Mathematical and Computer Modelling 48 (2008) 480–485 www.elsevier.com/locate/mcm
Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space Qing-bang Zhang ∗ , Cao-zong Cheng College of Applied Science, Beijing University of Technology, Beijing 100022, PR China Received 15 May 2007; received in revised form 3 September 2007; accepted 19 September 2007
Abstract A extension of Nakajo and Takahashi’s modification of Mann’s iterative process to the Ishikawa iterative process is given. The strong convergence of a modified Ishikawa iterative scheme to a common fixed point of a finite family of Lipschitz pseudocontractive self-mappings on a closed convex subset of a Hilbert space is proved. Our theorem extends several known results. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Lipschitz pseudocontractive mapping; Modified Ishikawa iterative process; Strong convergence
1. Introduction Let C be a closed convex subset of a Hilbert space H . A mapping T : C → C (see e.g. [1]) is said to be pseudocontractive if kT x − T yk2 ≤ kx − yk2 + k(I − T )x − (I − T )yk2 ,
∀x, y ∈ C.
The mapping T : C → C is said to be strictly pseudocontractive if there exists a constant 0 ≤ k < 1 such that kT x − T yk2 ≤ kx − yk2 + kk(I − T )x − (I − T )yk2 ,
∀x, y ∈ C;
T is said to be a nonexpansive mapping if kT x − T yk ≤ kx − yk, for all x, y ∈ C. It is obvious that all nonexpansive mappings and strictly pseudocontractive mappings are pseudocontractive mappings but the converse does not hold. In 1974, Ishikawa [5] introduced a new iterative scheme and proved the following theorem. Theorem 1.1. Let H be a real Hilbert space, C be a compact convex subset of H and T : C → C be a Lipschitz pseudocontractive mapping. Suppose: {αn } and {βn } are two positive sequences in [0, 1] such that for all n ≥ 1, ∗ Corresponding author.
E-mail addresses:
[email protected] (Q.-b. Zhang),
[email protected] (C.-z. Cheng). c 2007 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.09.014
Q.-b. Zhang, C.-z. Cheng / Mathematical and Computer Modelling 48 (2008) 480–485
0 ≤ αn ≤ βn ≤ 1, βn → 0 (n → ∞) and
P∞
n=1 αn βn
481
= ∞. The sequence {xn } defined by
yn = (1 − βn )xn + βn T xn , xn+1 = (1 − αn )xn + αn T yn , where the initial point x1 ∈ C is arbitrary, then {xn } converges strongly to u ∈ F(T ). Since the publication of Theorem 1.1 in 1974, it had remained an open question whether or not the Mann recursion formula [3], which is clearly simpler than the Ishikawa iterative scheme, converges under the setting of Theorem 1.1 to a fixed point of T if the operator T is pseudocontractive and continuous (or even Lipschitz). The problem for Lipschitz pseudocontractive mappings was resolved in 2001 in the negative by Chidume and Mutangadura [4]. They constructed a Lipschitz pseudocontractive mapping on a compact convex subset of a Hilbert space with a fixed point and showed that no Mann iterative scheme converges to the fixed point. Recently, Martinez-Yanes and Xu [7] extended the results of Nakajo and Takahashi [8] from the modified Mann’s iterative process to the Ishikawa iterative process. In fact, Martinez-Yanes and Xu proved the following theorem: Theorem 1.2. Let C be a closed convex subset of a Hilbert space H and let T : C → C be a nonexpansive mapping ∞ such that Fix(T ) 6= ∅. Assume that {αn }∞ n=0 and {βn }n=0 are sequences in [0, 1] such that βn ≥ δ for some δ ∈ (0, 1] ∞ and αn → 0. Define a sequence {xn }n=0 in C by the algorithm: x ∈ C chosen arbitrarily, 0 y = (1 − α )x + α T x , n n n n n z = (1 − β )x + β T y , n
n
n
n
n
Cn = {u ∈ C : kz n − uk2 ≤ kxn − uk2 + βn (kyn k2 − kxn k2 + 2hxn − yn , ui)}, Q n = {u ∈ C : hxn − u, x0 − xn i ≥ 0}, xn+1 = PCn ∩Q n (x0 ).
