Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 3151742, 13 pages
https://doi.org/10.1155/2018/3151742

Research Article

New Qualitative Results for Solutions of Functional Differential Equations of Second Order

Cemil Tunç and Sultan Erdur

Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, Campus, 65080, Van, Turkey

Correspondence should be addressed to Cemil Tunç; cemtunc@yahoo.com

Received 12 August 2018; Accepted 18 September 2018; Published 9 October 2018

Guest Editor: Abdul Qadeer Khan

Copyright © 2018 Cemil Tunç and Sultan Erdur. 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.

Abstract.

In this paper, we are concerned with the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second order by the second method of Lyapunov. We obtain sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations considered. We give an example for illustrations by MATLAB-Simulink, which shows the behaviors of the orbits. The findings of this paper extend and improve some results that can be found in the literature.

1. Introduction

Differential equations of second order with and without delay(s) can find a wide range of applications in atomic energy, biology, chemistry, control theory, economy, engineering technique fields, information theory, medicine, physics, population dynamics, and so forth (see Burton [1], El’sgol’ts [2], Hale [3], Krasovskii [4], Smith [5], and Yoshizawa [6]). During investigations, we would naturally be inclined to compute the solutions of differential equations of second order with and without delay(s) explicitly or numerically. However, as we know from practice, there are very few such equations, for example, linear equations with constant coefficients, but without delay(s), for which this can be effectively done. Further, it should be noted that finding analytical or explicit solutions of differential equations of second order with delay(s) is more difficult, even if, to the best of our knowledge, there is no general method in the literature to find the explicit solutions of those equations. In addition, most of the times, it is impossible to find analytical solutions for those equations. The problem therefore is to find convenient techniques that will be useful in obtaining some qualitative information such as stability, instability, convergence, global existence, integrability, boundedness of solutions, existence of periodic solutions, and so forth about the elusive solutions of ordinary or delay differential equations.

From the past till now, various methods have been constructed and are still discussed in order to investigate the various qualitative behaviors of solutions of ordinary or delay differential equations without solving those equations.

However, here, we would only like to summarize some works that can be found in the literature and methods used during the investigations of the existence of the periodic solutions, stability, asymptotic stability, square integrability, and boundedness of solutions of ordinary and functional differential equations of second order.

Yoshizawa [6] considered the following nonlinear differential equation of second order with constant delay:

and he investigated the existence of periodic solutions of this equation by using the second method of Lyapunov.

Zhao at al. [7] obtained sufficient conditions for the existence of periodic solutions of the below nonlinear differential equation of second order with constant delay

by the second method of Lyapunov.

Cong [8] considered a class of nonlinear differential equations of second order in the form

Cong [8] proved that there exists a unique periodic solution of those differential equations under Landesman–Lazer type conditions by applying the Leray–Schauder principle.

Guo and Xu [9] studied the existence of periodic solutions of a differential equation of second order with a deviating argument by means of Mawhin’s continuation theorem. In [9], a new result on the existence of periodic solutions is obtained.

Ji and Dong [10] discussed the existence and uniqueness of periodic solutions for a class of nonlinear differential equations of second order by using a comparison theorem and Leray-Schauder degree theory. The results obtained in [10] generalize and refine a recent work that can be found in the literature.

Tian and Zeng [11] studied the existence of periodic solutions to the second-order functional differential equation

by applying Mawhin’s continuation theorem of coincidence degree theory. In [11], some new results on the existence of at least two periodic solutions to this equation are obtained.

Li and Li [12] obtained existence results of positive periodic solutions for the following functional differential equation of second order with multiple variable delays:

In [12], the existence conditions concern the first eigenvalue of the associated linear periodic boundary problem and the discussion is based on the fixed-point index theory in cones.

Li and Zhang [13] established several criteria for the existence, multiplicity, and nonexistence of positive periodic solutions of the following system

by combining some new properties of Green’s function together with Krasnoselskii’s fixed-point theorem on the compression and expression of cones.

Zu [14] studied periodic solutions for the following nonlinear second-order ordinary differential equation:

By constructing upper and lower boundaries and using Leray-Schauder degree theory, the author presented a result about the existence and uniqueness of a periodic solution for the above second-order ordinary differential equation with some assumptions.

Tunç and Yazgan [15] took into consideration the following nonlinear differential equation of second order with multiple fixed delays:

and they obtained the sufficient conditions for the existence of periodic solutions of this delay equation by the second method of Lyapunov.

Ma and Lu [16] showed the existence of positive periodic solutions of the below second-order functional differential equation:

The approach in [16] is based on global bifurcation theorem.

Jia and Shao [17] established sufficient conditions for the existence and uniqueness of periodic solutions of an ordinary differential equation of second order by applying Mawhin’s continuation theorem of coincidence degree theory.

Ardjouni and Djoudi ([18, 19]) discussed the existence of periodic and positive periodic solutions for a class of nonlinear neutral differential equations of second order with variable delays by Burton-Krasnoselskii’s hybrid fixed-point theorem.

