The aim of this paper is to prove a fixed point theorem on asymptotic contractions with hypotheses slightly different from that of Chen [1], Theorem 2.2

G. V. R. Babu

K. P. R. Sastry

M. Balaiah

S. Ismail

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Mathematical Communications 12(2007), 191-194 191A fixed point theorem on asymptotic contractionsK.P.R. Sastry∗, G.V.R. Babu†, S. Ismail‡ and M. Balaiah§Abstract. The aim of this paper is to prove a ﬁxed point theoremon asymptotic contractions with hypotheses slightly diﬀerent from thatof Chen [1], Theorem 2.2.Key words: asymptotic contractions, complete metric space, ﬁxedpointAMS subject classifications: 54H25, 47H10Received March 8, 2007 Accepted July 17, 20071. IntroductionThroughout this paper, (M,d) denotes a complete metric space, R+ the set of allnonnegative reals, and Φ the class of all mappings ϕ : R+ → R+ satisfying (i) ϕ iscontinuous, and (ii) ϕ(t) < t for t > 0. For x ∈M , the orbit of x is {x, Tx, T 2x, · · · }and is denoted by O(x).In 2003, Kirk [2] introduced a new class of mappings namely asymptotic con-tractions and established the existence of ﬁxed points for such mappings by usingultra ﬁlter methods.Definition 1.1 (Kirk [2]). A mapping T : M →M is said to be an asymptoticcontraction, if there exists a sequence {ϕn}∞n=1, ϕn : R+ → R+ and ϕ ∈ Φ suchthat for all n ∈ Nd(T nx, T ny) ≤ ϕn(d(x, y)) (1)for all x, y ∈M, where ϕn → ϕ uniformly on the range of d.Theorem 1.2 (Kirk [2]). Suppose T : M → M is an asymptotic contractionfor which the mappings ϕn in (1) are also continuous. Assume that some orbit ofT is bounded. Then T has a unique ﬁxed point x∗ ∈ M, and moreover the Picardsequence {T nx}∞n=1 converges to x∗ for each x ∈M .∗8-28-8/1, Tamil street, Chinna Waltair, Visakhapatnam-530017, India, e-mail:kprsastry@hotmail.com†Department of Mathematics, Andhra University, Visakhapatnam-530003, India, e-mail:gvr babu@hotmail.com‡Department of Mathematics, Andhra University, Visakhapatnam-530003, India, e-mail:shaik 1949@hotmail.com§Department of Mathematics, Andhra University, Visakhapatnam-530003, India, e-mail:balaiah m19@hotmail.com192 K.P.R. Sastry, G.V.R. Babu, S. Ismail and M. BalaiahIn 2005, Chen [1] proved Theorem 1.2, under weaker assumptions without usingultra ﬁlter methods.Theorem 1.3 (Chen [1], Theorem 2.2). Let T :M →M be a mapping satisfying(1.3.1) d(T nx, T ny) ≤ ϕn(d(x, y))for all x, y ∈ M, where ϕn : R+ → R+ and limn→∞ϕn = ϕ uniformly onany bounded interval [0, b]. Suppose that ϕ is upper semicontinuous andϕ(t)< t for t > 0,(1.3.2) there exists a positive integer n∗ such that ϕn∗ is upper semicontinuousand ϕn∗(0) = 0, and(1.3.3) there exists x0 ∈M such that O(x0) is bounded.Then T has a unique ﬁxed point x∗ ∈ M such that limn→∞Tnx = x∗ for allx ∈M .The following example shows that T may not have a ﬁxed point if condition(1.3.2) is dropped in Theorem 1.3.Example 1.4. Let M = {0, 1, 2−1, 2−2, · · · } with the usual metric. DeﬁneT : M → M by T 0 = 1; T (2−n) = 2−(n+1) for n = 0, 1, 2, · · · . Deﬁne ϕn on R+by ϕn(t) = 2−1t + n−1 and ϕ on R+ by ϕ(t) = 2−1t, t ∈ R+. Then {ϕn} and ϕsatisfy conditions (1.3.1) and (1.3.3). Here we observe that ϕn(0) = 0 for every n,so