Convergence estimates and approximation solvability of nonlinear implicit variational inequalities (Q1608649)

Let \(H\) be a real Hilbert space with the inner product \(\langle \cdot,\cdot \rangle\) and norm \(\|\cdot\|\). Let \(K\) be a closed convex subset of \(H\) and \(T:K\times K\to H\) be a mapping. The author considers the following nonlinear implicit variational inequality (for short, NIVI) problem of finding \(x^*\in K\) such that \[ \langle T(x^*,x^*),x-x^*\rangle \geq 0 \] for all \(x\in K\). By using the projection method, the author constructs an algorithm for solving the (NIVI) and proves the approximation-solvability of the NIVI involving a combination of \(\gamma\)-partially relaxed monotone and monotone mappings in a Hilbert space setting. He also gives an application to the space \(\mathbb{R}^n\).