Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 189751, 7 pages doi:10.1155/2010/189751 Research Article On the Weak Relatively Nonexpansive Mappings in Banach Spaces Yongchun Xu 1 and Yongfu Su 2 1 Department of Mathematics, Hebei North University, Zhangjiakou 075000, China 2 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Correspondence should be addressed to Yongfu Su, suyongfu@tjpu.edu.cn Received 23 March 2010; Accepted 20 May 2010 Academic Editor: Billy Rhoades Copyright q 2010 Y. Xu and Y. Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In recent years, the definition of weak relatively nonexpansive mapping has been presented and studied by many authors. In this paper, we give some results about weak relatively nonexpansive mappings and give two examples which are weak relatively nonexpansive mappings but not relatively nonexpansive mappings in Banach space l 2 and L p 0, 11 <p<∞. 1. Introduction Let E be a smooth Banach space, and let C be a nonempty closed convex subset of E.We denote by φ the function defined by φ  x, y    x  2 − 2  x, Jy     y   2 for x, y ∈ E. 1.1 Following Alber 1, the generalized projection Π C from E onto C is defined by Π C  x   arg min y∈C φ  y, x  , ∀x ∈ E. 1.2 The generalized projection Π C from E onto C is well defined, single value and satisfies   x  −   y    2 ≤ φ  x, y  ≤   x     y    2 for x, y ∈ E. 1.3 If E is a Hilbert space, t hen φy, xy − x 2 ,andΠ C is the metric projection of E onto C. 2 Fixed Point Theory and Applications Let C be a closed convex subset of E,andletT be a mapping from C into itself. We denote by FT the set of fixed points of T.Apointp in C is said to be an asymptotic fixed point of T 2–4 if C contains a sequence {x n } which converges weakly to p such that lim n →∞ Tx n − x n   0. The set of asymptotic fixed point of T will be denoted by b  FT. Following Matsushita and Takahashi 2, a mapping T of C into itself is said to be relatively nonexpansive if the following conditions are satisfied: 1 FT is nonempty; 2 φu, Tx ≤ φu, x, for all u ∈ FT,x∈ C; 3  FTFT. The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications have been studied by many authors, for example 2–7 In recent years, the definition of weak relatively nonexpansive mapping has been presented and studied by many authors 5–8, but they have not given the example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. In this paper, we give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping in Banach space l 2 . Apointp in C is said to be a strong asymptotic fixed point of T 5, 6 if C contains a sequence {x n } which converges strongly to p such that lim n →∞ Tx n − x n   0. The set of strong asymptotic fixed points of T will be denoted by  FT. A mapping T from C into itself is called weak relatively nonexpansive if 1 FT is nonempty; 2 φu, Tx ≤ φu, x, for all u ∈ FT,x∈ C; 3  FTFT. Remark 1.1. In 6, the weak relatively nonexpansive mapping is also said to be relatively weak nonexpansive mapping. Remark 1.2. In 7, the authors have given the definition of hemirelatively nonexpansive mapping as follows. A mapping T from C into itself is called hemirelatively nonexpansive if 1 FT is nonempty; 2 φu, Tx ≤ φ u, x, for all u ∈ FT,x∈ C. The following conclusion is obvious. Conclusion 1. A mapping is closed hemi-relatively nonexpansive if and only if it is weak relatively nonexpansive. If E is strictly convex and reflexive Banach space, and A ⊂ E × E ∗ is a continuous monotone mapping with A −1 0 /  ∅, then it is proved in 2 that J r :J  rA −1 J,forr>0 is relatively nonexpansive. Moreover, if T : E → E is relatively nonexpansive, then using the definition of φ, one can show that FT is closed and convex. It is obvious that relatively nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping T : C → C, we have FT ⊂  FT ⊂  FT. Therefore, if T is relatively nonexpansive mapping, then FT  FT  FT. Fixed Point Theory and Applications 3 2. Results for Weak Relatively Nonexpansive Mappings in Banach Space Theorem 2.1. Let E be a smooth Banach space and C a nonempty closed convex and balanced subset of E.Let{x n } be a sequence in C such that {x n } converges weakly to x 0 /  0 and x n − x m ≥r>0 for all n /  m. Define a mapping T : C → C as follows: T  x   ⎧ ⎨ ⎩ n n  1 x n if x  x n  ∃n ≥ 1  , −x if x /  x n  ∀n ≥ 1  . 