Geodesic
The Schaefer method in the theory of Hammerstein integral equations
Sbornik. Mathematics, Tome 13 (1971) no. 3, pp. 451-471 
P. P. Zabreiko. The Schaefer method in the theory of Hammerstein integral equations. Sbornik. Mathematics, Tome 13 (1971) no. 3, pp. 451-471. http://geodesic.mathdoc.fr/item/SM_1971_13_3_a7/
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     title = {The {Schaefer} method in the theory of {Hammerstein} integral equations},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Hammerstein integral equation \begin{equation} x(t)=\int_\Omega k(t,s)f[s, x(s)]\,dt+g(t) \end{equation} is studied. It is assumed that the linear integral operator $K$ with symmetric kernel $k(t,s)$ acts and is completely continuous or the Hilbert space $H=L_2$. Furthermore, it is assumed that $E_0$ and $E$ ($E_0\subset E\subset H$) are ideal spaces for which the following conditions are fulfilled: a) the operator $K$ acts on the dual space $E'_0$; b) the eigenfunctions of $K$ lie in $E_0$; c) the linear span of the eigenfunctions of $K$ is dense in $E_0$ in the sense of $o$-covergence; d) the operator $~K$ acts from $E_0$ to $E'_0$ (and is completely continuous); e) the operator $f$ acts from $E_0$ to $E'_0$ and transforms bounded sets into $E_0$-weakly sequentially compact sets (acts from $E_0$ to $E'_0$). It is proved that under these hypotheses in the case of a positive definite $K$ a sufficient condition for the solvability of equation $(1)$ is the inequality \begin{equation} uf(s,u)\leqslant au^2+\omega(s,u) \end{equation} where $a\lambda<1$ ($\lambda$ is the largest eigenvalue of $K$) and $\omega (s,u)$ contains terms that grow at infinity more slowly than $u^2$. Bibliography: 10 titles.

[1] M. A. Krasnoselskii, Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, Gostekhizdat, Moskva, 1956 | MR

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

[3] P. P. Zabreiko, A. I. Povolotskii, “K teorii uravnenii Gammershteina”, Ukr. matem. zh., 22:2 (1970), 150–162 | MR | Zbl

[4] H. Schaefer, Neue Existenzsatze in der Theorie Nichtlineare Integralgleichungen, Berlin, 1955 | MR

[5] A. P. Makhmudov, “Primenenie printsipa X. Shefera k suschestvovaniyu reshenii nelineinykh integralnykh uravnenii”, Uchenye zapiski Azerbaidzh. un-ta, seriya fiz.-matem. i khim. nauk, 1901, no. 4, 31–35

[6] W. Petry, “Systeme nichtlinearer Integralgleichungen mit positiv definiten Kernen”, Z. Angew. Math, and Mech., 47:2 (1967), 97–108 | DOI | MR | Zbl

[7] P. P. Zabreiko, Nelineinye integralnye operatory, Trudy seminara po funkts. analizu, 8, Voronezh, 1966

[8] P. P. Zabreiko, Issledovaniya po teorii integralnykh operatorov v idealnykh prostranstvakh funktsii, Doktorskaya dissertatsiya, Voronezh, 1968

[9] M. A. Krasnoselskii, Ya. B. Rutitskii, R. M. Sultanov, Ob odnom nelineinom operatore, deistvuyuschem v prostranstvakh abstraktnykh funktsii, Izv. AN Azerbaidzh. SSR, seriya fiz.-matem. i tekhn. nauk, no. 3, 1959 | MR

[10] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, P. E. Sobolevskii, Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, Moskva, 1966 | MR