Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings.
This article deals with the generalized nonlinear mixed implicit quasi-variational inclusion with set-valued mappings \[ 0\leq N(x,y)+M (w,z),\;x\in Su,\;y\in Tu,\;z\in Gu,\;w\in Pu,\;w\in\text{dom} (M (\cdot,z)), \] where \(G,S,T,P:H\to 2^H\) are set-valued mappings, \(N:H\times H\to H\) a single-valued mapping, \(M:H\times H\to 2^H\) a set-valued mapping such that \(M(\cdot,t): H\to 2^H\) is a maximal monotone mapping and range \((P)\cap\text{dom}(M(\cdot,t)) \neq\emptyset\) \( (t\in H)\) and some its special variants. The problem above is equivalent to the problem \[ w= (I+\rho M ((w-\rho N(x,y) ),z) )^{-1} (w-\rho (N(x,y))) \] with a single-valued operator. For solving this problem, some iterative algorithms are considered. The main results are theorems about the existence of solutions and the convergence of iterations.
scientific article; zbMATH DE number 1902232