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Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces
Nonlinear Analysis 64 (2006) 971 – 978 www.elsevier.com/locate/na
Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces Tomonari Suzuki∗,1 Department of Mathematics, Kyushu Institute of Technology, 1-1, Sensuicho, Tobata-ku, Kitakyushu 804 8550, Japan Received 21 January 2005; accepted 25 April 2005
Abstract In this paper, we introduce the notion of asymptotic contraction of Meir–Keeler type, and prove a fixed-point theorem for such contractions, which is a generalization of fixed-point theorems of Meir–Keeler and Kirk. In our discussion, we use the characterization of Meir–Keeler contraction proved by Lim [On characterizations of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120]. We also give a simple proof of this characterization. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: Primary 54H25; Secondary 54E50 Keywords: Meir–Keeler contraction; L-function; Asymptotic contraction; Fixed point; Complete metric space
1. Introduction Throughout this paper we denote by N the set of all positive integers. In 1969, Meir and Keeler [7] proved the following very interesting fixed-point theorem, which is a generalization of the Banach contraction principle [2]. See also [8,9].
∗ Tel.: +81 93 884 3417; fax: +81 93 884 3417.
E-mail address: [email protected]. 1 The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of
Education, Culture, Sports, Science and Technology. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.04.054
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Theorem 1 (Meir and Keeler [7]). Let (X, d) be a complete metric space and let T be a mapping on X. Assume that for every > 0, there exists > 0 such that d(x, y) < +
implies d(T x, T y) <
for x, y ∈ X. Then T has a unique fixed point. On the other hand, in 2003, Kirk [5] introduced the notion of asymptotic contraction on a metric space, and proved a fixed-point theorem for such contractions. Asymptotic contraction is an asymptotic version of Boyd–Wong contraction [3]. See also [1]. Definition 1 (Kirk [5]). Let (X, d) be a metric space and let T be a mapping on X. Then T is called an asymptotic contraction on X if there exists a continuous function from [0, ∞) into itself and a sequence {n } of functions from [0, ∞) into itself such that (i) (ii) (iii) (iv)
(0) = 0, (r) < r for r ∈ (0, ∞), {n } converges to uniformly on the range of d, and for x, y ∈ X and n ∈ N, d(T n x, T n y)n (d(x, y)).
Theorem 2 (Kirk [5]). Let (X, d) be a complete metric space and let T be a continuous, asymptotic contraction on X with {n } and in Definition 1. Assume that there exists x ∈ X such that the orbit {T n x : n ∈ N} of x is bounded, and that n is continuous for n ∈ N. Then there exists a unique fixed point z ∈ X. Moreover, limn T n x = z for all x ∈ X. Jachymski and Jó´zwik showed that the continuity of T is needed in Theorem 2; see Example 1 in [4]. They also proved a result similar to Theorem 2. The assumption in [4] is that is upper semicontinuous with limt→∞ (t − (t)) = ∞ and T is uniformly continuous. In this paper, we introduce the notion of asymptotic contraction of Meir–Keeler type, and prove a fixed-point theorem for such contractions, which is a generalization of both Theorems 1 and 2. In our discussion, we use the characterization of Meir–Keeler contraction proved by Lim [6]. We also give a simple proof of this characterization. 2. Meir–Keeler contraction In this section, we discuss Meir–Keeler contraction. Definition 2. Let (X, d) be a metric space. Then a mapping T on X is said to be a Meir–Keeler contraction (MKC, for short) if for any > 0, there exists > 0 such that d(x, y) < +
implies d(T x, T y) <
for all x, y ∈ X. In [6], Lim introduced the notion of an L-function and characterized MKC.
