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

• Autorzy: Wojciech Mydlarczyk
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

singular initial value problems for ordinary differential equations; Volterra type integral equations; blowing up solutions

Kategorie tematyczne

• 45G10: Other nonlinear integral equations
• 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions
• 45D05: Volterra integral equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1998
• Tom: 68
• Numer: 2
• Strony: 177-189

Daty

wydano

1998

otrzymano

1997-06-12

Twórcy

autor

Wojciech Mydlarczyk

• Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Bibliografia

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