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Bifurcation theorems for Hammerstein nonlinear integral equations. (English) Zbl 1020.45002
The author studies the Hammerstein integral equation \[ u(x)=\lambda\int_{\Omega} k(x,y)f(y,u(y)) dy \] in \(L^\infty (\Omega)\) and shows the conditions when \(\lambda=0\) is a bifurcation point. An application to the Dirichlet boundary value problem \[ u^{\prime \prime}=-\lambda f(x,u), \qquad u(0)=u(1)=0 \] is also given.

MSC:
45G10 Other nonlinear integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
34L30 Nonlinear ordinary differential operators
47J15 Abstract bifurcation theory involving nonlinear operators
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