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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 758796, 5 pages
Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms
1Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, Van, Turkey
2Department of Electrical & Electronics Engineering, Yüzüncü Yıl University, Van, Turkey
Received 25 June 2013; Accepted 17 August 2013
Academic Editor: Thabet Abdeljawad
Copyright © 2013 Cemil Tunç and Muzaffer Ateş. 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.
This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.
In this paper, we study the boundedness of solutions to fourth-order nonlinear differential equation: where , , , , , , , and their first derivatives are continuous functions depending on the arguments shown. In addition, the functions and are oscillatory in the following sense: for each argument , there exist numbers such that where is either or , is either or , and all roots of the restoring term are isolated.
It should be noted that there exist many papers dealing with boundedness of solutions to certain nonlinear differential equations of third and fourth order in the literature [1–15]. For nonlinear differential equations of fourth order, Afuwape and Adesina  used the frequency-domain approach to discuss the stability and periodicity of solutions, while Tunç and Tiryaki [11, 12] used intrinsic method to study the boundedness and stability of solutions. On the same time, Tunç [13–15] used Lyapunov’s second method to investigate the stability and boundedness properties of solutions of certain fourth order nonlinear differential equations. Further, other papers in this connection include those of Andres , Ogundare , and Omeike [7, 8], where the Cauchy formula was applied to evaluate the boundedness of solutions to certain third and fourth order nonlinear differential equations with oscillatory restoring and forcing terms.
The aim of this work is to extend and improve the previous studies and make some contributions to the literature since there are only a few papers on the boundedness of solutions of fourth order differential equations with oscillatory restoring and forcing terms (see [6–8]). It should be noted that the equation considered here, (1), includes and extends that of Ogundare  and Omeike [7, 8].
2. Preliminary Results
We need the following lemmas in the proof of our main result.
Lemma 1. One assumes that there exist positive constants , , , , and , such that the following conditions hold for all and : (i),(ii),(iii), , ,(iv), , .
Then, each solution of (1) satisfies provided that
Note that the constants , , and satisfy the conditions ensuring that the auxiliary equation has negative real roots.
Proof. Substituting , we get from (1) that
with the solutions of the form
where is an arbitrary constant and is a great enough number. Let us assume that the assumptions (4) and (5) hold. Thus, by the conditions of Lemma 1, for , we have not only
This completes the proof of Lemma 1.
Lemma 2. In addition to the assumptions of Lemma 1, one assumes that the following conditions hold: (i),
where is a suitable constant. Then, every bounded solution of (1) either satisfies the relation or there exists a root of such that oscillates.
Proof. Let be a fixed bounded solution of (1). Substituting this solution into (1) and integrating the result from to (—a great enough number), we obtain the following:
By noting the assumption (ii) of Lemma 1 and the boundedness of solution , it follows that there exists a constant for such that
Now let us assume that does not converge to any root of , that is, and simultaneously,
Then, evidently is a composed monotone function with a finite or infinite limit for .
Since (13) implies that the “divergent case” can be disregarded, then it follows from (15) that not only but also holds, because otherwise if (i.e., ) (15) together with the fact that the roots of are isolated would yield which is a contradiction to (17).
Thus, the estimates (14) and (18) imply that and consequently, there exist such a sequence and a constant such that (in what follows, denotes the distance between and )(α), (β)
hold. Hence, the estimate implies that or
However, according to the assertion of Lemma 1, this case is impossible, and that is why necessarily oscillates. The remaining part of Lemma 2 is followed from the assertion where is a natural number and . This completes the proof.
Lemma 3. In addition to the assumptions of Lemma 2, one assumes that the following conditions hold: (i),
where , , and are suitable constants. Then, for every bounded solution of (1), there exists a root of such that oscillates.
Proof. If Lemma 3 does not hold, then according to Lemma 2, (11) holds and the fifth derivative of satisfies
Thus, by the assumptions of Lemmas 2 and 3, we have
Hence, by the boundedness of , , , and , it follows that there exists a constant such that which according to (24), gives the following estimates: or which is a contradiction to . This completes the proof of Lemma 3.
We now give the main result of this paper.
3. Main Result
Theorem 4. One supposes that there exist positive constants , , , , , and such that for and the following conditions hold:(i), ,(ii), , ,(iii), , (iv), , , ,(v),
where are roots of with and , denote the couple of adjacent roots of . Then, all solutions of (1) are bounded, and for each of them, there exists a root of such that oscillates.
Proof. Let us assume, on the contrary, that is an unbounded solution of (1), that is,. Then, Lemma 1 implies the existence of a number such that ,
with , , small enough constants.
Let be the last point with (-even) and the first point with . If we integrate (1) from to , , we come to
Therefore, on replacing and with and , respectively, for , we have . Multiplying (31) by , we obtain where is an arbitrary small constant, a contradiction to . The remaining part of the theorem follows from Lemma 3; therefore, we omit the details of the proof. The proof is complete.
Example 5. Consider the differential equation
where , , , , with and , for , being oscillatory, and the equation
has negative real roots. A simple calculation (with the earlier notation) gives that , , , , and . From the condition () of the theorem, since , then the roots of are , , and , , where and are the couple of adjacent roots of .
Since , then all the conditions of the theorem are satisfied; thus, all solutions of the above differential equation and their derivatives up to order three are bounded, and for each of them, there exists a root of such that oscillates.
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