### Necessary and Sufficient Conditions for Oscillations to a Second-Order Neutral Differential Equations with Impulses

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**Authors:**A. K. TRIPATHY AND S. S. SANTRA

**DOI:**10.46793/KgJMat2301.081T

**Abstract:**

In this work, we obtain necessary and suﬃcient conditions for oscillation of solutions of second-order neutral impulsive diﬀerential system

where z(t) = x(t) + p(t)x(τ(t)). Under the assumption
∫
_{0}^{∞}r(η)^{−1∕γ}dη = ∞, we consider two cases when γ > α_{i} and
γ < α_{i}. Our main tool is Lebesgue’s Dominated Convergence theorem.
Examples are given to illustrate our main results and we state an open
problem.

**Keywords:**

Oscillation, non-oscillation, neutral, delay, Lebesgue’s dominated convergence theorem, impulses.

**References:**

[1] D. D. Bainov and P. S. Simeonov, Systems with Impulse Eﬀect: Stability, Theory and Applications, Ellis Horwood, Chichester, 1989.

[2] D. D. Bainov and V. Covachev, Impulsive Diﬀerential Equations with a Small Parameter, World Scientiﬁc Publishers, Singapore, 1994.

[3] D. D. Bainov and P. S. Simeonov, Theory of Impulsive Diﬀerential Equations: Asymptotic Properties of the Solutions and Applications, World Scientiﬁc Publishers, Singapore, 1995.

[4] D. D. Bainov and V. Covachev, Impulsive Diﬀerential Qquations: Asymptotic Properties of the Solutions, World Scientiﬁc Publishers, Singapore, 1995.

[5] D. D. Bainov and M. B. Dimitrova, Oscillatory properties of the solutions of impulsive diﬀerential equations with a deviating argument and nonconstant coeﬃcients, Rocky Mountain J. Math. 27 (1997), 1027–1040.

[6] D. D. Bainov, M. B. Dimitrova and A. B. Dishliev, Oscillation of the solutions of impulsive diﬀerential equations and inequalities with a retarded argument, Rocky Mountain J. Math. 28 (1998), 25–40.

[7] L. Berezansky and E. Braverman, Oscillation of a linear delay impulsive diﬀerential equations, Comm. Appl. Nonlinear Anal. 3 (1996), 61–77.

[8] M.-P. Chen, J. S. Yu and J. H. Shen, The persistence of nonoscillatory solutions of delay diﬀerential equations under impulsive perturbations, Comput. Math. Appl. 27 (1994), 1–6.

[9] A. Domoshnitsky and M. Drakhlin, Nonoscillation of ﬁrst order impulse diﬀerential equations with delay, J. Math. Anal. Appl. 206 (1997), 254–269.

[10] A. Domoshnitsky, M. Drakhlin and E. Litsyn, On boundary value problems for n-th order functional diﬀerential equations with impulses, Adv. Math. Sci. Appl. 8(2) (1998), 987–996.

[11] M. B. Dimitrova and D. Mishev, Oscillation of the solutions of neutral impulsive diﬀerential-diﬀerence equations of ﬁrst order, Electron. J. Qual. Theory Diﬀer. Equ. 16 (2005), 1–11.

[12] M. B. Dimitrova and V. I. Donev, Suﬃcient conditions for the oscillation of solutions of ﬁrst-order neutral delay impulsive diﬀerential equations with constant coeﬃcients, Nonlinear Oscillations 13(1) (2010), 17–34.

[13] M. B. Dimitrova and V. I. Donev, Oscillatory properties of the solutions of a ﬁrst order neutral nonconstant delay impulsive diﬀerential equations with variable coeﬃcients, Int. J. Pure Appl. Math. 72(4) (2011), 537–554.

[14] M. B. Dimitrova and V. I. Donev, Oscillation criteria for the solutions of a ﬁrst order neutral nonconstant delay impulsive diﬀerential equations with variable coeﬃcients, Int. J. Pure Appl. Math. 73(1) (2011), 13–28.

[15] M. B. Dimitrova and V. I. Donev, On the nonoscillation and oscillation of the solutions of a ﬁrst order neutral nonconstant delay impulsive diﬀerential equations with variable or oscillating coeﬃcients, Int. J. Pure Appl. Math. 73(1) (2011), 111–128.

[16] A. Domoshnitsky, G. Landsman and S. Yanetz, About positivity of Green’s functions for impulsive second order delay equations, Opuscula Math. 34(2) (2014), 339–362.

[17] S. S. Santra and A. K. Tripathy, On oscillatory ﬁrst order nonlinear neutral diﬀerential equations with nonlinear impulses, J. Appl. Math. Comput. 59(1-2) (2019), 257–270.

[18] S. S. Santra, On oscillatory second order nonlinear neutral impulsive diﬀerential systems with variable delay, Adv. Dyn. Syst. Appl. 13(2) (2018), 176–192.

[19] A. K. Tripathy, Oscillation criteria for a class of ﬁrst order neutral impulsive diﬀerential-diﬀerence equations, J. Appl. Anal. Comput. 4 (2014), 89–101.

[20] A. K. Tripathy and S. S. Santra, Necessary and suﬃcient conditions for oscillation of a class of ﬁrst order impulsive diﬀerential equations, Funct. Diﬀer. Equ. 22(3-4) (2015), 149–167.

[21] A. K. Tripathy, S. S. Santra and S. Pinelas, Necessary and suﬃcient condition for asymptotic behaviour of solutions of a class of ﬁrst-order impulsive systems, Adv. Dyn. Syst. Appl. 11(2) (2016), 135–145.

[22] A. K. Tripathy and S. S. Santra, Oscillation properties of a class of second order impulsive systems of neutral type, Funct. Diﬀer. Equ. 23(1-2) (2016), 57–71.

[23] A. K. Tripathy and S. S. Santra, Characterization of a class of second order neutral impulsive systems via pulsatile constant, Diﬀer. Equ. Appl. 9(1) (2017), 87–98.