Viscoresolvent algorithms for monotone operators and nonexpansive mappings
Authors
Peize Li
 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.
Shin Min Kang
 Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660701, Korea.
LiJun Zhu
 School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China.
Abstract
Two new type of viscoresolvent algorithms for finding a zero of the sum of two monotone operators and a
fixed point of a nonexpansive mapping in a Hilbert space are investigated. The algorithms consist of the
zeros and the fixed points of the considered problems in which one operator is replaced with its resolvent
and a viscosity term is added. Strong convergence of the algorithms are shown. As special cases, we can
approach to the minimum norm common element of the zero of the sum of two monotone operators and the
fixed point of a nonexpansive mapping without using the metric projection. Some applications are included.
Share and Cite
ISRP Style
Peize Li, Shin Min Kang, LiJun Zhu, Viscoresolvent algorithms for monotone operators and nonexpansive mappings, Journal of Nonlinear Sciences and Applications, 7 (2014), no. 5, 325344
AMA Style
Li Peize, Kang Shin Min, Zhu LiJun, Viscoresolvent algorithms for monotone operators and nonexpansive mappings. J. Nonlinear Sci. Appl. (2014); 7(5):325344
Chicago/Turabian Style
Li, Peize, Kang, Shin Min, Zhu, LiJun. "Viscoresolvent algorithms for monotone operators and nonexpansive mappings." Journal of Nonlinear Sciences and Applications, 7, no. 5 (2014): 325344
Keywords
 Monotone operator
 nonexpansive mapping
 zero point
 fixed point
 resolvent.
MSC
References

[1]
K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, On a strongly nonexpansive sequence in Hilbert spaces, J. Nonlinear Convex Anal., 8 (2007), 471489.

[2]
H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review, 38 (1996), 367426.

[3]
H. H. Bauschke, P. L. Combettes, A Dykstralike algorithm for two monotone operators, Pacific J. Optim., 4 (2008), 383391.

[4]
H. H. Bauschke, P. L. Combettes, S. Reich , The asymptotic behavior of the composition of two resolvents , Nonlinear Anal., 60 (2005), 283301.

[5]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123145.

[6]
Y. Censor, S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, New York, USA (1997)

[7]
P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475504.

[8]
P. L. Combettes, S. A. Hirstoaga , Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117136.

[9]
P. L. Combettes, S. A. Hirstoaga , Approximating curves for nonexpansive and monotone operators, J. Convex Anal., 13 (2006), 633646.

[10]
P. L. Combettes, S. A. Hirstoaga, Viscopenalization of the sum of two monotone operators, Nonlinear Anal., 69 (2008), 579591.

[11]
F. Ding, T. Chen , Iterative least squares solutions of coupled Sylvester matrix equations, Systems Control Lett., 54 (2005), 95107.

[12]
J. Eckstein, D. P. Bertsekas, On the DouglasRachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293318.

[13]
Y. P. Fang, N. J. Huang, HMonotone operator resolvent operator technique for quasivariational inclusions, Appl. Math. Comput., 145 (2003), 795803.

[14]
K. Geobel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press (1990)

[15]
L. J. Lin, Variational inclusions problems with applications to Ekeland's variational principle, fixed point and optimization problems , J. Global Optim., 39 (2007), 509527.

[16]
P.L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964979.

[17]
X. Liu, Y. Cui, Common minimalnorm fixed point of a finite family of nonexpansive mappings, Nonlinear Anal., 73 (2010), 7683.

[18]
X. Lu, H. K. Xu, X. Yin , Hybrid methods for a class of monotone variational inequalities, Nonlinear Anal., 71 (2009), 10321041.

[19]
A. Moudafi, On the regularization of the sum of two maximal monotone operators, Nonlinear Anal., 42 (2009), 12031208.

[20]
G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383390.

[21]
J. W. Peng, Y. Wang, D. S. Shyu, J. C. Yao, Common solutions of an iterative scheme for variational inclusions, equilibrium problems and fixed point problems, J. Inequal. Appl., Article ID 720371, 2008 (2008), 15 pages.

[22]
S. M. Robinson, Generalized equation and their solutions, part I, basic theory, Math Program. Study, 10 (1979), 128141.

[23]
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209216.

[24]
R. T. Rockafellar , Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877898

[25]
A. Sabharwal, L. C. Potter, Convexly constrained linear inverse problems: iterative leastsquares and regularization, IEEE Trans. Signal Process, 46 (1998), 23452352.

[26]
T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 103123.

[27]
S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), 2741.

[28]
W. Takahashi, M. Toyoda , Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417428.

[29]
H. K. Xu, Iterative algorithms for nonlinear operators , J. London Math. Soc., 2 (2002), 117.

[30]
Y. Yao, R. Chen, H. K. Xu , Schemes for finding minimumnorm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 34473456.

[31]
Y. Yao, Y. C. Liou , Composite algorithms for minimization over the solutions of equilibrium problems and fixed point problems , Abstr. Appl. Anal., Article ID 763506, 2010 (2010), 19 pages.

[32]
S. S. Zhang, H. W. Lee Joseph, C. K. Chan, Algorithms of common solutions for quasi variational inclusion and fixed point problems , Appl. Math. Mech. (English Ed.), 29 (2008), 571581.