Bulletin of the Iranian Mathematical Society Vol. 39 No. 6 (2013), pp 1261-1272. FIXED POINTS FOR E-ASYMPTOTIC CONTRACTIONS AND BOYD-WONG TYPE E-CONTRACTIONS IN UNIFORM SPACES A. AGHANIANS, K. FALLAHI AND K. NOUROUZI∗ Communicated by Gholam Hossein Esslamzadeh Abstract. In this paper we discuss the fixed points of asymptotic contractions and Boyd-Wong type contractions in uniform spaces equipped with an E-distance. A new version of Kirk’s fixed point theorem is given for asymptotic contractions and Boyd-Wong type contractions is investigated in uniform spaces. 1. Introduction and preliminaries In 2003, Kirk [5] discussed the existence of fixed points for (not necessarily continuous) asymptotic contractions in complete metric spaces. Jachymski and Jóźwik [4] constructed an example to show that continuity of the self-mapping is essential in Kirk’s theorem. They also established a fixed point result for uniformly continuous asymptotic ϕcontractions in complete metric spaces. MSC(2010): Primary: 47H10; Secondary: 54E15, 47H09. Keywords: Separated uniform space, E-asymptotic contraction, Boyd-Wong type Econtraction, fixed point. Received: 1 June 2012, Accepted: 1 December 2012. ∗Corresponding author c 2013 Iranian Mathematical Society. 1261 1262 Aghanians, Fallahi and Nourouzi Motivated by [5, Theorem 2.1] and [4, Example 1], we aim to give a more general form of [5, Theorem 2.1] in uniform spaces where the selfmappings are assumed to be continuous. We also generalize the BoydWong fixed point theorem [3, Theorem 1] to uniform spaces equipped with an E-distance. We begin with some basics in uniform spaces which are needed in this paper. The reader can find an in-depth discussion in, e.g., [6]. A uniformity on a nonempty set X is a nonempty collection U of subsets of X × X (called the entourages of X) satisfying the following conditions: (1) Each entourage of X contains the diagonal {(x, x) : x ∈ X}; (2) U is closed under finite intersections; (3) For each entourage U in U, the set {(x, y) : (y, x) ∈ U } is in U; (4) For each U ∈ U, there exists an entourage V such that (x, y), (y, z) ∈ V implies (x, z) ∈ U for all x, y, z ∈ X; (5) U contains the supersets of its elements. If U is a uniformity on X, then (X, U) (shortly denoted by X) is called a uniform space. If d is a metric on a nonempty set X, then it induces a uniformity, called the uniformity induced by the metric d, in which the entourages of X are all the supersets of the sets  (x, y) ∈ X × X : d(x, y) < ε , where ε > 0. It is well-known that a uniformity U on a nonempty set X is separating if the intersection of all entourages of X coincides with the diagonal {(x, x) : x ∈ X}. In this case, X is called a separated uniform space. We next recall some basic concepts about E-distances. For more details and examples, the reader is referred to [1]. Definition 1.1. [1] Let X be a uniform space. A function p : X × X → R≥0 is called an E-distance on X if (1) for each entourage U in U, there exists a δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply (x, y) ∈ U for all x, y, z ∈ X; (2) p satisfies the triangular inequality, i.e., p(x, y) ≤ p(x, z) + p(z, y) (x, y, z ∈ X). If p is an E-distance on a uniform space X, then a sequence {xn } p in X is said to be p-convergent to a point x ∈ X, denoted by xn −→ x, whenever p(xn , x) → 0 as n → ∞, and X is p-Cauchy whenever E-asymptotic contractions and Boyd-Wong type E-contractions 1263 p(xm , xn ) → 0 as m, n → ∞. The uniform space X is called p-complete if every p-Cauchy sequence in X is p-convergent to some point of X. The next lemma contains an important property of E-distances on separated uniform spaces. The proof is straightforward and it is omitted here. Lemma 1.2. [1] Let {xn } and {yn } be two arbitrary sequences in a p separated uniform space X equipped with an E-distance p. If xn −→ x p and xn −→ y, then x = y. In particular, p(z, x) = p(z, y) = 0 for some z ∈ X implies x = y. Using E-distances, p-boundedness and p-continuity are defined in uniform spaces. Definition 1.3. [1] Let p be an E-distance on a uniform space X. Then (1) X is called p-bounded if  δp (X) = sup p(x, y) : x, y ∈ X < ∞. p (2) A mapping T : X → X is called p-continuous on X if xn −→ x p implies T xn −→ T x for all sequences {xn } and all points x in X. 2. E-asymptotic contractions In this section, we denote by Φ the class of all functions ϕ : R≥0 → R≥0 with the following properties: • ϕ is continuous on R≥0 ; • ϕ(t) < t for all t > 0. It is worth mentioning that if ϕ ∈ Φ, then 0 ≤ ϕ(0) = lim ϕ(t) ≤ lim t = 0, t→0+ t→0+ that is, ϕ(0) = 0. Following [5, Definition 2.1], we define E-asymptotic contractions. Definition 2.1. Let p be an E-distance on a uniform space X. We say that a mapping T : X → X is an E-asymptotic contraction if
(2.1) p(T n x, T n y) ≤ ϕn p(x, y) for all x, y ∈ X and n ≥ 1, where {ϕn } is a sequence of nonnegative functions on R≥0 converging uniformly to some ϕ ∈ Φ on the range of p. 1264 Aghanians, Fallahi and Nourouzi If (X, d) is a metric space, then replacing the E-distance p by the metric d in Definition 2.1, we get the concept of an asymptotic contraction introduced by Kirk [5, Definition 2.1]. So each asymptotic contraction on a metric space is an E-asymptotic contraction on the uniform space induced by the metric. But in the next example, we see that the converse is not generally true. Example 2.2. Uniformize the set X = [0, 1] with the uniformity induced from the Euclidean metric and put p(x, y) = y for all x, y ∈ X. It is easily verified that p is an E-distance on X. Define T : X → X and ϕ1 : R≥0 → R≥0 by  1   0 0 ≤ x < 1    16 0 ≤ t < 1  and ϕ1 (t) = Tx =     1  1 x=1 t≥1 8 8 for all x ∈ X and all t ≥ 0, and set ϕn = ϕ for n ≥ 2, where ϕ is any arbitrary fixed function in Φ. Clearly, ϕn → ϕ uniformly on R≥0 and T n = 0 for all n ≥ 2. To see that T is an E-asymptotic contraction on X, it suffices to check (2.1) for n = 1. To this end, given x, y ∈ [0, 1], if y = 1, then we have
1 p(T x, T 1) = T 1 = = ϕ1 (1) = ϕ1 p(x, 1) , 8 and for 0 ≤ y < 1, we have
1 p(T x, T y) = T y = 0 ≤ = ϕ1 (y) = ϕ1 p(x, y) . 16 But T fails to be an asymptotic contraction on the metric space X with the functions ϕn since  1 1 1
1  1 . = ϕ1 = ϕ1 1 − T1 − T = > 2 8 16 2 2 In the next example, we see that an E-asymptotic contraction need not be p-continuous. Example 2.3. Let X and p be as in Example 2.2. Define a mapping T : X → X by T x = 0 if 0 < x ≤ 1 and T 0 = 1. Note that T is fixed point free. Now, let ϕ1 be the constant function 1 and ϕ2 = ϕ3 = · · · = ϕ, where ϕ is an arbitrary fixed function in Φ. Then T satisfies (2.1) and since T 0 6= 0, it follows that T fails to be p-continuous on X. E-asymptotic contractions and Boyd-Wong type E-contractions 1265 Theorem 2.4. Let p be an E-distance on a separated uniform space X such that X is p-complete and let T : X → X be a p-continuous Easymptotic contraction for which the functions ϕn in Definition 2.1 are all continuous on R≥0 for large indices n. Then T has a unique fixed p point u ∈ X, and T n x −→ u for all x ∈ X. Proof. We divide the proof into three steps. Step 1: p(T n x, T n y) → 0 as n → ∞ for all x, y ∈ X. Let x, y ∈ X be given. Letting n → ∞ in (2.1), we get

0 ≤ lim sup p(T n x, T n y) ≤ lim ϕn p(x, y) = ϕ p(x, y) ≤ p(x, y) < ∞. n→∞ n→∞ Now, if lim sup p(T n x, T n y) = ε > 0, n→∞ then there exists a strictly increasing sequence {nk } of positive integers such that p(T nk x, T nk y) → ε, and so by the continuity of ϕ, one obtains
