A new class of monotone mappings and a new class of variational inclusions in Banach spaces
Let \(E\) be a Banach space, \(E^*\) its topological dual, and \(CB(E)\) denote the collection of all nonempty closed bounded subsets of \(E\). For \(k \geq 3,\) let \(A: E \to E^*,\) \(p,\) \(f_1,f_2,\dots,f_k: E \to E,\) \(F: E^k \to E^*\) be nonlinear maps and \(T_1,T_2,\dots,T_k: E \to CB(E),\) \(M: E^k \multimap E^*\) multimaps. Under some monotonicity assumptions on \(M,\) the author studies the following problem: for a given \(a \in E^*,\) find \(x \in E,\) \(t_1 \in T_1(x), t_2 \in T_2(x),\dots, t_k \in T_k(x)\) such that \[ a \in A(x - p(x)) + M(f_1(x),f_2(x),\dots,f_k(x)) - F(t_1,t_2,\dots,t_k). \] By using the technique of proximal maps, the author reduces this problem to a fixed point problem and suggests an iterative algorithm for its solving.
scientific article; zbMATH DE number 6136263