Research Paper Weak Convergence Theorem for Nonexpansive and Monotone, Lipschitz Continuous Mappings RENU CHUGH, REKHA RANI * and SANJAY KUMAR Department of Mathematics, Maharshi Dayanand University, Rohtak 124 001, India (Received on 13 December 2013, Revised on 5 March 2014; Accepted on 15 April 2014) In this paper, we shall prove a weak convergence theorem for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. With the help of a numerical example, we shall show the existence of a fixed point and find a solution of a variational inequality problem using C++. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of a monotone, Lipschitz continuous mapping. Key Words: Fixed Points; Monotone Mappings; Nonexpansive Mappings; Variational Inequalities *Author for Correspondence: E-mail: rekhadalal93@gmail.com Proc Indian Natn Sci Acad 80 No. 3 September 2014 pp. 583-588 Printed in India. Introduction Let H be a real Hilbert space with inner product .,.   and norm || ||  , respectively. Let C be a closed convex subset of H. The variational inequality problem is to find u C  such that , 0, Au v u v C      . The set of solutions of variational inequality problem VI(C, A) is denoted by . The variational inequality problem has been extensively studied in literature, see, for example, Browder and Petryshyn (1967), Liu and Nashed (1998), Takahashi (2000) and references therein. Definitions: Let A: C H be a mapping of C into H. 1. A is called monotone if , 0, , Au Av u v u v C       2. A is called -inverse-strongly-monotone (Browder and Petryshyn, 1967; Liu and Nashed, 1998) if there exists a positive real number such that 2 , || || , . Au Av u v Au Av u v C         It is easy to see that an -inverse strongly mapping A is monotone and Lipschitz continuous but converse is not true. 3. A mapping S : C C is called nonexpansive (Takahashi, 2000; Takahashi and Tamura, 1998) if || || || || , . Su Sv u v u v C      We denote by F(S) the set of fixed points of S. 4. A mapping S : C C is called Lipschitz continuous if there exists a real number L > 0 such that, || || || || , . Su Sv L u v u v C      Takahashi and Toyoda (2003) introduced the following iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an -inverse-strongly- monotone mapping in a real Hilbert space. DOI: 10.16943/ptinsa/2014/v80i3/55133