Fixed point theorem for F w -contractions in complete metric spaces

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1 Journal Nonlinear Analysis and Application 2013 (2013) 1-6 Available online at Volume 2013, Year 2013 Article ID jnaa-00211, 6 Pages doi: /2013/jnaa Research Article Fixed point theorem for F w -contractions in complete metric spaces Rakesh Batra 1, Sachin Vashistha 2 (1) Hans Raj College, University of Delhi,New Delhi, India, Pin code: (2) Hindu College, University of Delhi, New Delhi, India, Pin code: Copyright 2013 c Rakesh Batra and Sachin Vashistha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A new type of contractive mapping known as an F-contraction has been introduced for a metric space recently in Many results in the area of fixed point theory have been proved by different authors, using w-distance. In this paper we extend and generalize the concept of an F-contaction to an F w -contraction and prove a fixed point theorem for an F w -contraction in a complete metric space. Examples are given in support of usability of our results. Keywords: fixed point, complete metric space, w-distance, F-contraction, F w -contaction 1 Introduction Fixed point theory has variety of interesting applications in disciplines such as chemistry, economics, physics, biology and engineering. In dynamical systems it is used to prove several existence and stability results for the strict fixed points of a set-valued dynamic system F, as well as some conditions that guarantee each dynamic process converges and its limit is a strict fixed point of F. In theoretical economics, such as general equilibrium theory, there comes at point where one needs to know whether the solution to a system of equations necessarily exists; or, more specifically, under which conditions will a solution necessarily exist. The mathematical analysis of this question usually relies on fixed point theorems. In engineering, fixed point technique has been used in areas like image retrieval and signal processing. In game theory it is used to estblish the existence of Nash equilibrium. The studies of asymmetric structures and their applications in mathematics are important. One of the types of asymmetric structures on a metric space was introduced by Kada et. al. [7] in 1996 known as a w-distance and he proved some fixed point theorems using it. Since then, many fixed point results have been deveoped by different authors using w-distance on metric spaces or a generalized w-distance such as c-distance on cone metric spaces. For more study in this area one may refer to [1, 2, 4, 5, 6]. Recently in 2012, Wardowski [8] introduced the concept of F-contractive mapping on a metric space and proved a fixed point theorem for such a map on a complete metric space. In the present paper we extend the fixed point result due to Wardowski by introducing an F w -contraction which is the w-version of an F-contraction. Throughout the article, denoted by R is the set of all real numbers, by R + is the set of all positive real numbers and by N is the set of all natural numbers. Corresponding author. address: rakeshbatra.30@gmail.com, Tel:

