Iterative algorithms for finding approximate solutions of completely generalized strongly nonlinear quasivariational inequalities (Q1804431)

Let \(H\) be a Hilbert space whose norm and inner product are denoted by \(|\cdot|\) and \((\cdot, \cdot)\), respectively. Let \(K\) be a nonempty closed convex set in \(H\) and \(T, A: H\to 2^{H^*}\) be two multivalued mappings. Let \(g: H\to H\) be a continuous mapping. Consider the problem of finding \(x\in H\), \(u\in T(x)\), \(v\in A(x)\) such that \(g(x)\in K\) and \[ \langle u- v, y- g(x)\rangle\geq 0,\qquad\text{for all}\quad y\in K,\tag{1} \] where \(\langle\cdot, \cdot\rangle\) is the pairing between \(H^*\) (the dual space of \(H\)) and \(H\). The problem (1) is called the generalized strongly nonlinear variational inequality. If the convex set \(K\) also depends on the solution \(x\) itself, then problem (1) is called the generalized strongly nonlinear quasi-variational inequality problem, that is, find \(x\in H\), \(u\in T(x)\), \(v\in A(x)\) such that \(g(x)\in K(x)\) and \[ \langle u- v, y- g(x)\rangle\geq 0,\qquad\text{for all}\quad y\in K(x).\tag{2} \] The author has proposed and analyzed some iterative algorithm for finding the approximate solution of both the problems (1) and (2), when \(K(x)= m(x)+ K\), where \(m\) is a point-to-point mapping from \(H\) into itself under some artificial assumptions of the type \[ \text{Re}(m(x)- m(y), x- y- (g(x)- g(y)))\leq \gamma|x- y|^2,\tag{3} \] for all \(x, y\in H\) and some constant \(\gamma> 0\). We like to remark that the assumption (3) cannot be verified practically. Recently, the reviewer [Dyn. Syst. Appl. 4, 469-476 (1995)] has considered the more general problem and has shown that such type of iterative algorithms can be analyzed without the assumption (3).