 Research
 Open Access
 Published:
Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 45 (2012)
Abstract
The aim of this article is to give an answer to an interesting question proposed in Zhou. At the end of his article, he remarked that it was of great interest to extend his results to certain Banach spaces. So in this article, we extend the demiclosedness principle from Hilbert spaces to Banach spaces. A strong convergence theorem for asymptotical pseudocontractions in Banach spaces is established. The approaches are based on the extended demiclosedness principle, and the generalized projective operator, and the hybrid method in mathematical programming. Our results extend the previous known results from Hilbert spaces to Banach spaces.
MSC: 47H10; 47H09; 47H05.
1 Introduction
Zhou [1] proposed an interesting problem at the end of his article. He remarked that it was of great interest to extend his results to certain Banach spaces. Thus, this article essentially pursues two goals.

The first purpose of this article is to extend the demiclosedness principle from Hilbert spaces to Banach spaces.

The main aim is to establish a strong convergence theorem for asymptotical pseudocontractions in Banach spaces. The obtained theorem extends the main result in Zhou [1].
In 1972, Goebel and Kirk [2] introduced the concept of asymptotically nonexpansive mappings in the Hilbert space. Nineteen years later, the class of asymptotical pseudocontraction was introduced by Schu [3]. It is well known that asymptotical nonexpansive mappings form a subclass of asymptotical pseudocontractions.
Let H be a real Hilbert space with inner product 〈·,·〉, C be a nonempty closed convex subset of H, T be a mapping from C into itself, {k_{ n } } be a positive real sequence with k_{ n } → 1. T is said to be an asymptotical nonexpansive mapping, if the following inequality holds
for all x, y ∈ C and all n ≥ 1. T is called an asymptotical pseudocontraction if the following inequality holds
for all x, y ∈ C and all n ≥ 1.
Further, Schu proved the following convergence theorem in a Hilbert space.
Theorem 1.1[3]Let H be a Hilbert space; φ ≠ K ⊂ H closed bounded convex; L > 0;T : K → K completely continuous, uniformly LLipschitzian and asymptotically pseudocontractive with sequence {k_{ n } } ⊂ [1, ∞); q_{ n } = 2k_{ n } 1 for alln\ge 1\phantom{\rule{2.77695pt}{0ex}};{\sum}_{n=1}^{\infty}\left({q}_{n}^{2}1\right)<\infty ;\left\{{\alpha}_{n}\right\}, {β_{ n } } ⊂ [0, 1]; ε ≤ α_{ n } ≤ β_{ n } ≤ b for all n ≥ 1, some ε > 0 and someb\in \left(0,{L}^{2}\left[\sqrt{1+{L}^{2}}1\right]\right); x_{1} ∈ K; for all n ≥ 1, define x_{n+1}= (1  α_{ n } )x_{ n } + α_{ n }T^{n}y_{ n }, y_{ n } = (1  β_{ n } )x_{ n } + β_{ n }T^{n}x_{ n }. Then {x_{ n } } converges strongly to some fixed points of T.
Recently, Zhou [1] extended Schu's results by establishing a fixed point theorem for asymptotically pseudocontraction without any compact assumption on the mappings. By modifying the algorithm used in Theorem 1.2, Schu successfully proved a strong convergence theorem without any compact assumptions.
Theorem 1.2[1]Let C be a bounded and closed convex subset of a real Hilbert space H. Let T : C → C be a uniformly LLipschitzian and asymptotical pseudocontraction with a fixed point. Assume the control sequence {α_{ n } } is chosen so that α_{ n } ∈ [a, b] for some a, b ∈ \left(0,\frac{1}{1+L}\right). Let {x_{ n } } be a sequence generated by the following manner
Then the sequence {x_{ n } } generated by (1) converges strongly to P_{ F }_{(T)}x, where P_{ F }_{(T)}denotes the metric projection from H onto F(T), a closed convex subset of H.