ϕ p(T nk x, T nk y) → ϕ(ε) < ε. Therefore, there is an integer k0 ≥ 1 such that ϕ(p(T nk0 x, T nk0 y)) < ε. So (2.1) yields ε = lim sup p(T n x, T n y) n→∞ = lim sup p T n (T nk0 x), T n (T nk0 y) n→∞
≤ lim ϕn p(T nk0 x, T nk0 y) n→∞
= ϕ p(T nk0 x, T nk0 y) < ε,
which is a contradiction. Hence lim sup p(T n x, T n y) = 0. n→∞ Consequently, 0 ≤ lim inf p(T n x, T n y) ≤ lim sup p(T n x, T n y) = 0, n→∞ that is, p(T n x, T n y) n→∞ → 0. Step 2: The sequence {T n x} is p-Cauchy for all x ∈ X. Suppose that x ∈ X is arbitrary. If {T n x} is not p-Cauchy, then there exist ε > 0 and positive integers mk and nk such that mk > nk ≥ k and p(T mk x, T nk x) ≥ ε k = 1, 2, . . . . 1266 Aghanians, Fallahi and Nourouzi Keeping the integer nk fixed for sufficiently large k, say k ≥ k0 , and using Step 1, we may assume without loss of generality that mk > nk is the smallest integer with p(T mk x, T nk x) ≥ ε, that is, p(T mk −1 x, T nk x) < ε. Hence for each k ≥ k0 , we have ε ≤ p(T mk x, T nk x) ≤ p(T mk x, T mk −1 x) + p(T mk −1 x, T nk x) < p(T mk x, T mk −1 x) + ε. If k → ∞, since p(T mk x, T mk −1 x) → 0, it follows that p(T mk x, T nk x) → ε. We next show by induction that (2.2) lim sup p(T mk +i x, T nk +i x) ≥ ε, i = 1, 2, . . . . k→∞ To this end, note first that from Step 1, ε = lim p(T mk x, T nk x) = lim sup p(T mk x, T nk x) k→∞ k→∞ h ≤ lim sup p(T mk x, T mk +1 x) + p(T mk +1 x, T nk +1 x) i k→∞ + p(T nk +1 x, T nk x) ≤ lim sup p(T mk x, T mk +1 x) + lim sup p(T mk +1 x, T nk +1 x) k→∞ + lim sup p(T nk +1 x, T nk x) k→∞ k→∞ = lim sup p(T mk +1 x, T nk +1 x), k→∞ that is, (2.2) holds for i = 1. If (2.2) is true for some i, then ε ≤ lim sup p(T mk +i x, T nk +i x) k→∞ h ≤ lim sup p(T mk +i x, T mk +i+1 x) + p(T mk +i+1 x, T nk +i+1 x) i k→∞ + p(T nk +i+1 x, T nk +i x) ≤ lim sup p(T mk +i+1 x, T nk +i+1 x). k→∞ Consequently, we have ϕ(ε) = lim ϕ p(T mk x, T nk x) k→∞
E-asymptotic contractions and Boyd-Wong type E-contractions = lim lim ϕi p(T mk x, T nk x) k→∞ i→∞ = lim lim ϕi p(T mk x, T nk x) i→∞ k→∞ 1267

≥ lim sup lim sup p(T mk +i x, T nk +i x) i→∞ k→∞ ≥ ε, where the first equality holds because ϕ is continuous, the second equality holds because {ϕi } is pointwise convergent to ϕ on the range of p, the third equality holds because {ϕi } is uniformly convergent to ϕ on the range of p, and the last two inequalities hold by (2.1) and (2.2), respectively. Hence ϕ(ε) ≥ ε, which is a contradiction. Therefore {T n x} is p-Cauchy. Step 3: T has a unique fixed point. Because X is p-complete, it is concluded from Steps 1 and 2 that the family {{T n x} : x ∈ X} of Picard iterates of T is p-equiconvergent, p that is, there exists u ∈ X such that T n x −→ u for all x ∈ X. In p particular, T n u −→ u. We claim that u is the unique fixed point for T . To this end, first note that since T is p-continuous on X, it follows that p T n+1 u −→ T u, and so, by Lemma 1.2, we have u = T u. And if v ∈ X is a fixed point for T , then

p(u, v) = lim p(T n u, T n v) ≤ lim ϕn p(u, v) = ϕ p(u, v) , n→∞ n→∞ which is impossible unless p(u, v) = 0. Similarly p(u, u) = 0 and using Lemma 1.2 once more, we get v = u.  It is worth mentioning that the boundedness of some orbit of T is not necessary in Theorem 2.4 unlike [5, Theorem 2.1] or [2, Theorem 4.1.15]. As a consequence of Theorem 2.4, we have the following version of [1, Theorem 3.1]. Corollary 2.5. Let p be an E-distance on a separated uniform space X such that X is p-complete and p-bounded and let a mapping T : X → X satisfy
