a continuous function on X × X. Now we will show that X is g -complete.Let { (0 , x n ) } be a g -Cauchy sequence in X. So (cid:12)(cid:12)(cid:12) g (cid:16) (0 , x n ) , (0 , x m ) (cid:17)(cid:12)(cid:12)(cid:12) → n, m → ∞ = ⇒ | x n − x m | → n, m → ∞ . EST PROXIMITY POINT RESULTS IN TOPOLOGICAL SPACES 19
So the sequence { x n } is a Cauchy sequence in [ − , . Since [ − ,
1] is complete,so let x n → x ∈ [ − ,
1] as n → ∞ . Now, (cid:12)(cid:12)(cid:12) g (cid:16) (0 , x n ) , (0 , x ) (cid:17)(cid:12)(cid:12)(cid:12) = | x n − x | → n, m → ∞ . So the sequence { (0 , x n ) } is g -convergent to (0 , x ) ∈ X. So X is g -complete. Nowlet A = { }× [ − ,
0] and B = { }× [0 , . Then D g ( A, B ) = 0 . Now let (0 , x ) ∈ A g . Then there exists (0 , y ) ∈ B such that | g ((0 , x ) , (0 , y )) | = 0 . So | x − y | = 0 . Thisis satisfied only by x = 0 . This shows that A g = { (0 , } . Similarly, B g = { (0 , } . So, A g is compact, g -sequentially compact and g -closed.Now let us define H : X × X × [0 , → X by H (cid:16) (0 , y ) , (0 , y ) , β (cid:17) = (0 , βy + (1 − β ) y ) . It can be seen that the mapping H is topologically convex structure on X w.r.t g and the sets A and B are (0 , X. Also (cid:12)(cid:12)(cid:12) g (cid:16) (0 , , (0 , (cid:17)(cid:12)(cid:12)(cid:12) =0 = D g ( A, B ) . Also the pair (
A, B ) is topologically semi-sharp proximinal in X. Also here, | g ((0 , , x ) | + | g ( y, (0 , | = 0 = 2 D g ( A g , B g )for all x ∈ A g , y ∈ B g . So in this example the function g satisfies all the conditionsof Theorem 2.36. Now define f : A → B by f (0 , x ) = (0 , − x ) , (0 , x ) ∈ A. So, f (0 ,
0) = (0 ,
0) = ⇒ f ( A g ) ⊆ B g . Let (0 , p ) , (0 , p ) , (0 , u ) , (0 , u ) ∈ A suchthat (cid:12)(cid:12)(cid:12) g (cid:16) (0 , p ) , f (0 , u ) (cid:17)(cid:12)(cid:12)(cid:12) = 0and (cid:12)(cid:12)(cid:12) g (cid:16) (0 , p ) , f (0 , u ) (cid:17)(cid:12)(cid:12)(cid:12) = 0 . These two equations imply that p = − u and p = − u . Now (cid:12)(cid:12)(cid:12) g (cid:16) (0 , p ) , (0 , p ) (cid:17)(cid:12)(cid:12)(cid:12) = | p − p | = | u − u | = (cid:12)(cid:12)(cid:12) g (cid:16) (0 , u ) , (0 , u ) (cid:17)(cid:12)(cid:12)(cid:12) . This shows that the mapping f is topologically proximal Berinde non-expansivemapping with respect to g. So all the conditions of Theorem 2.36 are satisfied.So by the Theorem 2.36, the mapping f has a best proximity point in A g . Here p ∗ = (0 , ∈ A g is a best proximity point of f. In this example the best proximitypoint is unique.
Open question.
In Theorem 2.36 we use the condition | g ( r, x ) | + | g ( y, s ) | =2 D g ( A g , B g )( A g , B g = φ ) for all x ∈ B g , y ∈ A g , to prove the existence of bestproximity points for topologically proximal Berinde non-expansive mapping w.r.t g. Can Theorem 2.36 be proved without this condition?
Acknowledgement.
The first and third named authors are thankful to CSIR,Government of India, for their financial support (Ref. No. / / EMR-II).
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Kodai MathSemin Rep , 22, 142-149 (1970) Sumit Som, Department of Mathematics, National Institute of TechnologyDurgapur, India.
E-mail address : [email protected] Supriti Laha, Department of Mathematics, National Institute of Technol-ogy Durgapur, India.
E-mail address : [email protected] Lakshmi Kanta Dey, Department of Mathematics, National Institute ofTechnology Durgapur, India.
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