International Journal of Analysis and Applications ISSN 2291-8639 Volume 6, Number 1 (2014), 82-88 http://www.etamaths.com BEST PROXIMITY POINTS FOR K− PROXIMAL CONTRACTION AJAY SHARMA, BALWANT SINGH THAKUR∗ Abstract. In this paper, we define K− proximal contraction and prove best proximity point theorem for this contraction. We also provide an illustrative example. 1. Introduction and preliminaries Banach contraction principle is one of the pivotal results of analysis. It is widely considered as the source of metric fixed point theory. Due to its wide applications in various fields, a huge number of generalizations and extended versions of this principle appear in the literature. Some important and interesting generalizations of Banach’s principle is given in [2, 9, 10, 12, 18]. In 1997, Alber and Guerre [1] introduced the notion of weakly contractive self mapping. Definition 1.1. Let (X,d) be a metric space and A be a nonempty subset of X. A mapping T : A → A is said to be weakly contractive if d(Tx,Ty) ≤ d(x,y) −ψ(d(x,y)) , for all x,y ∈ A, where ψ : [0,∞) → [0,∞) is a continuous and nondecreasing function such that ψ is positive on (0,∞), ψ(0) = 0 and limt→∞ψ(t) = ∞. If A is bounded, then the infinity condition can be omitted [1]. They [1] further proved that, if A is a closed convex subset of a Hilbert space, then a weakly contractive self mapping T on A has a unique fixed point. Rhoades [15] extended and improved result of [1] to metric space and established the following: Theorem 1.1. Let (X,d) be a complete metric space, and suppose that T : X → X satisfies the following inequality (1) d(Tx,Ty) ≤ d(x,y) −ψ(d(x,y)) , for all x,y ∈ X where ψ : [0,∞) → [0,∞) is a continuous and nondecreasing function such that ψ(t) = 0 if and only if t = 0. Then T has a unique fixed point. 2010 Mathematics Subject Classification. 54H25. Key words and phrases. K− proximal contraction; fixed point. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 82 BEST PROXIMITY POINTS FOR K− PROXIMAL CONTRACTION 83 If we take ψ(t) = (1 − k)t, where 0 < k < 1, then the inequality (1) reduced to Banach contraction, hence the above Theorem 1.1 extends Banach’s contraction principle. The function ψ involved in the inequality (1) is known as alternating distance ( also called control function). It was initially used in 1984 by Khan and Sessa [11]. This function and its generalizations have been used in fixed point problems in metric and probabilistic metric spaces, see for example, [13, 14, 17] Motivated by Chatterjea [7] contraction, recently Choudhary [8] introduced weakly C-contraction mapping. Definition 1.2. Let (X,d) be a metric space. A mapping T : X → X is said to be weakly C−contraction if for all x,y ∈ X, d(Tx,Ty) ≤ 1 2 [d(x,Ty) + d(y,Tx)] −ψ(d(x,Ty),d(y,Tx)), where ψ : [0,∞) × [0,∞) → [0,∞) is a continuous mapping such that ψ(x,y) = 0 if and only if x = y = 0. If we take ψ(x,y) = k(x + y) where 0 < k < 1 2 then it reduces to Chatterjea contraction. Choudhary [8] further proved the following theorem: Theorem 1.2. Let (X,d) be a complete metric space. Then a weakly C−contraction T : X → X has a unique fixed point. Most of the extensions and generalizations of Banach’s contraction principle fo- cused on weakening the contractive condition of the operator or weakening the com- pleteness of the metric space. One method of weakening the contractive condition is to consider non self mapping. Sankar Raj [16] defined the notion of non-self-weakly contractive mappings as follows: Definition 1.3. Let A,B be nonempty subsets of a metric space (X,d). A mapping T : A → B is said to be a weakly contractive mapping if d(Tx,Ty) ≤ d(x,y) −ψ(d(x,y)) , for all x,y ∈ A, where ψ : [0,∞) → [0,∞) is