Eigenfunctions for a class of nonlinear differential equations

Abstract

Y (1.2) and olrw + bY(b) = 07 yy’(a) + Sy’(b) = 0. (1.3) A solutiony =y(X, x) of (1.1) + (1.2) [or (1.1) + (1.3)] will be called an nth eigenfunction of the boundary-value problem if it has 11 1 simple zeros in (a, b). In Sections 2 and 3 we treat the boundary conditions y(u) = y(b) = 0. (1.4) Moroney [7] examined this case and proved the existence of an infinite sequence of characteristic functions under certain conditions. We will show that these conditions can considerably be weakened (Section 3). In Sections 4 and 5 we prove analogous theorems for the boundary conditions (1.2) and (1.3). Finally, in Section 6 we show how the method of Section 4 may be extended to nonhomogeneous equations.