Abstract
In this paper, we introduce the concepts of the set of all the common attractive points (CAP(S, T)) and the set of all the common strongly attractive points (CSAP(S, T)) for the set-valued mappings S and T in a Hilbert space. Moreover, some fundamental properties related to the sets A(T), F(T) and CAP(S, T) are given. Furthermore, we generalize the Agarwal iteration for the case of two set-valued mappings and obtain a weak convergence theorem for two \((\alpha ,\beta )\)-generalized hybrid set-valued mappings in a Hilbert space. Moreover, an example of two \((\alpha ,\beta )\)-generalized hybrid set-valued mappings which have a common strongly attractive point is shown. Then we finish the work using the proposed algorithm to find a common element of the set of solutions of an equilibrium problem and the sets of fixed points of two \((\alpha ,\beta )\)-generalized hybrid set-valued mappings.
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Baillon, J.B.: Un th\(\acute{e}\)or\(\grave{e}\)me de type ergodique pour les contractions non lin\(\acute{e}\)ars dans un espaces de Hilbert. C. R. Acad. Sci. Paris Ser. A-B. 280, 1511–1541 (1975)
Reich, S.: Almost convergence and nonlinear ergodic theorems. J. Approx. Theory 24, 269–272 (1978)
Kocourek, P., Takahashi, W., Yao, J.C.: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwan. J. Math. 14, 2497–2511 (2010)
Takahashi, W., Takeuchi, Y.: Nonlinear ergodic theorem without convexity for generalized hybrid mappings in a Hilbert space. J. Nonlinear Convex Anal. 12, 399–406 (2011)
Hojo, M., Takahashi, W., Yao, J.C.: Weak and strong mean convergence theorems for super hybrid mappings in Hilbert spaces. Demonstr. Math. 38(1), 177–184 (2011)
Zheng, Y.: Attractive points and convergence theorems of generalized hybrid mapping. J. Nonlinear Sci. Appl. 8, 354–362 (2015)
Petruşel, A., Petruşel, G., Yao, J.C.: Variational analysis concepts in the theory of muti-valued coincidence problems. J. Nonlinear Convex Anal. 19, 935–958 (2018)
Petruşel, A., Petruşel, G.: On Reichs strict fixed point theorem for multi-valued operators in complete metric spaces. J. Nonlinear Var. Anal. 2, 103–112 (2018)
Altun, I., Sahin, H., Turkoglu, D.: Fixed point results for multivalued mappings of Feng-Liu type on M-metric spaces. J. Nonlinear Funct. Anal. 2018, 7 (2018)
Nguyen, L.V., Phuong, L.T., Hong, N.T., Qin, X.: Some fixed point theorems for multivalued mappings concerning F-contractions. J. Fixed Point Theory Appl. 20, 139 (2018)
Alsulami, S.M., Latif, A., Takahashi, W.: The split common fixed point problem and strong convergence theorems by hybrid methods for new demimetric mappings in Hilbert spaces. Appl. Anal. Optim. 2, 11–26 (2018)
Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)
Berinde, V.: A convergence theorem for some mean value fixed point iterations procedures. Demonstr. Math. 38(1), 177–184 (2005)
Chen, L., Gao, L., Zhao, Y.: A new iterative scheme for finding attractive points of \((\alpha,\beta )\)-generalized hybrid set-valued mappings. J. Nonlinear Sci. Appl. 10, 1228–1237 (2017)
Bruck, R.E., Reich, S.: Accretive operators, Banach limits, and dual ergodic theorems. Bull. Acad. Polon. Sci. 29, 585–589 (1981)
Kaewkhao, A., Inthakon, W., Kunwai, K.: Attractive points and convergence theorems for normally generalized hybrid mappings in CAT(0) spaces. Fixed Point Theory Appl. 2015, 96 (2015)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Takahashi, W.: Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 11, 79–88 (2010)
Reich, S.: Fixed points of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)
Reich, S.: Approximate selections, best approximations, fixed points, and invariant sets. J. Math. Anal. Appl. 62, 104–113 (1978)
Reich, S., Zaslavski, A.J.: Genericity in Nonlinear Analysis, vol. 34. Springer, New York (2014)
Reich, S.: Kannan’s fixed point theorem. Boll. Un. Mat. Ital. 4, 1–11 (1971)
Iemoto, S., Takahashi, W.: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 71, 2082–2089 (2009)
Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008)
Atsushiba, S., Takahashi, W.: Nolinear ergodic theorems without convexity for nonexpansive semigroups in Hilbert spaces. J. Nonlinear Convex Anal. 14(2), 209–219 (2013)
Khan, S.H.: Iterative approximation of common attractive points of further generalized hybrid mappings. Fixed Point Theory Appl. 8(1), 2018 (2018)
Takahashi, W., Yao, J.C.: Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces. Taiwan. J. Math. 15, 457–472 (2011)
Takahashi, W., Wong, N.C., Yao, J.C.: Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 13(4), 745–757 (2012)
Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118(2), 417–428 (2003)
Reich, S.: Nonlinear evolution equations and nonlinear ergodic theorems. Nonlinear Anal. 1, 319–330 (1977)
Agarwal, R.P., Regan, D.O., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61–79 (2007)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Acknowledgements
This work was supported by Training Program for Youth Innovation Talents of Heilongjiang Educational Committee under Grant UNPYSCT-2017078, Postdoctoral Science Foundation of Heilongjiang Province under Grant LBHQ18067 and Fundamental Research Foundation for Universities of Heilongjiang Province under Grant LGYC-2018JQ001.
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Chen, L., Zou, J., Zhao, Y. et al. Iterative approximation of common attractive points of \((\alpha ,\beta )\)-generalized hybrid set-valued mappings. J. Fixed Point Theory Appl. 21, 58 (2019). https://doi.org/10.1007/s11784-019-0692-0
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DOI: https://doi.org/10.1007/s11784-019-0692-0