Perturbed proximal point algorithms for generalized quasivariational inclusions
The following problem has been considered: Find \(x\in H\), \(u\in T(x)\) and \(v\in A(x)\) such that \(g(x)\in\text{dom }\partial \phi(\cdot, x)\) and \[ (u- v, y- f(x))\geq \phi(g(x), x)- \phi(y,x),\quad \forall y\in H, \] where \(H\) is a Hilbert space, \(T, A: H\to 2^H\) are multivalued mappings satisfying appropriate conditions of Lipschitz continuity and strong monotonicity, \(\phi(\cdot, y): H\to\mathbb{R}\cup \{+\infty\}\) is proper convex and lower semicontinuous for each fixed \(y\in H\). On the basis of some properties of the resolvent operator associated with a maximal monotone mapping the iterative algorithm for finding solutions of the corresponding quasivariational inequality problems has been constructed. The convergence of this algorithm has been shown under strong hypotheses that are rather difficult to be checked in the case of important practical problems.
scientific article; zbMATH DE number 1031637
scientific article