2 Page 2 of 6 2 Preliminaries and notations Definition 2.1. [8] Let F : R + R be a mapping satisfying (F1) F is strictly increasing. That is α < β F(α) < F(β) for all α,β R +. (F2) For every sequence {α n } in R + we have lim n α n = 0 if and only if lim n F(α n ) =. (F3) There exists a number k (0,1) such that lim α 0 + α k F(α) = 0. Let (X,d) be a metric space. A mapping T : X X is said to be an F-contraction if there exists a number τ > 0 such that for all x,y X with T x Ty. τ + F(d(T x,ty)) F(d(x,y)) (2.1) Remark 2.1. Clearly (2.1) of Definition 2.1 implies that d(t x,ty) < d(x,y) for all x,y X with T x Ty. Hence every F-contraction mapping is continuous. Next we give the notation of w-distance of Kada et. al. [7] with some properties. Definition 2.2. [7] Let (X,d) be a metric space. A function p : X X [0, ) is called a w-distance on X if the following conditions hold: (w1) p(x,z) p(x,y) + p(y,z) for all x,y,z X, (w2) p(x,.) is lower semi-continuous for all x X. That is, if x X and y n y in X then p(x,y) limin f n p(x,y n ). (w3) For all ε > 0, there exists δ > 0 such that p(z,x) δ and p(z,y) δ imply d(x,y) ε. Example 2.1. Let X = [0, ) and define a mapping d : X X R by d(x,y) = x y for all x,y X. Then (X,d) is a (complete) metric space. Define a mapping p : X X R by p(x,y) = y for all x,y X. Then p is a w-distance on X. Example 2.2. Let (X,d) be a metric space. Define a mapping p : X X X by p(x,y) = d(x,y) for all x,y X. Then, p is w-distance. Lemma 2.1. [7] Let (X,d) be a metric space and p be a w-distance on X. Let{x n } and {y n } be sequences in X and x,y,z X. Suppose that u n and v n are sequences in [0, ) converging to 0. Then the following hold: 1. If p(x n,y) u n and p(x n,z) v n,then y = z. In particular if p(x,y) = 0 and p(x,z) = 0 then y = z. 2. If p(x n,y n ) u n and p(x n,z) v n,then y n converges to z. 3. If p(x n,x m ) u n for m > n, then {x n } is a Cauchy sequence in X. 4. If p(y,x n ) u n, then {x n } is a Cauchy sequence in X. Remark 2.2. (i) p(x,y) = p(y,x) may not be true for all x,y X. (ii) p(x,y) = 0 is not necessarily equivalent to x = y for all x,y X. Definition 2.3. Let F be a mapping as defined in Definition 2.1 above. A mapping T : X X is said to be an F w -contraction if (i) p(x,y) = 0 p(t x,ty) = 0 (ii) There exists a number τ > 0 such that τ + F(p(T x,ty)) F(p(x,y)) f or all x,y X with p(t x,ty) > 0.

3 Page 3 of 6 Remark 2.3. Clearly, (ii) of Definition 2.3 implies that p(t x, Ty) < p(x, y) for all x, y X with p(t x, Ty) > 0. Example 2.3. Define F : R + R by F(α) = lnα. Then F satisfies (F1), (F2) and (F3) (for all k (0,1)) of Definition 2.1. A mapping T : X X satisfies p(t x,ty) λ p(x,y) (2.2) for all x,y X and some λ [0,1) if and only if T is an F w -contraction. Let us start with a mapping T : X X satisfying (2.2). If λ = 0 then (i) and (ii) in Definition 2.3 are vacuously satisfied. For 0 < λ < 1,(i) is obvious and (ii) is satisfied for τ = ln 1 λ. Thus T is an F w-contraction. Conversely, if T : X X is an F w -contraction then (ii) of Definition 2.3 implies that p(t x,ty) e τ p(x,y) for all x,y X with p(t x,ty) > 0. Clearly it is satisfied even for p(t x,ty) = 0. Thus p(t x,ty) λ p(x,y) for all x,y X, where λ = e τ [0,1). Example 2.4. Consider G(α) = lnα + α for all α > 0. Then G satisfies (F1), (F2) and (F3) of Definition 2.1. A mapping T : X X is an G w -contraction if and only if for all x,y X and some λ [0,1). Reason is similar to above example. p(t x,ty)e p(t x,ty) p(x,y) λ p(x,y) (2.3) Remark 2.4. For more examples of mappings like F in Definition 2.1 see [8]. Corresponding inequalities for F w - contractions can be obtained by working with a w-distance instead of a metric used in examples of F-contraction in [8]. Remark 2.5. From (F1) of Definition 2.1 and (ii) of Definition 2.3, it is clear that every F w -contraction T : X X satisfies p(t x,ty) < p(x,y) for all x,y X satisfying p(t x,ty) > 0. Remark 2.6. Let F, G : R + R be mappings satisfying (F1), (F2) and (F3) of Definition 2.1 together with F(α) G(α) for all α > 0. Let H = G F be nondecreasing. Then every F w -contraction T : X X is an G w -contraction. Indeed for any x, y X with p(t x, Ty) > 0, we have, from Remark 2.5 τ + G(p(T x,ty)) = τ + F(p(T x,ty)) + H(p(T x,ty)) F(p(x,y)) + H(p(x,y)) = G(p(x,y)) 3 Main section Theorem 3.1. Let (X,d) be a complete metric space and p be a w-distance on X. Let T : X X be an F w -contraction. Then T has a unique fixed point x in X and for every x 0 X, there is a sequence {T n x 0 } in X that converges to x. Further p(x,x ) = 0. Proof. For any two fixed points x and y of T in X with p(t x,ty ) > 0 we have τ F(p(x,y )) F(p(T x,ty )) = 0. Thus p(t x,ty ) = p(x,y ) = 0 for any two fixed points x and y of T in X. In particular p(t x,t x ) = p(x,x ) = 0. So by Lemma 2.1(1) we obtain x = y for any two fixed points x and y of T in X. Hence fixed point x of T if exists is unique and satisfies p(x,x ) = 0. Now we show the existence of a fixed point of T. Let x 0 X be arbitrary. Define a sequence {x n } in X by x n = T x n 1 for all n N. Let p n = p(x n 1,x n ) for all n N. If there exists k N with p(x k 1,x k ) = 0 then, by (i) of Definition 2.3, p(t x k 1,T x k ) = 0. That is p(x k,x k+1 ) = 0. Therefore p(x k 1,x k+1 ) p(x k 1,x k ) + p(x k,x k+1 ) = 0. By Lemma 2.1(1) we have x k = x k+1. Inductively we have x k = x k+i for all i N. This implies T i (x k ) = x k for all i N and in particular, for i = 1, T (x k ) = x k. Also lim n T n (x 0 ) = lim i T k+i (x 0 ) = lim i T i (x k ) = x k. Thus we can take x = x k in this case and settle the proof. Now assume that p n = p(x n 1,x n ) > 0 for all n N. Then by (ii) of Definition 2.3 we get F(p n ) F(p n 1 ) τ F(p n 2 ) 2τ F(p 0 ) nτ (3.4)

4 Page 4 of 6 From (3.4) we get lim n F(p n ) = and then by (F2) of Definition 2.1 we have Now,by (F3) of Definition 2.1, there exists k (0,1) such that By (3.4), following holds for all n N. Letting n in (3.7) and using (3.5) and (3.6) we have lim p n = 0 (3.5) n lim n pk nf(p n ) = 0 (3.6) p k nf(p n ) p k nf(p 0 ) = p k n(f(p n ) F(p 0 )) np k nτ (3.7) lim n npk n = 0 (3.8) By (3.8) there exists a positive integer n 0 such that np k n < 1 for all n n 0. Consequently p n < 1 n 1 k f or all n n 0. (3.9) Since the series n=1 1 we have n 1 k is convergent, therefore, by (3.9), the series n=1 p n is also convergent. Now for any m > n p(x n,x m ) p n+1 + p n p m < α n (3.10) where α n = i=n+1 p i 0 as n. Thus by Lemma 2.1(3) {x n } is a Cauchy sequence in X. By completeness of X, there exists x X such that lim n x n = x. From (3.10) and (ii) of Definition 2.2 we get Now for p(t x n 1,T x ) > 0, by Remark 2.5 and by (3.11) p(x n,x ) α n (3.11) p(x n,t x ) = p(t x n 1,T x ) Clearly (3.12) is satisfied even for p(t x n 1,T x ) = 0. Thus < p(x n 1,x ) α n 1 (3.12) p(x n,t x ) α n 1 f or all n N (3.13) From (3.11), (3.13) and by using Lemma 2.1(1) we get T x = x. Also we have seen above that x = lim n x n = lim n T n (x 0 ). Remark 3.1. From Example 2.2 it is clear that Theorem 2.1 of [8] is a particular case of our Theorem 3.1. Since every contraction T : X X satisfying (2.2) is an F w -contraction for F(α) = lnα, α > 0, F(α) < lnα +α = G(α) for all α > 0 and G F is non decreasing, therefore, by Remark 2.6, T is an G w -contraction and hence satisfies (2.3). In the following example we shall present a mapping T : X X which is an G w -contraction but not an F w - contraction and hence satisfies (2.3) but not (2.2). Thus our theorem deals with the fixed points of a more general class of contractions.

