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A hybrid iteration for asymptotically strictly pseudocontractive mappings
Journal of Inequalities and Applications volume 2014, Article number: 374 (2014)
Abstract
In this paper, we propose a new hybrid iteration for a finite family of asymptotically strictly pseudocontractive mappings. We also prove that such a sequence converges strongly to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings. Results in the paper extend and improve recent results in the literature.
MSC:47H09, 47H10.
1 Introduction
Let H be a real Hilbert space, C be a nonempty closed convex subset of H. A mapping T:C\to C is called Lipschitz or Lipschitz continuous if there exists L>0 such that
If L=1, then T is called nonexpansive, and if L<1, then T is called a contraction. It follows from (1.1) that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz.
A mapping T:C\to C is said to be a λstrictly pseudocontractive mapping in the sense of BrowderPetryshyn [1] if there exists a constant 0\le \lambda <1 such that
If \lambda =1, then T is said to be a pseudocontractive mapping, i.e.,
The class of strictly pseudocontractive mappings falls into the one between the class of nonexpansive mappings and that of pseudocontractive mappings. The class of strict pseudocontractive mappings has more powerful applications than nonexpansive mappings, see Scherzer [2].
A mapping T:C\to C is said to be an asymptotically λstrictly pseudocontractive mapping [3] if there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {lim}_{n\to \mathrm{\infty}}{k}_{n}=1, and a constant \lambda \in [0,1) such that
We now give an example to show that a λstrictly asymptotically pseudocontractive mapping is not necessarily a λstrictly pseudocontractive mapping.
Example 1.1 Consider H={\ell}_{2}=\{\overline{x}={\{{x}_{i}\}}_{i=1}^{\mathrm{\infty}}:{x}_{i}\in C,{\sum}_{i=1}^{\mathrm{\infty}}{{x}_{i}}^{2}<\mathrm{\infty}\}, and let \overline{B}=\{\overline{x}\in {\ell}_{2}:\parallel x\parallel \le 1\}. It is clear that {\ell}_{2} is a normed linear space with respect to the norm
Define T:\overline{B}\to {\ell}_{2} by the rule
where {\{{a}_{i}\}}_{i=1}^{\mathrm{\infty}} is a real sequence satisfying {a}_{2}>0, 0<{a}_{i}<1, i\ne 2 and {\prod}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}. By definition
and
Hence,
for all \lambda \in (0,1), n\ge 2. Since {lim}_{n\to \mathrm{\infty}}2{\prod}_{i=1}^{n}{a}_{i}=1, it follows that T is an asymptotically strictly pseudocontractive mapping.
We now show that the T is not a λstrictly pseudocontractive mapping. Choose \overline{x}=(\frac{1}{6},\frac{1}{6},\frac{1}{6},0,0,\dots ,0), \overline{y}=(0,0,0,\dots ,0) and {a}_{2}=2. Then
Hence T is not a λstrictly pseudocontractive mapping.
The study on iterative methods for strict pseudocontractive mappings was initiated by Browder and Petryshyn [1] in 1967, but the iterative methods for strict pseudocontractive mappings are far less developed than those for nonexpansive mappings. The probable reason is the second term appearing on the righthand side of (1.2), which impedes the convergence analysis. Therefore it is interesting to develop the iteration methods for strict pseudocontractive mappings.
Browder and Petryshyn [1] showed that if a λstrict pseudocontractive mapping T has a fixed point in C, then starting with an initial {x}_{0}\in C, the sequence \{{x}_{n}\} generated by the formula
where α is a constant such that \lambda <\alpha <1, converges weakly to a fixed point of T.
Marino and Xu [4] extended the above mentioned result of Browder and Petryshyn [1] by considering the sequence \{{x}_{n}\} generated by the following formula:
where \{{\alpha}_{n}\} is a sequence in (0,1). Iteration (1.5) is called the Mann iteration [5].
Another interesting problem is to find a common fixed point of a finite family of strict pseudocontractive mappings. One approach to study the problem is cyclic algorithm, in which sequence \{{x}_{n}\} is generated cyclically by
where {T}_{[n]}={T}_{i} with i=nmodN, 0\le i\le N1.
