D. Wardowski, N. Van Dung

Transcription

1 DEMONSTRATIO MATHEMATICA Vol XLVII No D Wardowski, N Van Dung FIXED POINTS OF F -WEAK CONTRACTIONS ON COMPLETE METRIC SPACES Abstract In this paper, we introduce the notion of an F -weak contraction and prove a fixed point theorem for F -weak contractions Examples are given to show that our result is a proper extension of some results known in the literature 1 Introduction and preliminaries Recently, many results of the fixed point problems for maps on metric spaces have been proved [1], [], [3], [6], [7], [9] In [11], Wardowski has introduced the concept of an F -contraction as follows Definition 11 ([11], Definition 1) Let F be the family of all functions F : p0, `8q ÝÑ R such that (F1) F is strictly increasing, that is, for all α, β P p0, `8q if α ă β then F pαq ă F pβq; (F) For each sequence tα n u of positive numbers, the following holds: lim α n 0 if and only if (F3) There exists k P p0, 1q such that lim αñ0` lim F pα n q 8; `αk F pαq 0 Let px, dq be a metric space A map T : X ÝÑ X is said to be an F - contraction on px, dq if there exist F P F and τ ą 0 such that for all x, y P X, (11) dpt x, T yq ą 0 ñ τ ` F `dpt x, T yq F `dpx, yq Example 1 [11] The following functions F : p0, `8q Ñ R are the elements of F: 010 Mathematics Subject Classification: Primary 47H10, 54H5; Secondary 54D99, 54E99 Key words and phrases: F -contraction, F -weak contraction, fixed point theorem, complete metric space This work is partly discussed at The Dong Thap Seminar on Mathematical Analysis DOI: 10478/dema c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology

2 (1) F α ln α, () F α ln α ` α, (3) F α? 1, α (4) F α lnpα ` αq Fixed points of F -weak contractions on complete metric spaces 147 By using the notion of F -contraction, the author has proved a fixed point theorem which generalizes Banach contraction principle in a different way than in the known results from the literature Theorem 13 ([11], Theorem 1) Let px, dq be a complete metric space and T : X ÝÑ X be an F -contraction Then we have (1) T has a unique fixed point x () For all x P X, the sequence tt n xu is convergent to x Remark 14 ([11], Remark 1) Let T be an F -contraction Then dpt x, T yq ă dpx, yq for all x, y P X such that T x T y Also, T is a continuous map In this paper, we introduce the notion of an F -weak contraction and prove a fixed point theorem for F -weak contractions, which generalizes some results known from the literature Examples are given to show that our result is a proper extension of [11, Theorem 1] Main results First we generalize the notion of an F -contraction into an F -weak contraction as follows Definition 1 Let px, dq be a metric space A map T : X ÝÑ X is said to be an F -weak contraction on px, dq if there exist F P F and τ ą 0 such that, for all x, y P X satisfying dpt x, T yq ą 0, the following holds: (1) τ ` F `dpt x, T yq F Remark ˆ max (1) Every F -contraction is an F -weak contraction () Let T be an F -weak contraction From (1) we have, for all x, y P X, T x T y F `dpt x, T yq ă τ ` F `dpt x, T yq F ˆ max

3 148 D Wardowski, N Van Dung Then, by (F1), we get dpt x, T yq ă max for all x, y P X, T x T y, The following example shows that the inverse implication of Remark (1) does not hold Example 3 Let T : r0, 1s ÝÑ r0, 1s be given by $ 1 &, if x P r0, 1q, T x 1 %, if x 1 4 Since T is not continuous, T is not an F -contraction by Remark 14 For x P r0, 1q and y 1, we have ˆ1 dpt x, T 1q d, ˇ 1 4ˇ 1 4 ą 0 and max dpx, 1q, dpx, T xq, dp1, T 1q, dpx, T 1q ` dp1, T xq dp1, T 1q 3 4 Therefore, by choosing F α ln α, α P p0, `8q and τ ln 3, we see that T is an F -weak contraction Now we state the main result of the paper Theorem 4 Let px, dq be a complete metric space and T : X ÝÑ X be an F -weak contraction If T or F is continuous, then we have (1) T has a unique fixed point x P X () For all x P X, the sequence tt n xu is convergent to x Proof (1) Let x P X be arbitrary and fixed We define x n`1 T x n for all n P NYt0u, where x 0 x If there exists n 0 P NYt0u such that x n0`1 x n0, then T x n0 x n0 This proves that x n0 is a fixed point of T Now we suppose that x n`1 x n for all n P NYt0u Then dpx n`1, x n q ą 0 for all n P N Y t0u It follows from (1) that for each n P N:

4 Fixed points of F -weak contractions on complete metric spaces 149 () F `dpx n`1, x n q F `dpt x n, T x n 1 q ˆ F max dpx n, x n 1 q, dpx n, T x n q, dpx n 1, T x n 1 q, dpx n, T x n 1 q ` dpx n 1, T x n q τ ˆ F max dpx n, x n 1 q, dpx n, x n`1 q, dpx n 1, x n q, dpx n 1, x n`1 q τ ˆ F max dpx n, x n 1 q, dpx n, x n`1 q, dpx n 1, x n`1 q τ ˆ F max dpx n, x n 1 q, dpx n, x n`1 q, dpx n 1, x n q ` dpx n, x n`1 q τ F ` max dpx n, x n 1 q, dpx n, x n`1 q ( τ If there exists n P N such that max dpx n, x n 1 q, dpx n, x n`1 q ( dpx n`1, x n q then () becomes F `dpx n`1, x n q F `dpx n`1, x n q τ ă F `dpx n`1, x n q It is a contradiction Therefore, max dpx n, x n 1 q, dpx n, x n`1 q ( dpx n, x n 1 q, for all n P N Thus, from (), we have for all n P N It implies that F `dpx n`1, x n q F `dpx n, x n 1 q τ, (3) F `dpx n`1, x n q F `dpx 1, x 0 q nτ, for all npn Taking the limit as in (3), we get that together with (F) gives lim F pdpx n`1, x n qq 8 (4) lim dpx n`1, x n q 0 From (F3), there exists k P p0, 1q such that (5) lim `dpxn`1, x n q k F `dpxn`1, x n q 0 It follows from (3) that (6) `dpxn`1, x n q k F `dpx n`1, x n q F `dpx 1, x 0 q `dpx n`1, x n q k nτ 0,

5 150 D Wardowski, N Van Dung for all n P N By using (4), (5) and taking the limit as n Ñ 8 in (6), we get (7) lim n`dpx n`1, x n q k 0 Then there exists n 1 P N such that n`dpx n`1, x n q k 1 for all n n1, that is, (8) dpx n`1, x n q 1, for all n n 1 For all m ą n n 1, by using (8) and the triangle inequality, we get (9) n 1 k dpx m, x n q dpx m, x m 1 q ` ` dpx n`1, x n q 8ÿ 8ÿ 1 ă dpx i`1, x i q i n i n i 1 k Since the series ř 8 n 1 1{n 1 k is convergent, taking the limit as n Ñ 8 in (9), we get lim n,mñ8 dpx m, x n q 0 This proves that tx n u is a Cauchy sequence in X Since X is complete, there exists x PX such that lim x n x We shall prove that x is a fixed point of T by two following cases Case 1 T is continuous We have dpx, T x q lim dpx n, T x n q lim dpx n, x n`1 q 0 This proves that x is a fixed point of T Case F is continuous In this case, we consider two following subcases Subcase 1 For each n P N, there exists i n P N such that x in`1 T x and i n ą i n 1 where i 0 1 Then we have x lim x in`1 lim T x T x This proves that x is a fixed point of T Subcase There exists n 0 P N such that x n`1 T x for all n n 0 That is dpt x n, T x q ą 0 for all n n 0 It follows from (1), (F1) and the triangle inequality that (10) τ ` F `dpx n`1, T x q τ ` F `dpt x n, T x q F ˆ max dpx n, x q, dpx n, T x n q, dpx, T x q, dpx n, T x q ` dpx, T x n q

