Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces
 YuanHeng Wang^{1}Email author and
 YongHui Xia^{1}
https://doi.org/10.1186/16871812201245
© Wang and Xia; licensee Springer. 2012
Received: 28 August 2011
Accepted: 23 March 2012
Published: 23 March 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.
Keywords
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.
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 all$n\ge 1\phantom{\rule{2.77695pt}{0ex}};{\sum}_{n=1}^{\infty}\left({q}_{n}^{2}1\right)<\infty ;\left\{{\alpha}_{n}\right\}$, {β_{ n } } ⊂ [0, 1]; ε ≤ α_{ n } ≤ β_{ n } ≤ b for all n ≥ 1, some ε > 0 and some$b\in \left(0,{L}^{2}\left[\sqrt{1+{L}^{2}}1\right]\right)$; 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.
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.
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 that$j\left({x}_{n}\right)\stackrel{\cdot}{\rightharpoonup}j\left(x\right)$, 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.
for all x, y ∈ C and all n ≥ 1.
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].
for all x, y ∈ E.
Obviously, we have
 (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.
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.
as n → ∞.
as n → ∞.
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.
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.
and which shows that u ∈ H_{ n } for all n ≥ 0. This proves that F(T) ⊂ H_{ n } for all n ≥ 0.
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→ ∞.
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 → ∞.
as n → ∞.
Step 6. x_{ n }  Tx_{ n }  → 0, as n → ∞.
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).
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.
Declarations
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).
Authors’ Affiliations
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