Strong convergence to common fixed points using Ishikawa and hybrid methods for mean-demiclosed mappings in Hilbert spaces
Abstract
In this paper, we establish a strong convergence theorem that approximates a common fixed point of two nonlinear mappings by comprehensively using an Ishikawa iterative method, a hybrid method, and a mean-valued iterative method. The shrinking projection method is also developed. The nonlinear mappings are a general type that includes nonexpansive mappings and other classes of well-known mappings. The two mappings are not assumed to be continuous or commutative. The main theorems in this paper generate a variety of strong convergence theorems including a type of “three-step iterative method”. An application to the variational inequality problem is also given.
Keyword : Ishikawa iteration, hybrid method, shrinking projection method, mean-valued iteration, mean-demiclosed mapping, common fixed point
How to Cite
Kondo, A. (2023). Strong convergence to common fixed points using Ishikawa and hybrid methods for mean-demiclosed mappings in Hilbert spaces. Mathematical Modelling and Analysis, 28(2), 285–307. https://doi.org/10.3846/mma.2023.15843
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Hojo, A. Kondo and W. Takahashi. Weak and strong convergence theorems for commutative normally 2-generalized hybrid mappings in Hilbert spaces. Linear Nonlinear Anal., 4(1):145–156, 2018.
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A. Kondo. Ishikawa type mean convergence theorems for finding common fixed points of nonlinear mappings in Hilbert spaces. Rend. Circ. Mat. Palermo, II. Ser, 2022. https://doi.org/10.1007/s12215-022-00742-x
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A. Kondo and W. Takahashi. Attractive point and weak convergence theorems for normally N-generalized hybrid mappings in Hilbert spaces. Linear Nonlinear Anal., 3(2):297–310, 2017.
A. Kondo and W. Takahashi. Strong convergence theorems of Halpern’s type for normally 2-generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal., 19(4):617–631, 2018.
A. Kondo and W. Takahashi. Strong convergence theorems for finding common attractive points of normally 2-generalized hybrid mappings and applications. Linear Nonlinear Anal., 6(3):421–438, 2020.
A. Kondo and W. Takahashi. Weak convergence theorems to common attractive points of normally 2-generalized hybrid mappings with errors. J. Nonlinear Convex Anal., 21(11):2549–2570, 2020.
W.R. Mann. Mean value methods in iteration. Proc. Amer. Math. Soc., 4(3):506– 510, 1953. https://doi.org/10.2307/2032162
T. Maruyama, W. Takahashi and M. Yao. Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal., 12(1):185–197, 2011.
K. Nakajo and W. Takahashi. Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279(2):372–378, 2003. https://doi.org/10.1016/S0022-247X(02)00458-4
M.A. Noor. New approximation schemes for general variational inequalities. J. Math. Anal. Appl., 251(1):217–229, 2000. https://doi.org/10.1006/jmaa.2000.7042
W. Phuengrattana and S. Suantai. On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math., 235(9):3006–3014, 2011. https://doi.org/10.1016/j.cam.2010.12.022
S. Reich. Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl., 67(2):274–276, 1979. https://doi.org/10.1016/0022-247X(79)90024-6
T. Shimizu and W. Takahashi. Strong convergence to common fixed points of families of nonexpansive mappings. J. Math. Anal. Appl., 211(1):71–83, 1997. https://doi.org/10.1006/jmaa.1997.5398
W. Takahashi. Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama, 2009.
W. Takahashi. Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal., 11(1):79–88, 2010. https://doi.org/10.1186/1687-1812-2013-116
W. Takahashi, Y. Takeuchi and R. Kubota. Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl., 341(1):276–286, 2008. https://doi.org/10.1016/j.jmaa.2007.09.062
W. Takahashi and M. Toyoda. Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl., 118(2):417–428, 2003. https://doi.org/10.1023/A:1025407607560
W. Takahashi, N.-C. Wong and J.-C. Yao. Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal., 13(4):745–757, 2012.
N.D. Truong, J.K. Kim and T.H.H. Anh. Hybrid inertial contraction projection methods extended to variational inequality problems. Nonlinear Funct. Anal. Appl., 27(1):203–221, 2022. https://doi.org/10.22771/nfaa.2022.27.01.13
I. Yamada. The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In D. Butnariu, Y. Censor and S. Reich(Eds.), Inherently parallel algorithms in feasibility and optimization and their applications, pp. 473–504, Amsterdam, Holland, 2001. North-Holland.
S. Atsushiba and W. Takahashi. Approximating common fixed points of two nonexpansive mappings in Banach spaces. Bull. Austral. Math. Soc., 57(1):117– 127, 1998. https://doi.org/10.1017/S0004972700031464
J.B. Baillon. Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert. C. R. Acad. Sci. Ser. A–B, 280:1511–1514, 1975.
F.E. Browder. Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA, 54(4):1041–1044, 1965.
R. Chugh, V. Kumar and S. Kumar. Strong convergence of a new three step iterative scheme in Banach spaces. American Journal of Computational Mathematics, 2(4):345–357, 2012. https://doi.org/10.4236/ajcm.2012.24048
M. Hojo, A. Kondo and W. Takahashi. Weak and strong convergence theorems for commutative normally 2-generalized hybrid mappings in Hilbert spaces. Linear Nonlinear Anal., 4(1):145–156, 2018.
S. Ishikawa. Fixed points by a new iteration method. Proc. Amer. Math. Soc., 44:147–150, 1974. https://doi.org/10.1090/S0002-9939-1974-0336469-5
S. Itoh and W. Takahashi. The common fixed point theory of singlevalued mappings and multivalued mappings. Pacific J. Math., 79(2):493–508, 1978.
P. Kocourek, W. Takahashi and J.-C. Yao. Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwanese J. Math., 14(6):2497–2511, 2010. https://doi.org/10.11650/twjm/1500406086
F. Kohsaka and W. Takahashi. Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math., 91(2):166–177, 2008. https://doi.org/10.1007/s00013-008-2545-8
A. Kondo. Convergence theorems using ishikawa iteration for finding common fixed points of demiclosed and 2-demiclosed mappings in Hilbert spaces. Adv. Oper. Theory, 7(3), 2022. https://doi.org/10.1007/s43036-022-00190-5
A. Kondo. Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces. Demonstr. Math., 55(1):752–759, 2022. https://doi.org/10.1515/dema-2022-0167
A. Kondo. Ishikawa type mean convergence theorems for finding common fixed points of nonlinear mappings in Hilbert spaces. Rend. Circ. Mat. Palermo, II. Ser, 2022. https://doi.org/10.1007/s12215-022-00742-x
A. Kondo. Mean convergence theorems using hybrid methods to find common fixed points of noncommutative nonlinear mappings in Hilbert spaces. J. Appl. Math. Comput., 68(1):217–248, 2022. https://doi.org/10.1007/s12190-021-01527-8
A. Kondo and W. Takahashi. Attractive point and weak convergence theorems for normally N-generalized hybrid mappings in Hilbert spaces. Linear Nonlinear Anal., 3(2):297–310, 2017.
A. Kondo and W. Takahashi. Strong convergence theorems of Halpern’s type for normally 2-generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal., 19(4):617–631, 2018.
A. Kondo and W. Takahashi. Strong convergence theorems for finding common attractive points of normally 2-generalized hybrid mappings and applications. Linear Nonlinear Anal., 6(3):421–438, 2020.
A. Kondo and W. Takahashi. Weak convergence theorems to common attractive points of normally 2-generalized hybrid mappings with errors. J. Nonlinear Convex Anal., 21(11):2549–2570, 2020.
W.R. Mann. Mean value methods in iteration. Proc. Amer. Math. Soc., 4(3):506– 510, 1953. https://doi.org/10.2307/2032162
T. Maruyama, W. Takahashi and M. Yao. Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal., 12(1):185–197, 2011.
K. Nakajo and W. Takahashi. Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279(2):372–378, 2003. https://doi.org/10.1016/S0022-247X(02)00458-4
M.A. Noor. New approximation schemes for general variational inequalities. J. Math. Anal. Appl., 251(1):217–229, 2000. https://doi.org/10.1006/jmaa.2000.7042
W. Phuengrattana and S. Suantai. On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math., 235(9):3006–3014, 2011. https://doi.org/10.1016/j.cam.2010.12.022
S. Reich. Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl., 67(2):274–276, 1979. https://doi.org/10.1016/0022-247X(79)90024-6
T. Shimizu and W. Takahashi. Strong convergence to common fixed points of families of nonexpansive mappings. J. Math. Anal. Appl., 211(1):71–83, 1997. https://doi.org/10.1006/jmaa.1997.5398
W. Takahashi. Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama, 2009.
W. Takahashi. Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal., 11(1):79–88, 2010. https://doi.org/10.1186/1687-1812-2013-116
W. Takahashi, Y. Takeuchi and R. Kubota. Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl., 341(1):276–286, 2008. https://doi.org/10.1016/j.jmaa.2007.09.062
W. Takahashi and M. Toyoda. Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl., 118(2):417–428, 2003. https://doi.org/10.1023/A:1025407607560
W. Takahashi, N.-C. Wong and J.-C. Yao. Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal., 13(4):745–757, 2012.
N.D. Truong, J.K. Kim and T.H.H. Anh. Hybrid inertial contraction projection methods extended to variational inequality problems. Nonlinear Funct. Anal. Appl., 27(1):203–221, 2022. https://doi.org/10.22771/nfaa.2022.27.01.13
I. Yamada. The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In D. Butnariu, Y. Censor and S. Reich(Eds.), Inherently parallel algorithms in feasibility and optimization and their applications, pp. 473–504, Amsterdam, Holland, 2001. North-Holland.