ACTA MATHEMATICA VIETNAMICA 299 Volume 29, Number 3, 2004, pp. 299-308 SOME FIXED POINT THEOREMS FOR MAPPINGS OF TWO VARIABLES TRAN QUOC BINH Abstract. The existence of fixed points x = T (x, x) and the convergence of implicit iteration x n = T (x n ,x n-1 ) to the fixed points for generalized nonex- pansive type mappings of two variables are investigated. 1. Introduction The aim of this paper is to study the existence of fixed points x = T (x, x) of mappings of two variables and to investigate the convergence of implicit iteration x n = T (x n ,x n−1 ) to fixed points of the mapping T (x, y). The implicit iteration x n = T (x n ,x n−1 ) was studied by Kurpel in [13]. It con- tains some iteration methods such as Picard and Seidel method as special cases. In [13], [17] and other works the implicit iteration was applied to nonlinear in- tegral equations, Volterra integral equations, differential equations of parabolic type, linear and nonlinear systems of integral equations, Cauchy problem, bound- ary value problems of linear itegro-differential equations, eigenvalue problems etc. In this paper we give some extensions of fixed point theorems of Meir-Keeler and Boyd-Wong contraction and nonlinear contraction theorems for mappings of two variables. The fixed point theorems for nonexpansive and condensing mappings with measure of weak noncompactness are also given. Under our as- sumptions the implicit iteration converges to the fixed points, while the Picard iteration may not. The paper is a continuation of [1, 2, 3]. 2. Main results Throughout the paper we denote by D a nonemty closed subset of a complete metric space X , T a mapping of D × D into D and for x,y,z,t ∈ D, Received February 9, 2004; in revised form April 26, 2004. 1991 Mathematics Subject Classification. Primary: 47H09, 54H25; Secondary: 65J15. Key words and phrases. Nonexpansive mappings, Meir-Keeler’s, Boyd-Wong’s contraction theorems, measure of weak noncompactness, implicit iteration, Picard iteration. This paper was supported by the National Fundamental Research Program in Natural Sci- ences of Vietnam.