However, the convergence of both algorithms (1.5) and (1.6) can only be weak in an infinite dimensional space. So, in order to have strong convergence, one must modify these algorithms.
One such modification of Mann’s algorithm for nonexpansive mappings is given by Nakajo and Takahashi [6], in which a modified algorithm is obtained by applying additional projections onto the intersection of two halfspaces and is guaranteed to have strong convergence. The sequence \{{x}_{n}\} is produced as follows:
here {P}_{C} is the metric projection of H onto C and {C}_{n}, {Q}_{n} are given by
where
and
Marino and Xu [4] proposed the following modification for strict pseudocontractive mappings in which the sequence \{{x}_{n}\} is given by the same formula (1.7) with {C}_{n} given by
where {y}_{n} and {Q}_{n} are given by formulas (1.9) and (1.10), respectively.
Thakur [7] extended the idea of Marino and Xu [4] to asymptotically strict pseudocontractive mappings. Recently, Yao and Chen [8] proposed a new hybrid method for strict pseudocontractive mappings, in which \{{x}_{n}\}, {C}_{n}, {Q}_{n} are given by the same formulas (1.7), (1.8), (1.10) respectively, and
where \delta \in (\lambda ,1), 0\le \lambda <1.
Takahashi et al. [9] introduced the idea of shrinking projection method for nonexpansive mappings, in which projection is applied on a single set. Here the sequence \{{x}_{n}\} is produced by the formula
where {C}_{n} is given by
and {y}_{n} is given by the same formula (1.9).
Inchan and Nammanee [10] modified the shrinking projection method for asymptotically strict pseudocontractive mappings, in which the sequence \{{x}_{n}\} is generated by the same formula (1.12) and
where
and
Motivated and inspired by the studies going on in this direction, we now propose the modified shrinking projection method for a finite family of asymptotically λstrict pseudocontractive mappings.
Let C be a bounded closed convex subset of a Hilbert space H and {\{{T}_{i}\}}_{i=1}^{N}:C\to C be a finite family of asymptotically ({\lambda}_{i},{k}_{n}^{(i)})strict pseudocontractive mappings with Lipschitz constant {L}_{n}^{(i)}\ge 1, i=1,2,\dots ,N, and for all n\in \mathbb{N} such that \mathcal{F}={\bigcap}_{i=1}^{N}F({T}_{i})\ne \mathrm{\varnothing}. Set \lambda =max\{{\lambda}_{i}\} and {k}_{n}=max\{{k}_{n}^{(i)}\}, i=1,2,\dots ,N. For arbitrarily chosen {x}_{0}\in C, let {C}_{1}=C and {x}_{1}={P}_{{C}_{1}}{x}_{0}, define a sequence \{{x}_{n}\} as
where \delta \in (\lambda ,1) is some constant and
also for each n\ge 1, it can be written as n=(h(n)1)N+i(n), where i(n)\in \{1,2,\dots ,N\} and h(n)\ge 1 is a positive integer with h(n)\to \mathrm{\infty} as n\to \mathrm{\infty}.
We shall prove that the iteration generated by (1.14) converges strongly to {z}_{0}={P}_{\mathcal{F}}{x}_{0}.
2 Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the next section.
We will use the following notation:

1.
⇀ for weak convergence and → for strong convergence.

2.
{\omega}_{w}({x}_{n})=\{x:\mathrm{\exists}{x}_{{n}_{j}}\rightharpoonup x\} denotes the weak ωlimit set of \{{x}_{n}\}.

3.
Fix(T) the set of fixed points of T.
Lemma 2.1 ([4])
The following identities hold in a Hilbert space H:

(i)
{\parallel x+y\parallel}^{2}={\parallel x\parallel}^{2}+{\parallel y\parallel}^{2}+2\u3008x,y\u3009 \mathrm{\forall}x,y\in H;

(ii)
{\parallel \alpha x+(1\alpha )y\parallel}^{2}=\alpha {\parallel x\parallel}^{2}+(1\alpha ){\parallel y\parallel}^{2}\alpha (1\alpha ){\parallel xy\parallel}^{2} \mathrm{\forall}\alpha \in [0,1].
