638 W. TAKAHASHI, N.-C. WONG, AND J.-C. YAO

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1 2013

2 638 W. TAKAHASHI, N.-C. WONG, AND J.-C. YAO theorems for such mappings on complete metric spaces. Let (X, d) be a metric space. A mapping T : X X is called contractively generalized hybrid [6] if there exist α, β R and r [0, 1) such that (1.1) αd(t x, T y)+(1 α)d(x, T y) r{βd(t x, y) + (1 β)d(x, y)} for all x, y X. Such a mapping T is also called contractively (α, β, r)-generalized hybrid; see also [3, 13]. For example, a ontractively (α, β, r)-generalized hybrid mapping is r-contractive for α = 1 and β = 0. It is contractively nonspreading for α = 1 + r and β = 1. Furthermore, it is ontractively hybrid for α = 1 + r 2 and β = 1 2 ; see Hasegawa, Komiya and Takahashi [6]. In this paper, motivated by w-distances and Hasegawa, Komiya and Takahashi [6], we first prove a generalized fixed point theorem for mappings without continuity in a complete metric space. Using this result, we obtain new and well-known fixed point theorems in a complete metric space. 2. Preliminaries Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively. Let X be a metric space with metric d. Then we denote by W (X) the set of all w-distances on X. A w-distance p on X is called symmetric if p(x, y) = p(y, x) for all x, y X. We denote by W 0 (X) the set of all symmetric w- distances on X. Note that the metric d is an element of W 0 (X). We also know that there are many important examples of w-distances on X; see [8, 16]. We denote by W C 1 (X) the set of all mappings T of X into itself such that there exist p W (X) and r [0, 1) satisfying p(t x, T y) rp(x, y) for all x, y X. Such a mapping T is called p-contractive. Shioji, Suzuki and Takahashi [14] also introduced the sets W C 2 (X), W C 0 (X), W K 1 (X), W K 2 (X) and W K 0 (X) of mappings of X into itself as follows: T W C 2 (X) if and only if there exist p W (X) and r [0, 1) such that p(t x, T y) rp(y, x) for all x, y X; T W C 0 (X) if and only if there exist p W 0 (X) and r [0, 1) such that p(t x, T y) rp(x, y) for all x, y X; T W K 1 (X) if and only if there exist p W (X) and α [0, 1/2) such that p(t x, T y) α{p(t x, x) + p(t y, y)} for all x, y X; T W K 2 (X) if and only if there exist p W (X) and α [0, 1/2) such that p(t x, T y) α{p(t x, x) + p(y, T y)} for all x, y X; T W K 0 (X) if and only if there exist p W 0 (X) and α [0, 1/2) such that p(t x, T y) α{p(t x, x) + p(t y, y)} for all x, y X. In particular, a mapping T W K 1 (X) is called p-kannan. The following lemma was proved in [14].

3 FIXED POINT THEOREMS FOR GENERAL CONTRACTIVE MAPPINGS 639 Lemma 2.1 ([14]). Let X be a metric space with metric d, let p be a w-distance on X, let T be a mapping of X into itself and let u be a point in X such that lim p(t m u, T n u) = 0. m,n Then for every x X, lim k p(t k u, x) and lim k p(x, T k u) exist. Moreover, let β and γ be functions from X to [0, ) defined by Then the following hold: β(x) = lim k p(t k u, x) and γ(x) = lim k p(x, T k u). (i) β is lower semicontinuous on X; (ii) for every ε > 0, there exists δ > 0 such that β(x) δ and β(y) δ imply d(x, y) ε. In particular, the set {x X : β(x) = 0} consists of at most one point; (iii) the functions q 0 and q 1 from X X to [0, ) defined by q 0 (x, y) = β(x) + β(y) and q 1 (x, y) = γ(x) + β(y) are w-distances on X. Shioji, Suzuki and Takahashi [14] proved the following theorem from Lemma 2.1. Theorem 2.2 ([14]). Let (X, d) be a metric space. Then W C 1 (X) = W C 0 (X) = W K 1 (X) = W K 0 (X) W C 2 (X) = W K 2 (X). Iemoto, Takahashi and Yingtaweesittikul [7] also introduced the following class of mappings of X into itself. Let p be a w-distance on X. A mapping T : X X is called p-contractively nonspreading if there exists α [0, 1/2) such that p(t x, T y) α{p(t x, y) + p(x, T y)}, x, y X. In [7], they proved the following result from Lemma 2.1. Theorem 2.3 ([7]). Let (X, d) be a metric space and let p be a w-distance on X such that p(x, x) = 0 for all x X. Let T be a p-contractively nonspreading mapping of X into itself. Then T is in W C 0 (X). Let l be the Banach space of bounded sequences with the supremum norm. A linear functional µ on l is called a mean if µ(e) = µ = 1, where e = (1, 1, 1,...). For x = (x 1, x 2, x 3,...), the value µ(x) is also denoted by µ n (x n ). A mean µ on l is called a Banach limit if it satisfies µ n (x n ) = µ n (x n+1 ). If µ is a Banach limit on l, then for x = (x 1, x 2, x 3,...) l, lim inf x n µ n (x n ) lim sup x n. n n In particular, if x = (x 1, x 2, x 3,...) l and x n a R, then we have µ(x) = µ n (x n ) = a. For details, we can refer [16].