Similarly, Lü et al. [20] and Tian [21] investigated the existence of multiple positive periodic solutions for certain ordinary differential equations of second order and a delay differential equation of second order, respectively. In addition, Zhang and Wang [22] studied the existence of periodic solutions for a class of second-order functional differential equations with deviating arguments by using the abstract continuation theorem of set contractive operator and some analysis techniques.

Zhou [23] considered the existence of periodic solutions for a class of semilinear second-order differential equations of the form

By applying the viscosity solutions method and the classical upper-lower solutions method, as well as the Leray-Schauder fixed-point principle, the author established the existence of periodic solutions. The result of Zhou [23] improves and generalizes many results on the ropes mechanics equations in the existing literature.

Wei [24] proved the existence and uniqueness of periodic solutions for second-order ordinary differential equation

under some assumptions on the function . The proofs in [24] are based on Schauder’s fixed-point theorem.

Finally, more recently, Zhu and Li [25] discussed the existence of periodic solutions for the below differential equation of second order with multiple delays

by using the monotone iterative method of upper and lower solutions.

Besides, for some other related papers, one can look at the book of Yoshizawa [26], the paper of Tunç and Çinar [27], and the references that can be found in the sources mentioned above.

In fact, through the papers or books presented above, it can be seen that the second method of Lyapunov has rarely been used to investigate the existence of periodic solutions of nonlinear differential equations of second order with and without delay(s) instead of the other mentioned methods. To the best of our knowledge, the basic reason for the lack of the papers by this method is to find suitable Lyapunov function(s) or functional(s), which give(s) meaningful results. In this paper, we study the existence of the periodic solutions by defining suitable new Lyapunov functionals. This is a contribution of this paper to the subject and literature.

On the other hand, the problems of the stability, asymptotic stability, convergence, integrability, and boundedness of solutions of linear and nonlinear differential equations of second order with and without delay(s) can appear in various physical, engineering, and many other scientific models. These kinds of differential equations are significant in describing fluid mechanical, nonlinear elastic mechanical phenomena, investigation of stability and instability of geodesic on Riemannian manifolds, dynamics process in electromechanical systems of physics and engineering, and so on. Many important theoretical and applied results related to these properties of solutions of differential equations of second order with and without delay(s) can be found in the literature (see Ahmad and Rama Mohana Rao [28], Burton [18], Burton and Hering [29], El’sgol’ts [2], Hale [3], Heidel [30], Kato [31, 32], Korkmaz and Tunç [33], Krasovskii [4], Liu and Huang [34, 35], Luk [36], Malyseva [37], Muresan [38], Mustafa and Tunç [39], Napoles Valdes [40], Amano [41], Sugie et al. [42], Tunç [43–52], Tunç and Dinç [53], Tunç and Tunç [54–57], Ye et al. [58], Yoshizawa [6], Yu and Xiao [59], Yu and Zhao [60], Zhang [61], Zhou and Xiang [62], and Zhou and Jiang [63] and their references).

In the sources mentioned, the second method of Lyapunov, perturbation theory, fixed-point method or theory, iterative techniques, the variation of constants formula, and some other tools are used to investigate the mentioned qualitative behaviors of solutions of linear and nonlinear differential equations of second order with and without delay(s). Here, for the sake of brevity, we would not like to give more details about these subjects. In addition, in view of the information given above, we would like to say that it is worthwhile to continue the investigation of the qualitative properties of the solutions of nonlinear differentials of second order with delay(s).

In this paper, we consider the following functional nonlinear differential equation of second order:

where , if fixed constant delay, ,   is a continuous differentiable positive and periodic function with ;  , and are continuous functions according to their related arguments and periodic in That is, ,  , and . Finally, and are continuous differentiable functions with .

The following system can be written from (13):

To the best of our knowledge from the literature, we did not find any paper on the existence of periodic solutions, stability, asymptotic stability, square integrability, and boundedness of solutions of a mathematical model like (13). The purpose of this paper is to give new sufficient hypotheses, five theorems, with an example by MATLAB-Simulink on existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of (13) by the second method of Lyapunov. By the results of this paper, we extend and improve some results that can be found in the references of this paper (see [1–63]). These are the contributions of this paper to the mentioned topics and relevant literature.

2. Existence of Periodic Solutions

We now establish our some basic assumptions.

(A) Hypotheses. We suppose that the following hypotheses hold:

where , , , , , and are some positive real constants with and .

Our first theorem for the existence of periodic solutions of system (14) can be given below.

Theorem 1. If hypotheses hold, then system (14) has a -periodic solution.

