ASYMPTOTIC FIXED POINTS FOR NONLINEAR CONTRACTIONS YONG-ZHUO CHEN Received 29 August 2004 and in re vised form 12 October 2004 Recently, W. A. Kirk proved an asymptotic fixed point theorem for nonlinear contractions by using ultrafilter methods. In this paper, we prove his theorem under weaker assump- tions. Furthermore, our proof does not use ultrafilter methods. 1. Introduction There are many papers in the literature that discuss the asy mptotic fixed point theory, in which the existence of the fixed points is deduced from the assumption on the iterates of an operator (e.g., [1, 6] and the references therein). Recently, Kirk [5] studied an asymp- totic fixed point theorem concerning nonlinear contractions. He proved the following theorem [5, Theorem 2.1] by appealing to ultrafilter methods. Theorem 1.1. Let (M,d) be a complete metr ic space. Let T : M → M beacontinuousmap- ping such that d  T n x, T n y  ≤ φ n  d(x, y)  (1.1) for all x, y ∈ M,whereφ n :[0,∞] → [0, ∞] and lim n→∞ φ n = φ uniformly on the range of d. Suppose that φ and all φ n are continuous, and φ(t) <tfor t>0.Ifthereexistsx 0 ∈ M which has a bounded orbit O(x 0 ) ={x 0 ,Tx 0 ,T 2 x 0 , }, then T has a unique fixed point x ∗ ∈ M such that lim n→∞ T n x = x ∗ for all x ∈ M. In this paper, we prove Theorem 1.1 under weaker assumptions without the use of ultrafilter methods. 2. Main results We need the following recursive inequality (cf. [2, Lemmas 2.1 and 3.1], [3, Lemmas 2.1 and 2.4], and [4, Lemma 1]). Lemma 2.1. Let φ : R + → R + be upper semicontinuous, that is, limsup t→t 0 φ(t) ≤ φ(t 0 ) for all t 0 ∈ R + ,andφ(t) <tfor t>0. Suppose that there exist two sequences of nonnegative real Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:2 (2005) 213–217 DOI: 10.1155/FPTA.2005.213 214 Asymptotic fixed points for nonlinear contractions numbers {u n } and { n } such that u 2n ≤ φ  u n  +  n , (2.1) where  n → 0 as n →∞. Then either supu n =∞or liminf u n = 0. Proof. Suppose that b = sup{u n } < ∞. Assume that liminf u n = 0. Then there exist m>0 and N 1 > 0suchthatu n >mfor all n>N 1 . Since φ is upper semicontinuous, φ(t)/t is upper semicontinuous on [m,b] and so that L m = max{φ(t)/t, t ∈ [m,b]} < 1 due to the facts that φ(t) <t for t>0 and that φ(t)/t achieves its maximum on [m,b]. Let  > 0. By (2.1), there exists N 2 >N 1 such that u 2n ≤ φ  u n  +  ≤ L m u n +  (2.2) for all n>N 2 . Note that the contraction mapping f (x) = L m x +  has a unique fixed point /(1 − L m )andlim n→∞ f n (x) = /(1 − L m ) for any real number x.Nowforanyn>N 2 , u 2 2 n ≤ φ  u 2n  +  ≤ L m u 2n +  ≤ L m f  u n  +  = f 2  u n  . (2.3) By induction, u 2 k n ≤ f k (u n )forallk,sothatm ≤ f k (u n ). Letting k →∞, m ≤ /(1 − L m ). This is impossible since  > 0 can be arbitrarily chosen.  