Geodesic
Nonlinear equations of Hammerstein type with potential and monotone operators in Banach spaces
Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 333-347 
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/
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     author = {M. M. Vainberg and I. M. Lavrent'ev},
     title = {Nonlinear equations of {Hammerstein} type with potential and monotone operators in {Banach} spaces},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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[2] M. Golomb, “Zur Theorie der nichtlinearen Integralgleichungen, Integralgleichungssysteme und algemeinen Funktionalgleichungen”, Math. Z., 39 (1934), 45–75 | DOI | MR | Zbl

[3] V. V. Nemytskii, “Teoremy suschestvovaniya i edinstvennosti dlya nelineinykh integralnykh uravnenii”, Matem. sb., 41 (1934), 421–452 | Zbl

[4] M. M. Vainberg, I. M. Lavrentev, “Uravneniya s monotonnymi i potentsialnymi operatorami v banakhovykh prostranstvakh”, DAN SSSR, 187:4 (1969), 711–714 | MR | Zbl

[5] M. M. Vainberg, I. M. Lavrentev, “O kvadratnom korne iz lineinogo operatora v nekotorykh banakhovykh prostranstvakh”, Vestnik MGU, matem., mekh., 1970, no. 6, 36–46 | MR | Zbl

[6] M. M. Vainberg, Variatsionnye metody issledovaniya nelineinykh operatorov, Gostekhizdat, Moskva, 1956

[7] N Danford, Dzh. T. Shvarts, Lineinye operatory. Obschaya teoriya, IL, Moskva, 1963

[8] F. E. Browder, “Nonlinear elliptic boundary value problems”, Bull. Amer. Math. Sos., 69:6 (1963), 862–874 | DOI | MR | Zbl

[9] G. J. Minty, “On a “monotoniicity method” for the solutions of nonlinear equations in Banach spaces”, Proc. Nat. Acad. Sci. USA, 50:6 (1963), 1038–1041 | DOI | MR | Zbl

[10] M. M. Vainberg, “Metodo variazionale e metodo di Caccioppoli nella teoria della equazioni funzionali non lineari”, Symposia Mathematica, v. II (Instituto Nazionale di Alta Matematica, Rome, 1968), Academic Press, London, 1969, 211–226 | MR

[11] M. M. Vainberg, “K primeneniyu variatsionnogo metoda v teorii differentsialnykh uravnenii”, Čas. Pěstov. Mat., 95:2 (1970), 119–145 | MR | Zbl