(2.3) p(T x, T y) ≤ ϕ p(x, y) for all x, y ∈ X, where ϕ : R≥0 → R≥0 is nondecreasing and continuous with ϕn (t) → 0 p for all t > 0. Then T has a unique fixed point u ∈ X, and T n x −→ u for all x ∈ X. 1268 Aghanians, Fallahi and Nourouzi Proof. Note first that ϕ(0) = 0; for if 0 < t < ϕ(0) for some t, then the monotonicity of ϕ implies that 0 < t < ϕ(0) ≤ ϕn (t) for all n ≥ 1, which contradicts the fact that ϕn (t) → 0. Next, since ϕ is nondecreasing, it follows that T satisfies
p(T n x, T n y) ≤ ϕn p(x, y) for all x, y ∈ X and n ≥ 1. Setting ϕn = ϕn for each n ≥ 1 in Definition 2.1, it is seen that {ϕn } converges pointwise to the constant function 0 on [0, +∞), and since n o

sup ϕn p(x, y) : x, y ∈ X = ϕn δp (X) → 0, it follows that {ϕn } converges uniformly to 0 on the range of p. Because the constant function 0 belongs to Φ, it is concluded that T is an Easymptotic contraction on X. Moreover, ϕn ’s are all continuous on R≥0 and (2.3) ensures that T is p-continuous on X. Consequently, the result follows immediately from Theorem 2.4.  The next corollary is a partial modification of Kirk’s theorem [5, Theorem 2.1] in uniform spaces. One can find it with an additional assumption, e.g., in [2, Theorem 4.1.15]. Corollary 2.6. Let X be a complete metric space and let T : X → X be a continuous asymptotic contraction for which the functions ϕn in Definition 2.1 are all continuous on R≥0 for large indices n. Then T has a unique fixed point u ∈ X, and T n x → u for all x ∈ X. 3. Boyd-Wong type E-contractions In this section, we denote by Ψ the class of all functions ψ : R≥0 → with the following properties: • ψ is upper semicontinuous on R≥0 from the right, i.e., R≥0 tn ↓ t ≥ 0 implies lim sup ψ(tn ) ≤ ψ(t); n→∞ • ψ(t) < t for all t > 0, and ψ(0) = 0. It might be interesting for the reader to be mentioned that the family Φ defined and used in Section 2 is contained in the family Ψ but these two families do not coincide. To see this, consider the function ψ(t) = 0 if 0 ≤ t < 1, and ψ(t) = 12 if t ≥ 1. Then ψ is upper semicontinuous from the right but it is not continuous on R≥0 . Furthermore, the upper semicontinuity of ψ on R≥0 from the right and the condition that ψ(t) < E-asymptotic contractions and Boyd-Wong type E-contractions 1269 t for all t > 0, do not imply that ψ vanishes at zero in general. In fact, the function ψ : R≥0 → R≥0 defined by the rule  a t=0        t 0<t<1 ψ(t) = 2        1 t≥1 2t for all t ≥ 0, where a is an arbitrary positive real number, confirms this claim. Theorem 3.1. Let p be an E-distance on a separated uniform space X such that X is p-complete and let T : X → X satisfy
(3.1) p(T x, T y) ≤ ψ p(x, y) for all x, y ∈ X, p where ψ ∈ Ψ. Then T has a unique fixed point u ∈ X, and T n x −→ u for all x ∈ X. Proof. We divide the proof into three steps as Theorem 2.4. Step 1: p(T n x, T n y) → 0 as n → ∞ for all x, y ∈ X. Let x, y ∈ X be given. Then for each nonnegative integer n, by the contractive condition (3.1) we have
(3.2) p(T n+1 x, T n+1 y) ≤ ψ p(T n x, T n y) ≤ p(T n x, T n y). Thus, {p(T n x, T n y)} is a nonincreasing sequence of nonnegative numbers and so it converges decreasingly to some α ≥ 0. Letting n → ∞ in (3.2), by the upper semicontinuity of ψ from the right, we get
α = lim p(T n+1 x, T n+1 y) ≤ lim sup ψ p(T n x, T n y) ≤ ψ(α), n→∞ n→∞ which is a contradiction unless α = 0. Consequently, p(T n x, T n y) → 0. Step 2: The sequence {T n x} is p-Cauchy for all x ∈ X. Let x ∈ X be arbitrary and suppose on the contrary that {T n x} is not p-Cauchy. Then similar to the proof of Step 2 of Theorem 2.4, it is seen that there exist an ε > 0 and sequences {mk } and {nk } of positive integers such that mk > nk for each k and p(T mk x, T nk x) → ε. On the other hand, for each k, by (3.1) we have p(T mk x, T nk x) ≤ p(T mk x, T mk +1 x) + p(T mk +1 x, T nk +1 x) + p(T nk +1 x, T nk x) 1270 Aghanians, Fallahi and Nourouzi ≤ p(T mk x, T mk +1 x) + ψ p(T mk x, T nk x) + p(T nk +1 x, T nk x).
