Nonlinear equations of Hammerstein type with potential and monotone operators in Banach spaces

We prove an existence and uniqueness theorem for solutions of equations of Hammerstein type
\begin{equation} x=SF(x) \end{equation}
in Banach spaces. The main difference between this study and previous ones is to be found in the assumptions that $S$ is a closed operator from one Banach space into another, and that bounds on $F$ are imposed only on certain subsets of the space in question. The proof of the basic results requires an extension of the nonlinear mappings; we do not assume continuity of these mappings. The concept of a generalized solution is introduced, and sufficient conditions are found for it to be unique, and to coincide with an exact solution.
Bibliography: 11 titles.