An optimization problem (Q1824188)

Consider the optimization problem: min \(\phi\) (x) for \(x\in S\), where S is the set of solutions to the Hammerstein equation (*) \(x+F_ 1F_ 2(x)=f_ 0\). Under some assumptions on the functional \(\phi\) (properly convex, lower semicontinuous, Gateaux differentiable) and on the operators \(F_ 1\), \(F_ 2\), the author proves that (*) can be regularized in such a way that the regularized problem has a unique solution tending to the solution of the optimization problem as the regularization parameters tend to zero.