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Convergence of an implicit iterative process for asymptotically pseudocontractive nonself-mappings
Nonlinear Analysis 74 (2011) 5851–5862
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Convergence of an implicit iterative process for asymptotically pseudocontractive nonself-mappings Xiaolong Qin a,b , Sun Young Cho c , Tianze Wang a , Shin Min Kang d,∗ a
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
b
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
c
Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea
d
Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
article
abstract
info
Article history: Received 19 April 2010 Accepted 11 April 2011 Communicated by Ravi Agarwal
In this work, an implicit iterative process is considered for asymptotically pseudocontractive nonself-mappings. Weak and strong convergence theorems for common fixed points of a family of asymptotically pseudocontractive nonself-mappings are established in the framework of Hilbert spaces. © 2011 Elsevier Ltd. All rights reserved.
MSC: 47H09 47H10 47J25 Keywords: Asymptotically pseudocontractive nonself-mapping Asymptotically nonexpansive nonself-mapping Fixed point Implicit iterative process
1. Introduction and preliminaries Throughout this work, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by
⟨·, ·⟩ and ‖ · ‖, respectively. Let C be a nonempty closed convex subset of H and T : C → C be a mapping. In this work, we denote the fixed point set of T by F (T ). Recall that T is said to be nonexpansive if
‖Tx − Ty‖ ≤ ‖x − y‖,
∀x, y ∈ C .
T is said to be asymptotically nonexpansive if there exists a sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ such that
‖T n x − T n y‖ ≤ kn ‖x − y‖,
∀x, y ∈ C , n ≥ 1.
(1.1)
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. It is known that if C is a nonempty bounded closed convex subset of a Hilbert space H, then every asymptotically nonexpansive self-mapping has
∗
Corresponding author. Fax: +82 557551917. E-mail addresses: [email protected], [email protected] (X. Qin), [email protected] (S.Y. Cho), [email protected] (T. Wang), [email protected] (S.M. Kang). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.04.031
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a fixed point. Since 1972, a host of authors have studied weak and strong convergence problems of the iterative processes for such a class of mappings. T is said to be strictly pseudocontractive if there exists a constant κ ∈ [0, 1) such that
‖Tx − Ty‖2 ≤ ‖x − y‖2 + κ‖(I − T )x − (I − T )y‖2 , ∀x, y ∈ C . For such a case, T is also said to be a κ -strict pseudocontraction. The class of strict pseudocontractions is introduced by Browder and Petryshyn [2] in 1967. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction. In [3], Marino and Xu established the demiclosed principle for the class of strict pseudocontractions; see also [4]. T is said to be an asymptotically strict pseudocontraction if there exist a sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ and a constant κ ∈ [0, 1) such that
‖T n x − T n y‖2 ≤ kn ‖x − y‖2 + κ‖(I − T n )x − (I − T n )y‖2 , ∀x, y ∈ C , n ≥ 1. (1.2) For such a case, T is also said to be an asymptotically κ -strict pseudocontraction. The class of asymptotically strict pseudocontractions was introduced by Liu [5] in 1996. It is clear that every asymptotically nonexpansive mapping is an asymptotical 0-strict pseudocontraction. In [6], Kim and Xu established the demiclosed principle for the class of asymptotically strict pseudocontractions; see also [7]. T is said to be pseudocontractive if
⟨Tx − Ty, x − y⟩ ≤ ‖x − y‖2 ,
∀x, y ∈ C .
(1.3)
It is easy to see that (1.3) is equivalent to
‖Tx − Ty‖2 ≤ ‖x − y‖2 + ‖(I − T )x − (I − T )y‖2 ,
∀ x, y ∈ C .
T is said to be asymptotically pseudocontractive if there exists a sequence kn ⊂ [1, ∞) with kn → 1 as n → ∞ such that
⟨T n x − T n y, x − y⟩ ≤ kn ‖x − y‖2 ,
∀x, y ∈ C , n ≥ 1.
(1.4)
The class of asymptotically pseudocontractive mapping was introduced by Schu [8]. It is easy to see that (1.4) is equivalent to
‖T n x − T n y‖2 ≤ (2kn − 1)‖x − y‖2 + ‖(I − T n )x − (I − T n )y‖2 ,
∀x, y ∈ C , n ≥ 1.
In [9], Rhoades gave the following example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings. Example R. For x ∈ [0, 1], define a mapping T : [0, 1] → [0, 1] by
2
Tx = 1 − x 3
23
.
Then T is asymptotically pseudocontractive but it is not asymptotically nonexpansive. T is said to be uniformly L-Lipschitz if there exists a positive constant L such that
‖T n x − T n y‖ ≤ L‖x − y‖,
∀x, y ∈ C , ∀n ≥ 1.
In 2001, Xu and Ori [10] introduced, in the framework of Hilbert spaces, the following implicit iteration process for a finite family of nonexpansive mappings {T1 , T2 , . . . , TN } with {αn } a real sequence in (0, 1) and an initial point x0 ∈ C : x1 = α1 x0 + (1 − α1 )T1 x1 , x2 = α2 x1 + (1 − α2 )T2 x2 ,
··· xN = αN xN −1 + (1 − αN )TN xN , xN +1 = αN +1 xN + (1 − αN +1 )T1 xN +1 ,
··· which can written in the following compact form: xn = αn xn−1 + (1 − αn )Tn xn ,
∀n ≥ 1,
(1.5)
where Tn = Tn(mod N ) (here mod N takes values in {1, 2, . . . , N }). They obtained the following weak convergence theorem. Theorem XO. Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and {Ti }Ni=1 : C → C be a finite family of nonexpansive mappings such that F = ∩Ni=1 F (Ti ) ̸= ∅. Let {xn } be defined by (1.5). If {αn } is chosen such that αn → 0 as n → ∞, then {xn } converges weakly to a common fixed point of the family of {Ti }Ni=1 . Subsequently, fixed point problems based on implicit iterative processes have been considered by many authors. In 2004, Osilike [11] reconsidered the implicit iterative process (1.5) for a finite family of strictly pseudocontractive mappings. To be more precise, he proved the following theorem.
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Theorem O. Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let {Ti }Ni=1 be N strictly pseudocontractive self-maps of C such that F = ∩ri=1 F (Ti ) ̸= ∅. Suppose that x0 ∈ C and let {αn } be a sequence in (0, 1) such that αn → 0 as n → ∞. Then the sequence {xn } defined by (1.5) converges weakly to a common fixed point of the mappings {Ti }Ni=1 . In 2003, Sun [12] introduced the following implicit iterative process for a finite family of asymptotically nonexpansive mappings {T1 , T2 , . . . , TN }, with {αn } a real sequence in (0, 1) and an initial point x0 ∈ C : x1 = α1 x0 + (1 − α1 )T1 x1 , x2 = α2 x1 + (1 − α2 )T2 x2 ,
.. . xN = αN xN −1 + (1 − αN )TN xN , xN +1 = αN +1 xN + (1 − αN +1 )T12 xN +1 ,
.. . x2N = α2N x2N −1 + (1 − α2N )TN2 x2N , x2N +1 = α2N +1 x2N + (1 − α2N +1 )T13 x2N +1 ,
.. .. For each n ≥ 1, we can write n = (h − 1)N + i, where i = i(n) ∈ {1, 2, . . . , N }, h = h(n) ≥ 1 is a positive integer and h(n) → ∞ as n → ∞. Hence the above table can be rewritten in the following compact form: h(n)
xn = αn xn−1 + (1 − αn )Ti(n) xn ,
∀n ≥ 1.
(1.6)
Strong convergence theorems for the implicit iterative process (1.6) for a finite family of asymptotically nonexpansive mappings were established. In 2004, Osilike and Akuchu [13] reconsidered the implicit iterative process (1.6) for a family of asymptotically pseudocontractive mappings. They also obtained a strong convergence theorem. Since there is no demiclosed principle, there is no weak convergence theorem for the implicit iterative process (1.6). Notice that, from the viewpoint of computation, the implicit iterative schemes (1.5) and (1.6) are often impractical. For each step, we must solve nonlinear operator equations exactly. Therefore, one of the interesting and important problems in the theory of implicit iterative algorithms is considering the iterative algorithm with errors. This is an efficient iterative algorithm for computing approximately fixed points of nonlinear mappings. Recently, Chang et al. [14] considered the following implicit iterative process for a finite family of asymptotically nonexpansive mappings {T1 , T2 , . . . , TN }, with {αn } a real sequence in (0, 1), {un } a bounded sequence in C and an initial point x0 ∈ C : x1 = α1 x0 + (1 − α1 )T1 x1 + u1 , x2 = α2 x1 + (1 − α2 )T2 x2 + u2 ,
.. . xN = αN xN −1 + (1 − αN )TN xN + uN , xN +1 = αN +1 xN + (1 − αN +1 )T1n xN +1 + uN +1 ,
.. . x2N = α2N x2N −1 + (1 − α2N )TN2 x2N + u2N , x2N +1 = α2N +1 x2N + (1 − α2N +1 )T13 x2N +1 + u2N +1 ,
.. .. For each n ≥ 1, we can write n = (h − 1)N + i, where i = i(n) ∈ {1, 2, . . . , N }, h = h(n) ≥ 1 is a positive integer and h(n) → ∞ as n → ∞. Hence the above table can be rewritten in the following compact form: h(n)
xn = αn xn−1 + (1 − αn )Ti(n) xn + un ,
∀n ≥ 1 .
