Summary
The equation to be considered is of the form (1) x(n)(t)+σp(t)x(g(t))=0 (t>a), where σ=±1, p(t) > 0 for t≧a and g(t) →∞ as t→∞. It is well- known that a nonoscillatory solution x(t) of (1) satisfies (2) x(t)x(i)(t)>0 (0≦i≦l), (−1)i−lx(t)x(i)(t)>0 (l≦i≦n) for some integer l, 0≦l≦n, (−1)n−l−1σ=1. In this paper, for a given l such that 0<l<n, (−1)n−l−1σ=1, necessary conditions and sufficient conditions are found for (1) to have a solution x(t) which satisfies (2), and a necessary and sufficient condition is established in order that for every λ>0 the equation x(n)(t)+λσp(t)x(g(t))=0 (t>a) has a solution x(t) which satisfies (2). Related results are also contained.
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Naito, M. Nonoscillatory solutions of linear differential equations with deviating arguments. Annali di Matematica pura ed applicata 136, 1–13 (1984). https://doi.org/10.1007/BF01773372
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DOI: https://doi.org/10.1007/BF01773372