Necessary conditions for optimality in variational inequality form for systems described by Hammerstein equations (Q1364117)

The paper considers the existence of an optimal control and necessary optimality conditions for an abstract problem in Banach spaces of the type \[ I(u, x)\to\min,\;u\in U,\;x\in K\subset X,\;x+BF(u,x)= 0,\;F_1(u, x)\in K_1, \] where the set \(U\) of admissible controls is convex and weakly compact, \(K\) is a convex closed set and \(K_1\) is a closed cone. Under some assumptions on differentiability of \(I\), \(F\), \(F_1\), monotonicity of \(B\), \(F\) (\(B\) is a linear operator) and weak continuity of involved maps there are proposed the existence of an optimal solution and the type of the necessary optimality conditions. In the reviewer's opinion the statements of the paper are not clearly formulated and a part of them (e.g., Theorem 2) have not sufficient proofs.