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Differential equations with several deviating arguments: Sturmian comparison method in oscillation theory. II. (English) Zbl 1016.34066
This is a continuation of the investigation on equations with two deviating arguments stated by the authors [Electron. J. Differ. Equ. 2001, Paper No. 40 (2001; Zbl 0982.34056)]. Consider the functional-differential equation of the mixed type $$\dot{x}(t)+a_1(t)x(r(t))+a_2(t)x(p(t))=0, \quad t\geq t_0,\tag 1$$ with nonnegative coefficients $a_i(t)$, and $r(t)\leq t,p(t)\geq t$ for $t\geq t_0$. Sufficient conditions for all solutions to (1) to be oscillating are obtained by using the Sturmian comparison method. The examples show that these conditions are rather sharp. The authors further obtain sufficient conditions for the existence of a nonoscillatory solution to (1). Similar nonoscillation results are obtained for (1) with nonpositive coefficients $a_i(t)\leq 0,i=1,2.$ The main idea of the Sturmian comparison method is to obtain the widest possible set of functional-differential inequalities (the “testing” equations) associated with (1), such that at least one of the solutions is a so called “slowly oscillating” solution. This together with the Sturmian comparison theorem (theorem 2.1 in this paper) yields that all solutions to (1) oscillate. The above proposed method is constructive. It also has the advantage that one needs to find only one solution which has the “slowly oscillating” property instead of checking the fact that all solutions to the “test” equation oscillate. It seems that this method has been used a lots in the references by the second author himself.

34K11Oscillation theory of functional-differential equations
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