Theorem 2.2. Let (M,d) be a complete metric space. Let T : M → M be such that d  T n x, T n y  ≤ φ n  d(x, y)  (2.4) for all x, y ∈ M,whereφ n :[0,∞] → [0,∞] and lim n→∞ φ n = φ uniformly on any bounded interval [0,b].Supposethatφ is upper semicontinuous and φ(t) <tfor t>0.Furthermore, suppose there exists a posit ive integer n ∗ such that φ n ∗ is upper semicontinuous and φ n ∗ (0) = 0.Ifthereexistsx 0 ∈ M which has a bounded orbit O(x 0 ) ={x 0 ,Tx 0 ,T 2 x 0 , }, then T has auniquefixedpointx ∗ ∈ M such that lim n→∞ T n x = x ∗ for all x ∈ M. Proof. First we establish the uniqueness of the fixed point. Assume that T has two differ- ent fixed points z 1 and z 2 .Thend(z 1 ,z 2 ) = d(T n z 1 ,T n z 2 ) ≤ φ n (d(z 1 ,z 2 )). Letting n →∞, d(z 1 ,z 2 ) ≤ φ(d(z 1 ,z 2 )) <d(z 1 ,z 2 ). This is a contradiction. Without loss of generality, we set φ n (0) = 0andφ(0) = 0. Let b be the diameter of clos{O(x 0 )}.Foragivenx ∈ clos{O(x 0 )}, denote a n = d(T n+1 x, T n x). Then a 2n = d  T 2n+1 x, T 2n x  ≤ φ n  d  T n+1 x, T n x   by (2.4)  = φ  a n  +  φ n  a n  − φ  a n  . (2.5) Let  n = φ n (a n ) − φ(a n ). Since φ n → φ uniformly on [0,b],  n → 0. By Lemma 2.1, liminf n→∞ a n = 0. Yong-Zhuo Chen 215 Assume that lim n→∞ a n = 0 is not t rue. Since liminf n→∞ a n = 0, there exists n 0 > 0such that a n 0 < limsup n→∞ a n . We choose a sequence n 0 <n 1 <n 3 < ··· with a n 0 <a n i for all i = 1,2, as can be done by choosing lim i→∞ a n i = limsup n→∞ a n .Then, a n 0 <a n i = d  T n i +1 x, T n i x  ≤ φ n i −n 0  d  T n 0 +1 x, T n 0 x   by (2.4)  = φ  a n 0  +  φ n i −n 0  a n 0  − φ  a n 0  . (2.6) Letting i →∞,wehavea n 0 ≤ φ(a n 0 ) <a n 0 , which is a contradiction. We conclude that lim n→∞ a n = 0. We next show that {T n x} is a Cauchy sequence. For if not, there exist p k and q k such that p k >q k for each k,and lim k→∞ d  T p k x, T q k x  = δ>0. (2.7) Without loss of generality, assume that d  T p k x, T q k x  > δ 2 ∀k. (2.8) Since s − φ(s) is lower semicontinuous, there exists  0 > 0suchthat s − φ(s) >  0 ∀s ∈  δ 2 ,b  (2.9) due to the fact that φ(s) <sfor s>0ands − φ(s) achieves its minimum on [δ/2,b]. Since lim n→∞ φ n = φ uniformly on [δ/2,b], there exists m 0 such that φ m 0 (s) <φ(s)+  0 <sfor all s ∈ [δ/2,b]. Now d  T p k x, T q k x  ≤ d  T p k x, T p k +1 x  + d  T p k +1 x, T p k +2 x  + ···+ d  T p k +m 0 −1 x, T p k +m 0 x  + d  T p k +m 0 x, T q k +m 0 x  + d  T q k +m 0 x, T q k +m 0 −1 x  + ···+ d  T q k +1 x, T q k x  ≤ a p k + a p k +1 + ···+ a p k +m 0 −1 + φ m 0  d  T p k x, T q k x  + a q k +m 0 −1 + ···+ a q k <a p k + a p k +1 + ···+ a p k +m 0 −1 + φ  d  T p k x, T q k x  +  0 + a q k +m 0 −1 + ···+ a q k . (2.10) Letting k →∞and using (2.7), lim n→∞ a n = 0, and the upper semicontinuity of φ, δ ≤ φ(δ)+ 0 <δ. (2.11) This is a contradiction. Hence {T n x} is a Cauchy sequence, there exists x ∗ ∈ M such that lim n→∞ T n x = x ∗ . Now for each n>0, d(T n ∗ +n x, T n ∗ x ∗ ) ≤ φ n ∗ (d(T n x, x ∗ )). Since limsup n→∞ φ n ∗ (d(T n x, x ∗ )) ≤ φ n ∗ (0) = 0, we have lim n→∞ T n x = T n ∗ x ∗ ,sothatT n ∗ x ∗ = x ∗ .NotethatT n ∗ (Tx ∗ ) = T(T n ∗ x ∗ ) = Tx ∗ . By the uniqueness of the fixed point of T n ∗ , Tx ∗ = x ∗ . 216 Asymptotic fixed points for nonlinear contractions For any y 0 ∈ M \{x 0 }, d  T n y 0 ,T n x 0  ≤ φ n  d  x 0 , y 0  −→ φ  d  x 0 , y 0  <d  x 0 , y 0  (2.12) as n →∞.Hence,O(y 0 ) is also bounded. By the previous argument, lim n→∞ T n y 0 = x ∗ due to the uniqueness of the fixed point.  