a continuous and nondecreasing function such that ψ is positive on (0,∞), ψ(0) = 0 and limt→∞ψ(t) = ∞. An element x ∈ A is said to be a fixed point of a map T : A → B if Tx = x. Clearly, T(A)∩A 6= ∅; is a necessary (but not sufficient) condition for the existence of a fixed point of T . If T(A) ∩A = ∅, then d(x,Tx) > 0 for all x ∈ A, that is, the set of fixed points of T is empty. In such a situation, one often attempts to find an element x which is in some sense closest to Tx. If such a point exists, we call it best proximity point, i.e., an element x ∈ A is called a best proximity point of T if d(x,Tx) = dist(A,B); where dist(A,B) = inf{d(x,y) : x ∈ A,y ∈ B}. Best proximity point analysis has been developed in this direction. The goal of the best proximity point theory is to furnish sufficient conditions that assure the existence of best proximity points. 84 SHARMA AND THAKUR Before proceeding further let us fix the following notations: A0 = {x ∈ A : d(x,y) = dist(A,B) for some y ∈ B}, B0 = {y ∈ B : d(x,y) = dist(A,B) for some x ∈ A}. If A and B are closed subsets of a normed linear space such that dist(A,B) > 0, then A0 and B0 are contained in the boundaries of A and B, respectively [6]. Let A and B be two nonempty subsets of a metric space (X,d) with A0 6= ∅. Then the pair (A,B) is said to have the P− property [16] if and only if d(x1,y1) = dist(A,B) d(x2,y2) = dist(A,B) } ⇒ d(x1,x2) = d(y1,y2) where x1,x2 ∈ A0 and y1,y2 ∈ B0. For any nonempty subset A of X, the pair (A,A) has the P−property. Sankar Raj [16] proved the following result: Theorem 1.3. Let A and B be two nonempty closed subsets of a metric space (X,d) such that A0 6= ∅. Let T : A → B be a weakly contractive mapping such that T(A0) ⊆ B0. Assume that the pair (A,B) has the P− property. Then there exists a unique x∗ ∈ A such that d(x∗,Tx∗) = dist(A,B). Basha [3] introduced the notion of proximal contraction as follows: Definition 1.4. A mapping T : A → B is said to be proximal contraction if there exists a non-negative number α < 1 such that, for all u1,u2,x1,x2 ∈ A d(u1,Tx1) = dist(A,B) d(u2,Tx2) = dist(A,B) } ⇒ d(u1,u2) ≤ αd(x1,x2) . Every self mapping that is a proximal contraction is essentially a contraction. Definition 1.5. [5] The set B is said to be approximatively compact with respect to A if every sequence {yn} of B satisfying the condition that d(x,yn) → d(x,B) for some x ∈ A has a convergent subsequence. Any compact set is approximatively compact, and that any set is approximatively compact with respect to itself. Further, if A is compact and B is approximatively compact with respect to A, then A0 and B0 are non-empty. Definition 1.6. [4] A mapping T : A → B is said to be a generalized proximal contraction if, for all u1,u2,x1,x2 ∈ A, d(u1,Tx1) = dist(A,B) d(u2,Tx2) = dist(A,B) } ⇒ d(u1,u2) ≤ d(x1,x2) −ψ(d(x1,x2)), where ψ : [0,∞) → [0,∞) is continuous and non-decreasing such that ψ is positive on (0,∞),ψ(0) = 0, and limt→∞ψ(t) = ∞. Motivated by above studies and Kannan contraction, we now introduce the notion of K−proximal contraction: Definition 1.7. Let A and B be two non-empty subsets of a metric space (X,d). Then T : A → B is said to be a K−proximal contraction if for all u1,u2,x1,x2 ∈ A, d(u1,Tx1) = dist(A,B) and d(u2,Tx2) = dist(A,B) BEST PROXIMITY POINTS FOR K− PROXIMAL CONTRACTION 85 implies that d(u1,u2) ≤ 1 2 (d(x1,u1) + d(x2,u2)) −ψ(d(x1,u1),d(x2,u2)) where ψ : [0,∞)× [0,∞) → [0,∞) is a continuous and nondecreasing function such that ψ(x1,x2) = 0 if and only if x1 = x2 = 0. We now establish necessary and sufficient condition for the existence and uniqueness of best proximity point of the K− proximal contraction. Theorem 1.4. Let (X,d) be a complete metric space, A and B