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

Annales Polonici Mathematici (1998)

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

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topWojciech 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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- [2] G. Gripenberg, Unique solutions of some Volterra integral equations, Math. Scand. 48 (1981), 59-67. Zbl0463.45002
- [3] R. K. Miller, Nonlinear Volterra Integral Equations, Benjamin, 1971. Zbl0448.45004
- [4] W. Mydlarczyk, The existence of nontrivial solutions of Volterra equations, Math. Scand. 68 (1991), 83-88. Zbl0701.45002
- [5] W. Mydlarczyk, An initial value problem for a third order differential equation, Ann. Polon. Math. 59 (1994), 215-223. Zbl0808.34004
- [6] W. Okrasiński, Nontrivial solutions to nonlinear Volterra integral equations, SIAM J. Math. Anal. 22 (1991), 1007-1015. Zbl0735.45005