NOTE ON ASYMPTOTIC CONTRACTIONS

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1 Applicable Analysis and Discrete Mathematics, , Available electronically at Presented at the conference: Topics in Mathematical Analysis and Graph Theory, Belgrade, September 1 4, NOTE ON ASYMPTOTIC CONTRACTIONS Ivan D. Arand - elović In 2003 W. A. Kirk introduced the notion of asymptotic contractions. In this paper we present one fixed point theorem of Kirk s type unifying and generalizing recent results of W. A. Kirk, J. Jachymski, I. Jóźwik and Y.-Z. Chen. 1. INTRODUCTION AND PRELIMINARIES W. A. Kirk [8] introduced the notion of asymptotic contractions and proved fixed point theorem for this class of mappings. In note [1] we present a new short and simple proof of Kirk s theorem. Further results on this class of mappings was obtained by: J. Jachymski, I. Jóźwik [7], Y.-Z. Chen [4], P. Gerhardy [5], [6], T. Suzuki [9], H. K. Xu [12], M. Arav, F. E. C. Santos, S. Reich, A. Zaslavski [2] and K. W lodarczyk, D. K, R. Plebaniak [10], [11]. The papers [10] and [11] presents some ideas for application of the theory of asymptotic contractions in the analysis of set-valued dynamic systems. In this paper we present one fixed point theorem of Kirk s type unifying and generalizing recent results of W. A. Kirk [8], J. Jachymski, I. Jóźwik [7] and Y.-Z. Chen [4]. Let X be a nonempty set and f : X X arbitrary mapping. x X is a fixed point for f if x = fx. If x 0 X, we say that a sequence x n defined by x n = f n x 0 is a sequence of Picard iterates of f at point x 0 or that x n is the orbit of f at point x 0. In [2] M. Arav, F. E. C. Santos, S. Reich and A. Zaslavski proved the following result: Proposition 1. Let X, d be a metric space, f : X X continuous function and ϕ i sequence of functions such that ϕ i : [0, [0, and for each x, y X d f i x, f i y ϕ i dx, y Mathematics Subject Classification. 54H25, 47H10 Key Words and Phrases. Fixed point, asymptotic contraction. 211

2 212 Ivan D. Arand - elović Assume also that there exists upper semicontinuous function ϕ : [0, [0, such that for any r > 0 ϕr < r, ϕ0 = 0 and ϕ i ϕ uniformly on any bounded interval [0, b]. If there exists y X such that y = fy then all sequences of Picard iterates defined by f converge to y, uniformly on each bounded subset of X. 2. MAIN RESULT S Now we present our results. Theorem 1. Let X, d be a complete metric space, f : X X continuous function and ϕ i sequence of functions such that ϕ i : [0, [0, and for each x, y X d f i x, f i y ϕ i dx, y. Assume also that there exists upper semicontinuous function ϕ : [0, [0, such that for any r > 0 ϕr < r, ϕ0 = 0 and ϕ i ϕ uniformly on any bounded interval[0, b]. If one of the following conditions is satisfying: 1 there exists x X such that orbit of f at x is bounded; or 2 t ϕt > 0; or ϕt 3 < 1. t then f has an unique fixed point y X and all sequences of Picard iterates defined by f converge to y, uniformly on each bounded subset of X. Proof. For any x, y X, x y, we have: d f n x, f n y ϕ n dx, y = ϕ dx, y < dx, y. Suppose that there exist x, y X and ε > 0 such that d f n x, f n y = ε. Then there exists sequence of integers m j, such that dx mj, y mj = dx m, y m. If d f k x, f k y ε, for each k m j, then from upper semicontinuity of ϕ follows dx mj, y mj ϕε < ε, which is a contradiction. So there exists k m j such that ϕ d f k x, f k y < ε. This implies that d f n x, f n y = n d f n f k x, f n f k y n ϕ n d f k x, f k y = ϕ d f k x, f k y < ε,

3 Note on asymptotic contractions 213 which is a contradiction. So we obtain that 1 d f n x, f n y = 0, for any x, y X, which implies that all sequences of Picard iterates defined by f, are equiconvergent. Now let a X be arbitrary, a n be a sequence of Picard iterates of f at point a, Y = a n and F n = {x Y : d x, f k x 1/n k = 1,...,n}. From 1 follows that F n is nonempty and since f is continuous F n is closed, for any n. Also, we have F n+1 F n. Let x n and y n be arbitrary sequences, such that x n, y n F n. Let n j be a sequence of integers, such that dx nj, y nj = dx n, y n. For any ε > 0 there exists positive integer k such that ϕt + ε ϕ k t for all t [0, + and m k, because ϕ n ϕ uniformly on the rang of d. Now we have: dx nj, y nj d x nj, f nj x nj + d f nj x nj, f nj y nj +d y nj, f nj y nj = d f nj x nj, f nj y nj ϕ nj dxnj, y nj ε + ϕ dx nj, y nj ε + ϕ dx nj, y nj, for n j k and so dx nj, y nj = ϕ dx nj, y nj dx nj, y nj {0, + }. Now we have following three cases: A Let dx nj, y nj = 0. Thus dx n, y n = 0 and so dx n, y n = 0. This implies that diamf n = 0. By completeness of Y follows that there exists z X such that F n = {z}. i=1 We remember that 1 A. Since d z, fz 1/n for any n, we have fz = z. From 1 follows that all sequences of Picard iterates defined by f converge to z. From Proposition 1 follows that this convergence is uniform on bounded subsets of X. B Let dx nj, y nj = + and follows t ϕt > 0. Then from dx nj, y nj d x nj, f nj x nj + d f nj x nj, f nj y nj + d y nj, f nj y nj d x nj, f nj x nj + d y nj, f nj y nj dx nj, y nj d f nj x nj, f nj y nj dx nj, y nj ϕ nj dxnj, y nj dx nj, y nj ϕ dx nj, y nj ε,

4 214 Ivan D. Arand - elović for n j k. Thus which is a contradiction. t ϕt < 0, n C Let dx nj, y nj = + and ϕt t < 1. Then from dx nj, y nj d x nj, f nj x nj + d f nj x nj, f nj y nj + d y nj, f nj y nj follows 1 d x nj, f nj x nj + d f nj x nj, f nj y nj + d y nj, f nj y nj dx nj, y nj d x nj, f nj x nj + ϕ nj dxnj, y nj + d y nj, f nj y nj dx nj, y nj d x nj, f nj x nj + ϕ dx nj, y nj + ε + d y nj, f nj y nj, dx nj, y nj for n j k. Thus which is a contradiction. 1 ϕ dx nj, y nj dx nj, y nj < 1 3. COMMENTS AND REMARKS The statement of W. A. Kirk [8] Theorem 2.1 has additional assumptions that all ϕ i are continuous, and so Kirk s result is include in our Theorem 1.1, as theorem Y.-Z. Chen [4]- Theorem 2.2 which has additional assumptions that one of ϕ i is upper semicontinuous. The statement of J. Jachymski, I. Jóźwik [7] Theorem 2 has additional assumptions f is uniformly continuous and condition t ϕt = + which is stronger then our condition t ϕt > 0. Thus this result is include in our Theorem 1.2. The statement of Y.-Z. Chen [4] Corollary 2.4 has additional assumptions that one of ϕ i is upper semicontinuous, and so it is include in our Theorem 1.3. In the statements of W. A. Kirk [8] Theorem 2.1 and Y.-Z. Chen [4] Theorem 2.2, the assumption f is continuous was inadvertently left out, but

5 Note on asymptotic contractions 215 it was used in the proofs of theorems. J. Jachymski, I. Jóźwik [7], give the following example for necessity of this condition. Example 1. Let X = [0, 1] an f : X X defined by fx = { 1, x = 0, x/2, x 0. So fx 0, 1 which implies f n X 0, 1/2 n 1. A sequence of functions ϕ n = 1/2 n 1 satisfies the conditions of Theorem 2.1 because ϕ n 0 uniformly, but f is fixed point free. Now we give the following example for necessity of uniformly convergence of sequence ϕ n. Example 2. Let x n be an arbitrary sequence, X = {x n }, and d : X 2 [0, + mapping defined by dx n, x n+k = k n + k + 1 nk, n {1, 2, 3,...}, k {0, 1, 2,...}. X, d is complete metric space, because each ball which radius is less then 1 contains only a finite number elements of X. d is discrete metric and all nonzero distance are distinct. Also we have: k dx n, x n+k = 1 and n dx n, x n+k = 0. Now define f : X X by fx n = x n+1. Let ϕt = t/2 and ϕ n dxi, x j { = max dx i+n, x j+n, dx i, x j 2 for i j. The function ϕ n is well defined because the number dx i, x j occurs only once in the rang of d. From n dx i+n, x j+n = 0 }, follows ϕ n ϕ on the rang of d. The inequality d f n x i, f n x j ϕ n dxi, x j is also satisfied, but f is fixed point free.

6 216 Ivan D. Arand - elović REFERENCES 1. I. Arand- elović: On a fixed point theorem of Kirk. J. Math. Anal. Appl., , M. Arav, F. E. C. Santos, S. Reich, A. Zaslavski: A note on asymptotic contractions. Fixed point theory and applications 2007 ID 39465, 6 pages, doi: /2007/ E. M. Briseid: A rate of convergence for asymptotic contractions. J. Math. Anal. Appl., 2006 doi: /jmaa Y.-Z. Chen:Asymptotic fixed points for nonlinear contractions. Fixed point Theory Appl., 2005:2 2005, P. Gerhardy:A quantitative version of Kirk s fixed point theorem for asymptotic contractions. J. Math. Anal. Appl., , P. Gerhardy:Applications of proof interpretations. PhD Dissertation, University of Aarhus J. Jachymski, I. Jóźwik:On Kirk s asymptotic contractions. J. Mat. Anal. Appl., , W. A. Kirk:Fixed points of asymptotic contractions. J. Math. Anal. Appl., , T. Suzuki :Fixed-point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces. Nonlinear Analysis, , K. W lodarczyk, D. K, R. Plebaniak: Existence and uniqueness of endpoints of closedset-valued asymptotic contractions in metric spaces. J. Math. Anal. Appl., , K. W lodarczyk, R. Plebaniak, Cezary Obczyński: Endpoints set-valued dynamical systems of asymptotic contractions of Meir-Keeler type and strict contractions in uniform spaces. Nonlinear Analysis, 2006, doi: /j.na H. K. Xu:Asymptotic and weakly asymptotic contractions. Indian J. pure appl. Math., , University of Belgrade, Received October 27, 2006 Faculty of Mechanical Engineering, Kraljice Marije 16, Beograd, Serbia E mail: iva@alfa.mas.bg.ac.yu