A singular initial value problem for the equation $u^{(n)}(x) = g(u(x))$

Volume 68 / 1998

Wojciech Mydlarczyk Annales Polonici Mathematici 68 (1998), 177-189 DOI: 10.4064/ap-68-2-177-189

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.

Authors

  • Wojciech Mydlarczyk

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