一般核的Hammerstein型积分方程

INTEGRAL EQUATIONS OF THE HAMMERSTEIN TYPE WITH GENERAL KERNEL

如果全连续对称核K的正特征值的个数与负特征值的个数没有任何限制,这种核称为一般核。本文在比已有存在性定理的条件更广的条件下,证明了一般核的Hammerstein型积分方程解的存在性。

We consider nonlinear integral equations of the Hammerstein type x=Kf(x) (1) Let H be a real Hilbert space, K:H→H a completely continuous selfadjoint operator with characteristic values ……<λ_(-2)λ_(-1)<0<λ_0<λ_1<λ_2<…… Let f:H→H be a continuous potential operator, i. c. f(x)=grad F(x), we suppose that (ⅰ)there exist numbers μ_N, μ_(N+1), such that λ_N<μ_N<μ_(N+1)<λ_(N+1), and μ_n||x||~2-c_1≤2F(x)≤μ_(N+1)||x||~2+c_2 for any x∈H, where c_1, c_2>0 are constants. Let H_1 be the subspace spanned by characteristic elements which correspond to characteristic values λ_(N+1), λ_(N+1),…, and H_2 the subspace spanned by characteristic elements which correspond to characteristic values λ_N, λ_(N-1),…, and elements x which satisfy Kx=0. Let P_1 and P_2 be the orthognal projection from H onto H_1 and H_2 respectively. We suppose that (ⅱ)for any z∈H_2, there exists v=v(z)∈H_2, v≠0, such that (z-P_2Kf(y+z), v)>0 for any y∈H_1 satisfying y=P_1Kf(y+z) and z≠P_2Kf(y+z). We have proved Theorem If conditions(ⅰ) (ⅱ)are satisfied, then equation (1) has at least one solution.