Covergence of hybrid algorithms for α-nonexpansive mappings in Hilbert spaces
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Abstract
In this paper, we introduce two hybrid algorithms and state their convergence theorems for α -nonexpansive mappings in Hilbert spaces. These results are generalizations of the main ones found in [2]. In addition, we provide illustrations for the obtained results.
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Keywords
hybrid algorithm, α-nonexpansive mapping, Hilbert space
References
[1]. K. Aoyama and F. Kohsaka (2011), “Fixed point theorem for α -nonexpansive mappings in Banach spaces”, Nonlinear Anal., (74), p. 4387-4391.
[2]. Q. Dong and Y. Lu (2015), “A new hybrid algorithm for a nonexpansive mapping”, Fixed Point Theory Appl., (2015:37), p. 1-7.
[3]. D. V. Hieu (2016), “An extension of hybrid method without extrapolation step to equilibrium problems”, J. Ind. Manag. Optim., p. 1-16, DOI:10.3934/jimo.2017015.
[4]. D. V. Hieu (2016), “Parallel extragradient-proximal methods for split equilibrium problems”, Math. Model. Anal., (21), p. 478-501.
[5]. D. V. Hieu (2017), “New subgradient extragradient methods for common solutions to equilibrium problems”, Comput. Optim. Appl., p. 1-24, DOI 10.1007/s10589-017-9899-4.
[6]. D. V. Hieu (2017), “Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings”, J. Appl. Math. Comput., (53), p. 531-554.
[7]. D. V. Hieu, P. K. Anh, and L. D. Muu (2017), “Modified hybrid projection methods for finding common solutions to variational inequality problems”, Comput. Optim. Appl., (66), p. 75-96.
[8]. D. V. Hieu, L. D. Muu, and P. K. Anh (2016), “Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings”, Numer. Algorithms., (73), p. 197-217.
[9]. F. Kohsaka and W. Takahashi (2008), “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces”, SIAM J. Optim., (19), p. 824-835.
[10]. D. Kong, L. Liu, and Y. Wu (2015), “Best proximity point theorems for α -nonexpansive mappings in Banach spaces”, Fixed Point Theory Appl., (2015:159), p. 1-10.
[11]. C. Matinez-Yanes and H. K. Xu (2006), “Strong convergence of the CQ method for fixed point processes”, Nonlinear Anal., (64), p. 2400-2411.
[12]. C. Mongkolkeha, Y. J. Cho, and P. Kumam (2014), “Weak convergence theorems of iterative sequences in Hilbert spaces”, J. Nonlinear Convex Anal., 15(6), p. 1303-1317.
[13]. K. Nakajo and W. Takahashi (2003), “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups”, J. Math. Anal. Appl., (279), p. 372-379.
[14]. S. Reich (1979), “Weak convergence theorems for nonexpansive mappings in Banach spaces”, J. Math. Anal. Appl., (67), p. 274-276.
[15]. W. Takahashi, Y. Takeuchi, and R. Kubota (2008), “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces”, J. Math. Anal. Appl., (341), p. 276-286.
[16]. W. Takahashi (2010), “Fixed point theorems for new nonlinear mappings in a Hilbert space”, J. Nonlinear Convex Anal., (11), p. 79-88.
[2]. Q. Dong and Y. Lu (2015), “A new hybrid algorithm for a nonexpansive mapping”, Fixed Point Theory Appl., (2015:37), p. 1-7.
[3]. D. V. Hieu (2016), “An extension of hybrid method without extrapolation step to equilibrium problems”, J. Ind. Manag. Optim., p. 1-16, DOI:10.3934/jimo.2017015.
[4]. D. V. Hieu (2016), “Parallel extragradient-proximal methods for split equilibrium problems”, Math. Model. Anal., (21), p. 478-501.
[5]. D. V. Hieu (2017), “New subgradient extragradient methods for common solutions to equilibrium problems”, Comput. Optim. Appl., p. 1-24, DOI 10.1007/s10589-017-9899-4.
[6]. D. V. Hieu (2017), “Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings”, J. Appl. Math. Comput., (53), p. 531-554.
[7]. D. V. Hieu, P. K. Anh, and L. D. Muu (2017), “Modified hybrid projection methods for finding common solutions to variational inequality problems”, Comput. Optim. Appl., (66), p. 75-96.
[8]. D. V. Hieu, L. D. Muu, and P. K. Anh (2016), “Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings”, Numer. Algorithms., (73), p. 197-217.
[9]. F. Kohsaka and W. Takahashi (2008), “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces”, SIAM J. Optim., (19), p. 824-835.
[10]. D. Kong, L. Liu, and Y. Wu (2015), “Best proximity point theorems for α -nonexpansive mappings in Banach spaces”, Fixed Point Theory Appl., (2015:159), p. 1-10.
[11]. C. Matinez-Yanes and H. K. Xu (2006), “Strong convergence of the CQ method for fixed point processes”, Nonlinear Anal., (64), p. 2400-2411.
[12]. C. Mongkolkeha, Y. J. Cho, and P. Kumam (2014), “Weak convergence theorems of iterative sequences in Hilbert spaces”, J. Nonlinear Convex Anal., 15(6), p. 1303-1317.
[13]. K. Nakajo and W. Takahashi (2003), “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups”, J. Math. Anal. Appl., (279), p. 372-379.
[14]. S. Reich (1979), “Weak convergence theorems for nonexpansive mappings in Banach spaces”, J. Math. Anal. Appl., (67), p. 274-276.
[15]. W. Takahashi, Y. Takeuchi, and R. Kubota (2008), “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces”, J. Math. Anal. Appl., (341), p. 276-286.
[16]. W. Takahashi (2010), “Fixed point theorems for new nonlinear mappings in a Hilbert space”, J. Nonlinear Convex Anal., (11), p. 79-88.