Lemma 2.2 ([11])
Assume that C is a closed and convex subset of a Hilbert space H, and let T:C\to C be an asymptotically λstrict pseudocontraction with Fix(T)\ne \mathrm{\varnothing}. Then:

(i)
For each n\ge 1, {T}^{n} satisfies the Lipschitz condition
\parallel {T}^{n}x{T}^{n}y\parallel \le {L}_{n}\parallel xy\parallelfor all x,y\in C, where {L}_{n}=\frac{\lambda +\sqrt{1+({k}_{n}^{2}1)(1\lambda )}}{1\lambda}.

(ii)
If \{{x}_{n}\} is a sequence in C with the properties {x}_{n}\rightharpoonup z and T{x}_{n}{x}_{n}\to 0, then (IT)z=0, i.e., IT is demiclosed at 0.

(iii)
The fixed point set Fix(T) of T is closed and convex so that the projection {P}_{Fix(T)} is well defined.
Lemma 2.3 ([12])
Let H be a real Hilbert space. Given a closed convex subset C\subset H and points x,y,z\in H. Given also a real number a. The set
is convex (and closed).
Lemma 2.4 Let C be a closed convex subset of a real Hilbert space H. Given x\in H and z\in C. Then z={P}_{K}x if and only if there holds the relation
where {P}_{K} is the nearest point projection from H onto C.
3 Main results
In this section, we prove a strong convergence theorem by the hybrid method for a finite family of asymptotically {\lambda}_{i}strictly pseudocontractive mappings in Hilbert spaces.
Theorem 3.1 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H. Let {\{{T}_{i}\}}_{i=1}^{N} be a finite family of asymptotically ({\lambda}_{i},{k}_{n}^{(i)})strictly pseudocontractive mappings of C into itself for some 0\le {\lambda}_{i}<1 with Lipschitz constant {L}_{n}^{(i)}\ge 1, i=1,2,\dots ,N and for all n\in \mathbb{N} such that \mathcal{F}={\bigcap}_{i=1}^{N}F({T}_{i})\ne \mathrm{\varnothing} and let {x}_{0}\in C. For {C}_{1}=C and {x}_{1}={P}_{{C}_{1}}{x}_{0}, assume that the control sequence {\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}} is chosen such that {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1. Then \{{x}_{n}\} generated by (1.14) converges strongly to {z}_{0}={P}_{\mathcal{F}}{x}_{0}.
Proof Suppose L=max\{{L}_{n}^{(i)}:1\le i\le N,n\in \mathbb{N}\}, {k}_{n}=max\{{k}_{n}^{(i)}:1\le i\le N\} and \lambda =max\{{\lambda}_{i}:1\le i\le N\}. By Lemma 2.3, set {C}_{n} is closed and convex.
Now, for every n\in \mathbb{N}, we prove that \mathcal{F}\subset {C}_{n} and \{{x}_{n}\} is well defined.
We use the method of mathematical induction. For any z\in \mathcal{F}, we have z\in C={C}_{1}. Hence \mathcal{F}\subset {C}_{1}. Now assume that \mathcal{F}\subset {C}_{k} for some k\in \mathbb{N}. Then, for any p\in \mathcal{F}\subset {C}_{k}, we have
It follows that p\in {C}_{k+1} and \mathcal{F}\subset {C}_{k+1}. Hence \mathcal{F}\subset {C}_{n} for all n\in \mathbb{N}. Since {C}_{n} is closed and convex for all n\in \mathbb{N}, this implies that \{{x}_{n}\} is well defined.
Now, we prove that \{{x}_{n}\} is bounded.
Since {x}_{n}={P}_{{C}_{n}}{x}_{0}, then by Lemma 2.4 we have
As \mathcal{F}\subset {C}_{n}, we have
So, for q\in \mathcal{F}, we have
This implies that
Hence \{{x}_{n}\} is bounded.
From {x}_{n}={P}_{{C}_{n}}{x}_{0} and {x}_{n+1}={P}_{{C}_{n+1}}{x}_{0}\in {C}_{n+1}\subset {C}_{n}, by Lemma 2.4 we have
So, for {x}_{n+1}\in {C}_{n}, we have, for n\in \mathbb{N},
This implies that
Hence {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}_{0}\parallel exists.
Next, we show that \parallel {x}_{n}{x}_{n+1}\parallel exists.
Using (3.1), we have
Since {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}_{0}\parallel exists, it follows that
Hence \{{x}_{n}\} is a Cauchy sequence, and so convergent.
Consequently,
Since {x}_{n+1}\in {C}_{n}, we have
By the definition of {y}_{n}, we have
Since {x}_{n+1}\in {C}_{n}, by (1.14) we have
which implies that
Using (3.3) and (3.4), we have
Since {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1, it follows from (3.5) that
Now, we prove that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{T}_{n}{x}_{n}\parallel =0.
Since, for any positive integer n\ge N, it can be written as n=(h(n)1)N+i(n) where i(n)\in \{1,2,\dots ,N\}, observe that
Since, for each n>N, i(n)=(nN)modN. Again since n=(h(n)1)N+i(n), we have
We observe that
and
Substituting (3.8), (3.9) in (3.7), we obtain
It follows from (3.2), (3.6) and (3.10) that
We also have
which gives that
For each i\in \{1,2,\dots ,N\}, by Lemma 2.2(ii), I{T}_{i} is demiclosed at zero. This together with the fact that \{{x}_{n}\} is bounded guarantees that every weak limit point of \{{x}_{n}\} is a fixed point of {T}_{i} (i\in \{1,2,\dots ,N\}). That is {\omega}_{w}({x}_{n})\subset \mathcal{F}={\bigcap}_{i=1}^{N}F({T}_{i}). Since for {z}_{0}={P}_{\mathcal{F}}({x}_{0}) we have \parallel {x}_{n}{x}_{0}\parallel \le \parallel {z}_{0}{x}_{0}\parallel for all n\ge 0, by the weak lower semicontinuity of the norm, we have
for all w\in {\omega}_{w}({x}_{n}). However, since {\omega}_{w}({x}_{n})\subset \mathcal{F}, we must have w={z}_{0} for all w\in {\omega}_{w}({x}_{n}). Thus {\omega}_{w}({x}_{n})=\{{z}_{0}\} and then {x}_{n}\rightharpoonup {z}_{0}. Hence, {x}_{n}\to {z}_{0}={P}_{\mathcal{F}}({x}_{0}) by
This completes the proof. □
If we take {\beta}_{n}=(1\delta ){\alpha}_{n}+\delta \in [\delta ,1) in (1.14), then we obtain the following.
Corollary 3.2 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H. Let {\{{T}_{i}\}}_{i=1}^{N} be a finite family of asymptotically {\lambda}_{i}strictly pseudocontractive mappings of C into itself for some 0\le {\lambda}_{i}<1 with Lipschitz constant {L}_{n}^{(i)}\ge 1, i=1,2,\dots ,N and for all n\in \mathbb{N} such that \mathcal{F}={\bigcap}_{i=1}^{N}F({T}_{i})\ne \mathrm{\varnothing} and let {x}_{0}\in C. For {C}_{1}=C and {x}_{1}={P}_{{C}_{1}}{x}_{0}, assume that the control sequence {\{{\beta}_{n}\}}_{n=1}^{\mathrm{\infty}} is chosen such that \delta \le {\beta}_{n}<1, define \{{x}_{n}\} as follows:
where
Then \{{x}_{n}\} generated by (3.11) converges strongly to {z}_{0}={P}_{\mathcal{F}}{x}_{0}.
