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

• Volume: 68, Issue: 2, page 177-189
• ISSN: 0066-2216

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## Abstract

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We consider the problem of the existence of positive solutions u to the problem ${u}^{\left(n\right)}\left(x\right)=g\left(u\left(x\right)\right)$, $u\left(0\right)={u}^{\text{'}}\left(0\right)=...={u}^{\left(n-1\right)}\left(0\right)=0$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition $\int {₀}^{\delta }1/s{\left[s/g\left(s\right)\right]}^{1/n}ds<\infty$ 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.

## How to cite

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Wojciech Mydlarczyk. "A singular initial value problem for the equation $u^{(n)}(x) = g(u(x))$." Annales Polonici Mathematici 68.2 (1998): 177-189. <http://eudml.org/doc/270540>.

@article{WojciechMydlarczyk1998,
abstract = {We consider the problem of the existence of positive solutions u to the problem $u^\{(n)\}(x) = g(u(x))$, $u(0) = u^\{\prime \}(0) = ... = u^\{(n-1)\}(0) = 0$ (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.},
author = {Wojciech Mydlarczyk},
journal = {Annales Polonici Mathematici},
keywords = {singular initial value problems for ordinary differential equations; Volterra type integral equations; blowing up solutions; singular initial value problems; blowing-up solutions},
language = {eng},
number = {2},
pages = {177-189},
title = {A singular initial value problem for the equation $u^\{(n)\}(x) = g(u(x))$},
url = {http://eudml.org/doc/270540},
volume = {68},
year = {1998},
}

TY - JOUR
AU - Wojciech Mydlarczyk
TI - A singular initial value problem for the equation $u^{(n)}(x) = g(u(x))$
JO - Annales Polonici Mathematici
PY - 1998
VL - 68
IS - 2
SP - 177
EP - 189
AB - We consider the problem of the existence of positive solutions u to the problem $u^{(n)}(x) = g(u(x))$, $u(0) = u^{\prime }(0) = ... = u^{(n-1)}(0) = 0$ (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.
LA - eng
KW - singular initial value problems for ordinary differential equations; Volterra type integral equations; blowing up solutions; singular initial value problems; blowing-up solutions
UR - http://eudml.org/doc/270540
ER -

## References

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1. [1] P. J. Bushell and W. Okrasiński, Uniqueness of solutions for a class of nonlinear Volterra integral equations with convolution kernel, Math. Proc. Cambridge Philos. Soc. 106 (1989), 547-552. Zbl0689.45013
2. [2] G. Gripenberg, Unique solutions of some Volterra integral equations, Math. Scand. 48 (1981), 59-67. Zbl0463.45002
3. [3] R. K. Miller, Nonlinear Volterra Integral Equations, Benjamin, 1971. Zbl0448.45004
4. [4] W. Mydlarczyk, The existence of nontrivial solutions of Volterra equations, Math. Scand. 68 (1991), 83-88. Zbl0701.45002
5. [5] W. Mydlarczyk, An initial value problem for a third order differential equation, Ann. Polon. Math. 59 (1994), 215-223. Zbl0808.34004
6. [6] W. Okrasiński, Nontrivial solutions to nonlinear Volterra integral equations, SIAM J. Math. Anal. 22 (1991), 1007-1015. Zbl0735.45005

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