Oscillation criteria for second order neutral impulsive differential equations with a forcing term

This paper considers a class of second order nonlinear neutral impulsive differential equations with a forcing term, having fixed moments of impulse actions. Conditions for forced oscillation of all solutions are obtained and examples are given to illustrate the relevance of the results.

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Introduction

This work aims at obtaining conditions for the oscillation of all solutions of the second-order nonlinear neutral impulsive differential equation with a forcing term of the form

We assume throughout this discussion and without further mention, the following conditions: C1.1: p, 0   and 0;

And u changes sign on   T, for any

Second order differential equations are generally the most important in applications. Same also applies to neutral second order delay impulsive differential equations which have been developed to model impulsive problems in physics, population dynamics, biotechnology, pharmacokinetics, industrial robotics, and so forth. A few recent results can be seen in [17][18][19][20][21] .

The introduction of oscillation theory has further boosted the concept and particularly helped in identifying more areas of applications both within and outside differential equations.

It is worth mentioning here that a lot of work has been done in the area of forced oscillations for neutral/non-neutral differential equations, with or without delay [13,15,14,8,9,10,16,6] . Some results have been obtained for forced oscillations of differential equations, with or without delay, subject to impulse actions [11,12,4] . However, there seems to be a dearth of the study of forced oscillation theory of second-order nonlinear impulsive delay ordinary differential equations of neutral type. This could be due to the complications that arise as a result of the momentary perturbations or impulsive jumps. Motivated by this, we seek sufficient conditions for the oscillation of all solutions of equation (1.1). The above ideology becomes more meaningful if we define other related terms and concepts that will continue to be useful as we progress.

In ordinary differential equations, the solutions are continuously differentiable, sometimes at least once, whereas impulsive differential equations generally possess non-continuous solutions. Since the continuity properties of the solutions play an important role in the analysis of the behaviour, the techniques used to handle the solutions of impulsive differential equations are fundamentally different, including the definitions of some of the basic terms. In this section, we examine some of these changes.

  be a strictly ascending sequence of the time moments of impulse effects and let E be a subscript set which can be the set of natural numbers N or the set of integers Z such that

Where 1k    . In order to simplify the statements of the assertions, we introduce the set of functions PC and r PC which are defined as follows:

  and let the set of impulse points S be fixed.

Definition 1.2:

From the studies in Bainov and Simeonov (1998), Lakshmikantham et al (1989) and Isaac et al. (2011) [1,7,4] , we define the function space r PC rN  :

To specify the points of discontinuity of functions belonging to PC and r PC , we shall sometimes Use the symbols PC( D,R;S ) and

Definition 1. 4 The solution y( t ) of the impulsive differential equation (1.1) is said to be 1. finally positive (finally negative) if there exist T0  such that y( t) is defined and is strictly positive (negative) for tT  [3] ; 2. non-oscillatory, if it is either finally positive or finally negative; and 3. Oscillatory, if it is neither finally positive nor finally negative [1,5] . Definition 1.5: An impulsive differential equation is said to be oscillatory if all its solutions are oscillatory. The following is a basic lemma that is essential in carrying out our investigation. It is an extension of Lemma 4.7.1 of the work by Erbe et al. [2] . It is noteworthy to mention here that, in the sequel, all functional inequalities are assumed to hold finally for all t large enough. Again, without loss of generality, we will deal only with the positive solutions of equation (2.1).

Lemma 1.1:

This completes the proof of Lemma 1.1

Main Results

The following theorems are an extension of their neutral delay versions as identified in the work by Erbe et al. [2] on pages 273 and 274. and

For every closed set E whose intersection with every segment of the form   Proof: Without loss of generality, let us assume by contradiction that y( t ) is a finally positive solution of equation (1.1) for

On the other hand, it is easy to show that     z t u t 0  finally. Then we have

By Lemma 1.1, for every * 0 t t 2  , there exists a set

Which contradicts assumption (2.1). This completes the proof of Theorem 3.1.

