General variational inclusions in Lp spaces

Variational inequalities are being used as a mathematical programming tool in modeling a wide class of problems arising in different branches of pure and applied sciences, see [1–26]. Variational inequalities have been extended and generalized in various directions using novel and innovative techniques. A useful and important generalization is called the general variational inclusion involving the sum of two nonlinear operators T and A. Moudafi and Noor [10] studied the sensitivity analysis of variational inclusions by using the technique of the resolvent equations. Recently, much attention has been given to develop iterative algorithms for solving the variational inclusions. It is known that such algorithms require an evaluation of the resolvent operator of the type (I + ρ(T +A))−1. The main difficulty with such problems is that the resolvent operator may be hard to invert. This difficulty has been overcome by using the resolvent operators (I + ρT)−1 and (I + ρA)−1 separately rather than (I + ρ(T +A))−1. Such a technique is called the splitting method. Such type of methods for solving variational inclusions has been studied extensively, see [4, 5, 7, 9–11, 17, 19, 23] and the references therein. In the context of the mixed variational inequalities (variational inclusions), Noor [15– 18] has used the resolvent operator and resolvent equations techniques to suggest and analyze a number of resolvent-type iterative methods. A useful feature of these splitting methods is that the resolvent step involves the subdifferential of the proper, convex, and lower semicontinuous function only and the other part facilitates the problem decomposition. Equally important is the concept of the resolvent equations, which has been introduced by Noor [16, 17] recently in connection with the mixed variational inequalities.