Some Fixed Point Theorems for Noncompact and Weakly Asymptotically Regular Set-Valued Mappings

Abstract

Some fixed point theorems of Browder, Petryshyn, and Williamson from Browder and Petryshyn (Browder, F. E., Petryshyn, W. V. ([1966]). The solution by interation of nonlinear functional equations in Banach spaces. Bull. Amer. Math. Soc. 72: 571–575); Petryshyn and Williamson (Petryshyn, W. V., Williamson, T. E. ([1973]). Strong and weak convergence of the sequence of successive approximation for quasinonexpansive mappings. J. Math. Anal. Appl. 43:459–497.) for noncompact and weakly asymptotically regular single-valued operators of one variable are extended to set-valued mappings of two variables. Some existence results are obtained by an approach, which is different from Browder and Petryshyn (Browder, F. E., Petryshyn, W. V. ([1966]). The solution by interation of nonlinear functional equations in Banach spaces. Bull. Amer. Math. Soc. 72:571–575); Petryshyn and Williamson (Petryshyn, W. V., Williamson, T. E. ([1973]). Strong and weak convergence of the sequence of successive approximation for quasinonexpansive mappings. J. Math. Anal. Appl. 43:459–497.), namely, by using an implicit iteration , which seems to be more efficient than the usual iteration, in many cases (see Tran Quoc Binh and Nguyen Minh Chuong (Tran Quoc Binh., Nguyen Minh Chuong. ([1996]). On a fixed point theorem. Functional Anal. Appl. 30:220–221 (English Transl.)); Tran Quoc Binh and Nguyen Minh Chuong (Tran Quoc Binh., Nguyen Minh Chuong. ([1999]). On a fixed point theorem for nonexpansive nonlinear operators. Acta Math. Vietnamica 24(1):1–8); Tran Quoc Binh and Nguyen Minh Chuong (Tran Quoc Binh., Nguyen Minh Chuong. ([2001]). Approximation of nonlinear operator equations. Number. Funct. Anal. And Optim. 22(7&8):831–844); Nguyen Minh Chuong et al. (Nguyen Minh Chuong, Ya. D., Mamedov, Khuat Van Ninh. ([1992]). Approximate solutions of operator equations. Sci. and Techn. Publ. House, Hanoi); Nguyen Minh Chuong and Nguyen Xuan Thuan (Nguyen Minh Chuong., Nguyen Xuan Thuan. ([2001]). Random fixed point theorems for multivalued nonlienear mappings. Rand. Oper. and Stoch. Equa. 9(3):235–245). Moreover, the problems are studied in the weak topology using the Hausdorff metric H ρ , where ρ is the metric induced by the wea topology.