Theorems of strong convergence of mixed iterative methods for obtaining strict pseudocontractions in Banach spaces

Applied Mathematics Letters 23 (2010) 791–795

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Theorems of strong convergence of mixed iterative methods for obtaining strict pseudocontractions in Banach spaces Liang-Gen Hu ∗ Department of Mathematics, Ningbo University, Zhejiang, 315211, PR China

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Article history: Received 4 June 2009 Received in revised form 3 November 2009 Accepted 22 March 2010 Keywords: Strict pseudocontractions Mixed iteration p-uniformly convex Strong convergence

abstract In this work, theorems of strong convergence of a mixed iteration scheme within the viscosity approximation methods for obtaining λ-strict pseudocontractions are established in p-uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. The main result extends various results existing in the current literature. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Let E be a real Banach space, and C a nonempty closed convex subset E. We denote by J the normalized duality map from ∗ E to 2E defined by J (x) = {x∗ ∈ E ∗ : hx, x∗ i = kxk2 = kx∗ k2 , ∀x ∈ E }. A mapping f : C → C is called a k-contraction if there exists a constant 0 < k < 1 such that kf (x) − f (y)k ≤ kkx − yk for all x, y ∈ C . A mapping T : C → C is said to be a λ-strictly pseudocontractive mapping (see, e.g., [1]) if there exists a constant 0 ≤ λ < 1 such that

kTx − Tyk2 ≤ kx − yk2 + λk(I − T )x − (I − T )yk2 ,

(1.1)

for all x, y ∈ C . Note that the class of λ-strict pseudocontractions strictly includes the class of nonexpansive mappings which are mappings of T onto C such that kTx − Tyk ≤ kx − yk, for all x, y ∈ C . That is, T is nonexpansive if and only if T is a 0-strict pseudocontraction. A mapping T : C → C is said to be a λ-strictly pseudocontractive mapping with respect to p if, for all x, y ∈ C , there exists a constant 0 ≤ λ < 1 such that

kTx − Tykp ≤ kx − ykp + λk(I − T )x − (I − T )ykp .

(1.2)

We denote by Fix(T ) the set of fixed points of T , i.e., Fix(T ) = {x ∈ C : Tx = x}. In the last twenty years or so, there have been many papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudocontractive mappings by using the Mann and Ishikawa iteration process. Results which had been established only for Hilbert spaces, and Lipschitz mappings have been extended to more general Banach spaces and more general classes of mappings (see, e.g., [1–4] and the references therein).

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Tel.: +86 15155932098. E-mail address: [email protected].

0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.03.011

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In 2007, Marino and Xu [3] proved that the Mann iterative sequence converges weakly to a fixed point of λ-strict pseudocontractions in Hilbert spaces, which extend Reich’s theorem [5, Theorem 2] from nonexpansive mappings to λ-strict pseudocontractions in Hilbert spaces. Recently, Zhou [4] obtained some theorems of weak and strong convergence for λ-strict pseudocontractions in Hilbert spaces by using Mann iteration and modified Ishikawa iteration which extend Marino and Xu’s convergence theorems [3]. More recently, Hu and Wang [2] obtained that the Mann iterative sequence converges weakly to a fixed point of λ-strict pseudocontractions with respect to p in p-uniformly convex Banach spaces. In this work, we introduce and study a mixed iterative sequence: Let C be a nonempty closed convex subset of E and f : C → C a k-contraction. For any x1 ∈ C , the sequence {xn } is defined by xn+1 = αn f (xn ) + βn xn + γn Txn ,

n ≥ 1,

(1.3)

where {αn }, {βn } and {γn } in (0, 1) satisfy αn + βn + γn = 1. The iterative sequence (1.3) is a natural generalization of well-known iterations:

