Parallel iterative methods for a finite family of sequences of nearly nonexpansive mappings in Hilbert spaces
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Parallel iterative methods for a finite family of sequences of nearly nonexpansive mappings in Hilbert spaces Truong Minh Tuyen1 · Nguyen Song Ha1
Received: 13 March 2017 / Accepted: 18 August 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this paper, we propose some new parallel iterative methods by the hybrid and shrinking projection to find a common fixed point of a finite family of sequences of nearly nonexpansive mappings in Hilbert spaces when the domain C satisfies diam(C) < ∞. We also give some applications of our main results for the problem of finding a common fixed point of nonexpansive mappings, nonexpansive semi-groups, the problem of finding a common zero point of monotone operators, the system of generalized mixed equilibrium problems and the system of variational inequalities without the condition diam(C) < ∞. Three numerical examples also are given to illustrate the effectiveness of the proposed algorithms. Keywords Sequences of nearly nonexpansive mapping · Common fixed point · Hybrid projection · Shrinking projection Mathematics Subject Classification 47H06 · 47H09 · 47J25 · 47J40 · 90C25
1 Introduction Many problems arising in different areas of mathematics and practical ones can be recast in terms of a fixed point problem for nonexpansive mappings. For instance, if the nonexpansive mappings are projections onto some nonempty closed and convex sets, then the fixed point problem becomes the famous convex feasibility problem. In order to find a common fixed point of a finite family of nonexpansive mappings in Hilbert spaces, Bauschke (1996) extended the result of Wittmann (1992) and he proved the following theorem.
Communicated by Nair Abreu.
B 1
Truong Minh Tuyen [email protected] Department of Mathematics and Informatics, Thainguyen University of Sciences, Thainguyen, Vietnam
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T. M. Tuyen, N. S. Ha
Theorem 1.1 (see Bauschke 1996) Let C be a nonempty closed convex subset of a real Hilbert space H . Let Ti , i = 1, 2, . . . , N be nonexpansive mappings from C into itself N such that S = i=1 F(Ti ) = ∅, S = F(TN TN −1 . . . T1 ) = F(T1 TN . . . T2 ) = · · · = F(TN −1 . . . T1 TN ). Let {αn } be a sequence in [0, 1) which satisfies the following conditions lim αn = 0,
n→∞
∞ n=0
αn = ∞,
∞
|αn+1 − αn | < ∞.
n=0
For arbitrary y, x0 ∈ C, let {xn } be a sequence in C generated by xn+1 = αn y + (1 − αn )Tn+1 xn , n ≥ 0, where Tn = T[n from C onto S.
mod N ] .
(1.1)
Then {xn } converges strongly to PS u, with PS is metric projection
The result of Bauschke has been studied and extended by many authors, for instance, Chang (2006); Chang et al. (2007), Ceng et al. (2007), Chidume and Ali (2008), Chidume et al. (2005), O’Hara et al. (2003), Jung (2005), Takahashi and Shimoji (2000), Kim and Tuyen (2011, 2016) and references therein. In 2012, inspired by Aoyama et al. (2007), Ceng et al. (2011), Sahu et al. (2011, 2012) introduce strong convergence theorem to find a solution of the variational inequality over the set of common fixed p
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