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This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Fixed point theorems for mappings with condition (B) Fixed Point Theory and Applications 2011, 2011:92 doi:10.1186/1687-1812-2011-92 Lai-Jiu Lin (maljlin@cc.ncue.edu.tw) Chih Sheng Chuang (cschuang1977@gmail.com) Zenn Tsun Yu (t106@nkut.edu.tw) ISSN 1687-1812 Article type Research Submission date 19 July 2011 Acceptance date 2 December 2011 Publication date 2 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/92 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Fixed point theorems for mappings with condition (B) Lai-Jiu Lin ∗1 , Chih-Sheng Chuang 1 and Zenn-Tsun Yu 2 1 Department of Mathematics, National Changhua University of Education, Changhua 50058, Taiwan 2 Department of Electronic Engineering, Nan Kai University of Technology, Nantour 54243, Taiwan ∗ Corresponding author: maljlin@cc.ncue.edu.tw Email address: C-SC: cschuang1977@gmail.com Z-TY: t106@nkut.edu.tw Abstract In this article, a new type of mappings that satisfies condition (B) is in- troduced. We study Pazy’s type fixed point theorems, demiclosed principles, and ergodic theorem for mappings with condition (B). Next, we consider the weak convergence theorems for equilibrium problems and the fixed points of mappings with condition (B). Keywords: fixed point; equilibrium problem; Banach limit; generalized hy- brid mapping; projection. 1 1 Introduction Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → H be a mapping, and let F(T ) denote the set of fixed points of T . A mapping T : C → H is said to be nonexpansive if ||T x − T y|| ≤ ||x − y|| for all x, y ∈ C. A mapping T : C → H is said to be quasi-nonexpansive mapping if F (T ) = ∅ and ||T x − y|| ≤ ||x − y|| for all x ∈ C and y ∈ F (T ). In 2008, Kohsaka and Takahashi [1] introduced nonspreading mapping, and obtained a fixed point theorem for a single nonspreading mapping, and a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. A mapping T : C → C is called nonspreading [1] if 2||T x − T y|| 2 ≤ ||T x − y|| 2 + ||T y − x|| 2 for all x, y ∈ C. Indeed, T : C → C is a nonspreading mapping if and only if ||T x − T y|| 2 ≤ ||x − y|| 2 + 2x − Tx, y − T y for all x, y ∈ C [2]. Recently, Takahashi and Yao [3] introduced two nonlinear mappings in Hilbert spaces. A mapping T : C → C is called a T J-1 mapping [3] if 2||T x − T y|| 2 ≤ ||x − y|| 2 + T x − y 2 for all x, y ∈ C. A mapping T : C → C is called a T J-2 [3] mapping if 3||T x − T y|| 2 ≤ 2||T x − y|| 2 + T y − x 2 for all x, y ∈ C. In 2010, Takahashi [4] introduced the hybrid mappings. A mapping T : C → C is hybrid [4] if ||T x − T y|| 2 ≤ ||x − y|| 2 + x − Tx, y − T y for each x, y ∈ C. Indeed, T : C → C is a hybrid mapping if and only if 3||T x − T y|| 2 ≤ ||x − y|| 2 + ||T x − y|| 2 + ||T y − x|| 2 2 for all x, y ∈ C [4]. In 2010, Aoyoma et al. [5] introduced λ-hybrid mappings in a Hilbert space. Note that the class of λ-hybrid mappings contain the classes of nonexpansive map- pings, nonspreading mappings, and hybrid mappings. Let λ be a real number. A mapping T : C → C is called λ-hybrid [5] if ||T x − T y|| 2 ≤ ||x − y|| 2 + 2λx − Tx, y − T y for all x, y ∈ C. In 2010, Kocourek et al. [6] introduced (α, β)-generalized hybrid mappings, and studied fixed point theorems and weak convergence theorems for such nonlinear mappings in Hilbert spaces. Let α, β ∈ R. A mapping T : C → H is (α, β)- generalized hybrid [6] if α||T x − T y|| 2 + (1 − α)||T y − x|| 2 ≤ β||T x − y|| 2 + (1 − β)||x − y|| 2 for all x, y ∈ C. In 2011, Aoyama and Kohsaka [7] introduced α-nonexpansive mapping on Ba- nach spaces. Let C be a nonempty closed convex subset of a Banach space E, and let α be a real number such that α < 1. A mapping T : C → E is said to be α-nonexpansive if ||T x − T y|| 2 ≤ α||T x − y|| 2 + α||T y − x|| 2 + (1 − 2α)||x − y|| 2 for all x, y ∈ C. Furthermore, we observed that Suzuki [8] introduced a new class of nonlinear mappings which satisfy condition (C) in Banach spaces. Let C be a nonempty subset of a Banach space E. Then, T : C → E is said to satisfy condition (C) if for all x, y ∈ C, 1 2 ||x − T x|| ≤ ||x − y|| ⇒ ||T x − T y|| ≤ ||x − y||. In fact, every nonexpansive mapping satisfies condition (C), but the converse may be false [8, Example 1]. Besides, if T : C → E satisfies condition (C) and F (T ) = ∅ , 3 then T is a quasi-nonexpansive mapping. However, the converse may be false [8, Example 2]. Motivated by the above studies, we introduced Takahashi’s ( 1 2 , 1 2 )-generalized hybrid mappings with Suzuki’s sense on Hilbert spaces. Definition 1.1. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → H be a mapping. Then, we say T satisfies condition (B) if for all x, y ∈ C, 1 2 ||x − T x|| ≤ ||x − y|| ⇒ ||T x − T y|| 2 + ||x − T y|| 2 ≤ ||T x − y|| 2 + ||x − y|| 2 . Remark 1.1. (i) In fact, if T is the identity mapping, then T satisfies condition (B). (ii) Every ( 1 2 , 1 2 )-generalized hybrid mapping satisfies condition (B). But the converse may be false. (iii) If T : C → C satisfies condition (B) and F (T ) = ∅, then T is a quasi- nonexpansive mapping, and this implies that F (T ) is a closed convex subset of C [9]. Remark 1.2 . Let H = R , let C be nonempty closed convex subset of H , and let T : C → H be a function. In fact, we have 1 2 ||x − T x|| ≤ ||x − y|| ⇔ (T x) 2 + x 2 − 2xT x ≤ 4x 2 + 4y 2 − 8xy ⇔ (T x) 2 − 2xT x ≤ 3x 2 + 4y 2 − 8xy ⇔ T x(T x − 2x) ≤ (3x − 2y)(x − 2y), and |T x − T y| 2 + |x − Ty| 2 ≤ |T x − y| 2 + |x − y| 2 ⇔ (T x) 2 +(T y) 2 −2T xT y+x 2 +(T y) 2 −2xT y ≤ (T x) 2 +y 2 −2yT x+x 2 +y 2 −2xy 4 ⇔ 2(T y) 2 − 2T y(T x + x) ≤ 2y 2 − 2y(T x + x) ⇔ 2(T y) 2 − 2y 2 ≤ 2(T y − y)(T x + x) ⇔ (T y − y)(T y + y) ≤ (T y − y)(T x + x) ⇔ (T y − y)[(T y + y) − (T x + x)] ≤ 0. Example 1.1. Let H = C = R, and let T : C → H be defined by T x := −x for each x ∈ C. Hence, we have the following conditions: (1) T is ( 1 2 , 1 2 )-generalized hybrid mapping, and T satisfies condition (B). (2) T is not a nonspreading mapping. Indeed, if x = 1 and y = −1, then 2||T x − T y|| 2 = 8 > 0 = ||T x − y|| 2 + ||T y − x|| 2 . (3) T is not a T J-1 mapping. Indeed, if x = 1 and y = −1, then 2||T x − T y|| 2 = 8 > 4 = 4 + 0 = ||x − y|| 2 + T x − y 2 . (4) T is not a T J-2 mapping. Indeed, if x = 1 and y = −1, then 3||T x − T y|| 2 = 12 > 0 = 2||T x − y|| 2 + T y − x 2 . (5) T is not a hybrid mapping. Indeed, if x = 1 and y = −1, then 3||T x − T y|| 2 = 12 > 4 = ||x − y|| 2 + ||T x − y|| 2 + ||T y − x|| 2 . (6) Now, we want to show that if α = 0, then T is not a α-nonexpansive mapping. For α > 0, let x = 1 and y = −1, ||T x − Ty|| 2 = 4 > 4 − 8α = α||T x − y|| 2 + α||T y − x|| 2 + (1 − 2α)||x − y|| 2 . For α < 0, let x = y = 1, ||T x − T y|| 2 = 0 > 8α = α||T x − y|| 2 + α||T y − x|| 2 + (1 − 2α)||x − y|| 2 . (7) Similar to (6), if α+β = 1, then T is not a (α, β)-generalized hybrid mapping. 5 Example 1.2. Let H = R, C = [−1, 1], and let T : C → C be defined by T (x) :=    x if x ∈ [−1, 0], −x if x ∈ (0, 1], for each x ∈ C. First, we consider the following conditions: (a) For x ∈ [−1, 0] and 1 2 ||x − T x|| ≤ ||x − y||, we know that (a) 1 if y ∈ [−1, 0], then T y = y and (T y − y)[(T y + y) − (T x + x)] = 0; (a) 2 if y ∈ (0, 1], then T y = −y and (T y −y)[(T y +y)−(T x +x)] = 4xy ≤ 0. (b) For x ∈ (0, 1] and 1 2 ||x − T x|| ≤ ||x − y||, we know that (b) 1 if y ≥ x, then x ≤ y − x, T x = −x, and T y = −y. So, (T y − y)[(T y + y) − (T x + x)] = 0; (b) 2 if y < x, then x ≤ x − y and this implies that y ≤ 0. So, (T y − y)[(T y + y) − (T x + x)] = 0. By these conditions and Remark 1.2, we know that T satisfies condition (B). In fact, T is ( 1 2 , 1 2 )-generalized hybrid mapping. Furthermore, we know that the following conditions: (1) T is a nonspreading mapping. Indeed, we know that the following conditions hold. (1) 1 If x > 0 and y > 0, then 2||T x − T y|| 2 = 2||x − y|| 2 ≤ 2||x + y|| 2 = ||T x − y|| 2 + ||T y − x|| 2 ; (1) 2 If x ≤ 0 and y ≤ 0, then 2||T x − T y|| 2 = 2||x − y|| 2 = ||T x − y|| 2 + ||T y − x|| 2 ; (1) 3 If x > 0 and y ≤ 0, then ||T x − T y|| 2 = ||T x − y|| 2 = ||x + y|| 2 , and ||T y − x|| 2 = ||x − y|| 2 . Hence, ||T x − y|| 2 + ||T y − x|| 2 − 2||T x − T y|| 2 = −4xy ≥ 0. 6 (2) Similar to the above, we know that T is a T J-1 mapping, a T J-2 map- ping, a hybrid mapping, (α, β)-generalized hybrid mapping, and T is a α- nonexpansive mapping. On the other hand, the following iteration process is known as Mann’s type iteration process [10] which is defined as x n+1 = α n x n + (1 − α n )T x n , n ∈ N, where the initial guess x 0 is taken in C arbitrarily and {α n } is a sequence in [0, 1]. In 1974, Ishikawa [11] gave an iteration process which is defined recursively by          x 1 ∈ C chosen arbitrary, y n := (1 − β n )x n + β n T x n , x n+1 := (1 − α n )x n + α n T y n , where {α n } and {β n } are sequences in [0, 1]. In 1995, Liu [12] introduced the following modification of the iteration method and he called Ishikawa iteration method with errors: for a normed space E, and T : E → E a given mapping, the Ishikawa iteration method with errors is the following sequence          x 1 ∈ E chosen arbitrary, y n := (1 − β n )x n + β n T x n + u n , x n+1 := (1 − α n )x n + α n T y n + v n , where {α n } and {β n } are sequences in [0, 1], and {u n } and {v n } are sequences in E with  ∞ n=1 ||u n || < ∞ and  ∞ n=1 ||v n || < ∞. In 1998, Xu [13] introduced an Ishikawa iteration method with errors which ap- pears to be more satisfactory than the one introduced by Liu [12]. For a nonempty convex subset C of E and T : C → C a given mapping, the Ishikawa iteration method with errors is generated by          x 1 ∈ C chosen arbitrary, y n := a n x n + b n T x n + c n u n , x n+1 := a  n x n + b  n T y n + c  n v n , 7 where {a n }, {b n }, {c n }, {a  n }, {b  n }, {c  n } are sequences in [0, 1] with a n +b n +c n = 1 and a  n + b  n + c  n = 1, and {u n } and {v n } are bounded sequences in C. Motivated by the above studies, we consider an Ishikawa iteration method with errors for mapping with condition (B). We also consider the following iteration for mappings with condition (B). Let C be a nonempty closed convex subset of a real Hilbert space H. Let G : C × C → R be a function. Let T : C → C be a mapping. Let {a n }, {b n }, and {θ n } be sequences in [0, 1] with a n + b n + θ n = 1. Let {ω n } be a bounded sequence in C. Let {r n } be a sequence of positive real numbers. Let {x n } be defined by u 1 ∈ H    x n ∈ C such that G(x n , y) + 1 r n y − x n , x n − u n  ≥ 0 ∀y ∈ C; u n+1 := a n x n + b n T x n + θ n ω n . Furthermore, we observed that Phuengrattana [14] studied approximating fixed points of for a nonlinear mapping T with condition (C) by the Ishikawa itera- tion metho d on uniform convex Banach space with Opal property. Here, we also consider the Ishikawa iteration method for a mapping T with condition (C) and improve some conditions of Phuengrattana’s result. In this article, a new type of mappings that satisfies condition (B) is introduced. We study Pazy’s type fixed point theorems, demiclosed principles, and ergodic theorem for mappings with condition (B). Next, we consider the weak convergence theorems for equilibrium problems and the fixed points of mappings with condition (B). 2 Preliminaries Throughout this article, let N be the set of positive integers and let R be the set of real numbers. Let H be a (real) Hilbert space with inner product ·, · and norm || · ||, respectively. We denote the strongly convergence and the weak convergence of {x n } to x ∈ H by x n → x and x n  x, respectively. From [15], for each x, y ∈ H and λ ∈ [0, 1], we have ||λx + (1 − λ)y|| 2 = λ||x|| 2 + (1 − λ)||y|| 2 − λ(1 − λ)||x − y|| 2 . 8 Hence, we also have 2x − y, u − v  = ||x − v|| 2 + ||y − u|| 2 − ||x − u|| 2 − ||y − v|| 2 for all x, y, u, v ∈ H. Furthermore, we know that ||αx+βy +γz|| 2 = α||x|| 2 +β||y|| 2 +γ||z|| 2 −αβ||x−y|| 2 −αγ||x−z|| 2 −βγ||y −z|| 2 for each x, y, z ∈ H and α, β, γ ∈ [0, 1] with α + β + γ = 1 [16]. Let  ∞ be the Banach space of bounded sequences with the supremum norm. Let µ be an element of ( ∞ ) ∗ (the dual space of  ∞ ). Then, we denote by µ(f) the value of µ at f = (x 1 , x 2 , x 3 , . . .) ∈  ∞ . Sometimes, we denote by µ n x n the value µ(f ). A linear functional µ on  ∞ is called a mean if µ(e) = ||µ|| = 1, where e = (1, 1, 1, . . .). For x = (x 1 , x 2 , x 3 , . . .), A Banach limit on  ∞ is an invariant mean, that is, µ n x n = µ n x n+1 for any n ∈ N. If µ is a Banach limit on  ∞ , then for f = (x 1 , x 2 , x 3 , . . .) ∈  ∞ , lim inf n→∞ x n ≤ µ n x n ≤ lim sup n→∞ x n . In particular, if f = (x 1 , x 2 , x 3 , . . .) ∈  ∞ and x n → a ∈ R, then we have µ(f) = µ n x n = a. For details, we can refer [17]. Lemma 2.1. [17] Let C be a nonempty closed convex subset of a Hilbert space H, {x n } be a bounded sequence in H, and µ be a Banach limit. Let g : C → R be defined by g(z) := µ n ||x n − z|| 2 for all z ∈ C. Then there exists a unique z 0 ∈ C such that g(z 0 ) = min z∈C g(z). Lemma 2.2. [17] Let C be a nonempty closed convex subset of a Hilbert space H. Let P C be the metric projection from H onto C. Then for each x ∈ H, we have x − P C x, P C x − y ≥ 0 for all y ∈ C. Lemma 2.3. [17] Let D be a nonempty closed convex subset of a real Hilbert space H. Let P D be the metric projection from H onto D, and let {x n } n∈N be a sequence in H. If x n  x 0 and P D x n → y 0 , then P D x 0 = y 0 . 9 [...]... Kohsaka, F, Takahashi, W: Fixed point and ergodic theorems for λ-hybrid mappings in Hilbert spaces J Nonlinear Convex Anal 11, 335–343 (2010) 23 [6] Kocourek, P, Takahashi, W, Yao, JC: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces Taiwanese J Math 14, 2497–2511 (2010) [7] Aoyama, K, Kohsaka, F: Fixed point theorem for α-nonexpansive mappings in Banach... S, Takahashi, W: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space Nonlinear Anal 71, e2082–e2089 (2009) [3] Takahashi, W, Yao, JC: Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces Taiwanese J Math 15, 457–472 (2011) [4] Takahashi, W: Fixed point theorems for new nonlinear mappings in a Hilbert spaces J Nonlinear Convex... [8] Suzuki, T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings J Math Anal Appl 340, 1088–1095 (2008) [9] Itoh, S, Takahashi, W: The common fixed point theory of single-valued mappings and multi-valued mappings Pacific J Math 79, 493–508 (1978) [10] Mann, WR: Mean value methods in iteration Proc Am Math Soc 4, 506– 510 (1953) [11] Ishikawa, S: Fixed points by a new... space H, and let T : C → C be a mapping with condition (B) Then F (T ) = ∅ The following theorem shows that Ballion’s type Ergodic’s theorem is also true for the mapping with condition (B) Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be a mapping with condition (B) Then the following conditions are equivalent: (i) for each x ∈ C, Sn x = 1 n n−1 T k x... responsible for problem resign, coordinator, discussion, revise the important part, and submit C-SC is responsible for the important results of this article, discuss, and draft Z-TY is responsible for giving the examples of this types of problems, discussion All authors read and approved the final manuscript References [1] Kohsaka, F, Takahashi, W: Fixed point theorems for a class of nonlinear mappings. .. demiclosed principle is true for mappings with condition (B) Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be a mapping with condition (B) Let {xn } be a sequence in C with xn x and lim ||xn − T xn || = 0 Then T x = x n→∞ Proof By Remark 3.1, we get: T xn − T x, x − T x ≤ xn − x, T x − x + ||xn − T xn || · ||x − T x|| for each n ∈ N By assumptions,... G(z, y) ≥ 0 foreach y ∈ C.(2.1) The solution set of equilibrium problem (2.1) is denoted by (EP ) For solving the equilibrium problem, let us assume that the bifunction G : C × C → R satisfies the following conditions: (A1) G(x, x) = 0 for each x ∈ C; (A2) G is monotone, i.e., G(x, y) + G(y, x) ≤ 0 for any x, y ∈ C; (A3) for each x, y, z ∈ C, lim G(tz + (1 − t)x, y) ≤ G(x, y); t↓0 (A4) for each x ∈... and Mann iteration processes with errors for nonlinear strongly accretive mappings in Banach spaces J Math Anal Appl 194, 114–125 (1995) [13] Xu, Y: Ishikawa and Mann iteration processes with errors for nonlinear strongly accretive operator equations J Math Anal Appl 224, 91–101 (1998) [14] Phuengrattana, W: Approximating fixed points of Suzuki-generalized nonexpansive mappings Nonlinear Anal Hybrid... ||Tr x − Tr y||2 ≤ Tr x − Tr y, x − y for each x, y ∈ C; (iii) (EP ) is a closed convex subset of C; (iv) (EP ) = F (Tr ) 3 Fixed point theorems Proposition 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C → C be a mapping with condition (B) Then for each x, y ∈ C, we have: (i) ||T x − T 2 x||2 + ||x − T 2 x||2 ≤ ||x − T x||2 ; (ii) ||T x − T 2 x|| ≤ ||x − T x|| and ||x... Osilike, MO, Igbokwe, DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations Comput Math Appl 40, 559–567 (2000) 24 [17] Takahashi, W: Nonlinear Functional Analysis -Fixed Point Theory and Its Applications Yokohama Publishers, Yokohama (2000) [18] Huang, S: Hybrid extragradient methods for asymptotically strict pseudocontractions in . appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Fixed point theorems for mappings with condition (B) Fixed Point Theory and Applications. fixed point theorems, demiclosed principles, and ergodic theorem for mappings with condition (B). Next, we consider the weak convergence theorems for equilibrium problems and the fixed points of mappings. article, a new type of mappings that satisfies condition (B) is in- troduced. We study Pazy’s type fixed point theorems, demiclosed principles, and ergodic theorem for mappings with condition (B). Next,