Then {xn } converges in norm to PFix(T ) x0 . The aim of this paper is to introduce a new iterative scheme (a modification of Ishikawa iterative process) and prove strong convergence of the scheme to a common fixed point of a finite family of Lipschitz pseudocontractive self-mappings defined on a closed convex subset of a Hilbert space. Our theorem extends the corresponding results of Nakajo and Takahashi [8]; and several other results recently announced (see e.g. [5–9] and references therein) to approximation of a common fixed point of a finite family of Lipschitz pseudocontractive self-mappings. Furthermore, the open problems posed in [9] are resolved. 2. Preliminaries The following notions should be used in the sequel: (1) * for weak convergence and → for strong convergence. (2) ω(xn ) = {x : ∃ xn j * x} denotes the weak ω-limit set of {xn }. F(T ) denotes the set of all fixed points of mapping T : A → A, where A ⊂ H . (3) Given a closed convex subset K of real Hilbert space H , PK denotes the nearest point projection from H onto K , that is, PK x is the unique point in K with the property kx − PK xk ≤ kx − yk, for all y ∈ K . We shall also need the following Lemmas which hold in Hilbert spaces (see e.g. [7]). Lemma 2.1. Let H be a real Hilbert space. There hold the following identities. (i) kx − yk2 = kxk2 − kyk2 − 2hx − y, yi, ∀x, y ∈ H . (ii) kt x + (1 − t)yk2 = tkxk2 + (1 − t)kyk2 − t (1 − t)kx − yk2 , ∀t ∈ [0, 1], ∀x, y ∈ H . (iii) If {xn } is a sequence in H weakly convergent to z, then lim sup kxn − yk2 = lim sup kxn − zk2 + ky − zk2 , n→∞
n→∞
∀y ∈ H.
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(iv) If C ⊂ H is a closed convex subset and x, y, z ∈ H , then for a given real number a ∈ R, the set {v ∈ C : ky − vk2 ≤ kx − vk2 + hz, vi + a} is convex and closed. Lemma 2.2. Let C be a closed convex subset of H. Let {xn } be a sequence in H and u ∈ H . Let q = PC u. Suppose {xn } is such that ω(xn ) ⊂ C and satisfies the condition kxn − uk ≤ ku − qk
for all n.
Then xn → q. Lemma 2.3. Let K be a closed convex subset of the real Hilbert space H . Given x ∈ H and z ∈ K , then z = PK x if and only if there holds the relation hx − z, y − zi ≤ 0,
∀y ∈ K .
Proposition 2.1. Let H be a real Hilbert space, C a closed convex subset of H and T : C → C a continuous pseudocontractive mapping, then (i) F(T ) is a closed convex subset of C. (ii) I −T is demiclosed at zero, i.e., if {xn } is a sequence in C such that xn * z and (I −T )xn → 0, then (I −T )z = 0. Proof. Define a mapping g : C → C by g(x) = (2I − T )−1 (x). From the Corollary 1 in [2], it follows that C ⊂ (2I − T )C and g is well-defined. Indeed, for any y ∈ C, define mapping A : C → C by A(x) = 21 y + 21 T x, then A is continuous and hAx1 − Ax2 , x1 − x2 i ≤ 12 kx1 − x2 k2 for all x1 , x2 ∈ C. Hence there exists xˆ ∈ C such that xˆ = A(x). ˆ It is obvious that g is nonexpansive and F(g) = F(T ). By the closedness and the convexity of F(g), we have that F(T ) is a closed convex subset of C. That is (i) holds. Suppose that {xn } is a sequence in C such that xn * z and (I − T )xn → 0. Let f (x) = lim sup kxn − xk2 ,
x ∈ H.
n→∞
By Lemma 2.1(iii), the weak convergence xn * x implies that f (x) = f (z) + kx − zk2 , for all x ∈ H . In particular, f (g(z)) = f (z) + kg(z) − zk2 . On the other hand, since C ⊂ (2I − T )C and T is a self-mapping, we have that kxn − g(xn )k = kgg −1 (xn ) − g(xn )k ≤ kg −1 (xn ) − xn k = kxn − T xn k → 0,
(n → ∞).
Hence f (gz) = lim sup kgxn − gzk2 ≤ lim sup kxn − zk2 = f (z). n→∞
n→∞
Therefore, g(z) = z and T z = z. That is (ii) holds.
