Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 189751, 7 pages doi:10.1155/2010/189751 ResearchArticleOntheWeakRelativelyNonexpansiveMappingsinBanach 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 weakrelativelynonexpansive mapping has been presented and studied by many authors. In this paper, we give some results about weakrelativelynonexpansivemappings and give two examples which are weakrelativelynonexpansivemappings but not relativelynonexpansivemappingsinBanach space l 2 and L p 0, 11 <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 φy, xy − 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 FT 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 FT. Following Matsushita and Takahashi 2, a mapping T of C into itself is said to be relativelynonexpansive if the following conditions are satisfied: 1 FT is nonempty; 2 φu, Tx ≤ φu, x, for all u ∈ FT,x∈ C; 3 FTFT. The hybrid algorithms for fixed point of relativelynonexpansivemappings and applications have been studied by many authors, for example 2–7 In recent years, the definition of weakrelativelynonexpansive mapping has been presented and studied by many authors 5–8, but they have not given the example which is weakrelativelynonexpansive mapping but not relativelynonexpansive mapping. In this paper, we give an example which is weakrelativelynonexpansive mapping but not relativelynonexpansive mapping inBanach 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 FT. A mapping T from C into itself is called weakrelativelynonexpansive if 1 FT is nonempty; 2 φu, Tx ≤ φu, x, for all u ∈ FT,x∈ C; 3 FTFT. Remark 1.1. In 6, theweakrelativelynonexpansive mapping is also said to be relativelyweaknonexpansive 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 FT is nonempty; 2 φu, Tx ≤ φ u, x, for all u ∈ FT,x∈ C. The following conclusion is obvious. Conclusion 1. A mapping is closed hemi-relatively nonexpansive if and only if it is weakrelatively 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 FT is closed and convex. It is obvious that relativelynonexpansive mapping is weakrelativelynonexpansive mapping. In fact, for any mapping T : C → C, we have FT ⊂ FT ⊂ FT. Therefore, if T is relativelynonexpansive mapping, then FT FT FT. Fixed Point Theory and Applications 3 2. Results for WeakRelativelyNonexpansiveMappingsinBanach 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 weakrelativelynonexpansive mapping but not relativelynonexpansive 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, FT{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, FT FT. 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 weakrelativelynonexpansive mapping. Onthe other hand, since x 0 is an asymptotic fixed point of T but not fixed point, hence T is not a relativelynonexpansive mapping. 2 For any x n / 0, we can take 0 ≤ λ m → 0 such that λ m x n ∈{x n } ∞ n1 , 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 } ∞ i1 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/n1δ n 1/1−δ n → ∞as n →∞, we know that T is not pseudo-contractive. 4 Inthe same as 2, we can take 1 ≤ δ n → 1 such that δ n x n ∈{x i } ∞ i1 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 1δ n 1/1 − δ n →−∞as n →∞, we know that T is not monotone accretive. 3. An Example inBanach Space l 2 In this section, we will give an example which is a weakrelativelynonexpansive mapping but not a relativelynonexpansive 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 , : ∞ n1 | x n | 2 < ∞ , ξ ∞ n1 | ξ n | 2 1/2 , ∀ξ ∈ l 2 , ξ, η ∞ n1 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 ∞ k2 ζ k ξ n,k ζ n1 −→ 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 weakrelativelynonexpansive mapping but not relativelynonexpansive 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 inBanach Space L p 0, 11 <p<∞ Let E L p 0, 11 <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 n1 − x n if x n ≤ x< x n1 x n 2 , −2 x n1 − x n if x n1 x n 2 ≤ x<x n1 , 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, 11 <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 n1 x n f n x − f m x p dx x m1 x m f n x − f m x p dx 1/p x n1 x n f n x p dx x m1 x m f m x p dx 1/p 2 x n1 − x n p x n1 − x n 2 x m1 − x m p x m1 − x m 1/p 2 p x n1 − x n p−1 2 p x m1 − 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 weakrelativelynonexpansive mapping but not relativelynonexpansive 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 inBanach 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 relativelynonexpansivemappingsin 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 relativelynonexpansivemappingsinBanach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009. 4 X. Qin and Y. Su, “Strong convergence theorems for relativelynonexpansivemappingsin 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 weakrelativelynonexpansivemappings 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 relativelyweaknonexpansive 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- relativelynonexpansive 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 FT FT FT. 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