However, all results above are obtained for Hilbert spaces. Motivated by the above mentioned studies, in this article, we first give the concepts of asymptotical pseudocontractions in Banach spaces. Then, we prove the demiclosedness principle in Banach space. Based on our extended demiclosedness principle, we establish a strong theorem for asymptotical pseudocontractions in Banach spaces. Therefore, we extend the main results of Zhou (see [1]) from Hilbert spaces to Banach spaces. Further, some other results are also improved (see [4, 5]).
2 Preliminaries
This section contains some definitions and lemmas which will be used in the proofs of our main results in the following section.
Throughout this article, let E be a real Banach space and E* be the dual of E. The normalized duality mapping J:E\to {2}^{E*} is defined by
where 〈·,·〉 denotes the duality pairing. It is well known (see e.g., [6]) that the operator J is well defined and J is identity mapping if and only if E is a Hilbert space. But in general, J is nonlinear and multiplevalued. So, We have the following definition.
Definition 2.1 The normalized duality mapping J of a Banach space E is said to be weakly sequential continuous, if∀{x_{ n } } ⊂ E, and x_{ n } ⇀ x, then there exist j(x_{ n } ) ∈ J(x_{ n } ), j(x) ∈ J(x) such thatj\left({x}_{n}\right)\stackrel{\cdot}{\rightharpoonup}j\left(x\right), where we denote weak convergence and weak star convergence by ⇀ and\stackrel{\cdot}{\rightharpoonup}respectively.
Naturally, the concept of asymptotical pseudocontraction can be extended from Hilbert spaces to Banach spaces.
Definition 2.2 Let C be a nonempty closed convex subset of E and let T be a mapping from C into itself. T is said to be an asymptotical pseudocontraction in Banach spaces if there exists a sequence {k_{ n } } with k_{ n } → 1 and j(x  y) ∈ J(x  y) for which the following inequality holds
for all x, y ∈ C and all n ≥ 1.
Definition 2.3[1]A mapping T : C → C is said to be uniform LLipschitzian if there exists some L > 0 such that
for all x, y ∈ C and for all n ≥ 1.
A Banach space E is said to be strictly convex if \frac{\u2225x+y\u2225}{2}<1 for x = y = 1 and x ≠ y; it is also said to be uniformly convex if lim_{n→∞}, x_{ n }  y_{ n }  = 0 for any two sequences {x_{ n } }, {y_{ n } } in E such that x_{ n }  = y_{ n }  = 1 and {\mathrm{lim}}_{n\to \infty}\frac{\u2225{x}_{n}+{y}_{n}\u2225}{2}=1. Let U = {x ∈ E : x = 1} be the unit sphere of E, then the Banach space E is said to be smooth provided {\mathrm{lim}}_{t\to 0}\frac{\u2225x+ty\u2225\u2225x\u2225}{t} exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attainted uniformly for each x, y ∈ U. It is well known that if E is reflexive and smooth, then the duality mapping J is single valued. It is also known that if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E. A Banach space E is said to have KadecKlee property if a sequence {x_{ n } } of E satisfies that x_{ n } ⇀ x ∈ E and x_{ n }  → x, then x_{ n } → x. It is known that if E is uniformly convex, then E has the KadecKlee property. Some more properties of the duality mapping have been given in [6, 7].
Definition 2.4[8]Let E be a reflexive and smooth Banach space. The function Φ : E×E → R is said to be a Lyapunov function defined by
for all x, y ∈ E.
Obviously, we have
(1) (x   y ) ^{2} ≤ φ(y, x) ≤ (y ^{2} + x ^{2} );

(2)
φ(x, y) = φ (x, z) + φ(z, y) + 2〈xz, Jz  Jx〉;

(3)
φ(x, y) = 〈x, Jx  Jy〉 + 〈y  x, Jy〉 ≤ x Jx  Jy + y  x y,
for all x, y, z ∈ E.