3. Main result N Theorem 3.1. Let H be a real Hilbert space and C a nonempty closed convex subset of H . Let {Ti }i=1 be N LipschitzT pseudocontractive self-mappings of C with constants L i (i = 1, . . . , N ) such that the common fixed point N set F = i=1 F(Ti ) 6= ∅. Let x1 ∈ C be any point and {xn }∞ n=1 be the sequence generated by the following modified Ishikawa iterative process: yn = (1 − αn )xn + αn T[n] xn , z n = (1 − βn )xn + βn T[n] yn , (MIIP1) Cn = {u ∈ C : kz n − uk2 ≤ kxn − uk2 − αn βn (1 − 2αn − L 2 αn2 )kxn − T[n] xn k2 }, Q = {u ∈ C : hxn − u, x1 − xn i ≥ 0}, n xn+1 = PCn ∩Q n (x1 ).
Q.-b. Zhang, C.-z. Cheng / Mathematical and Computer Modelling 48 (2008) 480–485
483
∞ Assume that {αn }∞ n=1 and {βn }n=1 are sequences of positive numbers satisfying the following conditions: for all n,
(i) 1 ≥ αn ≥ βn ≥ 0, (ii) there exists β > 0 such that βn ≥ β, √ 2 (iii) there exist α > 0 such that αn ≤ α < L L+1−1 . 2 Then {xn } converges strongly to PF (x1 ), where L = max{L i : 1 ≤ i ≤ N }, T[n] = Tn mod N . Proof. It is obvious that Q n is a closed convex subset of C. By Lemma 2.1(iv), Cn is a closed convex subset of C. Then Cn ∩ Q n is closed convex and {xn } is well-defined for all n ≥ 1. Also by the Proposition 2.1(i), F is closed and convex. Now we show that F ⊂ Cn ∩ Q n for all n ≥ 1. Take any p ∈ F, from the definition of pseudocontractive mapping, the modified Ishikawa iterative process and Lemma 2.1(ii), we have kz n − pk2 = (1 − βn )kxn − pk2 + βn kT[n] yn − pk2 − βn (1 − βn )kxn − T[n] yn k2 ≤ (1 − βn )kxn − pk2 − βn (1 − βn )kxn − T[n] yn k2 + βn (kT[n] yn − yn k2 + kyn − pk2 ), 2
kyn − T[n] yn k = = ≤ =
k(1 − αn )(xn − T[n] yn ) + αn (T[n] xn − T[n] yn )k (1 − αn )kxn − T[n] yn k2 + αn kT[n] (xn ) − T[n] (yn )k2 − αn (1 − αn )kxn − T[n] (xn )k2 (1 − αn )kxn − T[n] yn k2 + L 2 αn kxn − yn k2 − αn (1 − αn )kxn − T[n] (xn )k2 (1 − αn )kxn − T[n] yn k2 + αn (L 2 αn2 + αn − 1)kxn − T[n] (xn )k2 ,
(1)
2
(2)
where L = max{L i : 1 ≤ i ≤ N } and kyn − pk2 = k(1 − αn )(xn − p) + αn (T[n] (xn ) − p)k2 = (1 − αn )kxn − pk2 + αn kT[n] (xn ) − pk2 − αn (1 − αn )kT[n] (xn ) − xn k2 ≤ (1 − αn )kxn − pk2 − αn (1 − αn )kT[n] (xn ) − xn k2 + αn (kT[n] (xn ) − xn k2 + kxn − pk2 ) ≤ kxn − pk2 + αn2 kT[n] (xn ) − xn k2 .
(3)
By the Eqs. (1)–(3), we have that kz n − pk2 ≤ kxn − pk2 + βn (βn − αn )kxn − T[n] (yn )k2 − αn βn (1 − L 2 αn2 − 2αn )kxn − T[n] (xn )k2 . From the condition, we have βn (βn − αn ) ≤ 0. Hence kz n − pk2 ≤ kxn − pk2 − αn βn (1 − L 2 αn2 − 2αn )kxn − T[n] (xn )k2 . This implies that p ∈ Cn and F ⊂ Cn , for all n ≥ 0. Next we show that F ⊂ Q n , for all n ≥ 1, by induction. For n = 1, we have F ⊂ C = Q 1 . Assume that F ⊂ Q n for some n > 1. Since xn+1 = PCn ∩Q n (x1 ), by Lemma we have hxn+1 − z, x1 − xn+1 i ≥ 0,
∀z ∈ Cn ∩ Q n .