4 640 W. TAKAHASHI, N.-C. WONG, AND J.-C. YAO 3. Generalized fixed point theorem In this section, we prove a fixed point theorem for mappings with w-distances in complete metric spaces. Before proving it, we need the following lemma proved by Kada, Suzuki and Takahashi [8]; see also [16]. Lemma 3.1 ([8]). Let (X, d) be a complete metric space and let p be a w-distance on X. Let {x n } and {y n } be sequences in X. Let {α n } and {β n } be sequences in [0, ) converging to 0, and let x, y, z X. Then the following hold: (i) If p(x n, y) α n and p(x n, z) β n for all n N, then y = z. In particular, if p(x, y) = 0 and p(x, z) = 0, then y = z; (ii) if p(x n, y n ) α n and p(x n, z) β n for all n N, then the sequence {y n } converges to z; (iii) if p(x n, x m ) α n for all n, m N with m > n, then the sequence {x n } is a Cauchy sequence; (iv) if p(y, x n ) α n for all n N, then {x n } is a Cauchy sequence. Theorem 3.2. Let (X, d) be a complete metric space, let p W 0 (X) and let {x n } be a sequence in X such that {p(x n, x)} is bounded for some x X. Let T be a mapping of X into itself. Suppose that there exist a real number r [0, 1) and a mean µ on l such that µ n p(x n, T y) rµ n p(x n, y), y X. Then, the following hold: Proof. Since {p(x n, x)} is bounded for some x X, we have that, for any y X, {p(x n, y)} is bounded. In fact, we have that, for any n N, p(x n, y) p(x n, x) + p(x, y) sup p(x m, x) + p(x, y) <. m N Using a mean µ on l, we can define a function g : X R as follows: g(y) = µ n p(x n, y), y X. For any z X, consider a sequence {T n z} in X. We have that, for any m, n N, p(t m z, T m+1 z) p(t m z, x n ) + p(x n, T m+1 z). Since µ is a mean on l and p is symmetric, we have that, for any m N, (3.1) p(t m z, T m+1 z) µ n p(t m z, x n ) + µ n p(x n, T m+1 z) = µ n p(x n, T m z) + µ n p(x n, T m+1 z) rµ n p(x n, T m 1 z) + rµ n p(x n, T m z) r m µ n p(x n, z) + r m µ n p(x n, T z) r m µ n p(x n, z) + r m+1 µ n p(x n, z) = r m (1 + r)µ n p(x n, z) = r m (1 + r)g(z).