Lemma 2.1[8]Let E be a uniformly convex and smooth Banach space, and let {y_{ n } }, {z_{ n } } be two sequences of E. If φ (y_{ n }, z_{ n } ) → 0 and either {y_{ n } } or {z_{ n } } is bounded, then y_{ n }z_{ n }→ 0.
Lemma 2.2 If E is a strictly convex, reflexive, and smooth Banach space, then for x, y ∈ E, φ(x, y) = 0 if and only if x = y.
Proof. It is sufficient to show that if φ(x, y) = 0 then x = y. From (1), we have x = y. This implies 〈y, Jx〉 = y^{2} = Jx^{2}. From the definition of J, we have Jx = Jy. Since J is onetoone, we have x = y.
Definition 2.5 Let C be a closed convex subset of E. Suppose that E is reflexive, strictly convex, and smooth. Then, for any x ∈ E, there exists a point x_{ 0 } ∈ C such that φ (x_{ 0 }, x) = min_{ y∈C }φ(y, x). The mapping P_{ C } : E → C defined by P_{ C }x = x_{0}is called the generalized projection[7, 8].
The following are wellknown results.
Lemma 2.3[9]Let C be a closed convex subset of a smooth Banach space E and x ∈ E. Then, x_{0} = P_{ C } x if and only if
for all y ∈ C.
Lemma 2.4[10]Let E be a reflexive, strictly convex, and smooth Banach space and let C be a closed convex subset of E and x ∈ E. Then φ(y, P_{ C }x) + φ(P_{ C }x, x) ≤ φ(y, x) for all y ∈ C.
3 Main results
Theorem 3.1 (Demiclosedness principle) Let E be a reflexive smooth Banach space with a weakly sequential continuous duality mapping J. Let C be a nonempty bounded and closed convex subset of E and T : C → C be a uniformly LLipschitzian and asymptotical pseudocontraction. Then IT is demiclosed at zero, where I is the identical mapping.
Proof. Assume that {x_{ n } } ⊂ C with x_{ n } ⇀ x and x_{ n }  Tx_{ n } → 0 as n → ∞; we plan to show that x ∈ C and x = Tx. Since C is a closed convex subset of E, C is weakly closed, and hence x ∈ C. So, it is sufficient to show that x = Tx. To this end, we choose \alpha \in \left(0,\frac{1}{1+L}\right) and define y_{ m } = (1  α)x + αT^{m}x for arbitrary but fixed m ≥ 1. We first show that for fixed m ≥ 1,x_{ n }  T^{m}x_{ n } → 0, as n → ∞. In fact, in view of the uniform LLipschitz condition of T, we have
as n → ∞.
We next estimate 〈J(x  y_{ m } ), (I  T^{m} )y_{ m } 〉. By using the definition of T, we have
Since x_{ n } ⇀ x by our assumption and x_{ n }  Tx_{ n } → 0 as n → ∞, by x_{ n }  T^{m}x_{ n } → 0, as n → ∞ and since J is a weakly sequential continuous duality mapping, it follows that
as n → ∞.
Further, using the uniform LLipschitz condition of T and the definition of y_{ m } , we have
At this point, by the facts above, we have
which implies that
for all m ≥ 1.
Letting m → ∞, k_{ m } → 1 yields that T^{m} x → x, and hence T^{m+}^{1}x → Tx as m → ∞, since T : C → C is continuous. Consequently, we have x = Tx, completing the proof of Theorem 3.1.
Remark 3.1 Theorem 3.1 is useful in Banach spaces and a novel result which will play a very key role for establishing the strong convergence theorem of fixed points of asymptotical pseudocontractions in this article.
Theorem 3.2 Let E be a uniformly convex and uniformly smooth Banach space with a weakly sequential continuous convex duality mapping J. Let C be a nonempty bounded and closed convex subset of E, and let T : C → C be a uniform LLipschitzian, convex and asymptotical pseudocontraction with F(T) nonempty. Assume the control sequence {α_{ n } } is chosen so that α_{ n } ∈ [a, b] for somea,b\in \left(1+\frac{1}{1+L}\right). The sequence {x_{ n } } is given in the following manner
Then the sequence {x_{ n } } converges strongly to P_{ F }_{(T)}x, where P_{ F }_{(T)}is the generalized projection from E onto F (T).
Proof. We divide the proof into seven steps.
Step 1. P_{ F }_{(T)}is well defined for every x ∈ C.
Since T is uniform LLipschitzian continuous and convex, we know F(T) is closed and convex. Moreover, F(T) is nonempty by our assumption. Therefore, P_{ F }_{(T)}is well defined for every x ∈ C.
Step 2 Show that H_{ n } and W_{ n } are closed and convex for all n ≥ 0.
From the definitions of W_{ n } and H_{ n } , it is obvious that W_{ n } is closed and convex and H_{ n } is closed for each n ≥ 0. H_{ n } is convex for each n ≥ 0, which follows from the convexity of J. We omit the details.
Step 3. Show that F(T) ⊂ H_{ n }∩ W_{ n } for all n ≥ 0.
We first prove F(T) ⊂ H_{ n } . Let u ∈ F and let n ≥ 0. Then, using (2), the uniform LLipschitz continuity of T and the asymptotical pseudocontractiveness of T, we have
which implies that
and which shows that u ∈ H_{ n } for all n ≥ 0. This proves that F(T) ⊂ H_{ n } for all n ≥ 0.
Next we prove F(T) ⊂ W_{ n } for all n ≥ 0 by induction. For n = 0, we have F(T) ⊂ C = W_{0}. Assume that F(T) ⊂ W_{ n } . Since x_{ n+ }_{1} is the projection of x onto H_{ n } ∩ W_{ n } , by Lemma 2.3, we have
for any z ∈ H_{ n } ∩ W_{ n } . As F(T) ⊂ H_{ n } ∩ W_{ n } by the induction assumption, the last inequality holds, in particular, for all u ∈ F(T). This together with the definition of W_{ n+ }_{1} implies that F(T) ⊂ W_{ n+ }_{1}. Hence, F(T) ⊂ H_{ n } ∩ W_{ n } for all n ≥ 0.
Step 4. x_{ n+ }_{1} x_{ n }  → 0 as n→ ∞.
In view of (2) and Lemma 2.3, we have {x}_{n}={P}_{{W}_{n}}x, which means that for any z ∈ W_{ n }, φ(x_{ n }, x) ≤ φ (z, x). Since x_{ n+ }_{1} ∈ W_{ n } and u ∈ F(T) ⊂ W_{ n } , we obtain
for all n ≥ 0. Consequently, lim_{n→∞}φ (x_{ n }, x) exists and {x_{ n } } is bounded. By using Lemma 2.4, we have
as n→ ∞. By using Lemma 2.1, we obtain x_{ n+ }_{1} x_{ n }  → 0 as n → ∞.
Step 5. x_{ n }  T^{n}x_{ n }  → 0 as n → ∞.
It follows from step 4 that x_{ n+ }_{1} x_{ n }  → 0 as n → ∞. Since x_{ n+ }_{1} ⊂ H_{ n } , noting that {α_{ n } } is chosen so that α_{ n } ∈ [a, b] for some a,b\in \left(1+\frac{1}{1+L}\right), {y_{ n } } and {T^{n}y_{ n } } are bounded and J is a weakly sequential continuous duality mapping, from the definition of H_{ n } , we have
as n → ∞.
Step 6. x_{ n }  Tx_{ n }  → 0, as n → ∞.
Observing that
and using steps 4 and 5, we reach the desired conclusion.
Step 7. x_{ n } → P_{ F }_{(T)}x, as n → ∞.
