An iterative scheme for general mixed multivalued mildly nonlinear variational inequalities (Q1286978)

Let \(H\) be a real Hilbert space whose inner product is denoted by \(\langle\cdot,\cdot\rangle\) and let \(K\) be a closed convex subset of \(H\). Let \(g:H\to H\) be a mapping and let \(T,A:H\to C(H)\) be multivalued mappings, where \(C(H)\) is the family of all nonempty compact subset of \(H\). The problem of finding \(u\in H\), \(x\in T(u)\) and \(y\in A(u)\) such that \(g(u)\in K\) and \[ \bigl\langle x-y,g(v)-g(u)\bigr \rangle+b \bigl(u,g(v)\bigr) -b\bigl(u, g(u) \bigr)\geq 0,\;\forall g(v)\in K, \] called the general mixed multivalued mildly nonlinear variational inequality problem, is considered. An iterative scheme is given to obtain an approximate solution of this problem. It is shown that the approximate solutions obtained by the iterative scheme converge strongly to the exact solution. Some special cases are dicussed, too.