5 FIXED POINT THEOREMS FOR GENERAL CONTRACTIVE MAPPINGS 641 We have from (3.1) that, for any l, m N with m > l, (3.2) p(t l z, T m z) p(t l z, T l+1 z) + p(t l+1 z, T l+2 z) + + p(t m 1 z, T m z) r l (1 + r)g(z) + r l+1 (1 + r)g(z) + + r m 1 (1 + r)g(z) r l (1 + r)g(z) + r l+1 (1 + r)g(z) + + r m 1 (1 + r)g(z) + = r l (1 + r)g(z)(1 + r + r 2 + r 3 + ) = r l 1 (1 + r)g(z) 1 r and r l (1 + r)g(z) 1 1 r 0 as l. We have from Lemma 3.1 that {T m z} is a Cauchy sequence in X. Since X is complete, we have that {T m z} converges. Let T m z u. We know from the definition of p that, for any n N, y p(x n, y) is lower semicontinuous. Using this, we have that, for any n N, p(x n, u) lim inf p(x n, T m z) and hence ) (3.3) g(u) = µ n p(x n, u) µ n (lim inf p(x n, T m z). On the other hand, we have from (3.2) that, for any l, m, n N with m > l, p(x n, T m z) p(x n, T l z) + p(t l z, T m z) p(x n, T l z) + r l 1 (1 + r)g(z) 1 r and hence lim sup p(x n, T m z) p(x n, T l z) + r l 1 (1 + r)g(z) 1 r. Applying µ to both sides of the inequality, we have that ) µ n (lim sup p(x n, T m z) µ n p(x n, T l z) + r l 1 (1 + r)g(z) 1 r. Letting l, we get that ) (3.4) µ n (lim sup p(x n, T m z) lim inf l µ np(x n, T l z). We have from (3.3) and (3.4) that ) g(u) = µ n p(x n, u) µ n (lim inf p(x n, T m z) ) µ n (lim sup p(x n, T m z) (3.5) lim inf µ np(x n, T m z) Furthermore, from = lim inf g(t m z) lim sup g(t m z). g(t m z) = µ n p(x n, T m z) rµ n p(x n, T m 1 z) r m µ n p(x n, z) = r m g(z),

6 642 W. TAKAHASHI, N.-C. WONG, AND J.-C. YAO we have that (3.6) lim sup g(t m z) 0. Therefore, we obtain from (3.5) and (3.6) that g(u) 0. This implies that g(u) = µ n p(x n, u) = 0. We show that u is a fixed point of T. Since for all n N, we have p(t u, u) p(t u, x n ) + p(x n, u) p(t u, u) µ n p(x n, T u) + µ n p(x n, u) rµ n p(x n, u) + µ n p(x n, u) = r0 + 0 = 0 and hence p(t u, u) = 0. We also have that for all n N. From this, we have that p(t u, T u) p(t u, x n ) + p(x n, T u) p(t u, T u) µ n p(x n, T u) + µ n p(x n, T u) rµ n p(x n, u) + rµ n p(x n, u) = r0 + r0 = 0 and hence p(t u, T u) = 0. We have from Lemma 3.1 that T u = u. We show that such a fixed point u is unique. Let T u = u and T v = v. Since 0 r < 1 and µ n p(x n, u) = µ n p(x n, T u) rµ n p(x n, u), we obtain µ n p(x n, u) = 0. Similarly, we have µ n p(x n, v) = 0. Since for all n N, we have p(u, v) p(u, x n ) + p(x n, v) p(u, v) µ n p(x n, u) + µ n p(x n, v) = = 0 and hence p(u, v) = 0. Furthermore, since for all n N, we have p(u, u) p(u, x n ) + p(x n, u) p(u, u) µ n p(x n, u) + µ n p(x n, u) = = 0 and p(u, u) = 0. We have from Lemma 3.1 that u = v. This completes the proof. As a direct consequence of Theorem 3.2, we obtain the following theorem proved by Hasegawa, Komiya and Takahashi [6].