• If αn ≡ 0, for all n ≥ 1, in (1.3), then (1.3) reduces to the Mann iteration. • If βn ≡ 0, for all n ≥ 1, in (1.3), then (1.3) reduces to the viscosity approximation process. Our main aim is to prove theorems of strong convergence of the mixed iteration scheme for λ-strict pseudocontractions with respect to p in p-uniformly convex Banach spaces with uniformly Gâteaux differentiable norm by using viscosity approximation methods. Our theorems extend and improve on the comparable results in the following three aspects: (1) In contrast to weak convergence results in [2–4], our theorems of strong convergence of the mixed iterative sequence are obtained in the general setup of a Banach space; (2) in comparison to the results in [6,5], P∞the results with respect to nonexpansive mappings are extended to λ-strict pseudocontractions; (3) the restrictions n=1 |αn+1 − αn | < ∞ and P∞ n=1 |βn+1 − βn | < ∞ in [6, Theorem 3.1] are dropped. 2. Preliminaries The modulus of convexity of E is the function δE : [0, 2] → [0, 1] defined by



x + y

δE () = inf 1 − : kxk = 1, kyk = 1 and kx − yk ≥  , 2 

0 ≤  ≤ 2.

E is uniformly convex if and only if, for all 0 <  ≤ 2, δE () > 0. E is said to be p-uniformly convex if there exists a constant a > 0 such that δE () ≥ a p . A Banach space E is said to have Gâteaux differentiable norm if the limit lim

kx + tyk − kxk

t →0 t exists for each x, y ∈ U, where U = {x ∈ E : kxk = 1}. The norm of E is uniformly Gâteaux differentiable if for each y ∈ U, the limit is attained uniformly for x ∈ U. It is well-known that if E is a uniformly Gâteaux differentiable norm, then the duality mapping J is single-valued and norm-to-weak∗ uniformly continuous on each bounded subset of E.

Lemma 2.1 ([2]). Let E be a real p-uniformly convex Banach space and C a nonempty closed convex subset of E. Let T : C → C be a λ-strict pseudocontraction with respect to p, and {ξn } a real sequence in [0, 1]. If Tn : C → C is defined by Tn x := (1 − ξn )x + ξn Tx, ∀x ∈ C , then for all x, y ∈ C , the following inequality holds:

  kTn x − Tn ykp ≤ kx − ykp − wp (ξn )cp − ξn λ k(I − T )x − (I − T )ykp , where cp is a constant in [7, Theorem 1]. In addition, if 0 ≤ λ < min{1, 2−(p−2) cp }, ξ = 1 − 2p−2 λcp−1 , and ξn ∈ [0, ξ ], then kTn x − Tn yk ≤ kx − yk, for all x, y ∈ C . Lemma 2.2 ([8]). Let {xn } and {yn } be bounded sequences in a Banach space E such that xn+1 = αn xn + (1 − αn )yn ,

n ≥ 0,

where {αn } is a sequence in (0, 1) such that 0 < lim infn→∞ αn ≤ lim supn→∞ αn < 1. Assume





lim sup kyn+1 − yn k − kxn+1 − xn k ≤ 0. n→∞

Then limn→∞ kyn − xn k = 0. Lemma 2.3 ([9,10]). Let C be a nonempty closed convex subset of a Banach space E which has uniformly Gâteaux differentiable norm, T : C → C a nonexpansive mapping with Fix(T ) 6= ∅ and f : C → C a k-contraction. Assume that every nonempty closed convex bounded subset of C has the fixed point property for nonexpansive mappings. Then there exists a continuous path: t → xt , t ∈ (0, 1) satisfying xt = tf (xt ) + (1 − t )Txt , which converges to a fixed point of T as t → 0+ .

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Lemma 2.4. Let E be a real Banach space. Then for all x, y ∈ E and j(x + y) ∈ J (x + y), the following inequality holds:

kx + yk2 ≤ kxk2 + 2hy, j(x + y)i. Lemma 2.5 ([11]). Let {an } be a sequence of nonnegative real numbers such that an+1 ≤ (1 − δn )an + δn ηn ,

∀n ≥ 0,

where {δn } is a sequence in [0, 1] and {ηn } is a sequence in R satisfying limn→∞ an = 0.

P∞

n =1

δn = +∞ and lim supn→∞ ηn ≤ 0. Then

3. Main results Theorem 3.1. Let E be a real p-uniformly convex Banach space with uniformly Gâteaux differentiable norm, and C a nonempty closed convex subset of E which has the fixed point property for nonexpansive mappings. Let T : C → C be a λ-strict pseudocontraction with respect to p, λ ∈ [0, min{1, 2−(p−2) cp }) and Fix(T ) 6= ∅. Let f : C → C be a k-contraction with k ∈ (0, 1). Assume that real sequences {αn }, {βn } and {γn } in (0, 1) satisfy the following conditions: (i) limn→∞ αn = 0 and n=0 αn = +∞. (ii) 0 < lim infn→∞ γn ≤ lim supn→∞ γn < ξ , where ξ = 1 − 2p−2 λcp−1 .

P∞

For any x1 ∈ C , the sequence {xn } is generated by xn+1 = αn f (xn ) + βn xn + γn Txn ,

n ≥ 1.

(3.1)

Then the sequence {xn } converges strongly to a fixed point of T . γ

Proof. We take ξn ∈ (γn , ξ ] such that lim supn→∞ ξ n < 1 and limn→∞ |ξn+1 − ξn | = 0 (e.g., taking ξn ≡ ξ ). Rewrite the n iterative sequence (3.1) as follows: xn+1 = αn f (xn ) + βn0 xn + γn0 Tn xn , where βn0 = βn −

γn (1 ξn

n ≥ 1,

(3.2)

γn ξn

− ξn ), γn0 = and Tn := (1 − ξn )I + ξn T , I is the identity mapping. Taking any q ∈ Fix(T ). From (3.2) and Lemma 2.1, it is implied that kxn+1 − qk ≤ αn kf (xn ) − qk + βn0 kxn − qk + γn0 kTn xn − qk ≤ αn kkxn − qk + αn kf (q) − qk + (1 − αn )kxn − qk 1

= αn (1 − k) kf (q) − qk + (1 − αn (1 − k))kxn − qk 1−k   1 kf (p) − pk . ≤ max kx1 − pk, 1−k Therefore, the sequence {xn } is bounded, and so are the sequences {f (xn )}, {Tn xn }. Since Tn xn = (1 − ξn )xn + ξn Txn and lim infn ξn > 0, we know that {Txn } is bounded. The iterative sequence (3.2) can be expressed as follows: xn+1 = βn0 xn + (1 − βn0 )yn , where yn =

αn γn0 f (xn ) + Tn xn . 0 1 − βn 1 − βn0

(3.3)

We estimate from (3.3)

γn0+1 αn γn0 f (xn+1 ) + T n +1 x n +1 − f (xn ) − Tn xn 0

0 0 1 − βn+1 1 − βn 1 − βn n +1 αn+1 γn0+1 αn+1 αn ≤ kkxn+1 − xn k + kTn+1 xn+1 − Tn xn k + − kf (xn ) − Tn xn k. 0 0 0 1 − βn+1 1 − βn+1 1 − βn+1 1 − βn0

αn+1 kyn+1 − yn k =

1 − β0

(3.4)

Since Tn := (1 − ξn )I + ξn T , we have

kTn+1 xn+1 − Tn xn k ≤ k(1 − ξn+1 )xn+1 + ξn+1 Txn+1 − (1 − ξn+1 )xn − ξn+1 Txn k + |ξn+1 − ξn |kxn − Txn k ≤ kxn+1 − xn k + |ξn+1 − ξn |kxn − Txn k

(3.5)

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and limn→∞ αn = 0 and limn→∞ |ξn+1 − ξn | = 0 imply, from (3.4) and (3.5), that lim sup (kyn+1 − yn k − kxn+1 − xn k) ≤ 0. n→∞

Hence, by Lemma 2.2, we obtain lim kyn − xn k = 0.

(3.6)

n→∞

From (3.3), we get lim kyn − Tn xn k = lim

n→∞

n→∞

αn 1 − βn0

kf (xn ) − Tn xn k = 0,

(3.7)

and so it follows from (3.6) and (3.7) that limn→∞ kxn − Tn xn k = 0. Since xn − Tn xn = ξn (xn − Txn ) and lim infn ξn > 0, we have lim kxn − Txn k = lim

n→∞

n→∞

kxn − Tn xn k = 0. ξn

For any ρ ∈ (0, ξ ], defining Tρ := (1 − ρ)I + ρ T , we have lim kxn − Tρ xn k = lim ρkxn − Txn k = 0.

