General approximation solvability of a system of strongly \(g\)-\(r\)-pseudomonotonic nonlinear variational inequalities and projection methods (Q2489104)

Let \(H\) be a real Hilbert space, \(K\) be a closed convex subset of \(H\), and let \(T: K \to H\) and \(g: K \to K\) be nonlinear mappings. The present paper is concerned with the problem of finding elements \(x^*, y^* \in K\) such that \(\langle \rho T(y^*)+g(x^*)-g(y^*), g(x)- g(x^*) \rangle \geq 0\) for all \(g(x) \in K\) and \(\langle \eta T(x^*)+g(y^*)-g(x^*), g(x)- g(y^*) \rangle \geq 0\) for all \(g(x) \in K\) \((\rho, \eta>0)\). For solving the problem, two iterative projection algorithms are constructed. Under appropriate monotonicity and continuity assumptions on \(T\) and \(g\), the convergence of the suggested methods is established.