(1.7)
Weak and strong convergence theorems for the implicit iterative scheme (1.7) were obtained for a finite family of asymptotically nonexpansive mappings {T1 , T2 , . . . , TN }; see [14] for more details. Recently, fixed point problems for nonself-mappings have been studied by a number of authors; see, for example [15–23]. Next, we draw attention to nonself-mappings. Let S : C → H be a mapping and P be the metric projection from H onto C .
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Recall that S is said to be asymptotically nonexpansive with respect to P if there exists a sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ such that
‖(PS )n x − (PS )n y‖ ≤ kn ‖x − y‖,
∀x, y ∈ C , ∀n ≥ 1.
(1.8)
S is said to be an asymptotically strict pseudocontraction with respect to P if there exist a sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ and a constant κ ∈ [0, 1) such that
‖(PS )n x − (PS )n y‖2 ≤ kn ‖x − y‖2 + κ‖(I − (PS )n )x − (I − (PS )n )y‖2 ,
∀x, y ∈ C , n ≥ 1.
(1.9)
For such a case, S is also said to be an asymptotically κ -strict pseudocontraction with respect to P. It is clear that every asymptotically nonexpansive mapping with respect to P is an asymptotical 0-strict pseudocontraction with respect to P. S is said to be asymptotically pseudocontractive with respect to P if there exists a sequence kn ⊂ [1, ∞) with kn → 1 as n → ∞ such that
⟨(PS )n x − (PS )n y, x − y⟩ ≤ kn ‖x − y‖2 ,
∀ x, y ∈ C .
(1.10)
It is easy to see that (1.10) is equivalent to
‖(PS )n x − (PS )n y‖2 ≤ (2kn − 1)‖x − y‖2 + ‖(I − (PS )n )x − (I − (PS )n )y‖2 ,
∀ x, y ∈ C .
S is said to be uniformly L-Lipschitz with respect to P if there exists a positive constant L such that
‖(PS )n x − (PS )n y‖ ≤ L‖x − y‖,
∀x, y ∈ C , ∀n ≥ 1.
In this work, motivated by the above results, we introduced the following implicit iterative process for a finite family of Lipschitz asymptotically pseudocontractive nonself-mappings {S1 , S2 , . . . , SN } with respect to P: x1 = α1 x0 + β1 (PS1 )x1 + γ1 u1 , x2 = α2 x1 + β2 (PS2 )x2 + γ2 u2 ,
.. . xN = αN xN −1 + βN (PSN )xN + γN uN , xN +1 = αN +1 xN + βN +1 (PS1 )2 xN +1 + γN +1 uN +1 ,
.. . x2N = α2N x2N −1 + β2N (PSN )2 x2N + γ2N u2N , x2N +1 = α2N +1 x2N + β2N +1 (PS1 )3 x2N +1 + γ2N +1 u2N +1 ,
.. ., where x0 is the initial value, {un } is a bounded sequence in C and {αn }, {βn } and {γn } are sequences in [0, 1] such that αn +βn +γn = 1 for each n ≥ 1. For each n ≥ 1, we can write n = (h − 1)N + i, where i = i(n) ∈ {1, 2, . . . , N }, h = h(n) ≥ 1 is a positive integer and h(n) → ∞ as n → ∞. Hence the above table can be rewritten in the following compact form: xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
∀n ≥ 1.
(1.11)
Indeed, the implicit iterative sequence (1.11) can be employed for approximating common fixed points of a finite family of Lipschitz asymptotically pseudocontractive nonself-mappings with respect to P which can be obtained from the following. Define mappings Rn : C → C by Rn (x) = αn xn−1 + βn (PSi(n) )h(n) x + γn un ,
∀n ≥ 1.
Suppose that L = max{Li : 1 ≤ i ≤ N },
(1.12)
where Li is the Lipschitz constant of the mapping Si with respect to P. It follows that
‖Rn (x) − Rn (y)‖ = αn xn−1 + βn (PSi(n) )h(n) x + γn un − αn xn−1 + βn (PSi(n) )h(n) y + γn un = βn ‖(PSi(n) )h(n) x − (PSi(n) )h(n) y‖ ≤ βn L‖x − y‖, ∀x, y ∈ C . If βn L < 1 for each n ≥ 1, then we see that Rn is a contraction for each fixed n ≥ 1. By the Banach contraction principle, we see that there exists a unique fixed point xn ∈ C such that xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
∀n ≥ 1.
This shows that the implicit iterative process (1.11) is well defined.
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The purpose of this work is to study the weak and strong convergence of the implicit iteration process (1.11) for a finite family of asymptotically pseudocontractive nonself-mappings with respect to P. To prove our main results, we also need the following concepts and lemmas. Recall that a space X is said to satisfy Opial’s condition [24] if, for each sequence {xn } in X , the convergence xn → x weakly implies that lim inf ‖xn − x‖ < lim inf ‖xn − y‖, n→∞
n→∞
∀y ∈ X (y ̸= x).
Let C be a nonempty convex subset of a Hilbert space H. Then for x ∈ C , we define the inward set IC (x) as follows [21]: IC (x) := {y ∈ H : y = x + λ(u − x) : u ∈ C , λ ≥ 0}. A mapping S : C → H is said to satisfy the inward condition if Sx ∈ IC (x) for all x ∈ C . S is also said to satisfy the weakly inward condition if for each x ∈ C , then Sx ∈ cl[IC (x)], where cl[IC (x)] is the closure of IC (x). Clearly C ⊂ IC (x) and it is not hard to show that IC (x) is a convex set as C is. The following lemma has been proved by Xu and Yin [15]. For the sake of completeness, we give the proof here. Lemma 1.1. Let C be a nonempty closed convex subset of a real Hilbert space H, P be the metric projection from H onto C and S : C → H be a mapping satisfying the weakly inward condition. Then F (PS ) = F (S ). Proof. First, we show that F (PS ) ⊂ F (S ). Supposing that x ∈ F (PS ), we see that
⟨Sx − x, y − x⟩ ≤ 0,
∀y ∈ C .
(1.13)
In view of the weakly inward condition, we see that Sx ∈ cl[IC (x)]. Hence, there exist zn ∈ C and λn ≥ 0 such that yn → Sx, where yn = x + λn (un − x). It follows from (1.13) that ⟨Sx − x, yn − x⟩ ≤ 0. This implies that Sx = x, that is, x ∈ F (S ). This leads us to conclude that F (PS ) ⊂ F (S ). Next, we show that F (S ) ⊂ F (PS ). Supposing that x ∈ F (S ), we see that x = Sx = PSx, this is, x ∈ F (PS ). This completes F (S ) ⊂ F (PS ). This proof is completed. The following lemma is trivial. Lemma 1.2. Let (X , ⟨·, ·⟩) be a real inner product space. There holds the following equality:
‖α x + β y + γ z ‖2 = α‖x‖2 + β‖y‖2 + γ ‖z ‖2 − αβ‖x − y‖2 − αγ ‖x − z ‖2 − βγ ‖y − z ‖2 , for all x, y, z ∈ X , where α, β and γ are real numbers in [0, 1] such that α + β + γ = 1. Lemma 1.3 ([25]). Let {rn }, {sn } and {tn } be three nonnegative sequences satisfying the following condition: rn+1 ≤ (1 + sn )rn + tn , If
∑∞
n=1 sn
< ∞ and
∑∞
n =1 t n
for all n ∈ N .
< ∞, then limn→∞ rn exists.
2. The main results Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H and P be the metric projection from H onto C . Let Si : C → H be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with respect to P with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that Si satisfies the weakly inward condition for each 1 ≤ i ≤ N and that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions: (a) (b) (c) (d)
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β defined as (1.12), ∀n ≥ 1; ∑n L∞< 1, where L is ∑ ∞ n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }.
Then {xn } converges weakly to some point in F .
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Proof. Fixing p ∈ F , we see from Lemmas 1.1 and 1.2 that
‖xn − p‖2 = ‖αn (xn−1 − p) + βn (PSi(n) )h(n) xn − p + γn (un − p)‖2 = αn ‖xn−1 − p‖2 + βn ‖(PSi(n) )h(n) xn − p‖2 + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + ‖(PSi(n) )h(n) xn − xn ‖2 + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + βn αn ‖xn−1 − (PSi(n) )h(n) xn ‖ + γn ‖un − (PSi(n) )h(n) xn ‖ + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + 3γn M1 − αn βn (1 − αn )‖(PSi(n) )h(n) xn − xn−1 ‖2 ,
(2.1)
where M1 is an appropriate constant such that M1 ≥ max{supn≥1 {‖un −(PSi(n) )h(n) xn ‖2 }, supn≥1 {‖un −p‖2 }, supn≥1 {2‖xn−1 − (PSi(n) )h(n) xn ‖ ‖un − (PSi(n) )h(n) xn ‖}}. From the restriction (b), we see that there exists some N1 such that
βn (2kh(n) − 1) ≤ R1 , ∀n ≥ N1 , where R1 = (1 − a) 1 + 3(1a−a) . It follows from (2.1) that ‖x n − p ‖2 ≤
αn + γn 1 − βn (2kh(n) − 1)
≤
1+
≤
1+
‖xn−1 − p‖2 +
2βn (kh(n) − 1) 1 − R1 2(1 − a) 1 − R1
3M1 γn 1 − βn (2kh(n) − 1)
‖xn−1 − p‖2 +
3M1 1 − R1
(kh(n) − 1) ‖xn−1 − p‖2 +
γn
3M1 1 − R1
γn ,
∀n ≥ N 1 .
