Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

We consider the problem of the existence of positive solutions u to the problem $u^{(n)}(x) = g(u(x))$, $u(0) = u'(0) = ... = u^{(n-1)}(0) = 0$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition $∫₀^δ 1/s [s/g(s)]^{1/n} ds < ∞$ is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.