2.1 Then the following conclusions hold: 1 T is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping; 2 T is not continuous; 3 T is not pseudo-contractive; 4 if {x n }⊂intC,thenT is also not monotone (accretive), where intC is the interior of C. Proof. 1 It is obvious that T has a unique fixed point 0, that is, FT{0}. Firstly, we show that x 0 is an asymptotic fixed point of T. In fact since {x n } converges weakly to x 0 ,  Tx n − x n       n n  1 x n − x n      1 n  1  x n  −→ 0 2.2 as n →∞,so,x 0 is an asymptotic fixed point of T. Secondly, we show that T has a unique strong asymptotic fixed point 0, so that, FT  FT. In fact, for any strong convergent sequence, {z n }⊂C such that z n → z 0 and z n − Tz n →0asn →∞, from the conditions of Theorem 2.1, there exists sufficiently large nature number N such that z n /  x m , for any n, m > N. Then Tz n  −z n for n>N, it follows from z n −Tz n →0that2z n → 0,and hence z n → z 0  0. Observe that φ  0,Tx    Tx  2 ≤  x  2  φ  0,x  , ∀x ∈ C. 2.3 Then T is a weak relatively nonexpansive mapping. On the other hand, since x 0 is an asymptotic fixed point of T but not fixed point, hence T is not a relatively nonexpansive mapping. 2 For any x n /  0, we can take 0 ≤ λ m → 0 such that λ m x n ∈{x n } ∞ n1 , then we have  x n − λ m x n  −→ 0,m−→ ∞,  Tx n − T  λ m x n        n n  1 x n  λ m x n       n n  1  λ m   x n  ≥  n n  1   x n  > 0, 2.4 then T is not continuous. 4 Fixed Point Theory and Applications 3 Since x n − x m ≥r>0 for all n /  m, without loss of generality, we assume that x n /  0 for all n ≥ 1. In this case, we can take 1 ≥ δ n → 1 such that δ n x n ∈{x i } ∞ i1 for all n ≥ 1. Therefore we have  Tx n − T  δ n x n  ,J  x n − δ n x n     n n  1 x n  δ n x n ,J  x n − δ n x n     n n  1  δ n   x n ,J  1 − δ n  x n     n n  1  δ n  1 1 − δ n   1 − δ n  x n ,J  1 − δ n  x n     n n  1  δ n  1 1 − δ n  1 − δ n x n  2   n n  1  δ n  1 1 − δ n  x n − δ n x n  2 . 2.5 Since n/n1δ n 1/1−δ n  → ∞as n →∞, we know that T is not pseudo-contractive. 4 In the same as 2, we can take 1 ≤ δ n → 1 such that δ n x n ∈{x i } ∞ i1 for all n ≥ 1. Therefore we have  Tx n − T  δ n x n  ,J  x n − δ n x n     n n  1 x n  δ n x n ,J  x n − δ n x n     n n  1  δ n   x n ,J  1 − δ n  x n     n n  1  δ n  1 1 − δ n   1 − δ n  x n ,J  1 − δ n  x n     n n  1  δ n  1 1 − δ n  1 − δ n x n  2   n n  1  δ n  1 1 − δ n  x n − δ n x n  2 . 2.6 Since n/n  1δ n 1/1 − δ n  →−∞as n →∞, we know that T is not monotone accretive. 3. An Example in Banach Space l 2 In this section, we will give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping. Fixed Point Theory and Applications 5 Example 3.1. Let E  l 2 , where l 2   ξ   ξ 1 ,ξ 2 ,ξ 3 , ,ξ n ,  : ∞  n1 | x n | 2 < ∞  ,  ξ    ∞  n1 | ξ n | 2  1/2 , ∀ξ ∈ l 2 ,  ξ, η   ∞  n1 ξ n η n , ∀ξ   ξ 1 ,ξ 2 ,ξ 3 , ,ξ n ,  ,η  η 1 ,η 2 ,η 3 , ,η n ,  ∈ l 2 . 3.1 It is well known that l 2 is a Hilbert space, so that l 2  ∗  l 2 .Let{x n }⊂E be a sequence defined by x 0   1, 0, 0, 0,  , x 1   1, 1, 0, 0,  , x 2   1, 0, 1, 0, 0,  , x 3   1, 0, 0, 1, 0, 0,  , . . . x n   ξ n,1 ,ξ n,2 ,ξ n,3 , ,ξ n,k ,  , 3.2 where ξ n,k  ⎧ ⎨ ⎩ 1ifk  1,n 1, 0ifk /  1,k /  n  1, 3.3 for all n ≥ 1. Define a mapping T : E → E as follows: T  x   ⎧ ⎨ ⎩ n n  1 x n if x  x n  ∃n ≥ 1  , −x if x /  x n  ∀n ≥ 1  . 