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.

Keywords:

• 1991 Mathematics Subject Classification: Primary 47H10, 47H09
• Secondary 60H25

Acknowledgments

The authors thank the referee and the Editor in Chief for their helpful suggestions, due to which the article is improved. This article was supported in part by the National Fundamental Research Program in Natural Sciences, Vietnam.