The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. (English) Zbl 1189.49011

Summary: We discuss the strong convergence of the viscosity approximation method, in Hilbert spaces, relatively to the computation of fixed points of operators in the wide class of quasi-nonexpansive mappings. Our convergence results improve previously known ones obtained for the class of nonexpansive mappings.


49J40 Variational inequalities
65J15 Numerical solutions to equations with nonlinear operators
65K15 Numerical methods for variational inequalities and related problems
47J25 Iterative procedures involving nonlinear operators
Full Text: DOI


[1] Halpern, B., Fixed points of nonexpanding maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[2] Lions, P.L., Approximation de points fixes de contractions, C. R. acad. sci. ser. A-B Paris, 284, 1357-1359, (1977) · Zbl 0349.47046
[3] Moudafi, A., Viscosity approximations methods for fixed point problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039
[4] Wittman, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036
[5] Xu, H.K., Viscosity approximations methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060
[6] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publishers Yokohama, Japan
[7] Bauschke, H.H.; Combettes, P.L., A weak-to-strong convergence principle for Fejér monotone methods in Hilbert space, Math. oper. res., 26, 248-264, (2001) · Zbl 1082.65058
[8] Byrne, C.L., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems, 18, 441-453, (2004)
[9] Yang, Q.; Zhao, J., Generalized KM theorems and their applications, Inverse problems, 22, 833-844, (2006) · Zbl 1117.65081
[10] Hicks, T.L.; Kubicek, J.D., On the Mann iteration process in Hilbert spaces, J. math. anal. appl., 59, 498-504, (1977) · Zbl 0361.65057
[11] Maruster, S., The solution by iteration of nonlinear equations in Hilbert spaces, Proc. amer. math. soc., 63, 1, 69-73, (1997) · Zbl 0355.47037
[12] C. Moore, Iterative aproximation of fixed points of demicontractive maps, The Abdus Salam. Intern. Centre for Theoretical Physics, Trieste, Italy, Scientific Report, IC /98/214, November, 1998
[13] Yamada, I.; Ogura, N., Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. funct. anal. optim., 25, 7-8, 619-655, (2004) · Zbl 1095.47049
[14] Itoh, S.; Takahashi, W., The common fixed point theory of singlevalued mappings and multivalued mappings, Pacific J. math., 79, 2, 493-508, (1978) · Zbl 0371.47042
[15] Yamada, I.; Ogura, N.; Shirakawa, N.; Nashed, Z.; Scherzer, O., A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems, Inverse problems, image analysis and medical imaging, Contemp. math., 313, 269-305, (2002) · Zbl 1039.47051
[16] Goebel, K.; Kirk, W.A., ()
[17] Browder, F.E.; Petryshyn, W.V., Construction of fixed points of nonlinear mappings in Hilbert spaces, J. math. anal. appl., 20, 197-228, (1967) · Zbl 0153.45701
[18] Marino, G.; Xu, H.K., Weak and strong convergence theorems for strict pseudo-contractions, J. math. anal. appl., 329, 336-346, (2007) · Zbl 1116.47053
[19] Maingé, P.E., Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-valued anal., 16, 7-8, 899-912, (2008) · Zbl 1156.90426
[20] Chidume, C.E; Chidume, C.O., Iterative approximation of fixed point of nonexpansive mappings, J. math. anal. appl., 318, 288-295, (2006) · Zbl 1095.47034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.