We see this analogy in the following illustration:

We can see that condition (2.1) holds. Therefore, every solution of equation (2.3) is oscillatory. In fact, y(t ) sint  is such a solution.

We now consider equation (1.1) in the more general form as

Then every solution of equation (2.4) is oscillatory.

Proof: Without loss of generality, let us assume by contradiction that y( t ) is a finally positive solution of equation (2.4). Set

1. Bainov DD, Simeonov PS. Oscillation Theory of Impulsive Differential Equations, International Publications Orlando, Florida, 1998.
2. Erbe LH, Kong Q, Zhang BG. Oscillation Theory for Functional Differential Equations, Dekker, New York, 1995.
3. Isaac Lipcsey Z. Linearized Oscillations in Nonlinear Neutral Delay Impulsive Differential Equations, Journal of Modern Mathematics and Statistics. 2009; 3:1-7.
4. Isaac IO, Lipcsey Z. Oscillations in Linear Neutral Delay Impulsive Differential Equations with Constant Coefficients, Communications in Applied Analysis. 2010; 14:123-136.
5. Isaac IO, Lipcsey Z, Ibok UJ. Linearized Oscillations in Autonomous Delay Impulsive Differential Equations, British Journal of Mathematics and Computer Science. 2010; 4(21):3068-3076, 2014.
6. Jiang Y, Fu Y, Wang H, Li T. Oscillation of forced second-order neutral delay differential equations, Adv. Difference Equ., 2015, 1-5.
7. Lakshmikantham V, Bainov DD, Simeonov PS. Theory of Impulsive Differential Equations, World Scientific Publishing Co. Pte. Ltd. Singapore, 1989.
8. Li T, Han Z, Sun S, Zhang C. Forced oscillation of second-order nonlinear dynamic equations on time scales, Electron. J. Qual. Theory Differ. Equ. 2009, 1-8.
9. Li T, Rogovchenko YV, Zhang C. Oscillation of second-order neutral differential equations, Funkcial. Ekvac. 2013; 56:111120.
10. Li T, Rogovchenko YV. Oscillation of second-order neutral differential equations, Math. Nachr. 2015; 288:1150-1162.
11. Ozbekler A, Zafer A. Interval criteria for forced oscillation of super-half-linear differential equations under impulse effect, Math. Comput. Modelling. 2009; 50:59-65.
12. Ozbekler A, Zafer A. Oscillation of solutions of second order mixed nonlinear differential equations under impulsive perturbations, Comput. Math. Appl. 2011; 61:933-940.
13. Sun YG, Kong Q. Interval criteria for forced oscillation with nonlinearities given by riemann-stieltjes integrals, Computers and Mathematics with Applications. 2011; 62:243-252.
14. Sun YG, Meng FW. Interval criteria for oscillation of second order differential equations with mixed nonlinearities, Appl. Math. Comp. 2008; 198:375-381.
15. Sun YG, Wong JS. Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 2007; 334:549-560.
16. Wang Y, Li T, Thandapani E. Forced oscillation of second-order differential equations with mixed nonlinearities, Journal of Inequalities and Applications. 2014, 1-9.
17. Esuabana IM, Abasiekwere UA. On stability of first order linear impulsive differential equations, International Journal of Statistics and Applied Mathematics. 2018; 3(3C):231-236.
18. Abasiekwere UA, Esuabana IM, Isaac IO, Lipscey Z. Oscillations of Second order Impulsive Differential Equations with Advanced Arguments, Global Journal of Science Frontier Research: Mathematics and Decision Sciences (USA). 2018; 18(1):25-32.
19. Abasiekwere UA, Esuabana IM. Isaac IO, Lipscey Z. Classification of Non-Oscillatory Solutions of Nonlinear Neutral Delay Impulsive Differential Equations, Global Journal of Science Frontier Research: Mathematics and Decision Sciences (USA). 2018; 18(1):49-63.
20. Esuabana IM, Abasiekwere UA. Oscillation Theorem for Second Order Neutral Delay Differential Equations with Impulses, International Journal of Mathematics Trends and Technology. 2017; 52(5):330-333.