Oscillatory and asymptotic behaviour of a neutral differential equation with oscillating coefficients

J. G. Dix, Texas State University, San Marcos, Texas, U. S. A
N. Misra, Berhampur University, Berhampur, Orissa, India
L. N. Padhy, K.I.S.T., Bhubaneswar, Orissa, India
R. N. Rath, Khallikote Autonomous College, Berhampur, Orissa, India

E. J. Qualitative Theory of Diff. Equ., No. 19. (2008), pp. 1-10.

Abstract: In this paper, we obtain sufficient conditions so that every solution of
$$
\big(y(t)- \sum_{i=1}^n p_i(t) y(\delta_i(t))\big)'+\sum_{i=1}^m q_i(t) y(\sigma_i(t)) = f(t)
$$
oscillates or tends to zero as $t \to \infty$. Here the coefficients $p_i(t), q_i(t)$ and the forcing term $f(t)$ are allowed to oscillate; such oscillation condition in all coefficients is very rare in the literature. Furthermore, this paper provides an answer to the open problem 2.8.3 in [7, p. 57]. Suitable examples are included to illustrate our results.

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