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Definition 3 (Lim [6]). A function from [0, ∞) into itself is called an L-function if (0) = 0, (s) > 0 for s ∈ (0, ∞), and for every s ∈ (0, ∞) there exists > 0 such that (t)s for all t ∈ [s, s + ]. We give a simple proof of Lim’s characterization. Proposition 1 (Lim [6]). Let (X, d) be a metric space and let T be a mapping on X. Then T is an MKC if and only if there exists an (nondecreasing, right continuous) L-function such that d(T x, T y) < (d(x, y))
(1)
for all x, y ∈ X with x = y. Proof. We only show the necessity because we can prove the sufficiency easily; see [6]. Assume that T is an MKC. From the definition of MKC, we can define a function from (0, ∞) into itself such that d(x, y) < + 2() implies d(T x, T y) < for ∈ (0, ∞). Using such , we define a nondecreasing function from (0, ∞) into [0, ∞) by (t) = inf{ > 0 : t + ()} for t ∈ (0, ∞). Since t t + (t), we note that (t) t for t ∈ (0, ∞). Define a function 1 from [0, ∞) into itself by 0 if t = 0, 1 (t) = (t) if t > 0 and min{ > 0 : t + ()} exists, ((t) + t)/2 otherwise. It is clear that 1 (0) = 0 and 0 < 1 (s)s for s ∈ (0, ∞). Fix s ∈ (0, ∞). In the case of 1 (t)s for all t ∈ (s, s + (s)], we can put = (s). In the other case, there exists ∈ (s, s + (s)] with 1 () > s. Since s + (s), we have () s. If () = s, then 1 () = () = s < 1 (). This is a contradiction. Thus, () < s < 1 () = (() + )/2. We can choose u ∈ ((), s) with u + (u), and put = s − u > 0. Fix t ∈ [s, s + ]. Since t s + = 2s − u < 2(() + )/2 − () = u + (u), we have (t)u. Hence 1 (t)((t) + t)/2 (u + s + )/2 = s. Therefore 1 is an L-function. Fix x, y ∈ X with x = y. From the definition of 1 , for every t ∈ (0, ∞), there exists ∈ (0, 1 (t)] such that t + (). So, there exists
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∈ (0, 1 (d(x, y))] such that d(x, y) + (). Therefore d(T x, T y) < 1 (d(x, y)) holds. That is, 1 satisfies (1). Define functions 2 and 3 by 2 (t) = sup{1 (s) : s t} and 3 (t) = inf{2 (s) : s > t} for t ∈ [0, ∞). Then we have 0 < 1 (t)2 (t)3 (t)t for all t ∈ (0, ∞). Hence, 2 and 3 also satisfy (1). It is not difficult to verify that 2 is a nondecreasing L-function and 3 is a nondecreasing, right continuous L-function. This completes the proof. 3. ACMK In this section, we discuss the following notion, which is a generalization of both asymptotic contraction and MKC. Definition 4. Let (X, d) be a metric space. Then a mapping T on X is said to be an asymptotic contraction of Meir–Keeler type (ACMK, for short) if there exists a sequence {n } of functions from [0, ∞) into itself satisfying the following: (A1) lim supn n () for all 0. (A2) For each > 0, there exist > 0 and ∈ N such that (t) for all t ∈ [, + ]. (A3) d(T n x, T n y) < n (d(x, y)) for all n ∈ N and x, y ∈ X with x = y. We obtain the following. Proposition 2. Let (X, d) be a metric space. Let T be an MKC on X. Then T is also an ACMK on X. Proof. By Proposition 1, there exists an L-function from [0, ∞) into itself satisfying (1). Define a sequence {n } of functions by n = for all n ∈ N. It is obvious that {n } satisfies (A1) and (A2). Fix x, y ∈ X with x = y. We note d(T n+1 x, T n+1 y)d(T n x, T n y) · · · d(T x, T y)d(x, y) for all n ∈ N. So we obtain d(T n x, T n y)d(T x, T y) < (d(x, y)) = n (d(x, y)). This implies (A3). This completes the proof.
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Proposition 3. Let (X, d) be a metric space and let T be an asymptotic contraction on X. Then T is also an ACMK. Proof. We put E = {d(x, y) : x, y ∈ X}, that is, E is the range of d. Let and {n } be as in Definition 1. Define a sequence {n } of functions from [0, ∞) into itself by n (t) + t/n if t ∈ E, n (t) = 0 if t ∈ / E. We shall show that {n } satisfies (A1)–(A3). It is obvious that {n } satisfies (A1). Fix n ∈ N and x, y ∈ X with x = y. Since d(x, y) > 0, we have d(T n x, T n y) n (d(x, y)) < n (d(x, y)) + d(x, y)/n = n (d(x, y)). Therefore (A3) holds. Let us prove (A2). Fix > 0. Since () < and is continuous, we can choose such that 0 < < ( − ())/2 and (t) − () < ( − ())/2 for t ∈ [, + ]. For such , we also choose ∈ N such that ( + )/ < /2
and
sup{| (t) − (t)| : t ∈ E} < /2.
