| Let Aφ(x)=∫GK(x,y)f(y,φ(y))dy, where G is a bounded closed domain in Euclidean space, K(x,y) is continuous on G×G, f(x,u) is continuous on G×R, and f(x,0)≡0. Set Gx={x|x∈G,K(x,y)≠0},Gy={y|y∈G,K(x,y)≠0},G1=Gx∩Gy≠φ.Let K1(x,y) be the restriction of K(x,y) on G1×G1,f1(x,u)be the restriction of f(x, u) on G1× R, and A1φ=∫G1K1(x,y)f1(y,φ1(y))dy, The main result of this paper is Theorem λ≠0 is an eigenvalue of A, if and only if λ is an eigenvalue of A1. |
