非线性Hammerstein型积分方程的多重解及其应用

MULTIPLE SOLUTIONS OF NONLINEAR INTEGRAL EQUATIONS OF HAMMERSTEIN TYPE AND ITS APPLICATIONS

在讨论非线性Hammerstein型积分方程(*)φ(x)=integral from n=G to k(x,y)f(y,φ(y))dy,0<mesG<+∞时证明了:当f(x,u)满足文中假设(ii)—(iv)时,方程(*)具有三个互异解。作为其应用,还讨论了非线性Sturm-Liourille问题(**)d^2u/(dx^2)+f(x,u)=0,au(0)+bu'(0)=0,cu(1)+du'(1)=0,得到问题(**)三个互异C^2类解的存在性。本文使用变分方法,主要结果的证明基于文[1]中建立的“等”高”山路定理。

Nonlinear integral equations of Hammerstein type(*) (x) = ∫Gk(x, y)f(y, (y))dy, where O<mes G<C+∞are discussed in this paper, Under suitable hypotheses the author, with variational methods, obtained the existence of three solutions of Eq.(*), and gave an application to the nonlinear Sturm-Liouville problems (**) dx2d2u + f(x,u)=0, au(0)+bu'(0)=0, cu(1)+du'(1) = 0 . The proof of the main result is based on the Mountain Pass Theorem with 'zero height' (see[1]).