From the result of step 6, we know that if \left\{{x}_{{n}_{k}}\right\} is a subsequence of {x_{ n } } such that \left\{{x}_{{n}_{k}}\right\}\rightharpoonup \widehat{x}\epsilon C, then by the Theorem 3.1, we obtain \widehat{x}\in F\left(T\right). So we assume \left\{{x}_{{n}_{k}}\right\} be a subsequence of {x_{ n } } such that \left\{{x}_{{n}_{k}}\right\}\rightharpoonup \widehat{x}\in F\left(T\right) and ω = P_{ F }_{(T)}. For any n ≥ 1, from {x}_{n+1}={P}_{{H}_{n\cap {W}_{n}}}x and ω ∈ F(T) ⊂ H_{ n } ∩ W_{ n } , we have φ (x_{ n+ }_{1}, x) ≤ φ (ω, x).
On the other hand, from the weak lower semicontinuity of the norm, we have
From the definition of P_{ F }_{(T)}, we obtain \widehat{x}=\omega and hence {\mathrm{lim}}_{n\to \infty}\varphi \left({x}_{{n}_{k}},x\right)=\varphi \left(\omega ,x\right). So we have {\mathrm{lim}}_{k\to \infty}\u2225{x}_{{n}_{k}}\u2225=\u2225\omega \u2225. Using the Kadecklee property of E, we obtain that \left\{{x}_{{n}_{k}}\right\} converges strongly to P_{ F }_{(T)}x. Since \left\{{x}_{{n}_{k}}\right\} is an arbitrary weakly convergent sequence of {x_{ n } }, we can conclude that {x_{ n } } converges strongly to P_{ F }_{(T)}x.
Remark 3.2 Theorem 3.2 extends the main results of Zhou (see[1]) from Hilbert spaces to Banach spaces and improves some other results (see[4, 5]). Moreover, our method used in this article can be applied to other mappings, such as kstrict pseudocontractions (see[11]) for some k ∈ [0, 1) in Banach spaces. It should be pointed out that our extended demiclosedness principle plays a key role in the proof.
References
Zhou HY: Demiclosedness principle with applications for asymptotically pseudocontractions in Hilbert spaces. Nonlinear Anal 2009, 70: 3140–3145.
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Am Math Soc 1972, 35: 171–174.
Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J Math Anal Appl 1991, 158: 407–413.
Liu QH: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal 1996, 26: 1835–1842.
Kim TH, Xu HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal 2006, 64: 1140–1152.
Cioranescu I: Geometry of Banach Spaces. In Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers Dordrecht; 1990.
Li J: The generalized projection operator on reflexive Banach spaces and its applications. J Math Anal Appl 2005, 306: 55–71.
Kamimura S, Takahashi W: Strong convergence of a proximaltype algorithm in a Banach space. SIAM J Optim 2002, 13: 938–945.
Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operator of Accretive and Monotone Type,Lecture Notes in Pure and Appl Math. Volume 178. Edited by: Kartsatos AG. Marcel Dekker New York; 1996:15–50.
Wang YH, Zeng LC: Convergence of generalized projective modified iterative methods in Banach spaces (in Chinese). Chin Ann Math 2009, 30A: 55–62.
Ceng LC, AlHomidan S, Ansari QH, Yaod JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudocontraction mappings. J Comput Appl Math 2009, 223: 967–974.
Acknowledgements
The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the article. This study was supported by the National Natural Science Foundations of China (Grant Nos. 11071169 and 10901140), the Natural Science Foundations of Zhejiang Province of China (Grant No. Y6100696) and Zhejiang Innovation Project (Grant No. T200905).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wang, YH., Xia, YH. Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces. Fixed Point Theory Appl 2012, 45 (2012). https://doi.org/10.1186/16871812201245
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16871812201245
Keywords
 demiclosedness principle
 generalized projection
 strong convergence
 asymptotically pseudocontractions
 Banach space