(2.2)
In view of the restriction (d), we obtain from Lemma 1.3 that limn→∞ ‖xn − p‖2 exists. This in turn implies that {xn } is bounded. On the other hand, we see from (2.1) that
αn βn (1 − αn )‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + 3γn M1 − ‖xn − p‖2 ≤ αn (‖xn−1 − p‖2 − ‖xn − p‖2 ) + βn (2kh(n) − 2)‖xn − p‖2 + 3γn M1 . From the restriction (b), we see that βn is away from zero for each n ≥ 1. Therefore, we arrive at lim ‖(PSi(n) )h(n) xn − xn−1 ‖ = 0.
n→∞
(2.3)
Notice that
‖xn − xn−1 ‖ ≤ βn ‖(PSi(n) )h(n) xn − xn−1 ‖ + γn ‖un − xn−1 ‖. It follows from (2.3) and the restriction (d) that lim ‖xn − xn−1 ‖ = 0.
(2.4)
n→∞
On the other hand, we have
‖xn−1 − (PSi(n) )h(n) xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn ‖ + ‖(PSi(n) )h(n) xn − (PSi(n) )h(n) xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn ‖ + L‖xn − xn−1 ‖, where L is defined in (1.12). In view of (2.3) and (2.4), we obtain that lim ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ = 0.
n→∞∞
(2.5)
For any positive integer n > N, we can write n = (h(n) − 1)N + i(n), where i(n) ∈ {1, 2, . . . , N }. Observe that
‖xn−1 − (PSn )xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + ‖(PSi(n) )h(n) xn−1 − (PSn )xn−1 ‖ = ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + ‖(PSi(n) )h(n) xn−1 − (PSi(n) )xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + L‖(PSi(n) )h(n)−1 xn−1 − xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + L ‖(PSi(n) )h(n)−1 xn−1 − (PSi(n−N ) )h(n)−1 xn−N ‖ + ‖(PSi(n−N ) )h(n)−1 xn−N − x(n−N )−1 ‖ + ‖x(n−N )−1 − xn−1 ‖ .
(2.6)
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Since for each n > N , n = (n − N )(mod N ), on the other hand, we obtain from n = (h(n) − 1)N + i(n) that n − N = (h(n) − 1) − 1 N + i(n) = (h(n − N ) − 1)N + i(n − N ). That is, h(n − N ) = h(n) − 1
i(n − N ) = i(n).
and
Notice that
‖(PSi(n) )h(n)−1 xn−1 − (PSi(n−N ) )h(n)−1 xn−N ‖ = ‖(PSi(n) )h(n)−1 xn−1 − (PSi(n) )h(n)−1 xn−N ‖ ≤ L‖xn−1 − xn−N ‖
(2.7)
and
‖(PSi(n−N ) )h(n)−1 xn−N − x(n−N )−1 ‖ = ‖(PSi(n−N ) )h(n−N ) xn−N − x(n−N )−1 ‖.
(2.8)
Substituting (2.7) and (2.8) into (2.6), we arrive at
‖xn−1 − (PSn )xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + L L‖xn − xn−N ‖ + ‖(PSi(n−N ) )h(n−N ) xn−N − x(n−N )−1 ‖ + ‖x(n−N )−1 − xn−1 ‖ .
(2.9)
In view of (2.3)–(2.5), we obtain that lim ‖xn−1 − (PSn )xn−1 ‖ = 0.
(2.10)
n→∞
Notice that
‖xn − (PSn )xn ‖ ≤ ‖xn − xn−1 ‖ + ‖xn−1 − (PSn )xn−1 ‖ + ‖(PSn )xn−1 − (PSn )xn ‖ ≤ (1 + L)‖xn − xn−1 ‖ + ‖xn−1 − (PSn )xn−1 ‖.
(2.11)
From (2.4) and (2.10), we arrive at lim ‖xn − (PSn )xn ‖ = 0.
(2.12)
n→∞
On the other hand, we have
‖xn − (PSn+j )xn ‖ ≤ ‖xn − xn+j ‖ + ‖xn+j − (PSn+j )xn+j ‖ + ‖(PSn+j )xn+j − (PSn+j )xn ‖ ≤ (1 + L)‖xn − xn+j ‖ + ‖xn+j − (PSn+j )xn+j ‖, ∀j ∈ {1, 2, . . . , N }. It follows from (2.4) and (2.12) that lim ‖xn − (PSn+j )xn ‖ = 0,
∀j ∈ {1, 2, . . . , N }.
n→∞
Note that any subsequence of a convergent number sequence converges to the same limit. It follows that lim ‖xn − (PSl )xn ‖ = 0,
n→∞
∀l ∈ {1, 2, . . . , N }.
(2.13)
Since H is a Hilbert space, we see from the boundedness of {xn } that there exists a subsequence {xni } ⊂ {xn } such that {xni } converges weakly to p1 ∈ C . In view of (2.13), we have lim ‖xni − (PSl )xni ‖ = 0,
n→∞
∀l ∈ {1, 2, . . . , N }.
(2.14)
1 Next, we show, for each l ∈ {1, 2, . . . , N }, that p1 ∈ F (PSl ). Put ρ ∈ 0, 1+ , where L is defined in (1.12) and define L m yρ,m,l = (1 − ρ)p1 + ρ(PSl ) p1 for arbitrary but fixed m ≥ 1. We first show that, for fixed m ≥ 1, xni − (PSl )m xni → 0 as ni → ∞. Indeed, since Sl is uniformly Ll -Lipschitz with respect to P, we have
‖xni − (PSl )m xni ‖ ≤ ‖xni − (PSl )xni ‖ + ‖(PSl )xni − (PSl )2 xni ‖ + · · · + ‖(PSl )m−1 xni − (PSl )m xni ‖ ≤ mLl ‖xni − (PSl )xni ‖. It follows from (2.14) that lim ‖xni − (PSl )m xni ‖ = 0.
(2.15)
ni →∞
On the other hand, we have
p1 − yρ,m , I − (PSl )m yρ,m,l
= p1 − xni , I − (PSl )m yρ,m,l + xni − yρ,m,l , I − (PSl )m yρ,m,l = p1 − xni , I − (PSl )m yρ,m,l + xni − yρ,m,l , I − (PSl )m yρ,m,l − I − (PSl )m xni + xni − yρ,m,l , I − (PSl )m xni ≤ p1 − xni , I − (PSl )m yρ,m,l + (km − 1)Q + xni − yρ,m,l , I − (PSl )m xni ,
(2.16)
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where Q is an appropriate constant such that Q ≥ sup{‖yρ,m,l − xni ‖2 }. Notice that xni ⇀ p1 and (2.15). From (2.16), we arrive at
⟨p1 − yρ,m,l , (I − (PSl )m )yρ,m ⟩ ≤ (km − 1)Q .
(2.17)
On the other hand, since Sl is uniformly Ll -Lipschitz with respect to P, we see from the definition of yρ,m that
⟨p1 − yρ,m,l , (I − (PSl )m )p1 − (I − (PSl )m )yρ,m,l ⟩ ≤ (1 + Ll )‖p1 − yρ,m,l ‖2 = (1 + Ll )ρ 2 ‖p1 − (PSl )m p1 ‖2 .
(2.18)
Combining (2.17) with (2.18), we see that
‖p1 − (PSl )m p1 ‖2 = ⟨p1 − (PSl )m p1 , p1 − (PSl )m p1 ⟩ 1
=
ρ
=
ρ
⟨p1 − yρ,m,l , p1 − (PSl )m p1 ⟩
1 1 p1 − yρ,m,l , I − (PSl )m p1 − I − (PSl )m yρ,m,l + p1 − yρ,m , I − (PSl )m yρ,m,l
(km − 1)Q ≤ (1 + Ll )ρ‖p1 − (PSl )m p1 ‖2 + , ρ
ρ
which yields that
ρ 1 − ρ(1 + Ll ) ‖p1 − (PSl )m p1 ‖ ≤ (km − 1)Q ,
∀m ≥ 1.
(2.19)
Supposing that m → ∞ in (2.19) yields that (PSl )m p1 → p1 and hence (PSl )m+1 p1 → (PSl )p1 as m → ∞. Since Sl is Ll -Lipschitz continuous with respect to P, we can obtain that p1 = (PSl )p1 . This implies that p ∈ ∩Nl=1 F (PSl ) = ∩Nl=1 F (Sl ). Next we show {xn } converges weakly to p1 . Suppose the contrary; then there exists some subsequence {xnj } of {xn } such that {xnj } converges weakly to p2 ∈ C , where p1 ̸= p2 . Similarly, we can show that p2 ∈ F (PSl ) = F (Sl ) for each l ∈ {1, 2, . . . , N }. Notice that we have proved that limn→∞ ‖xn − p‖ exists for each p ∈ ∩Nl=1 F (PSl ) = ∩Nl=1 F (Sl ). Assume that limn→∞ ‖xn − p1 ‖ = d where d is a nonnegative number. By virtue of Opial’s condition for H, we see that d = lim inf ‖xni − p1 ‖ < lim inf ‖xni − p2 ‖ ni →∞
ni →∞
= lim inf ‖xnj − p2 ‖ < lim inf ‖xnj − p1 ‖ = d. nj →∞
nj →∞
This is a contradiction. Hence p1 = p2 . This completes the proof.
