On a General Projection Algorithm for Variational Inequalities

Abstract

Let H be a real Hilbert space with norm and inner product denoted by \(|| \cdot ||\) and \(\left\langle { \cdot, \cdot } \right\rangle\). Let K be a nonempty closed convex set of H, and let f be a linear continuous functional on H. Let A, T, g be nonlinear operators from H into itself, and let \(K(\cdot):H \to 2^H\) be a point-to-set mapping. We deal with the problem of finding uɛK such that g(u)ɛK(u) and the following relation is satisfied: \(\left\langle {A\left( {g\left( u \right)} \right),{\upsilon } - g\left( u \right)} \right\rangle \geqslant \left\langle {A\left( u \right),{\upsilon } - g\left( u \right)} \right\rangle - p\left\langle {T\left( u \right) - f,\upsilon - g\left( u \right)} \right\rangle \forall {\upsilon } \in K\left( u \right)\), where ρ>0 is a constant, which is called a general strong quasi-variational inequality. We give a general and unified iterative algorithm for finding the approximate solution to this problem by exploiting the projection method, and prove the existence of the solution to this problem and the convergence of the iterative sequence generated by this algorithm.