n→∞

n→∞

Since Tρ is a nonexpansive mapping, we have from Lemma 2.3 that the net {xt } generated by xt = tf (xt ) + (1 − t )Tρ xt converges strongly to v ∈ Fix(Tρ ) = Fix(T ), as t → 0+ . Obviously, xt − xn = (1 − t )(Tρ xt − xn ) + t (f (xt ) − xn ). In view of Lemma 2.4, we find

kxt − xn k2 ≤ (1 − t )2 kTρ xt − xn k2 + 2t hf (xt ) − xn , J (xt − xn )i ≤ (1 − 2t + t 2 )(kxt − xn k + kTρ xn − xn k)2 + 2t hf (xt ) − xt , J (xt − xn )i + 2t kxt − xn k2 and therefore

hf (xt ) − xt , J (xn − xt )i ≤

t 2

kxt − xn k2 +

(1 + t 2 )kxn − Tρ xn k

Since {xn }, {xt } and {Tρ xn } are bounded and limn→∞ lim suphf (xt ) − xt , J (xn − xt )i ≤ n→∞

t 2

2t kxn −Tρ xn k 2t

(2kxt − xn k + kxn − Tρ xn k).

= 0, we obtain

M,

(3.8)

where M = supn≥1,t ∈(0,1) {kxt − xn k2 }. We also know that

hf (v) − v, J (xn − v)i = hf (xt ) − xt , J (xn − xt )i + hf (v) − f (xt ) + xt − v, J (xn − xt )i + hf (v) − v, J (xn − v) − J (xn − xt )i.

(3.9)

From the facts that xt → v ∈ Fix(T ) as t → 0, {xn } is bounded and the duality mapping J is norm-to-weak uniformly continuous on bounded subsets of E, it follows that as t → 0, ∗

hf (v) − v, J (xn − v) − J (xn − xt )i → 0 and hf (v) − f (xt ) + xt − v, J (xn − xt )i → 0. Combining (3.8), (3.9) and the two results mentioned above, we get lim suphf (v) − v, J (xn − v)i ≤ 0.

(3.10)

n→∞

From (3.2) and Lemma 2.4, we get

kxn+1 − vk2 ≤ kαn (f (xn ) − f (v)) + βn0 (xn − v) + γn0 (Tn xn − v)k2 + 2αn hf (v) − v, J (xn+1 − v)i ≤ (1 − αn (1 − k))kxn − vk2 + 2αn hf (v) − v, J (xn+1 − v)i. Hence applying Lemma 2.5 to (3.11), we conclude that limn→∞ kxn − vk = 0.

(3.11) 

Corollary 3.1. Let E, C , T , {αn }, {βn } and {γn } be as in Theorem 3.1. For any u, x1 ∈ C , the sequence {xn } is generated by xn+1 = αn u + βn xn + γn Txn ,

n ≥ 1.

Then the sequence {xn } converges strongly to a fixed point of T .

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Remark 3.1. Theorem 3.1 and Corollary 3.1 improve on and extend the comparable results in [2,6,3,5,4] essentially since the following hold. (1) Theorem 3.1 and Corollary 3.1 give strong convergence results in p-uniformly convex Banach spaces for a mixed iteration scheme in contrast to the weak convergence result in [2, Theorem 3.2 and Theorem 3.3], [3, Theorem 3.1] and [4, Theorem 3.1 and Corollary 3.3]. (2) In comparison to the results in [6, Theorem 3.1] and [5, Theorem 2], these results with respect to nonexpansive mappings are extended to λ-strict pseudocontractions in p-uniformly convex Banach spaces. P∞ P∞ (3) In comparison to the case for the results in [6, Theorem 3.1], the restrictions n=1 |αn+1 − αn | < ∞ and n=1 |βn+1 − βn | < ∞ are removed. Remark 3.2. Corollary 3.1 extends the corresponding result [12, Theorem 3.1]. Acknowledgements The authors would like to thank the referees for careful reading of the manuscript and helpful suggestions. The work was supported partly by the National Science Foundation of China (60872095), the K.C. Wong Magna Fund of Ningbo University, Ningbo Natural Science Foundation (2008A610018) and the Scientific Research Fund of Zhejiang Provincial Education Department (Y200906210). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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