By the definition of Q n+1 , we know that z ∈ Q n+1 , and hence Cn ∩ Q n ⊂ Q n+1 . The induction assumption implies that F ⊂ Cn ∩ Q n , and then F ⊂ Q n+1 . Hence F ⊂ Cn ∩ Q n for all n ≥ 1. Since xn+1 = PCn ∩Q n (x1 ) and F ⊂ Cn ∩ Q n , we have that kxn+1 − x1 k ≤ kq − x1 k,
∀q ∈ F.
In particular, kxn+1 − x1 k ≤ kt − x1 k,
where t = PF (x1 ).
This implies that {xn } is bounded, hence {yn } and {z n } are bounded too. The fact that xn+1 ∈ Cn ∩ Q n ⊂ Q n implies that hxn − xn+1 , x1 − xn i ≥ 0,
∀n ≥ 1.
Therefore kxn − x1 k ≤ kxn+1 − x1 k,
∀n ≥ 1,
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and kxn+1 − xn k2 = k(xn+1 − x1 ) − (xn − x1 )k2 = kxn+1 − x1 k2 − kxn − x1 k2 − 2hxn − xn+1 , x1 − xn i ≤ kxn+1 − x1 k2 − kxn − x1 k2 . It follows that limn→∞ kxn − x1 k exists and limn→∞ kxn+1 − xn k = 0. By the fact xn+1 ∈ Cn , we get 0 ≤ kz n − xn k2 = kz n − xn+1 − (xn − xn+1 )k2 = kz n − xn+1 k2 − kxn − xn+1 k2 − 2hz n − xn , xn − xn+1 i ≤ kxn − xn+1 k2 − αn βn (1 − L 2 αn2 − 2αn )kxn − T[n] xn k2 − kxn − xn+1 k2 + 2kz n − xn k · kxn − xn+1 k. Hence, αn βn (1 − L 2 αn2 − 2αn )kxn − T[n] xn k2 ≤ 2kz n − xn k · kxn − xn+1 k. From the conditions (i)–(iii), it follows that β 2 (1 − L 2 α 2 − 2α)kxn − T[n] xn k2 ≤ 2kz n − xn k · kxn − xn+1 k → 0,
(n → ∞).
This means that kxn − T[n] xn k → 0,
(n → ∞).
(4)
Claim: ω(xn ) ⊂ F. Indeed, assume x¯ ∈ ω(xn ) and xn i * x¯ for some subsequence xn i of xn . We may further assume n i = l (mod N ) for all i. By limn→∞ kxn+1 − xn k = 0, we have limn→∞ kxn+ j − xn k = 0 for all j ∈ {1, 2, . . . , N }. From Eq. (4), we deduce that kxn i − Tl+ j xn i k = kxn i − T[n i + j] xn i k ≤ kxn i − xn i + j k + kxn i + j − T[ni + j] xn i + j k + kT[n i + j] xn i + j − T[ni + j] xn i k ≤ (1 + L)kxn i + j − xn i k + kxn i + j − T[ni + j] xn i + j k → 0
(i → ∞).
Then it follows from Proposition 2.1(ii) that x¯ ∈ F(Tl+ j ) for all j ∈ {1, 2, . . . , N }. Since for any j0 ∈ {1, 2, . . . , N }, there exists a j such that l + j = j0 (mod N ), we have that x¯ ∈ F(T j0 ) (∀ j0 ∈ {1, 2, . . . , N }), i.e., x¯ ∈ F. Therefore, by Lemma 2.2, we have that {xn } defined by the modified Ishikawa iterative process converges strongly to PF (x1 ). Remark 3.1. Observe that in Theorem 3.1, the compactness condition imposed on C in Theorem 1.1 is dispensed with. So, our theorem extends and generalizes Theorem 1.1 of Ishikawa to a closed and convex subset of a Hilbert space and from one self-mapping to a family of self-mappings. Furthermore, our theorem extends Theorem 1.2 of [7] from a nonexpansive mapping to a family of Lipschitz pseudocontractive mappings and Theorem 5.2 of [6] from a finite family of strictly pseudocontractive mappings to a finite family of Lipschitz pseudocontractive mappings. Our theorem also extends the corresponding results of [8,9]; and resolves the interesting open problems posed in [9]. Acknowledgement The authors are grateful to the referee for the careful reading of the manuscript and the constructive suggestions. References [1] [2] [3] [4]
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