Proof. Let be a Lyapunov functional defined by

It is obvious that the Lyapunov functional is positive definite.
By calculating the time derivative of the Lyapunov functional with respect to along system (14) and by usage of hypotheses of Theorem 1, we have

It is clear that

by hypothesis of Theorem 1. Then, from (18), we can write

by hypotheses of Theorem 1.
Let . Then, we have

by the last inequality. If and , then we can obtain

Case I. We assume that , and is a positive constant. Then, it is clear that

Suppose that and define the Lyapunov functional

Then, the time derivative of functional along system (14) is given by

In view of hypotheses of Theorem 1, we have

By hypotheses of Theorem 1, inequality (23), and the last estimate, we obtain

Since , it follows that

In view of the hypotheses for and for , it is obvious that

when .
Case II. We assume that

It is known that the function is bounded and . Then, inequality (23) holds.
We define a Lyapunov functional by

The time derivative of the Lyapunov functional along system (14) implies that

Hence, using inequality (18), we obtain

After that, in view of hypotheses of Theorem 1, it follows that

By taking into consideration the hypotheses of Theorem 1, it can be obtained that

Since for and for , it can be easily concluded that

Case III. Let

We now define a Lyapunov functional by

Then, calculating the time derivative of the functional with respect to , using inequality (18) and hypotheses of Theorem 1, we have

Case IV. Let

We define a Lyapunov functional by

Like before, we can obtain

by (18) and hypotheses of Theorem 1.
Case V. We suppose that for ,   or ,  .
We define a Lyapunov functional by

Case VI. Further, in the case of ,  , or ,  , we define Lyapunov functional by

In view of the above two cases, that is, Cases V and VI, and the hypotheses of Theorem 1, since is a positive constant, the time derivative of functional along system (14) leads to the following:

so that

Hence, it is obvious that the time derivative of functional , that is, , is negative semidefined for the above two cases. In addition, we can see that

Hence, “wedges” can be easily found (see Burton [1]), bounding the functional from bottom to top. Then, we can conclude that system (14) has a periodic solution of period (see Yoshizawa ([6, Theorem 37.2])).

3. Stability of Solutions

Let .

(B) Hypothesis. It is assumed that the following hypothesis holds:

where , , , and are some positive real constants with and

Theorem 2. If hypotheses and hold, then the zero solution of system (14) is stable.

Proof. Consider the Lyapunov functional defined by

It is clear that

and

where , by hypotheses and
Differentiating the Lyapunov functional along system (14) and considering hypotheses and , we obtain

Hence,

by hypotheses and of Theorem 2.
Let . Then, it follows that

This result completes that the zero solution of system (14) is stable.

Corollary 3. If hypotheses and hold, then the zero solution of system (14) is uniformly stable.

Theorem 4. If hypotheses and hold, then the zero solution of system (14) is asymptotically stable.

Proof. In the proof of this theorem, we benefit from the Lyapunov functional given in the proof of Theorem 2.
In the light of the hypotheses of Theorem 4, we can conclude that

Consider now the set defined by

When we apply LaSalle’s invariance principle, we observe that implies that , and hence is a constant). The last estimate and system (14), together, necessarily imply that

When , this equality can hold if and only if

Hence,

which implies that Therefore, we have . In fact, this result shows that the largest invariant set contained in is . Hence, we can conclude that the zero solution of system (14) is asymptotically stable. This completes the proof of Theorem 4.

Theorem 5. If hypotheses and hold, then the first derivatives of all solutions of (13) are square-integrable; that is, , where is the space of all Lebesgue square-integrable functions on .

Proof. Here, we also use the functional used in both theorems given just above. Notice the hypotheses of Theorem 5; we have

Integrating the last inequality from to , we have

From the above discussion, it can be seen that is positive definite and a decreasing functional. Therefore, we can say that

and hence, it is clear that

As the result of the above inequalities, we can conclude that

This result is the end of the proof of Theorem 5.

4. Boundedness of Solutions

Let .

(C) Hypothesis. It is assumed that the following hypothesis holds:

where is a nonnegative and continuous function for all such that is the space of all Lebesgue integrable functions on .

Theorem 6. If hypotheses , , and hold, then all solutions of system (13) are bounded as .

Proof. Here, once again, we use the functional just given above. Notice the hypotheses of Theorem 6; we can have

In view of hypothesis , we can get

Integrating the former inequality from to , we have

so that

By the Gronwall inequality, it follows that

Let

In addition, we also have

Hence, we can conclude that

This is the end of the proof of Theorem 6.

Example 7. We consider the following second-order nonlinear differential equation with constant delay:

When we compare this equation with (13), it follows that

It is obvious that the hypotheses of Theorem 1 are satisfied:

Therefore, the given differential equation satisfies all hypotheses of Theorem 1. Then, there exists a periodic solution of the above delay differential equation.

The orbits of the solutions of the considered delay differential equation are shown by Figures 1 and 2.

Figure 1: Trajectory of for Example 7.

Figure 2: Trajectory of for Example 7.

Remark 8. The differential equation with constant delay just given above by Example 7 can be modified and the related graphs of the orbits can be drawn by MATLAB-Simulink to verify the stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions. Here, we omit the details.

5. Conclusion

With the help of the second method of Lyapunov, the qualitative properties such as the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of a class of nonlinear differential equations of second order with constant delay are investigated. On the mentioned topics, five new theorems are proven. The obtained results include and improve some results in the literature. We give an example to verify the applicability of the results by MATLAB-Simulink, which shows the behaviors of the orbits (see Figures 1 and 2).

Data Availability

The data used to support the findings of this study are included within the article.

Disclosure

The authors confirm that the paper has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. The authors further confirm that the order of authors listed in the paper has been approved by all of them.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This research is supported by Van Yuzuncu Yil University, Van, Turkey (Grant No. FBA-2017-5868).

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