be two nonempty, closed subsets of X such that B is approximatively compact with respect to A. Suppose that A0 and B0 are non-empty and T : A → B is a non-self-mapping satisfying the following conditions: (i) T is a K−proximal contraction; (ii) T(A0) ⊆ B0. Then, there exists a unique element x ∈ A such that d(x,Tx) = dist(A,B). Further, the sequence {xn} converges to the best proximity point x, where for a fixed x0 ∈ A0 the sequence {xn} is given by d(xn+1,Txn) = dist(A,B) for all n ≥ 0. Proof. Let x0 be a fixed element in A0. Since T(A0) ⊆ B0, Tx0 is an element of B0. So by the definition of B0, there exists an element x1 ∈ A0 such that d(x1,Tx0) = dist(A,B). Again, since T(A0) ⊆ B0 we have Tx1 ∈ B0, it follows that there is x2 ∈ A0 such that d(x2,Tx1) = dist(A,B). Continuing this process, we can derive a sequence {xn} in A0, such that d(xn+1,Txn) = dist(A,B),(2) for every n ≥ 0. Since T is a K−proximal contraction, for each n ∈ N, we have d(xn,xn+1) ≤ 1 2 (d(xn−1,xn) + d(xn,xn+1)) −ψ(d(xn−1,xn),d(xn,xn+1)) ≤ 1 2 (d(xn−1,xn) + d(xn,xn+1)). Consequently, we get d(xn,xn+1) ≤ d(xn−1,xn). So the sequence {d(xn+1,xn)} is a monotone-decreasing sequence of nonnegative real numbers. Then there exists a number µ ≥ 0 such that d(xn,xn+1) → µ as n →∞.(3) Now, we will show that µ = 0. Since T is a K−proximal contraction, we get d(xn,xn+1) ≤ 1 2 (d(xn−1,xn) + d(xn,xn+1)) −ψ(d(xn−1,xn),d(xn,xn+1)).(4) Letting n →∞ in (4), by using (3) and continuity of ψ, we get µ ≤ 1 2 (µ + µ) −ψ(µ,µ) µ ≤ µ−ψ(µ,µ). 86 SHARMA AND THAKUR or ψ(µ,µ) ≤ 0, which is a contradiction unless µ = 0, that is lim n→∞ d(xn,xn+1) = 0.(5) Now, we show that {xn} is a Cauchy sequence. Suppose that {xn} is not a Cauchy sequence. Then, for every ε > 0 there exists nk > mk ≥ k with d(xmk,xnk ) ≥ ε and d(xmk,xnk−1) < ε, for each k ∈ N. Then, we have, d(xmk,xnk ) ≤ d(xmk,xnk−1) + d(xnk−1,xnk ) < ε + d(xnk−1,xnk ).(6) Taking k →∞ in (6) and using (5), we have lim k→∞ d(xmk,xnk ) = ε. By (2), we have (7) lim k→∞ d(xnk+1,Txnk ) = dist(A,B) , and (8) lim k→∞ d(xmk+1,Txmk ) = dist(A,B) . Since T is a K−proximal contraction, by (7), we have (9) d(xnk+1,xmk+1) ≤ 1 2 (d(xnk,xnk+1) + d(xmk,xmk+1)) −ψ (d(xnk,xnk+1),d(xmk,xmk+1)) . Since, (10) d(xnk,xmk ) ≤ d(xnk,xnk+1) + d(xnk+1,xmk+1) + d(xmk+1,xmk ) . From (9) and (10), we have d(xnk,xmk ) ≤ d(xnk,xnk+1) + d(xmk+1,xmk ) + 1 2 (d(xnk,xnk+1) + d(xmk,xmk+1)) −ψ (d(xnk,xnk+1),d(xmk,xmk+1)) . letting k →∞, we have ε ≤ 0 + 0 + 0 −ψ(0, 0) = 0 , a contradiction. Hence, {xn} is a Cauchy sequence. Since A is a closed subset of a complete metrics space, there exists x ∈ A such that limn→∞xn = x. Taking n → ∞ in (2) and continuity of T, we have d(x,Tx) = dist(A,B). This completes the proof. � Now, we give an example to illustrate Theorem 1.4. Example 1.8. Let X = R2 with the Euclidean metric. Suppose that A = {(1,x) : x ∈ R, 0 ≤ x ≤ 1}, and B = {(0,x) : x ∈ R,x ≥ 0}. BEST PROXIMITY POINTS FOR K− PROXIMAL CONTRACTION 87 Now we define a mapping T : A → B as below: T((1,x)) = ( 0, x 1 + x ) . It is easy to see that d(A,B) = 1, A0 = A and B0 = B. T is continuous, T(A0) ⊂ B0 and d(u1,Tx1) = d(A,B) = 1,d(u2,Tx2) = d(A,B) = 1. for some u1,u2,x1,x2 ∈ A. We will show that T is satisfy the condition (i) of Theorem 1.4 with ψ : [0,∞) × [0,∞) → [0,∞) defined by ψ(t1, t2) = max ( t1 1 + t1 , t2 1 + t2 ) for all (t1, t2) ∈ [0,∞) × [0,∞). Suppose u1 = u2 = (1, 0) and x1 = x2 = (0, 0), then d(x1,u1) = 1 , d(x2,u2) = 1 , d(u1,u2) = 0 and ψ (d(x1,u1),d(x2,u2)) = ψ(1, 1) = max ( 1 2 , 1 2 ) = 1 2 . We can see that d(u1,u2) ≤ 1 2 (d(x1,u1) + d(x2,u2)) −ψ(d(x1,u1),d(x2,u2)) , hence, T is a K−proximal contraction References [1] Ya. I. Alber and S. 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