that condition (1.3.2) does not hold and T has no ﬁxed points.In this paper, we prove that Theorem 1.3 holds well if (1.3.2) is replaced by(1.3.2)′: There exists x0 ∈M such that O(x0) is bounded and there exists y ∈ O(x0)such that O(Ty) is closed.2. Main resultTheorem 2.1. Let T :M →M be a mapping satisfying (1.3.1) and (1.3.2)′. ThenT has a unique ﬁxed point x∗ ∈M and limn→∞ Tnx0 = x∗. In fact limn→∞Tnx = x∗ forall x ∈M .Proof. For x0 ∈ M , the sequence {T nx0} is Cauchy, which follows from theproof of Theorem 1.3 (see [1], Theorem 2.2) and hence converges, say, to x∗. Thus(2.1.1) O(x0) = {x∗} ∪O(x0).Since O(x0) is bounded, there exists b > 0 such that d(T nx0, Tmx0) ≤ b for all n,m.Now, by using (1.3.1), we haved(T nx0, T nx∗) ≤ ϕn(d(x0, x∗)).Since ϕn → ϕ as n→∞, there exists a natural number N such thatϕn(d(x0, x∗)) < ϕ(d(x0, x∗)) + 1 for all n ≥ N.Takea = max1≤k≤N−1{d(T kx0, T kx∗), ϕ(d(x0, x∗)) + 1},A fixed point theorem on asymptotic contractions 193so that d(T nx0, T nx∗) ≤ a for all n. Nowd(T nx∗, Tmx∗) ≤ d(T nx∗, T nx0) + d(T nx0, Tmx0) + d(Tmx0, Tmx∗)≤ a+ b+ a = 2a+ b for all n,m.Hence O(x∗) is bounded. Thus {T nx∗} converges, say, to y∗.We now show that y∗ = x∗. If possible, suppose that y∗ = x∗.For a ﬁxed positive integer N , we haved(T n+Nx0, T n+Nx∗) ≤ ϕn(d(TNx0, TNx∗)).By letting n→∞, we getd(x∗, y∗) ≤ ϕ(d(TNx0, TNx∗)).Now, by letting N →∞, we haved(x∗, y∗) ≤ ϕ(d(x∗, y∗)) < d(x∗, y∗),a contradiction. Thus T nx∗ → x∗ as n→∞.By condition (1.3.2)′ and (2.1.1) the following two cases arise.Case (i): y = x∗Since T nx∗ → x∗, x∗ ∈ O(Tx∗). Therefore there exists a positive integer Nsuch that x∗ = TNx∗. Hence T nN+1x∗ = Tx∗ for n = 1, 2, . . . . Consequently, thesubsequence {T nN+1x∗} of {T nx∗} converges to Tx∗. Thus Tx∗ = x∗.Case (ii): y = x∗In this case y ∈ O(x0) by (2.1.1) and O(Ty) is closed by (1.3.2)′. Also observethat O(Ty) is bounded. Hence the conclusion follows by case (i).Now, for x ∈M we haved(T nx, x∗) = d(T nx, T nx∗) ≤ ϕn(d(x, x∗)) < ϕ(d(x, x∗)) + 1 for a large n.Hence {T nx} is bounded. Thus T nx→ x∗ as n→∞. This completes the proof ofthe theorem. ✷The following is an example which satisﬁes (1.3.2)′ but not (1.3.2).Example 2.2. Let M = {0, 1, 2−1, 3−1, · · · } with the usual metric. DeﬁneT : M → M by T 0 = 0; T (n−1) = (n + 1)−1 for n = 1, 2, . . . . For t ∈ R+,deﬁne ϕn(t) = n−1 for all n, and ϕ(t) = 0 for all t. Clearly ϕ(t) < t for t > 0and ϕn converges to ϕ uniformly on M . It is easy to verify the condition (1.3.1).Put x0 = 1. Then, O(x0) = {1, 2−1, 3−1, · · · } is bounded. We have 0 ∈ O(x0) andT kx0 → 0 as k → ∞. Hence 0 ∈ O(Tx0) so that O(Tx0) is closed. Consequently(1.3.2)′ holds. Thus T satisﬁes all the conditions of Theorem 2.1 and 0 is the onlyﬁxed point of T.However, we observe that ϕn(0) = 0 for every n, so that condition (1.3.2) failsto hold.194 K.P.R. Sastry, G.V.R. Babu, S. Ismail and M. BalaiahReferences[1] Y.Z. Chen, Asymptotic ﬁxed points for nonlinear contractions, Fixed PointTheory and Applications 2005(2005), 213-217.[2] W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal.Appl.277(2003), 645-650.

A fixed point theorem on asymptotic contractions

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