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Existence of solutions and bifurcation points to Hammerstein equations with essentially bounded kernel. (English) Zbl 1067.45001
This paper deals with an existence theorem of solutions and of bifurcation points for the Hammerstein integral equation \[ u(x)=\lambda\int_{\Omega}k(x,y)f(y,u(y))dy, \] where \(\lambda\in \mathbb R\), \(\Omega\) is a Lebesgue measurable subset of \(\mathbb R^{n}, \;k\in L^{\infty}(\Omega\times\Omega)\) and \(f:\Omega\times \mathbb R\to \mathbb R\) is a Carathéodory function. The proofs rely on the Tychonoff fixed point theorem.

45G10 Other nonlinear integral equations
Full Text: DOI
[1] Diestel, J.; Uhl, J.J., Vector measures, Math. survey, vol. 15, (1977), American Mathematical Society · Zbl 0369.46039
[2] Faraci, F., Bifurcation theorems for Hammerstein nonlinear integral equations, Glasgow math. J., 44, 471-481, (2002) · Zbl 1020.45002
[3] F. Faraci,V. Moroz, Solutions of Hammerstein integral equations via a variational principle, J. Integral Equations Appl., in press · Zbl 1060.45006
[4] Köthe, G., Topological vector spaces-I, (1969), Springer-Verlag
[5] Meehan, M.; O’Regan, D., Positive lp-solutions of Hammerstein integral equations, Arch. math., 76, 366-376, (2001) · Zbl 0981.45001
[6] Naselli, O.; Ricceri, B., An existence theorem for inclusions of the type ψ(u)(t)∈F(t,φ(u)(t)) and application to a multivalued boundary value problem, Appl. anal., 38, 259-270, (1990) · Zbl 0687.47044
[7] Moroz, V.; Zabreiko, P., On the Hammerstein equations with natural growth conditions, Z. anal. anwendungen, 18, 625-638, (1999) · Zbl 0942.45003
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