Corollary 3.3 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H. Let T be an asymptotically λstrictly pseudocontractive mapping of C into itself such that F(T)\ne \mathrm{\varnothing}, and let {x}_{0}\in C. For {C}_{1}=C and {x}_{1}={P}_{{C}_{1}}{x}_{0}, assume that the control sequence {\{{\beta}_{n}\}}_{n=1}^{\mathrm{\infty}} is chosen such that \delta \le {\beta}_{n}<1, define \{{x}_{n}\} as follows:
where
Then \{{x}_{n}\} generated by (3.12) converges strongly to {z}_{0}={P}_{F(T)}{x}_{0}.
Since asymptotically nonexpansive mappings are asymptotically 0strict pseudocontractions, we have the following consequence.
Corollary 3.4 Let H be a real Hilbert space, and let C be a nonempty bounded closed convex subset of H. Let T be an asymptotically nonexpansive mapping from C to itself such that F(T)\ne \mathrm{\varnothing}, and let {x}_{0}\in C. For {C}_{1}=C and {x}_{1}={P}_{{C}_{1}}{x}_{0}, assume that the control sequence {\{{\beta}_{n}\}}_{n=1}^{\mathrm{\infty}} is chosen such that \delta \le {\beta}_{n}<1. Then \{{x}_{n}\} generated by (3.12) converges strongly to {z}_{0}={P}_{F(T)}{x}_{0}.
Remark 3.5 From the main results, one can see that the corresponding results in Acedo and Xu [13], Inchan and Nammanee [10], Kim and Xu [11], Marino and Xu [4], MartinezYanez and Xu [12], Nakajo and Takahashi [6], Qin et al. [14], Yao and Chen [8] are all special cases of this paper.
References
Browder FE, Pertryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197228. 10.1016/0022247X(67)900856
Scherzer O: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 1991, 194: 911933.
Qihou L: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal. 1996, 26: 18351842. 10.1016/0362546X(94)00351H
Marino G, Xu HK: Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336349. 10.1016/j.jmaa.2006.06.055
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506510. 10.1090/S00029939195300548463
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279: 372379. 10.1016/S0022247X(02)004584
Thakur BS: Convergence of strictly asymptotically pseudocontractions. Thai J. Math. 2007, 5: 4152.
Yao Y, Chen R: Strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Appl. Math. Comput. 2010, 32: 6982. 10.1007/s121900090233x
Takahashi W, Takeuchi Y, Kubota R: Strong convergence by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2008, 341: 276286. 10.1016/j.jmaa.2007.09.062
Inchan I, Nammanee K: Strong convergence theorems by hybrid method for asymptotically k strict pseudocontractive mapping in Hilbert space. Nonlinear Anal. Hybrid Syst. 2009, 3: 380385. 10.1016/j.nahs.2009.02.002
Kim TH, Xu HK: Convergence of the modified Mann’s iteration method for asymptotically strict pseudocontractions. Nonlinear Anal. 2008, 68: 28282836. 10.1016/j.na.2007.02.029
MartinezYanez C, Xu HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 2006, 64: 24002411. 10.1016/j.na.2005.08.018
Acedo GL, Xu HK: Iterative methods for strict pseudocontractions in Hilbert spaces. Nonlinear Anal. 2009, 70: 19021911. 10.1016/j.na.2008.02.090
Qin XL, Cho YJ, Kang SM, Shang M: A hybrid iterative scheme for asymptotically k strict pseudocontractions in Hilbert spaces. Nonlinear Anal. 2009, 70: 19021911. 10.1016/j.na.2008.02.090
Acknowledgements
We are grateful to the referee for precise remarks which led to improvement of the paper. The second author is thankful to University Grants Commission of India for Project 411390/2012(SR).
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Dewangan, R., Thakur, B.S. & Postolache, M. A hybrid iteration for asymptotically strictly pseudocontractive mappings. J Inequal Appl 2014, 374 (2014). https://doi.org/10.1186/1029242X2014374
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DOI: https://doi.org/10.1186/1029242X2014374
Keywords
 asymptotically strictly pseudocontractive mappings
 Mann iteration
 shrinking projection method
 CQiteration
 strong convergence