On certain boundary value problems for second-order linear ordinary differential equations with singularities. (English) Zbl 0559.34012

Kiguradze, I. T.; Lomtatidze, A. G.

J. Math. Anal. Appl. 101, 325-347 (1984).

Authors’ summary: ”For the differential equation \(u''=p_ 1(t)u+p_ 2(t)u'+p_ 0(t)\) with locally integrable coefficients \(p_ k: | a,b| \to R\) \((k=0,1,2)\) and for each of the following three types of boundary conditions \(u(a+)=\alpha\), \(\lim_{t\to b-}u'(t)/\sigma (p_ 2)(t)=\beta\), \(u(a+)=\alpha\), \(u(b-)=\beta\), \(u(a+)=\alpha\), \(u(b- )=u(t_ 0)+\beta\), where \(-\infty <a<t_ 0<b<+\infty\) and \(\sigma (p_ 2)(t)=\exp (\int^{t}_{(a+b)/2}p_ 2(\tau)d\tau).\) The conditions for the existence and uniqueness of a solution are established.”

Reviewer: N.L.Maria

MSC:

34B05	Linear boundary value problems for ordinary differential equations
34A30	Linear ordinary differential equations and systems

Keywords:

locally integrable coefficients

References:

[1]	Bailey, P.B.; Shampine, L.F.; Waltman, P.E., Nonlinear two point boundary value problems, (1968), Academic Press New York · Zbl 0169.10502
[2]	Bellman, R., Stability theory of differential equations, (1953), New York/Toronto/London · Zbl 0052.31505
[3]	Bitsadze, A.V.; Samarskii, A.A., On some simple extensions of elliptic boundary value problems (Russian), Dokl. akad. nauk. SSSR, 185, 4, 739-740, (1969) · Zbl 0187.35501
[4]	Epheser, H., Über die existenz der Lösungen von radwertaufgaben MIN gewöhnlichen nichtlinearen differentialgleichungen zweiter ordnung, Math. Z., 61, 4, 435-454, (1955) · Zbl 0064.08602
[5]	Gogiberidze, N.V.; Kiguradze, I.T., Concerning the disconjugacy of second order singular linear differential equations (Russian), Differencial’nye uravnenija, 10, 4, 2064-2067, (1974)
[6]	Hartman, P., Ordinary differential equations, (1964), New York/London/Sydney · Zbl 0125.32102
[7]	Jamet, P., On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems, Numer. math., 14, 355-378, (1970) · Zbl 0179.22103
[8]	Kiguradze, I.T., On conditions of disconjugacy of second order singular linear differential equations (Russian), Mat. zametki, 6, 5, 633-639, (1969) · Zbl 0196.10203
[9]	Kiguradze, I.T., On a singular boundary value problem, J. math. anal. appl., 30, 475-489, (1970) · Zbl 0202.08903
[10]	Opial, Z., Sur une inégalité de C. de la valée poussin dans la théorie de l’équation différentielle linéaire du second ordre, Ann. polon. math., 6, 81-91, (1959) · Zbl 0084.08103
[11]	de la Vallée Poussin, Ch., Sur l’équation différentielle linéaire du second ordre. Détermination d’une intégrale par deux valeurs assignées. extension aux équations d’ordre n, J. math. pures appl., 8, 125-144, (1929) · JFM 55.0850.02