If Si is an asymptotically pseudocontractive self-mapping, then P is reduced to the identity mapping. The following theorem is not hard to derive. Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let Si : C → C be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: h(n)
xn = αn xn−1 + βn Si(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions: (a) αn + βn + γn = 1, ∀n ≥ 1; (b) there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; (c) β max{Li : 1 ≤ i ≤ N }, ∀n ≥ 1; ∑n L∞< 1, where L =∑ ∞ (d) n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. Then {xn } converges weakly to some point in F . Remark 2.3. Corollary 2.2 can be viewed as an improvement of Theorem O in Section 1. Note that the class of asymptotically pseudocontractive self-mappings contains properly the class of asymptotically nonexpansive self-mappings as a subclass. As some applications of Corollary 2.2, we have the following results. Corollary 2.4. Let C be a nonempty closed convex subset of a Hilbert space H. Let Si : C → C be an asymptotically nonexpansive mapping with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: h(n)
xn = αn xn−1 + βn Si(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions:
X. Qin et al. / Nonlinear Analysis 74 (2011) 5851–5862
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αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β supn≥1 {kn } < 1, ∑ ∀n ≥ 1; ∑n ∞ ∞ γ < ∞ and n n =1 n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. Then {xn } converges weakly to some point in F . (a) (b) (c) (d)
Next, we give a strong convergence theorem with the help of compactness. Recall that mapping S : C → H is semicompact with respect to P if any sequence {xn } in C satisfying limn→∞ ‖xn − (PS )xn ‖ = 0 has a convergent subsequence. Now, we are in a position to give the strong convergence. Theorem 2.5. Let C be a nonempty closed convex subset of a Hilbert space H and P be the metric projection from H on C . Let Si : C → H be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with respect to P with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that Si satisfies the weakly inward condition for each 1 ≤ i ≤ N and that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions:
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β defined as (1.12), ∀n ≥ 1; ∑n L∞< 1, where L is ∑ ∞ n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. If one of {S1 , S2 , . . . , SN } is semicompact with respect to P, then {xn } converges weakly to some point in F .
(a) (b) (c) (d)
Proof. We may, without loss of generality, assume that S1 is semicompact with respect to P. From (2.13), we see that lim ‖xn − (PS1 )xn ‖ = 0.
(2.20)
n→∞
Since S1 is semicompact with respect to P, we see from (2.20) that there exists a subsequence {xni } of {xn } such that xni → x∗ ∈ C . Notice that
‖x∗ − (PSl )x∗ ‖ ≤ ‖x∗ − xni ‖ + ‖xni − (PSl )xni ‖. From (2.13), we obtain that x∗ ∈ F (PSl ) = F (Sl ) for each l ∈ {1, 2, . . . , N }, that is, x∗ ∈ F . For each p ∈ F , we have that limn→∞ ‖xn − p‖ exists. Supposing that p = x∗ , we see that limn→∞ ‖xn − x∗ ‖ exists. This implies that xn → x∗ as n → ∞. This completes the proof. For asymptotically pseudocontractive self-mappings, we have the following. Corollary 2.6. Let C be a nonempty closed convex subset of a Hilbert space H. Let Si : C → C be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: h(n)
xn = αn xn−1 + βn Si(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions:
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β max{Li : 1 ≤ i ≤ N }, ∀n ≥ 1; ∑n L∞< 1, where L =∑ ∞ n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. If one of {S1 , S2 , . . . , SN } is semicompact with respect to P, then {xn } converges weakly to some point in F . (a) (b) (c) (d)
Finally, we give a strong convergence criterion. Theorem 2.7. Let C be a nonempty closed convex subset of a Hilbert space H and P be the metric projection from H on C . Let Si : C → H be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with respect to P with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that Si satisfies the weakly inward condition for each 1 ≤ i ≤ N and that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions:
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X. Qin et al. / Nonlinear Analysis 74 (2011) 5851–5862
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β defined as (1.12), ∀n ≥ 1; ∑n L∞< 1, where L is ∑ ∞ n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. n=1 γn < ∞ and
Then {xn } converges strongly to some point in F if and only if lim infn→∞ d(xn , F ) = 0. Proof. The necessity is obvious. We only show the sufficiency. Assume that lim inf d(xn , F ) = 0.
(2.21)
n→∞
Notice that {xn } is bounded. Fixing p ∈ F , we see from Lemmas 1.1 and 1.2 that
‖xn − p‖2 = αn ‖xn−1 − p‖2 + βn ‖(PSi(n) )h(n) xn − p‖2 + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + ‖(PSi(n) )h(n) xn − xn ‖2 + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 2 + βn αn ‖xn−1 − (PSi(n) )h(n) xn ‖ + γn ‖un − (PSi(n) )h(n) xn ‖ + γn (‖un − xn ‖ + ‖xn − p‖)2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 2 + βn αn ‖xn−1 − (PSi(n) )h(n) xn ‖ + γn ‖un − (PSi(n) )h(n) xn ‖ + γn (2‖un − xn ‖2 + 2‖xn − p‖2 ) − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1) + 2γn ‖xn − p‖2 + 3γn M2 − αn βn (1 − αn )‖(PSi(n) )h(n) xn − xn−1 ‖2 ,
(2.22) h(n)
where M2 is an appropriate constant such that M2 ≥ max{supn≥1 {‖un − (PSi(n) ) xn ‖ }, supn≥1 2{‖un − xn ‖2 }, supn≥1 {2‖xn−1 − (PSi(n) )h(n) xn ‖ ‖un − (PSi(n) )h(n) xn ‖}}. From the restrictions (b) and (d), we see that there exists some N2 such that 2
βn (2kh(n) − 1) + 2γn ≤ R2 , ∀n ≥ N2 , where R2 = (1 − a) 1 + 4(1a−a) + 4a . It follows from (2.22) that αn + γn
‖x n − p ‖2 ≤
1 − βn (2kh(n) − 1)
≤
1+
≤
1+
‖xn−1 − p‖2 +
2βn (kh(n) − 1)
1 − R2 2(1 − a) 1 − R2
3M2 γn 1 − βn (2kh(n) − 1)
‖xn−1 − p‖2 +
3M2 1 − R2
(kh(n) − 1) ‖xn−1 − p‖2 +
γn
3M2 1 − R2
γn ,
∀n ≥ N 2 .
(2.23)
Therefore, we see that, for any p ∈ F ,
d(xn , F ) ≤ 2
1+
2(1 − a) 1 − R2
(kh(n) − 1) d(xn−1 , F )2 +
3M2 1 − R2
γn ,
∀n ≥ N2 .
(2.24)
In view of the restriction (d), we obtain from Lemma 1.3 that limn→∞ d(xn , F ) exists. This implies from (2.21) that lim d(xn , F ) = 0.
(2.25)
n→∞
Next, we show that {xn } is a Cauchy sequence in C . Put λn = 1−R2 γn . For any positive integers m, n, where m > n > N2 , 2 we see from (2.23) that 3M
‖x m − p ‖2 ≤ R 3 ‖x n − p ‖2 + R 3
∞ − i=n+1
λi + λm ,
X. Qin et al. / Nonlinear Analysis 74 (2011) 5851–5862
∑
∞ 2(1−a) n=1 1−R2 (kh(n)
5861
− 1) . It follows that ∞ − ‖xm − p‖ ≤ R3 ‖xn − p‖ + R3 λi + λm .
where R3 = exp
i=n+1
This in turn implies that
‖xn − xm ‖ ≤ ‖xn − p‖ + ‖xm − p‖ ≤ (1 +
∞ − R3 )‖xn − p‖ + R3 λi + λm . i=n+1
Taking the infimum over all p ∈ F , we see that
‖xn − xm ‖ ≤ (1 +
∞ − R3 )d(xn , F ) + R3 λi + λm . i=n+1
In view of (2.25), we see from n=1 λn < ∞ that {xn } is a Cauchy sequence in C and so {xn } converges strongly to some q¯ ∈ C . Since F = ∩Ni=1 F (Si ) = ∩Ni=1 F (PSi ) is closed, we obtain from (2.25) that q¯ ∈ F . This completes the proof.
∑∞
For asymptotically pseudocontractive self-mappings, we have the following. Corollary 2.8. Let C be a nonempty closed convex subset of a Hilbert space H. Let Si : C → C be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: h(n)
xn = αn xn−1 + βn Si(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions: (a) (b) (c) (d)
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β max{Li : 1 ≤ i ≤ N }, for each n ≥ 1; ∑n L∞< 1, where L =∑ ∞ n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }.
Then {xn } converges strongly to some point in F if and only if lim infn→∞ d(xn , F ) = 0. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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Convergence of an implicit iterative process for asymptotically pseudocontractive nonself-mappings Xiaolong Qin a,b , Sun Young Cho c , Tianze Wang a , Shin Min Kang d,∗ a
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
b
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
c
Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea
d
Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
article
abstract
info
Article history: Received 19 April 2010 Accepted 11 April 2011 Communicated by Ravi Agarwal
In this work, an implicit iterative process is considered for asymptotically pseudocontractive nonself-mappings. Weak and strong convergence theorems for common fixed points of a family of asymptotically pseudocontractive nonself-mappings are established in the framework of Hilbert spaces. © 2011 Elsevier Ltd. All rights reserved.