6 Fixed points of F -weak contractions on complete metric spaces 151 ˆ F max dpx n, x q, dpx n, x n`1 q, dpx, T x q, dpx n, T x q ` dpx, x n`1 q ˆ F max dpx n, x q, dpx n, x n`1 q, dpx, T x q, If dpx, T x q ą 0 then by the fact dpx n, x q ` dpx, T x q ` dpx, x n`1 q lim dpx n, x q lim dpx, x n`1 q 0, there exists n 1 P N such that for all n n 1, we have max dpx n, x q, dpx n, x n`1 q, dpx, T x q, dpx n, x q ` dpx, T x q ` dpx, x n`1 q dpx, T x q By (10), we get (11) τ ` F `dpx n`1, T x q F `dpx, T x q, for all n maxtn 0, n 1 u Since F is continuous, taking the limit as n Ñ 8 in (11), we obtain τ ` F `dpx, T x q F `dpx, T x q It is a contradiction Therefore, dpx, T x q 0, that is, x is a fixed point of T By two above cases, T has a fixed point x Now, we prove that the fixed point of T is unique Let x 1, x be two fixed points of T Suppose to the contrary that x 1 x Then T x 1 T x It follows from (1) that τ ` F `dpx 1, x q τ ` F `dpt x 1, T x q ˆ dpx 1, T x q ` dpx, T x 1 q F max dpx 1, x q, dpx 1, T x 1q, dpx, T x q, F ˆ max F `dpx 1, x q dpx 1, x q, dpx 1, x 1q, dpx, x q, dpx 1, x q ` dpx, x 1 q It is a contradiction Then dpx 1, x q 0, that is, x 1 x This proves that the fixed point of T is unique () It follows from the proof of (1) that lim T n x lim x n`1 x From Theorem 4, we get the following corollaries

7 15 D Wardowski, N Van Dung Corollary 5 Let px, dq be a complete metric space and T : X ÝÑ X satisfies (1) dpt x, T yq ą 0 ñ τ ` F `dpt x, T yq F `adpx, yq ` bdpx, T xq ` cdpy, T yq ` e, for all x, y P X where a, b, c 0 and a ` b ` c ` e ă 1 continuous then (1) T has a unique fixed point x P X () For all x P X, the sequence tt n xu is convergent to x Proof For all x, y P X, we have If T or F is adpx, yq ` bdpx, T xq ` cdpy, T yq ` e pa ` b ` c ` eq max max Then, by (F1) we see that (1) is a consequence of (1) Then the corollary is proved Remark 6 Since (1) is a consequence of (11) and T is continuous by Remark 14, we get Theorem 13 from Corollary 5 The following example shows that Theorem 4 is a proper extension of Theorem 13 Example 7 Let T be given as in Example 3 Since T is not an F - contraction for any F, Corollary 13 is not applicable to T On the other hand, let F be given as in Example 3 Then T is an F -weak contraction Therefore, Theorem 4 can be applicable to T and the unique fixed point of T is 1 Remark 8 When we consider the different types of F -weak contractions then we obtain the variety of known contractions in the literature For example, see the following (1) For all x, y P X and a, b, c 0, a ` b ` c ă 1, we have that dpt x, T yq adpx, yq ` bdpx, T xq ` cdpy, T yq implies dpt x, T yq pa`b`cq max

