Second-Order Differential Equation: Oscillation Theorems and Applications

Santra, Shyam S.; Bazighifan, Omar; Ahmad, Hijaz; Chu, Yu-Ming

doi:https://doi.org/10.1155/2020/8820066

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Applied Mathematics and Statistical Mechanics and their Applications

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Research Article | Open Access

Volume 2020 | Article ID 8820066 | https://doi.org/10.1155/2020/8820066

Second-Order Differential Equation: Oscillation Theorems and Applications

Shyam S. Santra,¹Omar Bazighifan,^2,3Hijaz Ahmad,^4,5and Yu-Ming Chu^6,7

Academic Editor: Fateh Mebarek-Oudina

Received12 Sept 2020

Revised08 Oct 2020

Accepted12 Oct 2020

Published31 Oct 2020

Abstract

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form under the assumption . Two cases are considered for and , where and are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.

1. Introduction

Differential equations (DEs) have received a lot of attention, and it is an active research area among scientists and engineers [1–8]. The DEs have ability to formulate many complex phenomena in various fields such as biology, fluid mechanics, plasma physics, fluid mechanics, and optics; many exact and numerical schemes have been being derived such as [9–15]. Differential equation of second order appears in models as well as in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations and quantum mechanics, biological, physical and chemical phenomena, optimization, mathematics of networks, and dynamical systems (see [16–24]).

In this article, we consider the differential equationwhere and are the quotient of two positive odd integers, and the functions are continuous that satisfy the conditions stated below: (A1) , , . (A2) , ; , for all ; is not identically zero in any interval . (A3) with . (A4) the existence of a differentiable function such that , for .

In [25, 26], Baculǐkovǎ and Džurina have consideredand obtained oscillation criteria for the solutions of (2) using comparison techniques when , , and . In the same technique, Džurina and Dzurina [27] have studied the oscillatory behavior of the solutions of (2) under the assumptions and . In [28], Bohner et al. have studied the oscillatory behavior of solutions of (2) under , , and . Grace et al. [29] have studied the oscillatory behavior of (2) when and and and . In [30], Ali has studied the oscillatory behavior of the solutions of (2), under the assumptions and . Karpuz and Santra [31] have studied the oscillatory behavior ofby considering the assumptions and for different ranges of the neutral coefficient .

For further work on the oscillation of this type of equations, we refer the readers to the references. Note that the majority of works consider only sufficient conditions, and merely a few consider both necessary and sufficient conditions. Hence, the objective of this work is to establish both necessary and sufficient conditions for the oscillation of solutions of (1) without using the comparison and the Riccati techniques. In this paper, we restrict our attention to the study (1), which includes the class of functional differential equations of neutral type.

Remark 1. When the domain is not specified explicitly, all functional inequalities considered in this paper are assumed to hold eventually, i.e., they are satisfied for all large enough.

2. Necessary and Sufficient Conditions

Lemma 1. Let (A1)–(A3) hold and that is an eventually positive solution of (1). Then, there exist and such thatfor .

Proof. Let be an eventually positive solution of (1). Then, by (A1), there exists a such that and for all . From (1) it follows thatTherefore, is nonincreasing for . Next, we show that is positive. By contradiction, assume that at a certain time . Using that is not identically zero on any interval and by (6), there exists such thatRecall that is the quotient of two positive odd integers. Then,Integrating from to , we haveBy (A3), the right-hand side approaches ; then, . This is a contradiction to the fact that . Therefore, for all . From being nonincreasing, we haveIntegrating this inequality from to and using that is continuous,Since , there exists a positive constant such that (4) holds.
Since is positive and nonincreasing, exists and is nonnegative. Integrating (1) from to , we haveLetting limit as , we getThen,Since , integrating the above inequality yieldsSince the integrand is positive, we can increase the lower limit of integration from to and then use the definition of to obtainwhich yields (5).

Theorem 1. Assume that there exists a constant , the quotient of two positive odd integers, such that .

If (A1)–(A3) hold, then each solution of (1) is oscillatory if and only if

Proof. On the contrary, we assume that is eventually positive solution. So, Lemma 1 holds, and then there exists such thatwhereComputing the derivative of , we haveThus, is nonnegative and nonincreasing. Since , by (A2), it follows that cannot be identically zero in any interval ; thus, cannot be identically zero, and cannot be constant on any interval . Therefore, for . Computing the derivative, we haveIntegrating (21) from to and using that , we haveNext, we find a lower bound for the right-hand side of (25), independent of the solution . By (4) and (19), we haveSince is nonincreasing, , and , it follows thatGoing back to (22), we haveSince , by (17) the right-hand side approaches as . This contradicts (25) and completes the proof of sufficiency for eventually positive solutions.
The eventually negative solution can be dealt similarly by introducing the variables .
Next, we show the necessity part by a contrapositive argument. If (17) does not hold, then for each there exists such thatfor all . We define the set of continuous functionsWe define an operator on byNote that when is continuous, is also continuous on . If is a fixed point of , i.e., , then is a solution of (1).
First, we estimate from below. By (A3), we haveNow, we estimate from above. For in , we have . Then, by (26),Therefore, maps to .
Next, we find a fixed point for in . Let us define a sequence of functions in by the recurrence relationNote that for each fixed , we have . Using mathematical induction, we can show that . Therefore, the sequence converges pointwise to a function . Using the Lebesgue dominated convergence theorem, we can show that is a fixed point of in . This shows under assumption (26), there is a nonoscillatory solution that does not converge to zero. This completes the proof.

Theorem 2. Assume that there exists a constant , the quotient of two positive odd integers such that . If (A1)–(A4) hold and is nondecreasing, then each solution of (1) is oscillatory if and only if

Proof. On the contrary, we assume that is an eventually positive solution that does not converge to zero. Using the same argument as in Lemma 1, there exists such that and is positive and nonincreasing. Since , is increasing for . Using , we haveand henceUsing (34) and , from (13), we havefor . From being nonincreasing and , we haveWe use this in the left-hand side of (35). Then, dividing by and raising both sides to the power, we havefor . Multiplying the left-hand side by and integrating from to , we haveOn the left-hand side, since , integrating, we haveOn the right-hand side of (38), we use that to conclude that (32) implies the right-hand side approaching , as , which is a contradiction. Hence, the solution cannot be eventually positive.
For eventually negative solutions, we use the same change of variables as in Theorem 1 and proceed as above.
To prove the necessity part, we assume that (32) does not hold and obtain an eventually positive solution that does not converge to zero. If (32) does not hold, then for each there exists such thatWe define the set of continuous functionThen, we define the operatorNote that if is continuous, is also continuous at . Also, note that if , then is solution of (1).
First, we estimate from below. Let , we have , on .
Now, we estimate from above. Let . Then, and by (40), we haveTherefore, maps to . To find a fixed point for in , we define a sequence of functions by the recurrence relationNote that for each fixed , we have . Using mathematical induction, we can prove that . Therefore, converges pointwise to a function in . Then, is a fixed point of and a positive solution of (1). The proof is completed.

Example 1. Consider the differential equationsHere, , , , , and . For , we have . To check (17), we haveSo, every conditions of Theorem 1 hold true, and therefore, all solutions of (45) are oscillatory or converge to zero.

Example 2. Consider the differential equationsHere, , , , and . For , we have . To check (32), we haveSo, every conditions of Theorem 2 hold true. Thus, all solutions of (47) are oscillatory or converge to zero.

3. Conclusion

The aim of this work is to establish necessary and sufficient conditions for the oscillation of solution to second-order half-linear differential equation. The obtained oscillation theorems complement the well-known oscillation results present in the literature. This work, as well as [31–41], leads us to pose an open problem: Can we find necessary and sufficient conditions for the oscillation of solutions to second-order differential equation

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).

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Copyright © 2020 Shyam S. Santra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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