An extension of Jentzsch's theorem to nonlinear Hammerstein operators

Abstract

A classical theorem of Jentzsch (J. Reine Angew. Math. 141, 235-44 (1912)) states that the Fredholm linear integral operator Ax(t) = integral K(t,s) x (s)ds, with continuous kernel K: (a,b) x (a,b) ..-->.. (0, integral ), has a positive eigenvalue lambda corresponding to a positive eigenfunction x: (a,b) ..-->.. R. Of particular interest is an important nonlinear analog of the above, the Hammerstein integral operator Ax(t) = integral K(t,s)f(s,x(s))ds. Here K is nonnegative and the nonlinearity f is continuous and nonnegative on (a,b) x (0,c) for some c greater than 0. Also, f(s,0) is identically 0, so that the zero function is an eigenvector of A. It is of interest to find conditions on K and f under which A has positive eigenpairs, that is, a positive eigenvalue lambda corresponding to a nonnegative eigenfunction x which is not identically zero. This article proves the existence of positive eigenpairs for the Hammerstein integral operator for an extensive class of locally positive kernels which satisfy a weak uniformity condition, with no growth restrictions whatever on f. In fact, all that is assumed is that f is positive on certain subsets of (a,b) x (0, infinity). The class of kernels considered here includesmore » and is far less restrictive than the class of kernels of Jentzsch's theorem. (RWR)« less