Letting k → ∞ and using Step 1 and the upper semicontinuity of ψ from the right, we obtain ε = lim p(T mk x, T nk x) = lim sup p(T mk x, T nk x) k→∞ k→∞ h
mk mk +1 ≤ lim sup p(T x, T x) + ψ p(T mk x, T nk x) i k→∞ + p(T nk +1 x, T nk x) ≤ lim sup p(T mk x, T mk +1 x) + lim sup ψ p(T mk x, T nk x) k→∞ + lim sup p(T nk +1 x, T nk x) k→∞
= lim sup ψ p(T mk x, T nk x) k→∞
k→∞ ≤ ψ(ε), which is a contradiction. Therefore, {T n x} is p-Cauchy. Step 3: T has a unique fixed point. Since X is p-complete, it follows from Steps 1 and 2 that the family {{T n x} : x ∈ X} is p-equiconvergent to some u ∈ X. In particular, p T n u −→ u. Since (3.1) implies the p-continuity of T on X, it follows p that T n+1 u −→ T u and so, by Lemma 1.2, we have u = T u, that is, u is a fixed point for T . If v ∈ X is a fixed point for T , then
p(u, v) = p(T u, T v) ≤ ψ p(u, v) , which is impossible unless p(u, v) = 0. Similarly p(u, u) = 0. Therefore, using Lemma 1.2 once more, one gets v = u.  As an immediate consequence of Theorem 3.1, we have the BoydWong’s theorem [3] in metric spaces: Corollary 3.2. Let X be a complete metric space and let a mapping T : X → X satisfy
(3.3) d(T x, T y) ≤ ψ d(x, y) for all x, y ∈ X, where ψ ∈ Ψ. Then T has a unique fixed point u ∈ X, and T n x → u for all x ∈ X. E-asymptotic contractions and Boyd-Wong type E-contractions 1271 In the following example, we see that Theorem 3.1 guarantees the existence and uniqueness of a fixed point while Corollary 3.2 cannot be applied. Example 3.3. Let the set X = [0, 1] be endowed with the uniformity induced by the Euclidean metric and define a mapping T : X → X by T x = 0 if 0 ≤ x < 1, and T 1 = 41 . Then T does not satisfy (3.3) for any ψ ∈ Ψ since it is not continuous on X. In fact, if ψ ∈ Ψ is arbitrary, then  1 3 1
3  . =ψ 1− T1 − T = > ψ 4 4 4 4 Now set p(x, y) = max{x, y}. Then p is an E-distance on X and T satisfies (3.1) for the function ψ : R≥0 → R≥0 defined by the rule ψ(t) = t 4 for all t ≥ 0. It is easy to check that this ψ belongs to Ψ, and the hypotheses of Theorem 3.1 are fulfilled. Remark 3.4. In Theorem 2.4 (Corollary 2.6), assume that for some index k the function ϕk belongs to Φ. Then Theorem 3.1 (Corollary 3.2) p implies that T k and so T has a unique fixed point u and T kn x −→ u for all x ∈ X. So, it is concluded by the p-continuity of T that the family {{T n x} : x ∈ X} is p-equiconvergent to u. Hence the significance of Theorem 2.4 (Corollary 2.6) is whenever none of ϕn ’s satisfy ϕn (t) < t for all t > 0, that is, whenever for each n ≥ 1 there exists a tn > 0 such that ϕn (tn ) ≥ tn . References [1] M. Aamri and D. El Moutawakil, Common fixed point theorems for E-contractive or E-expansive maps in uniform spaces, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 20 (2004), no. 1, 83–91. [2] R. P. Agarwal, D. O’Regan and D. R. Sahu, Fixed Point Theory for LipschitzianType Mappings with Applications, Springer, New York, 2009. [3] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464. [4] J. Jachymski and I. Jóźwik, On Kirk’s asymptotic contractions, J. Math. Anal. Appl. 300 (2004), no. 1, 147–159. [5] W. A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003), no. 2, 645–650. [6] S. Willard, General Topology, Addison-Wesley Publishing Co., Mass.-LondonDon Mills, Ont., 1970. 1272 Aghanians, Fallahi and Nourouzi Aris Aghanians Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 163151618, Tehran, Iran Email: a.aghanians@dena.kntu.ac.ir Kamal Fallahi Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 163151618, Tehran, Iran Email: k fallahi@dena.kntu.ac.ir Kourosh Nourouzi Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 163151618, Tehran, Iran Email: nourouzi@kntu.ac.ir