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Regularly varying solutions to functional differential equations with deviating argument. (English) Zbl 1199.34329

Plenty of interesting results concerning the existence of regularly varying solutions for differential equations of the type \(x''(t)=q(t)x(g(t))\) are given. We quote here the following result (Theorem A): Let \(c>0\) and denote by \(\lambda_i\), \(i=0,1\); \(\lambda_0<\lambda_1\), the two roots of the equation \(\lambda^2-\lambda-c=0\). Suppose that \(\lim_{t\to\infty}g(t)/t=1\). Then the equation above has two regularly varying solutions of the form \(x_i(t)=t^{\lambda_i}L_i(t)\), \(i=0,1\), where \(L_i(t)\) are some normalized slowly varying functions, if and only if \(t\int_t^\infty q(s)\,ds\to c\) when \(t\to\infty\).

MSC:

34K11 Oscillation theory of functional-differential equations
34K06 Linear functional-differential equations
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