MSC: 47H09 47H10 47J25 Keywords: Asymptotically pseudocontractive nonself-mapping Asymptotically nonexpansive nonself-mapping Fixed point Implicit iterative process
1. Introduction and preliminaries Throughout this work, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by
⟨·, ·⟩ and ‖ · ‖, respectively. Let C be a nonempty closed convex subset of H and T : C → C be a mapping. In this work, we denote the fixed point set of T by F (T ). Recall that T is said to be nonexpansive if
‖Tx − Ty‖ ≤ ‖x − y‖,
∀x, y ∈ C .
T is said to be asymptotically nonexpansive if there exists a sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ such that
‖T n x − T n y‖ ≤ kn ‖x − y‖,
∀x, y ∈ C , n ≥ 1.
(1.1)
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. It is known that if C is a nonempty bounded closed convex subset of a Hilbert space H, then every asymptotically nonexpansive self-mapping has
∗
Corresponding author. Fax: +82 557551917. E-mail addresses: [email protected], [email protected] (X. Qin), [email protected] (S.Y. Cho), [email protected] (T. Wang), [email protected] (S.M. Kang). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.04.031
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a fixed point. Since 1972, a host of authors have studied weak and strong convergence problems of the iterative processes for such a class of mappings. T is said to be strictly pseudocontractive if there exists a constant κ ∈ [0, 1) such that
‖Tx − Ty‖2 ≤ ‖x − y‖2 + κ‖(I − T )x − (I − T )y‖2 , ∀x, y ∈ C . For such a case, T is also said to be a κ -strict pseudocontraction. The class of strict pseudocontractions is introduced by Browder and Petryshyn [2] in 1967. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction. In [3], Marino and Xu established the demiclosed principle for the class of strict pseudocontractions; see also [4]. T is said to be an asymptotically strict pseudocontraction if there exist a sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ and a constant κ ∈ [0, 1) such that
‖T n x − T n y‖2 ≤ kn ‖x − y‖2 + κ‖(I − T n )x − (I − T n )y‖2 , ∀x, y ∈ C , n ≥ 1. (1.2) For such a case, T is also said to be an asymptotically κ -strict pseudocontraction. The class of asymptotically strict pseudocontractions was introduced by Liu [5] in 1996. It is clear that every asymptotically nonexpansive mapping is an asymptotical 0-strict pseudocontraction. In [6], Kim and Xu established the demiclosed principle for the class of asymptotically strict pseudocontractions; see also [7]. T is said to be pseudocontractive if
⟨Tx − Ty, x − y⟩ ≤ ‖x − y‖2 ,
∀x, y ∈ C .
(1.3)
It is easy to see that (1.3) is equivalent to
‖Tx − Ty‖2 ≤ ‖x − y‖2 + ‖(I − T )x − (I − T )y‖2 ,
∀ x, y ∈ C .
T is said to be asymptotically pseudocontractive if there exists a sequence kn ⊂ [1, ∞) with kn → 1 as n → ∞ such that
⟨T n x − T n y, x − y⟩ ≤ kn ‖x − y‖2 ,
∀x, y ∈ C , n ≥ 1.
(1.4)
The class of asymptotically pseudocontractive mapping was introduced by Schu [8]. It is easy to see that (1.4) is equivalent to
‖T n x − T n y‖2 ≤ (2kn − 1)‖x − y‖2 + ‖(I − T n )x − (I − T n )y‖2 ,
∀x, y ∈ C , n ≥ 1.
In [9], Rhoades gave the following example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings. Example R. For x ∈ [0, 1], define a mapping T : [0, 1] → [0, 1] by
2
Tx = 1 − x 3
23
.
Then T is asymptotically pseudocontractive but it is not asymptotically nonexpansive. T is said to be uniformly L-Lipschitz if there exists a positive constant L such that
‖T n x − T n y‖ ≤ L‖x − y‖,
∀x, y ∈ C , ∀n ≥ 1.
In 2001, Xu and Ori [10] introduced, in the framework of Hilbert spaces, the following implicit iteration process for a finite family of nonexpansive mappings {T1 , T2 , . . . , TN } with {αn } a real sequence in (0, 1) and an initial point x0 ∈ C : x1 = α1 x0 + (1 − α1 )T1 x1 , x2 = α2 x1 + (1 − α2 )T2 x2 ,
··· xN = αN xN −1 + (1 − αN )TN xN , xN +1 = αN +1 xN + (1 − αN +1 )T1 xN +1 ,
··· which can written in the following compact form: xn = αn xn−1 + (1 − αn )Tn xn ,
∀n ≥ 1,
(1.5)
where Tn = Tn(mod N ) (here mod N takes values in {1, 2, . . . , N }). They obtained the following weak convergence theorem. Theorem XO. Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and {Ti }Ni=1 : C → C be a finite family of nonexpansive mappings such that F = ∩Ni=1 F (Ti ) ̸= ∅. Let {xn } be defined by (1.5). If {αn } is chosen such that αn → 0 as n → ∞, then {xn } converges weakly to a common fixed point of the family of {Ti }Ni=1 . Subsequently, fixed point problems based on implicit iterative processes have been considered by many authors. In 2004, Osilike [11] reconsidered the implicit iterative process (1.5) for a finite family of strictly pseudocontractive mappings. To be more precise, he proved the following theorem.
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Theorem O. Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let {Ti }Ni=1 be N strictly pseudocontractive self-maps of C such that F = ∩ri=1 F (Ti ) ̸= ∅. Suppose that x0 ∈ C and let {αn } be a sequence in (0, 1) such that αn → 0 as n → ∞. Then the sequence {xn } defined by (1.5) converges weakly to a common fixed point of the mappings {Ti }Ni=1 . In 2003, Sun [12] introduced the following implicit iterative process for a finite family of asymptotically nonexpansive mappings {T1 , T2 , . . . , TN }, with {αn } a real sequence in (0, 1) and an initial point x0 ∈ C : x1 = α1 x0 + (1 − α1 )T1 x1 , x2 = α2 x1 + (1 − α2 )T2 x2 ,
.. . xN = αN xN −1 + (1 − αN )TN xN , xN +1 = αN +1 xN + (1 − αN +1 )T12 xN +1 ,
.. . x2N = α2N x2N −1 + (1 − α2N )TN2 x2N , x2N +1 = α2N +1 x2N + (1 − α2N +1 )T13 x2N +1 ,
.. .. For each n ≥ 1, we can write n = (h − 1)N + i, where i = i(n) ∈ {1, 2, . . . , N }, h = h(n) ≥ 1 is a positive integer and h(n) → ∞ as n → ∞. Hence the above table can be rewritten in the following compact form: h(n)
xn = αn xn−1 + (1 − αn )Ti(n) xn ,
∀n ≥ 1.
(1.6)
Strong convergence theorems for the implicit iterative process (1.6) for a finite family of asymptotically nonexpansive mappings were established. In 2004, Osilike and Akuchu [13] reconsidered the implicit iterative process (1.6) for a family of asymptotically pseudocontractive mappings. They also obtained a strong convergence theorem. Since there is no demiclosed principle, there is no weak convergence theorem for the implicit iterative process (1.6). Notice that, from the viewpoint of computation, the implicit iterative schemes (1.5) and (1.6) are often impractical. For each step, we must solve nonlinear operator equations exactly. Therefore, one of the interesting and important problems in the theory of implicit iterative algorithms is considering the iterative algorithm with errors. This is an efficient iterative algorithm for computing approximately fixed points of nonlinear mappings. Recently, Chang et al. [14] considered the following implicit iterative process for a finite family of asymptotically nonexpansive mappings {T1 , T2 , . . . , TN }, with {αn } a real sequence in (0, 1), {un } a bounded sequence in C and an initial point x0 ∈ C : x1 = α1 x0 + (1 − α1 )T1 x1 + u1 , x2 = α2 x1 + (1 − α2 )T2 x2 + u2 ,
.. . xN = αN xN −1 + (1 − αN )TN xN + uN , xN +1 = αN +1 xN + (1 − αN +1 )T1n xN +1 + uN +1 ,
.. . x2N = α2N x2N −1 + (1 − α2N )TN2 x2N + u2N , x2N +1 = α2N +1 x2N + (1 − α2N +1 )T13 x2N +1 + u2N +1 ,
.. .. For each n ≥ 1, we can write n = (h − 1)N + i, where i = i(n) ∈ {1, 2, . . . , N }, h = h(n) ≥ 1 is a positive integer and h(n) → ∞ as n → ∞. Hence the above table can be rewritten in the following compact form: h(n)
xn = αn xn−1 + (1 − αn )Ti(n) xn + un ,
∀n ≥ 1 .
(1.7)
Weak and strong convergence theorems for the implicit iterative scheme (1.7) were obtained for a finite family of asymptotically nonexpansive mappings {T1 , T2 , . . . , TN }; see [14] for more details. Recently, fixed point problems for nonself-mappings have been studied by a number of authors; see, for example [15–23]. Next, we draw attention to nonself-mappings. Let S : C → H be a mapping and P be the metric projection from H onto C .
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Recall that S is said to be asymptotically nonexpansive with respect to P if there exists a sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ such that
‖(PS )n x − (PS )n y‖ ≤ kn ‖x − y‖,
∀x, y ∈ C , ∀n ≥ 1.