8 Fixed points of F -weak contractions on complete metric spaces 153 Then if dpt x, T yq ą 0, we get τ ` lnpdpt x, Tˆ yq ln max, where τ ln 1{pa ` b ` cq ą 0 Then the contraction condition in [10] becomes the condition (1) with F α ln α for all α ą 0 This proves that Theorem 4 is a generalization of the main result of [10] () For all x, y P X and k P r0, 1q, we have that dpt x, T yq k max dpx, T xq, dpy, T yq ( implies dpt x, T yq k max Then if dpt x, T yq ą 0, we get τ ` ln `dpt x, T yq ˆ ln max, where τ ln 1{k ą 0 Then the contraction condition in [4] becomes the condition (1) with F α ln α, for all α ą 0 This proves that Theorem 4 is a generalization of the main result of [4] (3) For all x, y P X and non-negative numbers qpx, yq, rpx, yq, spx, yq and tpx, yq with ( qpx, yq ` rpx, yq ` spx, yq ` tpx, yq λ ă 1 sup x,ypx and dpt x, T yq ą 0, we have that dpt x, T yq qpx, yqdpx, yq ` rpx, yqdpx, T xq ` spx, yqdpy, T yq ` tpx, yq implies dpt x, T yq λ max Then if dpt x, T yq ą 0, we get ln 1 ` ln dpt x, T yq λ ˆ ln max,

9 154 D Wardowski, N Van Dung where τ ln 1{λ ą 0 Then the contraction condition (C) in [5, page 0] becomes the condition (1) with F α ln α for all α ą 0 This proves that Theorem 4 is a generalization of [5, Theorem 5(i)] (4) For all x, y P X and non-negative numbers a, b, c, e, f with a ` b ` c ` e ` f ă 1, we have that dpt x, T yq a ` b dpx, T xq ` dpy, T yq ` c ` e ` fdpx, yq implies dpt x, T yq pa ` b ` c ` e ` fq ˆ max Then if dpt x, T yq ą 0, we get 1 ln ` ln dpt x, T yq a ` b ` c ` e ` f ˆ ă ln max, where τ ln 1{pa ` b ` c ` e ` fq ą 0 Then the contraction condition () in [8, page 0] becomes the condition (1) with F α ln α for all α ą 0 This proves that Theorem 4 is a generalization of [8, Theorem 1(1)(a)] References [1] M A Alghamdi, A Petrusel, N Shahzad, A fixed point theorem for cyclic generalized contractions in metric spaces, Fixed Point Theory Appl 1 (01), 10 pages [] T V An, K P Chi, E Karapinar, T D Thanh, An extension of generalized pψ, ϕq-weak contractions, Int J Math Math Sci 01 (01), 11 pages [3] V Berinde, F Vetro, Common fixed points of mappings satisfying implicit contractive conditions, Fixed Point Theory Appl 105 (01), 16 pages [4] R M T Bianchini, Su un problema di S Reich aguardante la teoria dei punti fissi, Boll Un Mat Ital 5 (197), [5] L B Ćirić, A generalization of Banach s contraction principle, Proc Amer Math Soc 45 (1974), [6] H-S Ding, L Li, S Radenovic, Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces, Fixed Point Theory Appl 96 (01), 17 pages [7] W S Du, S X Zheng, Nonlinear conditions for coincidence point and fixed point theorems, Taiwanese J Math 16(3) (01),

10 Fixed points of F -weak contractions on complete metric spaces 155 [8] G E Hardy, T D Rogers, A generalization of a fixed point theorem of Reich, Canad Math Bull 16() (1973), [9] A Latif, W A Albar, Fixed point results in complete metric spaces, Demonstratio Math 41 (008), [10] S Reich, Some remarks concerning contraction mappings, Canad Math Bull 14(1) (1971), [11] D Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl 94 (01), 11 pages D Wardowski (corresponding author) UNIVERSITY OF ŁÓDŹ FACULTY OF MATHEMATICS AND COMPUTER SCIENCE DEPARTMENT OF NONLINEAR ANALYSIS Banacha ŁÓDŹ, POLAND wardd@mathunilodzpl Nguyen Van Dung DONG THAP UNIVERSITY DEPARTMENT OF MATHEMATICS 783 Pham Huu Lau Street WARD 6, CAO LANH CITY DONG THAP PROVINCE, VIETNAM, POSTAL CODE: 84 nvdung@dthueduvn, nguyendungtc@yahoocom Received September 5, 01; revised version September 10, 01