We fix t ∈ [, + ]. In the case of t ∈ / E, we have (t) = 0 . In the case of t ∈ E, we have (t) = (t) + t/ (t) + /2 + ( + )/ () + ( − ())/2 + 2 /2 () + 2( − ())/2 = . We have shown (A2). This completes the proof.
Remark. We only use the right upper semicontinuity of . So this proposition is the affirmative answer of the problem raised by Jachymski and Jó´zwik; see Remark 4 in [4]. 4. Fxed-point theorems In this section, we prove a fixed-point theorem which is a generalization of both Theorems 1 and 2. Theorem 3. Let (X, d) be a complete metric space. Let T be an ACMK on X. Assume that T is continuous for some ∈ N. Then there exists a unique fixed point z ∈ X. Moreover, limn T n x = z for all x ∈ X. Proof. Let {n } be as in Definition 4. We note d(T n x, T n y) n (d(x, y))
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for all x, y ∈ X and n ∈ N. We first show lim d(T n x, T n y) = 0
n→∞
for all x, y ∈ X.
(2)
Fix x, y ∈ X. Let T 0 be the identity mapping on X. In the case of T x = T y for some ∈ N ∪ {0}, (2) clearly holds. In the other case of T x = T y for all ∈ N ∪ {0}, we assume := lim supn d(T n x, T n y) > 0. From (A2), we can choose 1 ∈ N satisfying 1 (d(x, y))d(x, y). We have d(T 1 x, T 1 y) < 1 (d(x, y))d(x, y). By (A1), we have = lim sup d(T n ◦ T 1 x, T n ◦ T 1 y) n→∞
lim sup n (d(T 1 x, T 1 y)) n→∞
d(T 1 x, T 1 y) < d(x, y). By a similar argument, we obtain < d(T x, T y) for all ∈ N∪{0}. Thus, {d(T n x, T n y)} converges to . Since 0 < < d(x, y) < ∞, there exist 2 > 0 and 2 ∈ N such that 2 (t) for all t ∈ [, + 2 ]. We choose 3 ∈ N with d(T 3 x, T 3 y) < + 2 . Then we have d(T 2 +3 x, T 2 +3 y) = d(T 2 ◦ T 3 x, T 2 ◦ T 3 y) < 2 (d(T 3 x, T 3 y)) . This is a contradiction. Therefore we have shown (2). Let u ∈ X and define a sequence {un } in X by un = T n u for n ∈ N. From (2), we have limn d(un , un+1 ) = 0. We shall show that lim sup d(un , um ) = 0.
(3)
n→∞ m>n
Let > 0 be fixed. Then there exist 4 ∈ (0, ) and 4 ∈ N such that 4 (t) for all t ∈ [, + 4 ]. For such 4 , there exists 5 ∈ N such that d(un , un+1 ) < 4 /4 for every n 5 . Arguing by contradiction, we assume that there exist , m ∈ N with m > 5 and d(u , um ) > 2. Then we put k = min{j ∈ N : < j, + 4 d(u , uj )}. It is obvious that k m. Since 24 < + 4 d(u , uk )
k−1
d(uj , uj +1 )
j =
k−1
4 /4 = (k − ) 4 /4 ,
j =
we have 24 < k − and hence < k − 24 < k − 4 . We have d(u , uk−4 ) d(u , uk ) − d(uk−4 , uk ) d(u , uk ) −
4 −1
d(uk−j −1 , uk−j )
j =0
+ 4 − 4 4 /4 = .
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Since d(u , uk−4 ) < + 4 , we have d(u+4 , uk ) = d(T 4 u , T 4 uk−4 ) < 4 (d(u , uk−4 )) . Hence d(u , uk )
4
d(u+j −1 , u+j ) + d(u+4 , uk ) < 4 4 /4 + = 4 + .
j =1
This contradicts the definition of k. Therefore m > n5 implies d(un , um ) 2 and hence (3) holds. So {un } is a Cauchy sequence. Since X is complete, there exists z ∈ X such that {un } converges to z. Then from the continuity of T , we have z = lim T +n u = lim T ◦ T n u = T lim T n u = T z. n→∞
n→∞
n→∞
That is, z is a fixed point of T . Since lim d(T n+1 u, T z) = lim d(T n+1 u, T n+1 z) = lim d(T n u, T n z) = 0 n→∞
n→∞
n→∞
by (2), we have T z = lim T n+1 u = lim T n u = z. n→∞
n→∞
That is, z is a fixed point of T. If T x = x, then d(z, x) = lim d(T n z, T n x) = 0 n→∞
by (2), and hence x = z. Therefore a fixed point of T is unique. Since u is arbitrary, limn T n x = z holds for every x ∈ X. This completes the proof. We can state Theorem 2 as follows: Theorem 4 (Kirk [5]). Let (X, d) be a complete metric space and let T be a continuous, asymptotic contraction on X. Then there exists a unique fixed point z ∈ X. Moreover, limn T n x = z for all x ∈ X. Acknowledgements The author wishes to express his gratitude to Professor W. A. Kirk for giving the historical comment. References [1] I.D. Arandelovi´c, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384–385. [2] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922) 133–181.
978 [3] [4] [5] [6] [7] [8] [9]
T. Suzuki / Nonlinear Analysis 64 (2006) 971 – 978 D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464. J.R. Jachymski, I. Jó´zwik, On Kirk’s asymptotic contractions, J. Math. Anal. Appl. 300 (2004) 147–159. W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650. T.C. Lim, On characterizations of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120. A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969) 326–329. T. Suzuki, Several fixed point theorems in complete metric spaces, Yokohama Math. J. 44 (1997) 61–72. T. Suzuki, Several fixed point theorems concerning -distance, Fixed Point Theory Appl. 2004 (2004) 195–209.
Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces Tomonari Suzuki∗,1 Department of Mathematics, Kyushu Institute of Technology, 1-1, Sensuicho, Tobata-ku, Kitakyushu 804 8550, Japan Received 21 January 2005; accepted 25 April 2005
Abstract In this paper, we introduce the notion of asymptotic contraction of Meir–Keeler type, and prove a fixed-point theorem for such contractions, which is a generalization of fixed-point theorems of Meir–Keeler and Kirk. In our discussion, we use the characterization of Meir–Keeler contraction proved by Lim [On characterizations of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120]. We also give a simple proof of this characterization. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: Primary 54H25; Secondary 54E50 Keywords: Meir–Keeler contraction; L-function; Asymptotic contraction; Fixed point; Complete metric space
1. Introduction Throughout this paper we denote by N the set of all positive integers. In 1969, Meir and Keeler [7] proved the following very interesting fixed-point theorem, which is a generalization of the Banach contraction principle [2]. See also [8,9].
∗ Tel.: +81 93 884 3417; fax: +81 93 884 3417.
E-mail address: [email protected]. 1 The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of
Education, Culture, Sports, Science and Technology. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.04.054
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T. Suzuki / Nonlinear Analysis 64 (2006) 971 – 978
Theorem 1 (Meir and Keeler [7]). Let (X, d) be a complete metric space and let T be a mapping on X. Assume that for every > 0, there exists > 0 such that d(x, y) < +
implies d(T x, T y) <
for x, y ∈ X. Then T has a unique fixed point. On the other hand, in 2003, Kirk [5] introduced the notion of asymptotic contraction on a metric space, and proved a fixed-point theorem for such contractions. Asymptotic contraction is an asymptotic version of Boyd–Wong contraction [3]. See also [1]. Definition 1 (Kirk [5]). Let (X, d) be a metric space and let T be a mapping on X. Then T is called an asymptotic contraction on X if there exists a continuous function from [0, ∞) into itself and a sequence {n } of functions from [0, ∞) into itself such that (i) (ii) (iii) (iv)
(0) = 0, (r) < r for r ∈ (0, ∞), {n } converges to uniformly on the range of d, and for x, y ∈ X and n ∈ N, d(T n x, T n y)n (d(x, y)).