(1.8)
S is said to be an asymptotically strict pseudocontraction with respect to P if there exist a sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ and a constant κ ∈ [0, 1) such that
‖(PS )n x − (PS )n y‖2 ≤ kn ‖x − y‖2 + κ‖(I − (PS )n )x − (I − (PS )n )y‖2 ,
∀x, y ∈ C , n ≥ 1.
(1.9)
For such a case, S is also said to be an asymptotically κ -strict pseudocontraction with respect to P. It is clear that every asymptotically nonexpansive mapping with respect to P is an asymptotical 0-strict pseudocontraction with respect to P. S is said to be asymptotically pseudocontractive with respect to P if there exists a sequence kn ⊂ [1, ∞) with kn → 1 as n → ∞ such that
⟨(PS )n x − (PS )n y, x − y⟩ ≤ kn ‖x − y‖2 ,
∀ x, y ∈ C .
(1.10)
It is easy to see that (1.10) is equivalent to
‖(PS )n x − (PS )n y‖2 ≤ (2kn − 1)‖x − y‖2 + ‖(I − (PS )n )x − (I − (PS )n )y‖2 ,
∀ x, y ∈ C .
S is said to be uniformly L-Lipschitz with respect to P if there exists a positive constant L such that
‖(PS )n x − (PS )n y‖ ≤ L‖x − y‖,
∀x, y ∈ C , ∀n ≥ 1.
In this work, motivated by the above results, we introduced the following implicit iterative process for a finite family of Lipschitz asymptotically pseudocontractive nonself-mappings {S1 , S2 , . . . , SN } with respect to P: x1 = α1 x0 + β1 (PS1 )x1 + γ1 u1 , x2 = α2 x1 + β2 (PS2 )x2 + γ2 u2 ,
.. . xN = αN xN −1 + βN (PSN )xN + γN uN , xN +1 = αN +1 xN + βN +1 (PS1 )2 xN +1 + γN +1 uN +1 ,
.. . x2N = α2N x2N −1 + β2N (PSN )2 x2N + γ2N u2N , x2N +1 = α2N +1 x2N + β2N +1 (PS1 )3 x2N +1 + γ2N +1 u2N +1 ,
.. ., where x0 is the initial value, {un } is a bounded sequence in C and {αn }, {βn } and {γn } are sequences in [0, 1] such that αn +βn +γn = 1 for each n ≥ 1. For each n ≥ 1, we can write n = (h − 1)N + i, where i = i(n) ∈ {1, 2, . . . , N }, h = h(n) ≥ 1 is a positive integer and h(n) → ∞ as n → ∞. Hence the above table can be rewritten in the following compact form: xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
∀n ≥ 1.
(1.11)
Indeed, the implicit iterative sequence (1.11) can be employed for approximating common fixed points of a finite family of Lipschitz asymptotically pseudocontractive nonself-mappings with respect to P which can be obtained from the following. Define mappings Rn : C → C by Rn (x) = αn xn−1 + βn (PSi(n) )h(n) x + γn un ,
∀n ≥ 1.
Suppose that L = max{Li : 1 ≤ i ≤ N },
(1.12)
where Li is the Lipschitz constant of the mapping Si with respect to P. It follows that
‖Rn (x) − Rn (y)‖ = αn xn−1 + βn (PSi(n) )h(n) x + γn un − αn xn−1 + βn (PSi(n) )h(n) y + γn un = βn ‖(PSi(n) )h(n) x − (PSi(n) )h(n) y‖ ≤ βn L‖x − y‖, ∀x, y ∈ C . If βn L < 1 for each n ≥ 1, then we see that Rn is a contraction for each fixed n ≥ 1. By the Banach contraction principle, we see that there exists a unique fixed point xn ∈ C such that xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
∀n ≥ 1.
This shows that the implicit iterative process (1.11) is well defined.
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The purpose of this work is to study the weak and strong convergence of the implicit iteration process (1.11) for a finite family of asymptotically pseudocontractive nonself-mappings with respect to P. To prove our main results, we also need the following concepts and lemmas. Recall that a space X is said to satisfy Opial’s condition [24] if, for each sequence {xn } in X , the convergence xn → x weakly implies that lim inf ‖xn − x‖ < lim inf ‖xn − y‖, n→∞
n→∞
∀y ∈ X (y ̸= x).
Let C be a nonempty convex subset of a Hilbert space H. Then for x ∈ C , we define the inward set IC (x) as follows [21]: IC (x) := {y ∈ H : y = x + λ(u − x) : u ∈ C , λ ≥ 0}. A mapping S : C → H is said to satisfy the inward condition if Sx ∈ IC (x) for all x ∈ C . S is also said to satisfy the weakly inward condition if for each x ∈ C , then Sx ∈ cl[IC (x)], where cl[IC (x)] is the closure of IC (x). Clearly C ⊂ IC (x) and it is not hard to show that IC (x) is a convex set as C is. The following lemma has been proved by Xu and Yin [15]. For the sake of completeness, we give the proof here. Lemma 1.1. Let C be a nonempty closed convex subset of a real Hilbert space H, P be the metric projection from H onto C and S : C → H be a mapping satisfying the weakly inward condition. Then F (PS ) = F (S ). Proof. First, we show that F (PS ) ⊂ F (S ). Supposing that x ∈ F (PS ), we see that
⟨Sx − x, y − x⟩ ≤ 0,
∀y ∈ C .
(1.13)
In view of the weakly inward condition, we see that Sx ∈ cl[IC (x)]. Hence, there exist zn ∈ C and λn ≥ 0 such that yn → Sx, where yn = x + λn (un − x). It follows from (1.13) that ⟨Sx − x, yn − x⟩ ≤ 0. This implies that Sx = x, that is, x ∈ F (S ). This leads us to conclude that F (PS ) ⊂ F (S ). Next, we show that F (S ) ⊂ F (PS ). Supposing that x ∈ F (S ), we see that x = Sx = PSx, this is, x ∈ F (PS ). This completes F (S ) ⊂ F (PS ). This proof is completed. The following lemma is trivial. Lemma 1.2. Let (X , ⟨·, ·⟩) be a real inner product space. There holds the following equality:
‖α x + β y + γ z ‖2 = α‖x‖2 + β‖y‖2 + γ ‖z ‖2 − αβ‖x − y‖2 − αγ ‖x − z ‖2 − βγ ‖y − z ‖2 , for all x, y, z ∈ X , where α, β and γ are real numbers in [0, 1] such that α + β + γ = 1. Lemma 1.3 ([25]). Let {rn }, {sn } and {tn } be three nonnegative sequences satisfying the following condition: rn+1 ≤ (1 + sn )rn + tn , If
∑∞
n=1 sn
< ∞ and
∑∞
n =1 t n
for all n ∈ N .
< ∞, then limn→∞ rn exists.
2. The main results Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H and P be the metric projection from H onto C . Let Si : C → H be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with respect to P with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that Si satisfies the weakly inward condition for each 1 ≤ i ≤ N and that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions: (a) (b) (c) (d)
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β defined as (1.12), ∀n ≥ 1; ∑n L∞< 1, where L is ∑ ∞ n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }.
Then {xn } converges weakly to some point in F .
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Proof. Fixing p ∈ F , we see from Lemmas 1.1 and 1.2 that
‖xn − p‖2 = ‖αn (xn−1 − p) + βn (PSi(n) )h(n) xn − p + γn (un − p)‖2 = αn ‖xn−1 − p‖2 + βn ‖(PSi(n) )h(n) xn − p‖2 + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + ‖(PSi(n) )h(n) xn − xn ‖2 + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + βn αn ‖xn−1 − (PSi(n) )h(n) xn ‖ + γn ‖un − (PSi(n) )h(n) xn ‖ + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + 3γn M1 − αn βn (1 − αn )‖(PSi(n) )h(n) xn − xn−1 ‖2 ,
(2.1)
where M1 is an appropriate constant such that M1 ≥ max{supn≥1 {‖un −(PSi(n) )h(n) xn ‖2 }, supn≥1 {‖un −p‖2 }, supn≥1 {2‖xn−1 − (PSi(n) )h(n) xn ‖ ‖un − (PSi(n) )h(n) xn ‖}}. From the restriction (b), we see that there exists some N1 such that
βn (2kh(n) − 1) ≤ R1 , ∀n ≥ N1 , where R1 = (1 − a) 1 + 3(1a−a) . It follows from (2.1) that ‖x n − p ‖2 ≤
αn + γn 1 − βn (2kh(n) − 1)
≤
1+
≤
1+
‖xn−1 − p‖2 +
2βn (kh(n) − 1) 1 − R1 2(1 − a) 1 − R1
3M1 γn 1 − βn (2kh(n) − 1)
‖xn−1 − p‖2 +
3M1 1 − R1
(kh(n) − 1) ‖xn−1 − p‖2 +
γn
3M1 1 − R1
γn ,
∀n ≥ N 1 .
