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Solvability of a nonlinear boundary value problem in a fixed set of functions. (Russian) Zbl 0743.34070

The author considers the boundary value problem \(x^{(m)}(t)=F(x(\centerdot))(t)\), \(t\in[a,b]\), \(B_ k(x^{(k)}(\centerdot))=0\), \(k=0,\ldots,m-1\), where \(F:C^{m-1}\to L_ 1\), \(B_ k:CL_ 1^{m-1}\to\mathbb{R}^ n\) are continuous and \(CL^ m_ 1\) is the set of all functions \(x\in C^{m-1}\) having absolutely continuous derivatives of the order \(m-1\). Using the Schauder fixed point theorem and the Arzelà-Ascoli theorem the author proves the existence of at least one solution of this problem in the given fixed set of functions. A multiplicity result for a second order differential equation with nonlinear functional conditions is stated here.

MSC:

34K10 Boundary value problems for functional-differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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