Theorem 2 (Kirk [5]). Let (X, d) be a complete metric space and let T be a continuous, asymptotic contraction on X with {n } and in Definition 1. Assume that there exists x ∈ X such that the orbit {T n x : n ∈ N} of x is bounded, and that n is continuous for n ∈ N. Then there exists a unique fixed point z ∈ X. Moreover, limn T n x = z for all x ∈ X. Jachymski and Jó´zwik showed that the continuity of T is needed in Theorem 2; see Example 1 in [4]. They also proved a result similar to Theorem 2. The assumption in [4] is that is upper semicontinuous with limt→∞ (t − (t)) = ∞ and T is uniformly continuous. In this paper, we introduce the notion of asymptotic contraction of Meir–Keeler type, and prove a fixed-point theorem for such contractions, which is a generalization of both Theorems 1 and 2. In our discussion, we use the characterization of Meir–Keeler contraction proved by Lim [6]. We also give a simple proof of this characterization. 2. Meir–Keeler contraction In this section, we discuss Meir–Keeler contraction. Definition 2. Let (X, d) be a metric space. Then a mapping T on X is said to be a Meir–Keeler contraction (MKC, for short) if for any > 0, there exists > 0 such that d(x, y) < +
implies d(T x, T y) <
for all x, y ∈ X. In [6], Lim introduced the notion of an L-function and characterized MKC.
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Definition 3 (Lim [6]). A function from [0, ∞) into itself is called an L-function if (0) = 0, (s) > 0 for s ∈ (0, ∞), and for every s ∈ (0, ∞) there exists > 0 such that (t)s for all t ∈ [s, s + ]. We give a simple proof of Lim’s characterization. Proposition 1 (Lim [6]). Let (X, d) be a metric space and let T be a mapping on X. Then T is an MKC if and only if there exists an (nondecreasing, right continuous) L-function such that d(T x, T y) < (d(x, y))
(1)
for all x, y ∈ X with x = y. Proof. We only show the necessity because we can prove the sufficiency easily; see [6]. Assume that T is an MKC. From the definition of MKC, we can define a function from (0, ∞) into itself such that d(x, y) < + 2() implies d(T x, T y) < for ∈ (0, ∞). Using such , we define a nondecreasing function from (0, ∞) into [0, ∞) by (t) = inf{ > 0 : t + ()} for t ∈ (0, ∞). Since t t + (t), we note that (t) t for t ∈ (0, ∞). Define a function 1 from [0, ∞) into itself by 0 if t = 0, 1 (t) = (t) if t > 0 and min{ > 0 : t + ()} exists, ((t) + t)/2 otherwise. It is clear that 1 (0) = 0 and 0 < 1 (s)s for s ∈ (0, ∞). Fix s ∈ (0, ∞). In the case of 1 (t)s for all t ∈ (s, s + (s)], we can put = (s). In the other case, there exists ∈ (s, s + (s)] with 1 () > s. Since s + (s), we have () s. If () = s, then 1 () = () = s < 1 (). This is a contradiction. Thus, () < s < 1 () = (() + )/2. We can choose u ∈ ((), s) with u + (u), and put = s − u > 0. Fix t ∈ [s, s + ]. Since t s + = 2s − u < 2(() + )/2 − () = u + (u), we have (t)u. Hence 1 (t)((t) + t)/2 (u + s + )/2 = s. Therefore 1 is an L-function. Fix x, y ∈ X with x = y. From the definition of 1 , for every t ∈ (0, ∞), there exists ∈ (0, 1 (t)] such that t + (). So, there exists
