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Multipoint boundary value problems at resonance. (English) Zbl 0715.34034
Differential equations and their applications, Proc. 7th Conf., Equadiff 7, Prague/Czech. 1989, Teubner-Texte Math. 118, 106-109 (1990).
[For the entire collection see Zbl 0704.00019.]
We investigate the multipoint BVPs, where the number of points is greater than the order of a differential equation. For the differential equation of the second order such problem can have the form \[ (1)\quad u''=f(t,u,u'),\quad (2)\quad u(a)=c_ 1,\quad u(b)=u(t_ 0)+c_ 2, \] where \(a,t_ 0,b,c_ 1,c_ 2\in R\), \(a<t_ 0<b\). The questions of the existence of solutions of problem (1), (2) were studied by H. Dörner [Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math. Naturwiss. Reihe 32, No.1, 37-40 (1983; Zbl 0521.34018)] and by I. T. Kiguradze and A. G. Lomtatidze [J. Math. Anal. Appl. 101, 325- 347 (1984; Zbl 0559.34012)] for linear differential equations, the nonlinear case was considered by A. G. Lomtatidze [Tr. Inst. Prikl. Mat. Im. I. N. Vekua 17, 122-134 (1986; Zbl 0632.34011)]. It is worth mentioning that the similar problem but for partial differential equations, which is known now as the Bitsadze-Samarskij problem, was first stated and solved by A. V. Bitsadze and A. A. Samarskij [Sov. Math., Dokl. 10, 398-400 (1969); translation from Dokl. Akad. Nauk SSSR 185, No.4, 739-740 (1969; Zbl 0187.355)].
MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations