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An application of A-mapping theory to boundary value problems for ordinary differential equations. (English) Zbl 0725.34023

The main purpose of this paper is to present the advantages of the A- mapping theory in the study of the existence of solutions of certain boundary value problems for second-order ordinary differential equations. This approach, dealing with an approximation technique and a version of the topological degree, differs from the usual methods employed in these problems in seeking solutions of boundary value problems as zeros of functional equations rather than as fixed points. The main result concerns problems of the form \(x''=f(t,x,x',x'')\), \(t\in [0,\infty)\), \(x\in {\mathbb{R}}^ N\), \(x\in {\mathbb{B}}\), where f: \([0,\infty)\times {\mathbb{R}}^{3N}\to {\mathbb{R}}^ N\) is continuous and \({\mathbb{B}}\) denotes one of the following boundary conditions \[ \lim_{t\to \infty}x(t)=m\in {\mathbb{R}}^ N,\quad \lim_{t\to \infty}x'(t)=0;\quad x(0)=c\in {\mathbb{R}}^ N,\quad \lim_{t\to \infty}x'(t)=m\in {\mathbb{R}}^ N. \] Other existence problems are also discussed.
Reviewer: P.Pucci (Modena)

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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