Nonlinear oscillation of first order delay differential equations. (English) Zbl 0852.34064
In the first section of the paper, the author considers the nonlinear delay differential equation (1) \(y'(t)+ p(t) f(y(\sigma(t)))= q(t)\) under the following hypotheses: \(p(t)\in C([t_0, \infty); [0, \infty))\), \(q(t)\in C([t_0, \infty); \mathbb{R}^1)\); \(f(s)\in C(\mathbb{R}^1; \mathbb{R}^1)\), \(f(s)\geq 0\), for \(s\geq 0\), \(f(- s)= - f(s)\) for \(s> 0\), and \(f(s)\) is nondecreasing for \(s\geq 0\); \(\sigma(t)\in C([t_0, \infty); \mathbb{R}^1)\), \(\lim_{t\to \infty}\sigma(t)= \infty\), \(\sigma(t)\leq t\) for \(t\geq t_0\), and \(\sigma(s)\) is nondecreasing for \(s\geq 0\). Sufficient conditions are obtained under which all solutions of (1) are oscillatory.
In the second section the more general equation \[ y'(t)+ a(t) y(t)+ \sum^k_{i= 1} b_i(t) y(\rho_i(t))+ p(t) f(y(\sigma(t)))= q(t)\tag{2} \] is considered. Applying the results on equation (1), oscillation criteria for equation (2) are established. Examples are also given that illustrate the main results of the author.
In the second section the more general equation \[ y'(t)+ a(t) y(t)+ \sum^k_{i= 1} b_i(t) y(\rho_i(t))+ p(t) f(y(\sigma(t)))= q(t)\tag{2} \] is considered. Applying the results on equation (1), oscillation criteria for equation (2) are established. Examples are also given that illustrate the main results of the author.
Reviewer: V.Petrov (Plovdiv)
MSC:
34K11 | Oscillation theory of functional-differential equations |
34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
References:
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