Lalli, B. S.; Yu, Y. H.; Cui, B. T. Oscillation of hyperbolic equations with functional arguments. (English) Zbl 0798.35151 Appl. Math. Comput. 53, No. 2-3, 97-110 (1993). The oscillation of hyperbolic pde’s with functional arguments of the form \[ {\partial^ 2 u\over \partial t^ 2}+ p(x,t)u(x,t)+\sum^ k_{i=1} p_ i(x,t)u(x,\tau_ i(t))= a(t)\Delta u+ \sum^ m_{j=1} a_ j(t)\Delta u(x,\sigma_ j(t))\tag{1} \] is considered. First, the problem is reduced, by avaraging, to the oscillation of ordinary delay inequalities of the form \[ y''(t)+ q(t)y(t)+ \sum^ n_{i=1} q_ i(t)y(g_ i(t))\leq 0. \] It turns out that all solutions of (1) oscillate if, in addition to some continuity conditions, we have \[ \liminf_{t\to\infty} \int^ t_{\tau_ i(t)} p_ i(s)\tau_ i(s)\exp\left(\int^ s_{\tau_ i(s)} rp(r)dr\right) ds>{1\over e} \] for some \(i\in \{1,\dots,k\}\), where \(p(t)= \min_ x p(x,t)\), \(p_ i(t)= \min_ x p_ i(x,t)\). Reviewer: S.P.Banks (Sheffield) Cited in 17 Documents MSC: 35R10 Partial functional-differential equations 35L10 Second-order hyperbolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs Keywords:hyperbolic functional pde’s; oscillation of ordinary delay inequalities PDFBibTeX XMLCite \textit{B. S. Lalli} et al., Appl. Math. Comput. 53, No. 2--3, 97--110 (1993; Zbl 0798.35151) Full Text: DOI References: [1] Georgou, D.; Kreith, K., Functional characteristic initial value problems, J. Math. Anal. Appl., 107, 2, 414-424 (1985) · Zbl 0585.35067 [2] Ladde, G. S.; Lakshmikantham, V.; Zhang, B. G., Oscillation Theory of Differential Equations With Deviating Arguments (1987), Marcel Dekker Inc: Marcel Dekker Inc New York and Basel · Zbl 0622.34071 [3] Mishev, D. P., Oscillatory properties of the solutions of hyperbolic differential equations with “maximum”, Hiroshima Math. J., 16, 77-83 (1986) · Zbl 0609.35054 [4] Mishev, D. P.; Bainov, D. D., Oscillation properties of the solutions of a class of hyperbolic equations of neutral type, Funkcial. Ekvac., 29, 2, 213-218 (1986) · Zbl 0651.35052 [5] Mishev, D. P.; Bainov, D. D., Oscillation properties of solutions of a class of neutral hyperbolic equations, (Proceedings of Colloquium on Qualitative Theory of Differential Equations. Proceedings of Colloquium on Qualitative Theory of Differential Equations, Szeged, Hungary (1984)), 771-780 · Zbl 0651.35052 [6] Mishev, D. P.; Bainov, D. D., Oscillation of the solutions of parabolic differential equations of neutral type, Appl. Math. Comput., 28, 97-111 (1988) · Zbl 0673.35037 [7] Yoshida, N., On the zeros of solutions of hyperbolic equations of neutral type, Differential Integral Equations, 3, 155-160 (1990) · Zbl 0749.35006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.