## On oscillation and asymptotic behaviour of solutions of forced first-order neutral differential equations.(English)Zbl 0995.34058

Here, sufficient conditions are obtained under which every solution to $[y(t)\pm y(t-\tau)]'\pm Q(t)G(y(t-\sigma))=f(t), \quad t\geq 0,$ oscillates or tends to zero or to $$\pm \infty$$ as $$t\longrightarrow \infty$$. Usually, these conditions are stronger than $\int_{0}^{\infty}Q(t) dt=\infty.\tag{1}$ An example is given to show that the condition (1) is not enough to arrive at the above conclusion. The existence of a positive (or negative) solution to $[y(t)-y(t-\tau)]'+Q(t)G(y(t-\sigma))=f(t)$ is considered.

### MSC:

 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations 34K25 Asymptotic theory of functional-differential equations
Full Text:

### References:

 [1] Chuanxi, Q.; Ladas, G., Oscillation of neutral differential equations with variable coefficients, Appl. Anal., 32, 215-228 (1989) · Zbl 0682.34049 [2] Gyori, I.; Ladas, G., Oscillation Theory of Delay Differential Equations with Applications (1991), Oxford: Clarendon Press, Oxford · Zbl 0780.34048 [3] Ivanov, A. F.; Kusano, T., Oscillations of solutions of a class of first order functional differential equations of neutral type, Ukrain. Mat. Z., 51, 1370-1375 (1989) · Zbl 0694.34056 [4] Jaros, J.; Kusano, T., Oscillation properties of first order nonlinear differential equations of neutral type, Diff. Integral Eq., 5, 425-436 (1991) · Zbl 0729.34054 [5] Kitamura, Y.; Kusano, T., Oscillation and asymptotic behaviour of solutions of first order functional differential equations of neutral type, Funkcial Ekvac, 33, 325-343 (1990) · Zbl 0722.34064 [6] Ladas, G.; Sficas, Y. G., Oscillation of neutral delay differential equations, Can. Math. Bull., 29, 438-445 (1986) · Zbl 0566.34054 [7] Liu, X. Z.; Yu, J. S.; Zhang, B. G., Oscillation and nonoscillation for a class of neutral differential equations, Diff. Eq. Dynamical Systems, 1, 197-204 (1993) · Zbl 0873.34056 [8] Parhi, N.; Rath, R. N., On oscillation criteria for a forced neutral differential equation, Bull. Inst. Math. Acad. Sinica, 28, 59-70 (2000) · Zbl 0961.34059 [9] Parhi, N.; Rath, R. N., Oscillation criteria for forced first order neutral differential equations with variable coefficients, J. Math. Anal. Appl., 256, 525-541 (2001) · Zbl 0982.34057 [10] Piao, D., On an open problem by Ladas, Ann. Diff. Eq., 13, 16-18 (1997) · Zbl 0878.34066 [11] Yu, J. S., The existence of positive solutions of neutral delay differential equations, The Proceeding of Conference of Ordinary Differential Equations, 263-269 (1991), Beijing: Science Press, Beijing [12] Yu, J. S.; Wang, Z. C.; Chuanxi, Q., Oscillation of neutral delay differential equations, Bull. Austral. Math. Soc., 45, 195-200 (1992) · Zbl 0729.34052 [13] Yu, J. S.; Wang, Z. C., Asymptotic behaviour and oscillation in neutral delay difference equations, Funkcial. Ekvac., 37, 241-248 (1994) · Zbl 0808.34083
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.