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Existence of solutions to second order ordinary differential equations having finite limits at \(\pm\) infinity. (English) Zbl 1053.34027
The authors study the boundary value problem \[ {\ddot x}(t)=a(t,x,{\dot x}){\dot x}+b(t,x,{\dot x})x+c(t), \] \[ \lim_{t\to-\infty}x(t)=\lim_{t\to\infty}x(t),\,\,\, \lim_{t\to-\infty}{\dot x}(t)=\lim_{t\to\infty}{\dot x}(t), \] where \(a,b\: {\mathbb R}^3\to {\mathbb R}\), and \(c\: {\mathbb R}\to {\mathbb R}\) are continuous functions. Under suitable assumptions on the growth at infinity of \(a,b,c\), the authors prove the existence of solutions of the above boundary value problem by using the Bohnenblust-Karlin fixed-point theorem for multivalued mappings and a compactness criterion developed by the first author in [Ann. Mat. Pura Appl., IV. Ser. 81, 147–168 (1969; Zbl 0196.10701); Abstr. Appl. Anal. 7, No. 1, 1–27 (2002; Zbl 1009.34025)].

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
54C60 Set-valued maps in general topology
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