Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces

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 pseudo-contractions 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.

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 pseudo-contractions 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 pseudo-contraction was introduced by Schu [3]. It is well known that asymptotical nonexpansive mappings form a subclass of asymptotical pseudo-contractions.

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 pseudo-contraction 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 L-Lipschitzian and asymptotically pseudo-contractive with sequence {k_n} ⊂ [1, ∞); q_n = 2k_n- 1 for all n ≥ 1 ; ∑ n = 1 ∞ ( q n 2 - 1 ) < ∞ ; { α n } , {β_n} ⊂ [0, 1]; ε ≤ α_n ≤ β_n ≤ b for all n ≥ 1, some ε > 0 and some b ∈ ( 0 , L - 2 [ 1 + L 2 - 1 ] ) ; x₁ ∈ K; for all n ≥ 1, define x_n+1 = (1 - α_n)x_n + α_nTⁿy_n, y_n = (1 - β_n)x_n + β_nTⁿx_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 pseudo-contraction 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 L-Lipschitzian and asymptotical pseudo-contraction with a fixed point. Assume the control sequence {α_n} is chosen so that α_n ∈ [a, b] for some a, b ∈ 0 , 1 1 + L . 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 pseudo-contractions 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 pseudo-contractions 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]).

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 → 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 multiple-valued. 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 ( x n ) ⇀ ⋅ j ( x ) , where we denote weak convergence and weak star convergence by ⇀ and ⇀ ⋅ 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 L-Lipschitzian 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 x + y 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 lim n → ∞ x n + y n 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 lim t → 0 x + t y - x 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 norm-to-norm continuous on each bounded subset of E. A Banach space E is said to have Kadec-Klee 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 Kadec-Klee 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〈x-z, 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 one-to-one, 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 x₀ ∈ C such that φ (x₀, x) = min_y∈Cφ(y, x). The mapping P_C : E → C defined by P_Cx = x₀ is called the generalized projection [7,8].

The following are well-known results.

Lemma 2.3 [9]Let C be a closed convex subset of a smooth Banach space E and x ∈ E. Then, x₀ = 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_Cx) + φ(P_Cx, x) ≤ φ(y, x) for all y ∈ C.

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 L-Lipschitzian and asymptotical pseudo-contraction. Then I-T 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 α ∈ 0 , 1 1 + L and define y_m = (1 - α)x + αT^mx for arbitrary but fixed m ≥ 1. We first show that for fixed m ≥ 1,x_n - T^mx_n → 0, as n → ∞. In fact, in view of the uniform L-Lipschitz 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^mx_n → 0, as n → ∞ and since J is a weakly sequential continuous duality mapping, it follows that

as n → ∞.

Further, using the uniform L-Lipschitz 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+1x → 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 pseudo-contractions 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 L-Lipschitzian, convex and asymptotical pseudo-contraction with F(T) nonempty. Assume the control sequence {α_n} is chosen so that α_n ∈ [a, b] for some a , b ∈ 1 + 1 1 + L . 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 L-Lipschitzian 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 L-Lipschitz continuity of T and the asymptotical pseudo-contractiveness 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 = W₀. 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 - Tⁿx_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 ∈ 1 + 1 1 + L , {y_n} and {Tⁿy_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 { x n k } is a subsequence of {x_n} such that { x n k } ⇀ x ^ ε C , then by the Theorem 3.1, we obtain x ^ ∈ F ( T ) . So we assume { x n k } be a subsequence of {x_n} such that { x n k } ⇀ x ^ ∈ F ( T ) and ω = P_F(T). For any n ≥ 1, from x n + 1 = P H n ∩ 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 x ^ = ω and hence lim n → ∞ ϕ ( x n k , x ) = ϕ ( ω , x ) . So we have lim k → ∞ x n k = ω . Using the Kadec-klee property of E, we obtain that { x n k } converges strongly to P_F(T)x. Since { x n k } 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 k-strict pseudo-contractions (see [11]) for some k ∈ [0, 1) in Banach spaces. It should be pointed out that our extended demiclosed-ness principle plays a key role in the proof.

The authors declare that they have no competing interests.

All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.

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).

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