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∈ (0, 1 (d(x, y))] such that d(x, y) + (). Therefore d(T x, T y) < 1 (d(x, y)) holds. That is, 1 satisfies (1). Define functions 2 and 3 by 2 (t) = sup{1 (s) : s t} and 3 (t) = inf{2 (s) : s > t} for t ∈ [0, ∞). Then we have 0 < 1 (t)2 (t)3 (t)t for all t ∈ (0, ∞). Hence, 2 and 3 also satisfy (1). It is not difficult to verify that 2 is a nondecreasing L-function and 3 is a nondecreasing, right continuous L-function. This completes the proof. 3. ACMK In this section, we discuss the following notion, which is a generalization of both asymptotic contraction and MKC. Definition 4. Let (X, d) be a metric space. Then a mapping T on X is said to be an asymptotic contraction of Meir–Keeler type (ACMK, for short) if there exists a sequence {n } of functions from [0, ∞) into itself satisfying the following: (A1) lim supn n () for all 0. (A2) For each > 0, there exist > 0 and ∈ N such that (t) for all t ∈ [, + ]. (A3) d(T n x, T n y) < n (d(x, y)) for all n ∈ N and x, y ∈ X with x = y. We obtain the following. Proposition 2. Let (X, d) be a metric space. Let T be an MKC on X. Then T is also an ACMK on X. Proof. By Proposition 1, there exists an L-function from [0, ∞) into itself satisfying (1). Define a sequence {n } of functions by n = for all n ∈ N. It is obvious that {n } satisfies (A1) and (A2). Fix x, y ∈ X with x = y. We note d(T n+1 x, T n+1 y)d(T n x, T n y) · · · d(T x, T y)d(x, y) for all n ∈ N. So we obtain d(T n x, T n y)d(T x, T y) < (d(x, y)) = n (d(x, y)). This implies (A3). This completes the proof.
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Proposition 3. Let (X, d) be a metric space and let T be an asymptotic contraction on X. Then T is also an ACMK. Proof. We put E = {d(x, y) : x, y ∈ X}, that is, E is the range of d. Let and {n } be as in Definition 1. Define a sequence {n } of functions from [0, ∞) into itself by n (t) + t/n if t ∈ E, n (t) = 0 if t ∈ / E. We shall show that {n } satisfies (A1)–(A3). It is obvious that {n } satisfies (A1). Fix n ∈ N and x, y ∈ X with x = y. Since d(x, y) > 0, we have d(T n x, T n y) n (d(x, y)) < n (d(x, y)) + d(x, y)/n = n (d(x, y)). Therefore (A3) holds. Let us prove (A2). Fix > 0. Since () < and is continuous, we can choose such that 0 < < ( − ())/2 and (t) − () < ( − ())/2 for t ∈ [, + ]. For such , we also choose ∈ N such that ( + )/ < /2
and
sup{| (t) − (t)| : t ∈ E} < /2.
We fix t ∈ [, + ]. In the case of t ∈ / E, we have (t) = 0 . In the case of t ∈ E, we have (t) = (t) + t/ (t) + /2 + ( + )/ () + ( − ())/2 + 2 /2 () + 2( − ())/2 = . We have shown (A2). This completes the proof.
Remark. We only use the right upper semicontinuity of . So this proposition is the affirmative answer of the problem raised by Jachymski and Jó´zwik; see Remark 4 in [4]. 4. Fxed-point theorems In this section, we prove a fixed-point theorem which is a generalization of both Theorems 1 and 2. Theorem 3. Let (X, d) be a complete metric space. Let T be an ACMK on X. Assume that T is continuous for some ∈ N. Then there exists a unique fixed point z ∈ X. Moreover, limn T n x = z for all x ∈ X. Proof. Let {n } be as in Definition 4. We note d(T n x, T n y) n (d(x, y))
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for all x, y ∈ X and n ∈ N. We first show lim d(T n x, T n y) = 0
n→∞
for all x, y ∈ X.