(2.2)
In view of the restriction (d), we obtain from Lemma 1.3 that limn→∞ ‖xn − p‖2 exists. This in turn implies that {xn } is bounded. On the other hand, we see from (2.1) that
αn βn (1 − αn )‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + 3γn M1 − ‖xn − p‖2 ≤ αn (‖xn−1 − p‖2 − ‖xn − p‖2 ) + βn (2kh(n) − 2)‖xn − p‖2 + 3γn M1 . From the restriction (b), we see that βn is away from zero for each n ≥ 1. Therefore, we arrive at lim ‖(PSi(n) )h(n) xn − xn−1 ‖ = 0.
n→∞
(2.3)
Notice that
‖xn − xn−1 ‖ ≤ βn ‖(PSi(n) )h(n) xn − xn−1 ‖ + γn ‖un − xn−1 ‖. It follows from (2.3) and the restriction (d) that lim ‖xn − xn−1 ‖ = 0.
(2.4)
n→∞
On the other hand, we have
‖xn−1 − (PSi(n) )h(n) xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn ‖ + ‖(PSi(n) )h(n) xn − (PSi(n) )h(n) xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn ‖ + L‖xn − xn−1 ‖, where L is defined in (1.12). In view of (2.3) and (2.4), we obtain that lim ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ = 0.
n→∞∞
(2.5)
For any positive integer n > N, we can write n = (h(n) − 1)N + i(n), where i(n) ∈ {1, 2, . . . , N }. Observe that
‖xn−1 − (PSn )xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + ‖(PSi(n) )h(n) xn−1 − (PSn )xn−1 ‖ = ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + ‖(PSi(n) )h(n) xn−1 − (PSi(n) )xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + L‖(PSi(n) )h(n)−1 xn−1 − xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + L ‖(PSi(n) )h(n)−1 xn−1 − (PSi(n−N ) )h(n)−1 xn−N ‖ + ‖(PSi(n−N ) )h(n)−1 xn−N − x(n−N )−1 ‖ + ‖x(n−N )−1 − xn−1 ‖ .
(2.6)
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Since for each n > N , n = (n − N )(mod N ), on the other hand, we obtain from n = (h(n) − 1)N + i(n) that n − N = (h(n) − 1) − 1 N + i(n) = (h(n − N ) − 1)N + i(n − N ). That is, h(n − N ) = h(n) − 1
i(n − N ) = i(n).
and
Notice that
‖(PSi(n) )h(n)−1 xn−1 − (PSi(n−N ) )h(n)−1 xn−N ‖ = ‖(PSi(n) )h(n)−1 xn−1 − (PSi(n) )h(n)−1 xn−N ‖ ≤ L‖xn−1 − xn−N ‖
(2.7)
and
‖(PSi(n−N ) )h(n)−1 xn−N − x(n−N )−1 ‖ = ‖(PSi(n−N ) )h(n−N ) xn−N − x(n−N )−1 ‖.
(2.8)
Substituting (2.7) and (2.8) into (2.6), we arrive at
‖xn−1 − (PSn )xn−1 ‖ ≤ ‖xn−1 − (PSi(n) )h(n) xn−1 ‖ + L L‖xn − xn−N ‖ + ‖(PSi(n−N ) )h(n−N ) xn−N − x(n−N )−1 ‖ + ‖x(n−N )−1 − xn−1 ‖ .
(2.9)
In view of (2.3)–(2.5), we obtain that lim ‖xn−1 − (PSn )xn−1 ‖ = 0.
(2.10)
n→∞
Notice that
‖xn − (PSn )xn ‖ ≤ ‖xn − xn−1 ‖ + ‖xn−1 − (PSn )xn−1 ‖ + ‖(PSn )xn−1 − (PSn )xn ‖ ≤ (1 + L)‖xn − xn−1 ‖ + ‖xn−1 − (PSn )xn−1 ‖.
(2.11)
From (2.4) and (2.10), we arrive at lim ‖xn − (PSn )xn ‖ = 0.
(2.12)
n→∞
On the other hand, we have
‖xn − (PSn+j )xn ‖ ≤ ‖xn − xn+j ‖ + ‖xn+j − (PSn+j )xn+j ‖ + ‖(PSn+j )xn+j − (PSn+j )xn ‖ ≤ (1 + L)‖xn − xn+j ‖ + ‖xn+j − (PSn+j )xn+j ‖, ∀j ∈ {1, 2, . . . , N }. It follows from (2.4) and (2.12) that lim ‖xn − (PSn+j )xn ‖ = 0,
∀j ∈ {1, 2, . . . , N }.
n→∞
Note that any subsequence of a convergent number sequence converges to the same limit. It follows that lim ‖xn − (PSl )xn ‖ = 0,
n→∞
∀l ∈ {1, 2, . . . , N }.
(2.13)
Since H is a Hilbert space, we see from the boundedness of {xn } that there exists a subsequence {xni } ⊂ {xn } such that {xni } converges weakly to p1 ∈ C . In view of (2.13), we have lim ‖xni − (PSl )xni ‖ = 0,
n→∞
∀l ∈ {1, 2, . . . , N }.
(2.14)
1 Next, we show, for each l ∈ {1, 2, . . . , N }, that p1 ∈ F (PSl ). Put ρ ∈ 0, 1+ , where L is defined in (1.12) and define L m yρ,m,l = (1 − ρ)p1 + ρ(PSl ) p1 for arbitrary but fixed m ≥ 1. We first show that, for fixed m ≥ 1, xni − (PSl )m xni → 0 as ni → ∞. Indeed, since Sl is uniformly Ll -Lipschitz with respect to P, we have
‖xni − (PSl )m xni ‖ ≤ ‖xni − (PSl )xni ‖ + ‖(PSl )xni − (PSl )2 xni ‖ + · · · + ‖(PSl )m−1 xni − (PSl )m xni ‖ ≤ mLl ‖xni − (PSl )xni ‖. It follows from (2.14) that lim ‖xni − (PSl )m xni ‖ = 0.
(2.15)
ni →∞
On the other hand, we have
p1 − yρ,m , I − (PSl )m yρ,m,l
= p1 − xni , I − (PSl )m yρ,m,l + xni − yρ,m,l , I − (PSl )m yρ,m,l = p1 − xni , I − (PSl )m yρ,m,l + xni − yρ,m,l , I − (PSl )m yρ,m,l − I − (PSl )m xni + xni − yρ,m,l , I − (PSl )m xni ≤ p1 − xni , I − (PSl )m yρ,m,l + (km − 1)Q + xni − yρ,m,l , I − (PSl )m xni ,
(2.16)
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where Q is an appropriate constant such that Q ≥ sup{‖yρ,m,l − xni ‖2 }. Notice that xni ⇀ p1 and (2.15). From (2.16), we arrive at
⟨p1 − yρ,m,l , (I − (PSl )m )yρ,m ⟩ ≤ (km − 1)Q .
(2.17)
On the other hand, since Sl is uniformly Ll -Lipschitz with respect to P, we see from the definition of yρ,m that
⟨p1 − yρ,m,l , (I − (PSl )m )p1 − (I − (PSl )m )yρ,m,l ⟩ ≤ (1 + Ll )‖p1 − yρ,m,l ‖2 = (1 + Ll )ρ 2 ‖p1 − (PSl )m p1 ‖2 .
(2.18)
Combining (2.17) with (2.18), we see that
‖p1 − (PSl )m p1 ‖2 = ⟨p1 − (PSl )m p1 , p1 − (PSl )m p1 ⟩ 1
=
ρ
=
ρ
⟨p1 − yρ,m,l , p1 − (PSl )m p1 ⟩
1 1 p1 − yρ,m,l , I − (PSl )m p1 − I − (PSl )m yρ,m,l + p1 − yρ,m , I − (PSl )m yρ,m,l
(km − 1)Q ≤ (1 + Ll )ρ‖p1 − (PSl )m p1 ‖2 + , ρ
ρ
which yields that
ρ 1 − ρ(1 + Ll ) ‖p1 − (PSl )m p1 ‖ ≤ (km − 1)Q ,
∀m ≥ 1.
(2.19)
Supposing that m → ∞ in (2.19) yields that (PSl )m p1 → p1 and hence (PSl )m+1 p1 → (PSl )p1 as m → ∞. Since Sl is Ll -Lipschitz continuous with respect to P, we can obtain that p1 = (PSl )p1 . This implies that p ∈ ∩Nl=1 F (PSl ) = ∩Nl=1 F (Sl ). Next we show {xn } converges weakly to p1 . Suppose the contrary; then there exists some subsequence {xnj } of {xn } such that {xnj } converges weakly to p2 ∈ C , where p1 ̸= p2 . Similarly, we can show that p2 ∈ F (PSl ) = F (Sl ) for each l ∈ {1, 2, . . . , N }. Notice that we have proved that limn→∞ ‖xn − p‖ exists for each p ∈ ∩Nl=1 F (PSl ) = ∩Nl=1 F (Sl ). Assume that limn→∞ ‖xn − p1 ‖ = d where d is a nonnegative number. By virtue of Opial’s condition for H, we see that d = lim inf ‖xni − p1 ‖ < lim inf ‖xni − p2 ‖ ni →∞
ni →∞
= lim inf ‖xnj − p2 ‖ < lim inf ‖xnj − p1 ‖ = d. nj →∞
nj →∞
This is a contradiction. Hence p1 = p2 . This completes the proof.
