Iterative Methods for Some Classes of Equilibrium Problems and Fxed Point Problems

The present thesis entitled “Iterative methods for some classes of equilibrium problems and fixed point problems” is an outcome of the studies made by the author at Department of Mathematics, Aligarh Muslim University, Aligarh, India. It is well known that the theory of variational inequalities and the theory of equilibrium problems provide powerful techniques for studying a large number of problems arising in optimization, transportation, economic equilibrium and other problems of practical interest. The theory of variational inequalities has been initiated, independently by Stampacchia [38] and Fichera [14] in the early 1960’s to study the boundary value problems arising in the elasticity and potential theory, respectively. This theory has enjoyed a vigorous growth for decades. Variational inequalities have been extended and generalized in several directions using innovative techniques, to study a wide range of unrelated problems in a unified and general framework. One of the important generalizations of variational inequality problem is monotone variational inclusion, initially studied by Brézis [6], Martinet [23] and Rockafellar [35]. Theory of equilibrium problems was introduced by Zuhovickii et al. [43] and Fan [11,12], perhaps motivated by minimax problems appearing in economic equilibrium. But, in 1994, the terminology of equilibrium problem was adopted by Blum and Oettli [5]. Since then various extensions and generalizations of equilibrium problem have been studied by many authors. The equilibrium problem includes many mathematical problems as particular cases, i.e., mathematical programming problems, complementary problems, variational inequality problems, saddle point problems, Nash equilibrium problems in noncooperative games, minimax inequality problems, minimization problems and fixed point problems. It is well known that many problems arising in different areas of applied mathematics such as optimization, variational analysis and differential equations, can be modeled by fixed point problem. There is a substantial literature available for studying a number 1 Ma ula na Az ad Li rar y, Ali ga rh Mu sli m Un ive rsi ty of iterative methods for fixed point problem of various classes of nonlinear mappings. In recent past, some important generalizations of fixed point problems such as split common fixed point problem [26], split equality fixed point problem [28], multiple-sets split equality fixed point problem [8] and hierarchical fixed point problem [29] have been studied. Let C be a nonempty, closed and convex subset of a real Hilbert space H and let T : C → H be a nonlinear mapping. The classical variational inequality problem (in short, VIP) introduced by Hartmann and Stampacchia [16] is given as: Find x ∈ C such that 〈Tx, y − x〉 ≥ 0, ∀y ∈ C. (1) The solution set of classical variational inequality problem (1) is denoted by Sol(VIP(1)). An important generalization of variational inequality is a monotone variational inclusion problem. Monotone variational inclusion problems are inclusion problems for maximal monotone mappings and occur in different fields of applied mathematics including optimization, mathematical programming, optimal control, mathematical economics, game theory, complementarity problems, equilibrium theory and variational inequalities. The monotone variational inclusion problem is to find x ∈ H such that 0 ∈ m ∑