(2)
Fix x, y ∈ X. Let T 0 be the identity mapping on X. In the case of T x = T y for some ∈ N ∪ {0}, (2) clearly holds. In the other case of T x = T y for all ∈ N ∪ {0}, we assume := lim supn d(T n x, T n y) > 0. From (A2), we can choose 1 ∈ N satisfying 1 (d(x, y))d(x, y). We have d(T 1 x, T 1 y) < 1 (d(x, y))d(x, y). By (A1), we have = lim sup d(T n ◦ T 1 x, T n ◦ T 1 y) n→∞
lim sup n (d(T 1 x, T 1 y)) n→∞
d(T 1 x, T 1 y) < d(x, y). By a similar argument, we obtain < d(T x, T y) for all ∈ N∪{0}. Thus, {d(T n x, T n y)} converges to . Since 0 < < d(x, y) < ∞, there exist 2 > 0 and 2 ∈ N such that 2 (t) for all t ∈ [, + 2 ]. We choose 3 ∈ N with d(T 3 x, T 3 y) < + 2 . Then we have d(T 2 +3 x, T 2 +3 y) = d(T 2 ◦ T 3 x, T 2 ◦ T 3 y) < 2 (d(T 3 x, T 3 y)) . This is a contradiction. Therefore we have shown (2). Let u ∈ X and define a sequence {un } in X by un = T n u for n ∈ N. From (2), we have limn d(un , un+1 ) = 0. We shall show that lim sup d(un , um ) = 0.
(3)
n→∞ m>n
Let > 0 be fixed. Then there exist 4 ∈ (0, ) and 4 ∈ N such that 4 (t) for all t ∈ [, + 4 ]. For such 4 , there exists 5 ∈ N such that d(un , un+1 ) < 4 /4 for every n 5 . Arguing by contradiction, we assume that there exist , m ∈ N with m > 5 and d(u , um ) > 2. Then we put k = min{j ∈ N : < j, + 4 d(u , uj )}. It is obvious that k m. Since 24 < + 4 d(u , uk )
k−1
d(uj , uj +1 )
j =
k−1
4 /4 = (k − ) 4 /4 ,
j =
we have 24 < k − and hence < k − 24 < k − 4 . We have d(u , uk−4 ) d(u , uk ) − d(uk−4 , uk ) d(u , uk ) −
4 −1
d(uk−j −1 , uk−j )
j =0
+ 4 − 4 4 /4 = .
T. Suzuki / Nonlinear Analysis 64 (2006) 971 – 978
977
Since d(u , uk−4 ) < + 4 , we have d(u+4 , uk ) = d(T 4 u , T 4 uk−4 ) < 4 (d(u , uk−4 )) . Hence d(u , uk )
4
d(u+j −1 , u+j ) + d(u+4 , uk ) < 4 4 /4 + = 4 + .
j =1
This contradicts the definition of k. Therefore m > n5 implies d(un , um ) 2 and hence (3) holds. So {un } is a Cauchy sequence. Since X is complete, there exists z ∈ X such that {un } converges to z. Then from the continuity of T , we have z = lim T +n u = lim T ◦ T n u = T lim T n u = T z. n→∞
n→∞
n→∞
That is, z is a fixed point of T . Since lim d(T n+1 u, T z) = lim d(T n+1 u, T n+1 z) = lim d(T n u, T n z) = 0 n→∞
n→∞
n→∞
by (2), we have T z = lim T n+1 u = lim T n u = z. n→∞
n→∞
That is, z is a fixed point of T. If T x = x, then d(z, x) = lim d(T n z, T n x) = 0 n→∞
by (2), and hence x = z. Therefore a fixed point of T is unique. Since u is arbitrary, limn T n x = z holds for every x ∈ X. This completes the proof. We can state Theorem 2 as follows: Theorem 4 (Kirk [5]). Let (X, d) be a complete metric space and let T be a continuous, asymptotic contraction on X. Then there exists a unique fixed point z ∈ X. Moreover, limn T n x = z for all x ∈ X. Acknowledgements The author wishes to express his gratitude to Professor W. A. Kirk for giving the historical comment. References [1] I.D. Arandelovi´c, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384–385. [2] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922) 133–181.
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T. Suzuki / Nonlinear Analysis 64 (2006) 971 – 978 D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464. J.R. Jachymski, I. Jó´zwik, On Kirk’s asymptotic contractions, J. Math. Anal. Appl. 300 (2004) 147–159. W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650. T.C. Lim, On characterizations of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120. A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969) 326–329. T. Suzuki, Several fixed point theorems in complete metric spaces, Yokohama Math. J. 44 (1997) 61–72. T. Suzuki, Several fixed point theorems concerning -distance, Fixed Point Theory Appl. 2004 (2004) 195–209.