5 Page 5 of 6 Example 3.1. Consider the sequence a n = n(n 1) 2 for n N. Let X = {a n : n N} and d(x,y) = x y for all x,y X. Then (X,d) is a complete metric space. Define p : X X R + by p(x,y) = y for all x,y X. It is easy to verify that p is a w-distance on X. Define a mapping T : X X by Ta 1 = a 1, Ta n = a n 1 f or n > 1. Take F as in Example 2.3 and G as in Example 2.4. T is not an F w -contraction as lim n p(ta 1,Ta n ) p(a 1,a n ) = lim n a n 1 a n = 1. But T is an G w -contraction. Before we explain it let us first observe that p(ta m,ta n ) > 0 Ta n > 0 n > 2. Now for n > 2 we have p(ta m,ta n ) p(a m,a n ) ep(ta m,ta n ) p(a m,a n ) = a n 1 e a n 1 a n = (1 2 n )e1 n < e 1 n < e 1 a n Thus T is an G w -contraction for τ = 1. Clearly a 1 = 0 is a fixed point of T, p(a 1,a 1 ) = a 1 = 0 and for any a m X, lim n T n a m = lim n T n+m a m = lim n T n (T m a m ) = lim n T n a 1 = a 1 Example 3.2. Let X = [0, ), d(x,y) = x y for all x,y X and p(x,y) = y for all x,y X. Then (X,d) is a complete metric space and p is a w-distance on X. Define T : X X by { x 2 T x = 2 if 0 x 1 0 if x > 1 Since T is not continuous, therefore it is not an F-contraction for any mapping F as described in Definition 2.1. Now consider the mapping F as described in Example 2.3. We note that p(t x,ty) = Ty > 0 if and only if 0 < y 1. For x,y X with 0 < y 1 we have p(t x,ty) p(x, y) = Ty y2 y = 2 y = y So p satisfies (2.2) for all x,y X and for λ = 1 2. Thus T is an F w-contraction which is not an F-contraction for any F. Consequently Theorem 2.1 of [8] can not applied though this mapping has a fixed point by Theorem 3.1. We note that 0 is the unique fixed point of T and p(0,0) = 0. References [1] R. Batra, S. Vashistha, Coupled coincidence point theorems for nonlinear contractions under c-distance in cone metric spaces, Annals of Functional Analysis, 4 (1) (2013) [2] R. Batra, S. Vashistha, Coupled coincidence point theorems for nonlinear contractions under (F, g)-invariant set in cone metric spaces, Journal of Nonlinear Sciences and Applications, 6 (2) (2013) [3] R. Batra, S. Vashistha, Some coupled coincidence point results under c-distance in cone metric spaces, Engineering Mathematics Letters, 2 (2) (2013) [4] Z. M. Fadail, A. G. B. Ahmad, Common coupled fixed point theorems of single valued mapping for c-distance in cone metric spaces, Abstract and Applied Analysis, Article ID , 2012 (2012) 24. [5] Z. M. Fadail, A. G. B. Ahmad, Coupled fixed point theorems of single valued mapping for c-distance in cone metric spaces, Journal of Applied Mathematics, Article ID , 2012 (2012) 20. [6] Z. M. Fadail, A. G. B. Ahmad, New fixed point results of single valued mapping for c-distance in cone metric spaces, Abstract and Applied Analysis, Article ID , 2012 (2012) 12. [7] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Mathematica Japonica, 44 (2) (1996)

6 Page 6 of 6 [8] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory and Applications, (2012) 2012: