Geodesic
A singular initial value problem for the equation $u^{(n)}(x) = g(u(x))$
Annales Polonici Mathematici, Tome 68 (1998) no. 2, pp. 177-189.

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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.
DOI : 10.4064/ap-68-2-177-189
Keywords: singular initial value problems for ordinary differential equations, Volterra type integral equations, blowing up solutions

Wojciech Mydlarczyk 1

1 
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Wojciech Mydlarczyk. A singular initial value problem for the equation $u^{(n)}(x) = g(u(x))$. Annales Polonici Mathematici, Tome 68 (1998) no. 2, pp. 177-189. doi : 10.4064/ap-68-2-177-189. http://geodesic.mathdoc.fr/articles/10.4064/ap-68-2-177-189/

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