Existence of Solutions to Nonselfadjoint Boundary Value Problems for Ordinary Differential Equations

where the coefficients p,(t) are continuous functions on [0, a], pa(t) > 0, and the ah,, b, are constants. Our results show that, even for nonselfadjoint boundary value problems, a limited quantitative form of the Landesman and Lazer sufficient condition for existence holds. Moreover, further extensions are obtained of the recent form proposed by Shaw of the Landesman and Laxer theorem. The point of departure of the present paper is the theorem of Landesman and Lazer [ll], which represents a necessary and sufficient condition in order that a selfadjoint real elliptic partial differential equation of order 2, Ex = f(r) + g(x), r E G, with x = 0 on aG, has a weak solution x E Wk2(G), where G is a bounded domain in EC!“, g: R + R is continuous with finite limits g(-x) f g(+m), andf E L,(G). This remarkable theorem was extended by Williams [15] and by De Figueiredo [lo] to elliptic problems of order 2n and nonlinearity g depending on derivatives of orders <2n 1. The same theorem was then extended by Shaw [14] to nonselfadjoint boundary value problems for partial differential equations with p = q < =, p = dim ker E, q = dim ker E*, where E* is the adjoint of E, provided corresponding elements of ker E and ker E*, share the same regions of positivity and negativity. In particular, for ordinary differential equations with n = 1, p = q = 1, this condition is always satisfied, and the Landesman and Lazer theorem holds. For their results, Landesman and Laxer, as well as Williams and Shaw made use of ideas from the alternative method (see, e.g. [6]).