3.4 Conclusion 1. {x n } converges weakly to x 0 . Proof . For any f ζ 1 ,ζ 2 ,ζ 3 , ,ζ k ,  ∈ l 2 l 2  ∗ , we have f  x n − x 0   f, x n − x 0   ∞  k2 ζ k ξ n,k  ζ n1 −→ 0, 3.5 as n →∞.Thatis,{x n } converges weakly to x 0 . The following conclusion is obvious. 6 Fixed Point Theory and Applications Conclusion 2. x n − x m   √ 2 for any n /  m. It follows from Theorem 2.1 and the above two conclusions that T is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping. We have also the following conclusions: 1 T is not continuous; 2 T is not pseudo-contractive; 3 T is also not monotone accretive. 4. An Example in Banach Space L p 0, 11 <p<∞ Let E  L p 0, 11 <p<∞,and x n  1 − 1 2 n ,n 1, 2, 3, . 4.1 Define a sequence of functions in L p 0, 1 by the following expression: f n  x   ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 x n1 − x n if x n ≤ x< x n1  x n 2 , −2 x n1 − x n if x n1  x n 2 ≤ x<x n1 , 0 otherwise 4.2 for all n ≥ 1. Firstly, we can see, for any x ∈ 0, 1,that  x 0 f n  t  dt −→ 0   x 0 f 0  t  dt, 4.3 where f 0 x ≡ 0. It is wellknown that the above relation 4.3 is equivalent to {f n x} which converges weakly to f 0 x in uniformly smooth Banach space L p 0, 11 <p<∞.Onthe other hand, for any n /  m, we have   f n − f m      1 0   f n  x  − f m  x    p dx  1/p    x n1 x n   f n  x  − f m  x    p dx   x m1 x m   f n  x  − f m  x    p dx  1/p    x n1 x n   f n  x    p dx   x m1 x m   f m  x    p dx  1/p   2 x n1 − x n  p  x n1 − x n    2 x m1 − x m  p  x m1 − x m   1/p   2 p  x n1 − x n  p−1  2 p  x m1 − x m  p−1  1/p ≥  2 p  2 p  1/p > 0. 4.4 Fixed Point Theory and Applications 7 Let u n  x   f n  x   1, ∀n ≥ 1. 4.5 It is obvious that u n converges weakly to u 0 x ≡ 1and  u n − u m     f n − f m   ≥  2 p  2 p  1/p > 0, ∀n ≥ 1. 4.6 Define a mapping T : E → E as follows: T  x   ⎧ ⎨ ⎩ n n  1 u n if x  u n  ∃n ≥ 1  , −x if x /  u n  ∀n ≥ 1  . 4.7 Since 4.6 holds, by using Theorem 2.1,weknowthatT : L p 0, 1 → L p 0, 1 is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping. We have also the following conclusions: 1 T is not continuous; 2 T is not pseudo-contractive; 3 T is also not monotone accretive. Acknowledgments This project is supported by the Zhangjiakou City Technology Research and Development Projects Foundation 0811024B-5, Hebei Education Department Research Projects Founda- tion 2009103, and Hebei North University Research Projects Foundation 2009008. References 1 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996. 2 S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005. 3 W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009. 4 X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 6, pp. 1958–1965, 2007. 5 Y. Su, J. Gao, and H. Zhou, “Monotone CQ algorithm of fixed points for weak relatively nonexpansive mappings and applications,” Journal of Mathematical Research and Exposition, vol. 28, no. 4, pp. 957–967, 2008. 6 H. Zegeye and N. Shahzad, “Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 7, pp. 2707–2716, 2009. 7 Y. Su, D. Wang, and M. Shang, “Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings,” Fixed Point Theory and Applications, Article ID 284613, 8 pages, 2008. 8 Y. Su, Z. Wang, and H. Xu, “Strong convergence theorems for a common fixed point of two hemi- relatively nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5616–5628, 2009. . two conclusions that T is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping. We have also the following conclusions: 1 T is not continuous; 2 T is not pseudo-contractive;. relatively nonexpansive mapping, then FT  FT  FT. Fixed Point Theory and Applications 3 2. Results for Weak Relatively Nonexpansive Mappings in Banach Space Theorem 2.1. Let E be a smooth Banach. 1  . 2.1 Then the following conclusions hold: 1 T is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping; 2 T is not continuous; 3 T is not pseudo-contractive; 4