Remark 2.3. Kirk’ s paper [5] assumes the continuity for φ and all φ n . We only assume the upper semicontinuity of φ and one of the φ n ’s, which is weaker and easier to check. If we have limsup t→∞ (φ(t)/t) < 1, then the assumption of the existence of a bounded orbit in Theorem 2.2 can be removed. This observation is formulated as the following corollary . Corollar y 2.4. Let (M,d) be a complete metric space. Let T : M → M be such that d  T n x, T n y  ≤ φ n  d(x, y)  (2.13) for all x, y ∈ M,whereφ n :[0,∞] → [0,∞] and lim n→∞ φ n = φ uniformly on any bounded interval [0,b].Supposethatφ is upper semicontinuous, φ(t) <tfor t>0,andlimsup t→∞ (φ(t)/t) < 1. If there exists a posit ive integer n ∗ such that φ n ∗ is upper semicontinuous and φ n ∗ (0) = 0, then T has a unique fixed point x ∗ ∈ M such that lim n→∞ T n x = x ∗ for all x ∈ M. Proof. Examining the proofs of Lemma 2.1 and Theorem 2.2, one can find that the boundedness of the orbit O(x) is only used to guarantee that sup  φ(t) t : t ∈ [m,b]  < 1, inf  t − φ(t):t ∈ [m,b]  > 0 (2.14) for some b>0andallm>0 satisfying 0 <m<b<∞.Ifwehavelimsup t→∞ (φ(t)/t) < 1, then there exists b>0suchthatsup t∈[b,∞] (φ(t)/t) < 1. For otherwise, there will be t n → ∞ with lim t→∞ (φ(t n )/t n ) = 1, which implies that limsup t→∞ (φ(t)/t) = 1andleadstoa contradiction. Hence, limsup t→∞ (φ(t)/t) < 1 combined with the upper semicontinuity of φ can guar antee (2.14)forallm>0with0<m<b<∞.  Acknowledgment The author would like to thank the referees for their valuable comments and suggestions which helped to improve this paper. References [1] F. E. Browder, Asymptotic fixed point theorems, Math. Ann. 185 (1970), 38–60. [2] Y Z. Chen, Inhomogeneous iterates of contraction mappings and nonlinear ergodic theorems, Nonlinear Anal. Ser. A: Theory Methods 39 (2000), no. 1, 1–10. [3] , Path stability and nonlinear weak ergodic theorems,Trans.Amer.Math.Soc.352 (2000), no. 11, 5279–5292. Yong-Zhuo Chen 217 [4] J. R. Jachymski, An extension of A. Ostrowski’s theorem on the round-off stability of iterations, Aequationes Math. 53 (1997), no. 3, 242–253. [5] W.A.Kirk,Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003), no. 2, 645– 650. [6] R. D. Nussbaum, Some asymptotic fixed point theorems,Trans.Amer.Math.Soc.171 (1972), 349–375. Yong-Zhuo Chen: Department of Mathematics, Computer Science and Engineering, University of Pittsburgh at Bradford, Bradford, PA 16701, USA E-mail address: yong@pitt.edu . ASYMPTOTIC FIXED POINTS FOR NONLINEAR CONTRACTIONS YONG-ZHUO CHEN Received 29 August 2004 and in re vised form 12 October 2004 Recently, W. A. Kirk proved an asymptotic fixed point theorem for. Tx ∗ . By the uniqueness of the fixed point of T n ∗ , Tx ∗ = x ∗ . 216 Asymptotic fixed points for nonlinear contractions For any y 0 ∈ M {x 0 }, d  T n y 0 ,T n x 0  ≤ φ n  d  x 0 , y 0  −→. Point Theory and Applications 2005:2 (2005) 213–217 DOI: 10.1155/FPTA.2005.213 214 Asymptotic fixed points for nonlinear contractions numbers {u n } and { n } such that u 2n ≤ φ  u n  +  n ,