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Existence of solutions of boundary value problems for a second order equation. (Russian) Zbl 0715.34041

The BVP \(x''=f(t,x,x')\), \(H_ ix=h_ i\), \(i=1,2\), is considered where f is a Carathéodory function and \(H_ i\) are continuous functionals. Using the notion of lower and upper function introduced by himself [Differ. Equations 18, 925-931 (1983); translation from Differ. Uravn. 18, 1323-1330 (1982; Zbl 0521.34025)], the author gives a number (several tens!) of sufficient conditions for existence of a solution of the BVP (e.g.: for every solution x of the above differential equation on [a,b], the identities \(x(a)=\alpha (a)\), \(x(b)=\alpha (b)\), where \(\alpha\) is the lower function, imply \(H_ 1x\leq h_ 1\) or \(H_ 2x\leq h_ 2)\).
Reviewer: J.Jarník

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 0521.34025
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