If Si is an asymptotically pseudocontractive self-mapping, then P is reduced to the identity mapping. The following theorem is not hard to derive. Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let Si : C → C be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: h(n)
xn = αn xn−1 + βn Si(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions: (a) αn + βn + γn = 1, ∀n ≥ 1; (b) there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; (c) β max{Li : 1 ≤ i ≤ N }, ∀n ≥ 1; ∑n L∞< 1, where L =∑ ∞ (d) n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. Then {xn } converges weakly to some point in F . Remark 2.3. Corollary 2.2 can be viewed as an improvement of Theorem O in Section 1. Note that the class of asymptotically pseudocontractive self-mappings contains properly the class of asymptotically nonexpansive self-mappings as a subclass. As some applications of Corollary 2.2, we have the following results. Corollary 2.4. Let C be a nonempty closed convex subset of a Hilbert space H. Let Si : C → C be an asymptotically nonexpansive mapping with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: h(n)
xn = αn xn−1 + βn Si(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions:
X. Qin et al. / Nonlinear Analysis 74 (2011) 5851–5862
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αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β supn≥1 {kn } < 1, ∑ ∀n ≥ 1; ∑n ∞ ∞ γ < ∞ and n n =1 n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. Then {xn } converges weakly to some point in F . (a) (b) (c) (d)
Next, we give a strong convergence theorem with the help of compactness. Recall that mapping S : C → H is semicompact with respect to P if any sequence {xn } in C satisfying limn→∞ ‖xn − (PS )xn ‖ = 0 has a convergent subsequence. Now, we are in a position to give the strong convergence. Theorem 2.5. Let C be a nonempty closed convex subset of a Hilbert space H and P be the metric projection from H on C . Let Si : C → H be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with respect to P with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that Si satisfies the weakly inward condition for each 1 ≤ i ≤ N and that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions:
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β defined as (1.12), ∀n ≥ 1; ∑n L∞< 1, where L is ∑ ∞ n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. If one of {S1 , S2 , . . . , SN } is semicompact with respect to P, then {xn } converges weakly to some point in F .
(a) (b) (c) (d)
Proof. We may, without loss of generality, assume that S1 is semicompact with respect to P. From (2.13), we see that lim ‖xn − (PS1 )xn ‖ = 0.
(2.20)
n→∞
Since S1 is semicompact with respect to P, we see from (2.20) that there exists a subsequence {xni } of {xn } such that xni → x∗ ∈ C . Notice that
‖x∗ − (PSl )x∗ ‖ ≤ ‖x∗ − xni ‖ + ‖xni − (PSl )xni ‖. From (2.13), we obtain that x∗ ∈ F (PSl ) = F (Sl ) for each l ∈ {1, 2, . . . , N }, that is, x∗ ∈ F . For each p ∈ F , we have that limn→∞ ‖xn − p‖ exists. Supposing that p = x∗ , we see that limn→∞ ‖xn − x∗ ‖ exists. This implies that xn → x∗ as n → ∞. This completes the proof. For asymptotically pseudocontractive self-mappings, we have the following. Corollary 2.6. Let C be a nonempty closed convex subset of a Hilbert space H. Let Si : C → C be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: h(n)
xn = αn xn−1 + βn Si(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions:
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β max{Li : 1 ≤ i ≤ N }, ∀n ≥ 1; ∑n L∞< 1, where L =∑ ∞ n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. If one of {S1 , S2 , . . . , SN } is semicompact with respect to P, then {xn } converges weakly to some point in F . (a) (b) (c) (d)
Finally, we give a strong convergence criterion. Theorem 2.7. Let C be a nonempty closed convex subset of a Hilbert space H and P be the metric projection from H on C . Let Si : C → H be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with respect to P with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that Si satisfies the weakly inward condition for each 1 ≤ i ≤ N and that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: xn = αn xn−1 + βn (PSi(n) )h(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions:
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X. Qin et al. / Nonlinear Analysis 74 (2011) 5851–5862
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β defined as (1.12), ∀n ≥ 1; ∑n L∞< 1, where L is ∑ ∞ n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }. n=1 γn < ∞ and
Then {xn } converges strongly to some point in F if and only if lim infn→∞ d(xn , F ) = 0. Proof. The necessity is obvious. We only show the sufficiency. Assume that lim inf d(xn , F ) = 0.
(2.21)
n→∞
Notice that {xn } is bounded. Fixing p ∈ F , we see from Lemmas 1.1 and 1.2 that
‖xn − p‖2 = αn ‖xn−1 − p‖2 + βn ‖(PSi(n) )h(n) xn − p‖2 + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 + ‖(PSi(n) )h(n) xn − xn ‖2 + γn ‖un − p‖2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 2 + βn αn ‖xn−1 − (PSi(n) )h(n) xn ‖ + γn ‖un − (PSi(n) )h(n) xn ‖ + γn (‖un − xn ‖ + ‖xn − p‖)2 − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1)‖xn − p‖2 2 + βn αn ‖xn−1 − (PSi(n) )h(n) xn ‖ + γn ‖un − (PSi(n) )h(n) xn ‖ + γn (2‖un − xn ‖2 + 2‖xn − p‖2 ) − αn βn ‖(PSi(n) )h(n) xn − xn−1 ‖2 ≤ αn ‖xn−1 − p‖2 + βn (2kh(n) − 1) + 2γn ‖xn − p‖2 + 3γn M2 − αn βn (1 − αn )‖(PSi(n) )h(n) xn − xn−1 ‖2 ,
(2.22) h(n)
where M2 is an appropriate constant such that M2 ≥ max{supn≥1 {‖un − (PSi(n) ) xn ‖ }, supn≥1 2{‖un − xn ‖2 }, supn≥1 {2‖xn−1 − (PSi(n) )h(n) xn ‖ ‖un − (PSi(n) )h(n) xn ‖}}. From the restrictions (b) and (d), we see that there exists some N2 such that 2
βn (2kh(n) − 1) + 2γn ≤ R2 , ∀n ≥ N2 , where R2 = (1 − a) 1 + 4(1a−a) + 4a . It follows from (2.22) that αn + γn
‖x n − p ‖2 ≤
1 − βn (2kh(n) − 1)
≤
1+
≤
1+
‖xn−1 − p‖2 +
2βn (kh(n) − 1)
1 − R2 2(1 − a) 1 − R2
3M2 γn 1 − βn (2kh(n) − 1)
‖xn−1 − p‖2 +
3M2 1 − R2
(kh(n) − 1) ‖xn−1 − p‖2 +
γn
3M2 1 − R2
γn ,
∀n ≥ N 2 .
(2.23)
Therefore, we see that, for any p ∈ F ,
d(xn , F ) ≤ 2
1+
2(1 − a) 1 − R2
(kh(n) − 1) d(xn−1 , F )2 +
3M2 1 − R2
γn ,
∀n ≥ N2 .
(2.24)
In view of the restriction (d), we obtain from Lemma 1.3 that limn→∞ d(xn , F ) exists. This implies from (2.21) that lim d(xn , F ) = 0.
(2.25)
n→∞
Next, we show that {xn } is a Cauchy sequence in C . Put λn = 1−R2 γn . For any positive integers m, n, where m > n > N2 , 2 we see from (2.23) that 3M
‖x m − p ‖2 ≤ R 3 ‖x n − p ‖2 + R 3
∞ − i=n+1
λi + λm ,
X. Qin et al. / Nonlinear Analysis 74 (2011) 5851–5862
∑
∞ 2(1−a) n=1 1−R2 (kh(n)
5861
− 1) . It follows that ∞ − ‖xm − p‖ ≤ R3 ‖xn − p‖ + R3 λi + λm .
where R3 = exp
i=n+1
This in turn implies that
‖xn − xm ‖ ≤ ‖xn − p‖ + ‖xm − p‖ ≤ (1 +
∞ − R3 )‖xn − p‖ + R3 λi + λm . i=n+1
Taking the infimum over all p ∈ F , we see that
‖xn − xm ‖ ≤ (1 +
∞ − R3 )d(xn , F ) + R3 λi + λm . i=n+1
In view of (2.25), we see from n=1 λn < ∞ that {xn } is a Cauchy sequence in C and so {xn } converges strongly to some q¯ ∈ C . Since F = ∩Ni=1 F (Si ) = ∩Ni=1 F (PSi ) is closed, we obtain from (2.25) that q¯ ∈ F . This completes the proof.
∑∞
For asymptotically pseudocontractive self-mappings, we have the following. Corollary 2.8. Let C be a nonempty closed convex subset of a Hilbert space H. Let Si : C → C be an Li -Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {kn,i } ⊂ [1, ∞) such that kn,i → 1 as n → ∞ for each 1 ≤ i ≤ N, where N ≥ 1 is some positive integer. Assume that the common fixed point set F = ∩Ni=1 F (Si ) is nonempty. Let {un } be a bounded sequence in C and suppose that x0 ∈ C . Let {xn }∞ n=0 be a sequence generated by the following implicit iterative process: h(n)
xn = αn xn−1 + βn Si(n) xn + γn un ,
n ≥ 1.
Assume that the control sequences {αn }, {βn } and {γn } in [0, 1] satisfy the following restrictions: (a) (b) (c) (d)
αn + βn + γn = 1, ∀n ≥ 1; there exist constants a, b ∈ (0, 1) such that a ≤ αn ≤ b, ∀n ≥ 1; β max{Li : 1 ≤ i ≤ N }, for each n ≥ 1; ∑n L∞< 1, where L =∑ ∞ n=1 γn < ∞ and n=1 (kn − 1) < ∞, where kn = max{kn,i : 1 ≤ i ≤ N }.
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