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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, Nemytskiĭ, Uryson, etc.) 34L30 Nonlinear ordinary differential operators 47J15 Abstract bifurcation theory involving nonlinear operators
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