7 FIXED POINT THEOREMS FOR GENERAL CONTRACTIVE MAPPINGS 643 Theorem 3.3 ([6]). Let (X, d) be a complete metric space and let T be a mapping of X into itself. Suppose that there exist a real number r with 0 r < 1 and an element x X such that {T n x} is bounded and µ n d(t n x, T y) rµ n d(t n x, y), y X for some mean µ on l. Then, the following hold: Proof. We know that the metric d is one of symmetric w-distances on X; see [8, 16]. We also have that {d(t n x, x)} is bounded because {T n x} is bounded. Thus we have the desired result from Theorem Applications In this section, using Theorem 3.2, we prove new and well-known fixed point theorems in a complete metric space. We first prove a fixed point theorem for generalized hybrid mappings with w-distances in a metric space. Let (X, d) be a metric space and let p be a w-distance on X. A mapping T : X X is called p-contractively generalized hybrid if there exist α, β R and r [0, 1) such that (4.1) αp(t x, T y) + (1 α)p(x, T y) r{βp(t x, y) + (1 β)p(x, y)} for all x, y X. We call such a mapping T a p-contractively (α,β,r)-generalized hybrid mapping. We know that the class of the mappings above covers well-known mappings in a metric space. For example, a p-contractively (α,β,r)-generalized hybrid mapping T is p-contractive for α = 1 and β = 0, i.e., there exists r [0, 1) such that p(t x, T y) rp(x, y), x, y X. Theorem 4.1. Let (X, d) be a complete metric space and let p be a symmetric w-distance on X. Let T : X X be a p-contractively generalized hybrid mapping. Then T has a fixed point in X if and only if {p(t n x, x)} is bounded for some x X. In this case, the following hold: Proof. Since T : X X is a p-contractively generalized hybrid mapping, there exist α, β R and r [0, 1) such that (4.2) αp(t x, T y) + (1 α)p(x, T y) r{βp(t x, y) + (1 β)p(x, y)} for all x, y X. If F (T ), then {T n u} = {u} for u F (T ). So, {p(t n u, u)} = {p(u, u)} is bounded. We show the reverse. Take x X such that {p(t n x, x)} is bounded. Then we have that, for any y X and n N, {p(t n x, y)} is bounded. In fact, we have that (4.3) p(t n x, y) p(t n x, x) + p(x, y) sup p(t m x, x) + p(x, y) <. m N We also have from (4.2) that, for any y X, αp(t n+1 x, T y)+(1 α)p(t n x, T y)

8 644 W. TAKAHASHI, N.-C. WONG, AND J.-C. YAO r{βp(t n+1 x, y) + (1 β)p(t n x, y)}. Applying a Banach limit µ to both sides of the inequality, we have Then, we obtain and hence µ n (αp(t n+1 x, T y) + (1 α)p(t n x, T y)) µ n (r{βp(t n+1 x, y) + (1 β)p(t n x, y)}). αµ n p(t n+1 x, T y)+(1 α)µ n p(t n x, T y) rβµ n p(t n+1 x, y) + r(1 β)µ n p(t n x, y) αµ n p(t n x, T y)+(1 α)µ n p(t n x, T y) rβµ n p(t n x, y) + r(1 β)µ n p(t n x, y). This implies that µ n p(t n x, T y) rµ n p(t n x, y) for all y X. By Theorem 3.2, T has a unique fixed point u in X. Furthermore, for any z X, the sequence {T n z} converges to u. Using Theorem 4.1, we prove a fixed point theorem for p-contractive mappings in a complete metric space. Theorem 4.2. Let (X, d) be a complete metric space and let p be a w-distance on X. Let T : X X be a p-contractive mapping, i.e., there exists a real number r with 0 r < 1 such that p(t x, T y) rp(x, y) for all x, y X. Then, the following hold: Proof. Putting α = 1 and β = 0 in (4.2), we have that p(t x, T y) rp(x, y) for all x, y X. From Theorem 2.2, we have W C 1 (X) = W C 0 (X). Then there exist a symmetric q W 0 (X) and a real number λ [0, 1) such that (4.4) q(t x, T y) λq(x, y), x, y X. Take x X and n N. Replacing x by T n 1 x and y by T n x in (4.4), we have that (4.5) q(t n x, T n+1 x) λq(t n 1 x, T n x). Thus we have that, for any n N, q(x, T n x) q(x, T x) + q(t x, T 2 x) + + q(t n 1 x, T n x) q(x, T x) + λq(x, T x) + + λ n 1 q(x, T x) q(x, T x) + λq(x, T x) + + λ n 1 q(x, T x) +... = q(x, T x)(1 + λ + + λ n ) 1 = q(x, T x) 1 λ

9 FIXED POINT THEOREMS FOR GENERAL CONTRACTIVE MAPPINGS 645 and hence {q(x, T n x)} = {q(t n x, x)} is bounded. We also have that (4.6) µ n q(t n x, T y) = µ n q(t n+1 x, T y) λµ n q(t n x, y), y X. Therefore, we have the desired result from Theorem 4.1. The following is a fixed point theorem for p-kannan mappings in a complete metric space. Theorem 4.3. Let (X, d) be a complete metric space and let p be a w-distance on X. Let T W K 1 (X), i.e., there exists α [0, 1/2) such that Then, the following hold: p(t x, T y) α{p(t x, x) + p(t y, y)}, x, y X. Proof. From Theorem 2.2, we have W K 1 (X) = W C 0 (X). From Theorem 4.2, we have the desired result. Using Theorems 4.2 and 2.3, we also have the following fixed point theorem. Theorem 4.4. Let (X, d) be a complete metric space and let p be a w-distance on X such that p(x, x) = 0 for all x X. Let T : X X be p-contractively nonspreading, i.e., there exists a real number γ with 0 γ < 1 2 such that p(t x, T y) γ{p(t x, y) + p(x, T y)} for all x, y X. Then, the following hold: Proof. We know from Theorem 2.3 that the mapping T is in W C 0 (X). So, we have the desired result from Theorem 4.2. Concerning that {p(t n x, x)} is bounded for some x X in Theorem 4.1, we have the following lemma. Lemma 4.5. Let (X, d) be a complete metric space and let p be a w-distance on X such that p(x, x) = 0 for all x X. Let T : X X be a p-contractively (α,β,r)-generalized hybrid mapping, i.e., there exist α, β R and r [0, 1) such that (4.7) αp(t x, T y) + (1 α)p(x, T y) r{βp(t x, y) + (1 β)p(x, y)} for all x, y X. Furthermore, α, β and r satisfy β 0, α rβ > 0 and r < Then, {p(t n x, x)} is bounded for all x X. α 1 + β.

10 646 W. TAKAHASHI, N.-C. WONG, AND J.-C. YAO Proof. Take x X and n N. Replacing x by T n x and y by T n 1 x in (4.7), we have (4.8) αp(t n+1 x, T n x)+(1 α)p(t n x, T n x) From β 0 and (4.8), we have (4.9) and hence r{βp(t n+1 x, T n 1 x) + (1 β)p(t n x, T n 1 x)}. αp(t n+1 x, T n x) r{β(p(t n+1 x, T n x) + p(t n x, T n 1 x)) + (1 β)p(t n x, T n 1 x)} (4.10) (α rβ)p(t n+1 x, T n x) rp(t n x, T n 1 x). From α rβ > 0 we have (4.11) p(t n+1 x, T n x) From r < Putting λ = r α rβ p(t n x, T n 1 x). α 1+β, we have r < α rβ and r 0 α rβ < 1. r α rβ, we have that for any n N, p(t n x, x) p(t n x, T n 1 x) + p(t n 1 x, T n 2 x) + + p(t 2 x, T x) + p(t x, x) λ n 1 p(t x, x) + λ n 2 p(t x, x) + + λp(t x, x) + p(t x, x) p(t x, x)(1 + λ + + λ n ) 1 = p(t x, x) 1 λ. Then the sequence {p(t n x, x)} is bounded. Using Theorem 4.1 and Lemma 4.5, we prove the following fixed point theorem proved by Hasegawa, Komiya and Takahashi [6]. Theorem 4.6 ([6]). Let (X, d) be a complete metric space and let T : X X be an (α,β,r)-contractively generalized hybrid mapping, i.e., there exist α, β R and r [0, 1) such that αd(t x, T y) + (1 α)d(x, T y) r{βd(t x, y) + (1 β)d(x, y)} for all x, y X. Furthermore, α, β and r satisfy Then, the following hold: β 0, α rβ > 0 and r < α 1 + β. Proof. Since d(x, y) = d(y, x) and d(x, x) = 0 for all x, y X, we have the desired result from Theorem 4.1 and Lemma 4.5.

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12 648 W. TAKAHASHI, N.-C. WONG, AND J.-C. YAO Wataru Takahashi Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan; Keio Research and Education Center for Natural Sciences, Keio University, Yokohama ; and Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo address: Ngai-Ching Wong Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424,Taiwan; and Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan address: Jen-Chih Yao Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan; and Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia address: