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On a boundary value problem of periodic type for a linear second order ordinary differential equation. (Russian) Zbl 0607.34016

The boundary value problem
(1) \(u''=f(t,u,u'),\)
(2) \(u(a)=c_{11}u(b)+c_{12}u'(b)\), \(u'(a)=c_{21}u(b)+c_{22}u'(b),\)
is considered; the main result is the following: Suppose that all solutions of (1) are prolongable on [a,b], that \(f(t,x,y)\quad sgn x=- p(t)h(| x|,| y|)+q(t)(| x| +| y|),\) for \(a\leq t\leq b\), \(x^ 2+y^ 2\geq 1\), with suitable p and q and for \(\epsilon\in]0,\pi /2[\), \(\lim \quad (1/\rho)h(\rho \quad \cos \phi,\quad \rho \quad \sin \phi)]=\infty\) uniformly with respect to \(\phi\in [0,\pi /2-\epsilon]\). Then the problem (1),(2) has at least one solution.
Reviewer: A.Haimovici

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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