Geodesic
Nonlinear equations of Hammerstein type with potential and monotone operators in Banach spaces
Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 333-347

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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. 
@article{SM_1972_16_3_a1,
     author = {M. M. Vainberg and I. M. Lavrent'ev},
     title = {Nonlinear equations of {Hammerstein} type with potential and monotone operators in {Banach} spaces},
     journal = {Sbornik. Mathematics},
     pages = {333--347},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1972_16_3_a1/}
}
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M. M. Vainberg; I. M. Lavrent'ev. Nonlinear equations of Hammerstein type with potential and monotone operators in Banach spaces. Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 333-347. http://geodesic.mathdoc.fr/item/SM_1972_16_3_a1/