Oscillation Criteria of Second-Order Nonlinear Differential Equations with Variable Coefficients

Abstract

Some new oscillation criteria are given for second-order nonlinear differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literatures. Some examples are considered to illustrate the main results.

1. Introduction

Since many problems in physics, chemically reacting systems, celestial mechanics, and others fields are modeled by second-order nonlinear differential equations, the oscillatory and asymptotic behaviors of solutions of such differential equations have been investigated by many authors [1–9]. Investigation of nonlinear differential equation in this paper is motivated by [1], where some of the conditions required in the theorems contain the unknown solution . It seems that any verification of such conditions is questionable.

In this paper, we consider the oscillation of second-order nonlinear forced differential equation where and and is a continuous function on .

Consider the following with respect to (1):(A₁), ;(A₂), for all ;(A₃), for ;(A₄) for all ; and .We say that a nonzero function , is called a nontrivial solution of (1) if satisfies (1) for all . A solution of (1) is called oscillatory if there exists a sequence of points in the interval , so that and , ; otherwise it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.

We note that if and , then (1) becomes the differential equation of the form The oscillation of solutions of (2) has been studied by El-sheikh [1], Manojlovic [10], and Ohriska and Zulova [5] under different conditions. Also, if , and , then (1) becomes the differential equation of the form

The oscillation of solutions of (3) has been studied by Remili [7, 8], where new results with additional suitable weighted function are found. Recently, Temtek and Tiryaki [9] obtained several new oscillation criteria for (3) with by using generalized Riccati transformation and well-known techniques, such as averaging technique.

The purpose of this research is to remove the above mentioned conditions which depend on the solution and extend some results presented in [1, 5, 7, 8].

Although Grace [2] considered the function defined by , for some constant and there is a positive continuous function on such that , Kirane and Rogovchenko [11] and Tiryaki and Ayanlar [12] also mentioned that they can obtain new oscillation criteria with the a particular choice of the functions and with , and , , and Li [13] defined , , ,  . However, according to Ohriska and Zulova [5], research which involves oscillatory criteria based on the weight function has increased. These results have great theoretical value but they are less effective in applications. On the other hand, the results which contain the requirements only on the functions occurring in differential equation are usually better applicable. In particular, our weight function is not strictly Philos-type function because in the present paper, we ignored Philos’s conditions (see [2, 6, 10, 11, 14–17]. This paper contains only results of the latter kind by restricting only the functions in the differential equation (1). The relevance of the theorems in this paper is illustrated by examples.

2. Main Results

In this section we give some new sufficient conditions for all solutions of (1) to be oscillatory. We introduce the following notation: By Theorems 1 and 3, the given conditions include Kamenev’s integral criteria.

Theorem 1. One assumes that there are positive constants and satisfying Furthermore let, for some integer , Then (1) is oscillatory.

Proof. Assume, for the sake of a contradiction, that (1) has an eventually positive solution , for all . The case being eventually negative can be similarly discussed. Set Then by (A₄), (1), and (5), we have Hence, for all , it can be written as Then (9) implies that Then we have It is clear that, for , it holds Combining (11) and (12), it follows that It is observable that . Thus which contradicts condition (6). The proof of the theorem is completed.

Example 2. Consider the differential equation Clearly, and, for any integer , we have By Theorem 1, this equation is oscillatory.

Theorem 3. Assume that (5) holds and suppose that for some . Then (1) is oscillatory.

Proof. Assume that (1) has an eventually positive solution , for all . Taking as it is defined in (7), we obtain (8).
Hence by (A₃), for , we can write That is, Hence it follows that Using the inequality (21), for , we obtained Thus, and it contradicts (18).

Example 4. Consider the differential equation which satisfies the conditions of Theorem 3, since we have and, for any integer , we have It thus follows that every solution of the given equation oscillates.

Theorem 5. Suppose that Let be a positive continuously differentiable function over such that over , Then all solutions of (1) are oscillatory.

Proof. Let be a nonoscillatory solution on , of (1). We assume that is positive on . A similar argument holds for the case when is negative. Let Then Using (1), we obtain Since and condition (A₄) we obtained, From (27) and (30), Therefore, Multiplying by and integrating from to , we obtained Let By integrating by parts, we obtained The inequality presented by (38) can be written as Let , Taking the limit for both sides of (40) and using (29), Hence, there exists such that Condition (29) also implies that , and there exists such that Multiplying (1) by and integrating by parts on , we obtained which implies Now, integrating by parts, we get where .
Therefore, From (27), we have Finally, from (28), , which is a contradiction.

Example 6. Let us consider the following equation: We note that Hence, Let For every , we obtained Then, Theorem 5 ensures that every solution in this example is oscillatory.

Theorem 7. If conditions (27)–(29) hold and thus all solutions of (1) are oscillatory.

Remark 8. Condition (55) implies that and ; hence, (56) takes the form of , for all large .

Proof. Let be a nonoscillatory solution on , of (1). Let us assume that is positive on and consider the following three cases for the behavior of .
Case 1. Consider for for some ; then, from (40), we have From (27), we obtained Hence, for all , Using (57), we obtained This contradicts condition (56).
Case 2. If is oscillatory, then there exists a sequence on such that . Let us assume that is sufficiently large, so that Then, from (27) and (40), we have Thus, By (55), we obtained which contradicts the fact that oscillates.
Case 3. Let for for some ; then, for any , there exists such that for all . Choosing , and multiplying (1) by and integrating by parts, we obtained where .
Thus, From (27), we obtained From (28), it follows that , as , which is a contradiction.

Example 9. Consider the following equation: We note that hence, Let ; then. Because , Thus, from Theorem 7, it follows that the equation is oscillatory.

Remark 10. If we let and in our Theorems 5 and 7, we will get Theorems 1 and 2 of Remili [7] which extends the results of Graef et al. [3].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research has been completed with the support of these Grants: FRGS/1/2012/SG04/UKM/01/1, DIP-2012-31, and FRGS/2/2013/SG04/UKM/02/3.