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Hille-type oscillation and nonoscillation criteria for neutral equations with positive and negative coefficients. (English) Zbl 1051.34053

Summary: Hille-type oscillation and nonoscillation criteria are established for the neutral differential equation with positive and negative coefficients \[ \bigl[x(t)- R(t)x(t-r)\bigr]'+ P(t)x(t-\tau)-Q(t)x(t-\sigma)= 0,\;t\geq t_0, \] with \(r\in(0,\infty)\), \(\tau,\sigma\in [0,\infty)\), \(P,Q,R \in C([t_0,\infty), \mathbb{R}^+)\). These criteria essentially improve many known oscillation results in the literature. Necessary and sufficient conditions for the existence of bounded positive solutions and sufficient conditions for the existence of unbounded positive solutions are obtained, too.

MSC:

34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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