A new class of generalized set-valued implicit variational inclusions in Banach spaces with an application (Q5948763)

The author introduces and studies the generalized set--valued implicit variational inclusion in the real Banach space \(E\) defined as follows:NEWLINENEWLINENEWLINEfind \(u\in E\) and \(w\in T(u)\) such that \(0 \in w+A(g(u))\), where \(T:E\to CB(E)\), \(g:E\to E\), and \(A:E\to 2^E\) is \(m\)-accretive.NEWLINENEWLINENEWLINEBy using the resolvent operator technique, the equivalence between the previous implicit variational inclusion and a suitable (single-valued) equation involving the resolvent operator \(R^A_{\rho}(u)=(I+\rho A)^{-1}, \rho>0\), associated to \(A\) is proved. Then, by Nadler's contraction principle, sequences of approximate solutions of the single--valued equation are constructed and it is showed, under suitable assumptions on \(T\) and \(g\), their convergence to the solution of the initial inclusion.