Oscillation theorems for second order nonlinear forced differential equations
© Salhin et al.; licensee Springer. 2014
Received: 4 April 2014
Accepted: 16 May 2014
Published: 18 June 2014
In this paper, a class of second order forced nonlinear differential equation is considered and several new oscillation theorems are obtained. Our results generalize and improve those known ones in the literature.
KeywordsOscillation Forced nonlinear differential equations of second order
where r, q ∈ C([t0, ∞), ℝ), and f, ψ, g ∈ C(ℝ, ℝ) and H is a continuous function on [t0, ∞) × ℝ2,
α is a positive real number. Throughout the paper, it is assumed that the following conditions are satisfied:
(A1) r(t) > 0, t ≥ 0;
(A2) xg(x) > 0, g ∈ C1(ℝ) for x ≠ 0;
We restrict our attention only to the solutions of the differential equations (1.1) and (1.2) that exist on some ray [t0, ∞), where t0 ≥ t, to may depend on the particular solutions. Such a solution is said to be oscillatory if it has arbitrarily large zeros, and otherwise, it is said to be nonoscillatory. Equations (1.1) and (1.2) are called oscillatory if all its solutions are oscillatory.
The problem of finding oscillation criteria for second order nonlinear ordinary differential equations, which involve the average of integral of the alternating coefficient, has received the attention of many authors because in the fact there are many physical systems are modeled by second order nonlinear ordinary differential equations; for example, the so called Emden – Fowler equation arises in the study of gas dynamics and fluid mechanics. This equation appears also in the study of relativistic mechanics, nuclear physics and in the study of chemically reacting systems.
The oscillatory theory as a part of the qualitative theory of differential equations has been developed rapidly in the last decades, and there has been a great deal of work on the oscillatory behavior of differential equations; see e.g. (Agarwal et al. 2010; Beqiri and Koci 2012; Bihari 1963; Elabbasy and Elsharabasy 1997; Elabbasy and Elhaddad 2007; Grace et al. 1984, 1988; Grace and Lalli 1987, 1989, 1990; Grace 1989, 1990, 1992; Greaf and Spikes 1986; Graef et al. 1978; Lee and Yeh 2007; Kamenev 1978; Kartsatos 1968; Li and Agarwal 2000; Meng 1996; Nagabuchi and Yamamoto 1988; Ohriska and Zulova 2004; Ouyang et al. 2009; Philos 1983, 1984, 1985; Remili 2010; Salhin 2014; Tiryaki and Basci 2008; Tiryaki 2009; Temtek and Tiryaki 2013; Yan 1986; Yibing et al. 2013a, [b]; Zhang and Wang 2010).
and its special cases by using generalized Riccati transformation and well known techniques.
In this paper, we continue in this direction the study of oscillatory properties of equations (1.1) and (1.2). The purpose of this paper is to improve and extend the above mentioned results. Our results are more general than the previous results. The relevance of our results becomes clear due to some carefully selected examples.
In this section we prove our main results.
Then all solutions of equation (1.1) are oscillatory.
Taking the limit for both sides of (2.9) and using (2.4), we find w(t) → − ∞. Hence, there exists T1 ≥ T such that f(x′(t)) < 0 ⇒ x′(t) < 0, ∀t ≥ T1.
Condition (2.4) also implies that and there exists T2 ≥ T1such that
From (2.3) and 0 < x(t) ≤ x(T2), this implies that is lower bounded, but the right side of it tends to mines infinity. Then, this is a contradiction.
Evidently, if we take Then all conditions of Theorem 2.1 are satisfied, hence, all the solutions are oscillatory.
Thus all solutions of Eq. (1.1) are oscillatory.
Proof. Let x(t) be a non-oscillatory solution on [T, ∞), T ≥ T0 of Eq. (1.1). Let us assume that x(t) is positive on [T, ∞) and consider the following three cases for the behavior of x′(t).
which contradicts to the condition (2.13).
which contradicts to the assume that x′(t) oscillates.
Case 3: Let x′(t) < 0 for t ≥ T1. Condition (2.11) implies that for any t0 ≥ T1 such that
for all t ≥ T1.
The remaining part of the proof is similar to that of Theorem 2.1 then will be omitted.
Evidently, if we take Then the equation given in Example 2.2 is oscillatory by Theorem 2.2.
Remark 2.1. Condition (2.10) implies that and hence (2.11) takes the form of for all large T.
Remark 2.2. when α = 1, ψ(x(t)) = 1 and f(x′(t)) = x′(t), Theorem 2.1 and 2.2 reduce to Theorem 1 and 2 Remili (2010) and Theorems 2.1 and 2.3 are obtained by analogy with Theorems 2.1 and 2.2 from (Temtek and Tiryaki 2013).
Then the differentia Eq. (1.2) is oscillatory.
which is again a contradiction. This completes proof the Theorem 2.3.
Then the differential equation (1.2) is oscillatory.
which is again a contradiction. This completes proof the Theorem 2.6.
then, Theorem 2.4 ensures that every solution of the equation given oscillates.
This research has been completed with the support of these grants: DIP-2012-31, FRGS/2/2013/SG04/UKM/02/3 and FRGS/1/2012/SG04/UKM/01/1.
- Agarwal RP, Avramescu C, Mustafa OG: On the oscillation theory of a second-order strictly sublinear differential equation. Can Math Bull 2010, 53(2):193-203. 10.4153/CMB-2010-001-2View ArticleGoogle Scholar
- Beqiri XH, Koci E: Oscillation criteria for second order nonlinear differential equations. British Journal of Science 2012, 6(2):73-80.Google Scholar
- Bihari I: An oscillation theorem concerning the half linear differential equation of the Second order. Magyar Tud Akad Mat Kutato Int Kozl 1963, 8: 275-280.Google Scholar
- Elabbasy EM, Elhaddad WW: Oscillation of second order nonlinear differential equations with damping term. Electron J Qual Theor Differ Equat 2007, 25: 1-19.View ArticleGoogle Scholar
- Elabbasy EM, Elsharabasy MA: Oscillation properties for second order nonlinear differential equations. Kyungpook Math J 1997, 37: 211-220.Google Scholar
- Grace SR: Oscillation theorems for second order nonlinear differential equations with damping. Math Nachr 1989, 141: 117-127. 10.1002/mana.19891410114View ArticleGoogle Scholar
- Grace SR: Oscillation criteria for second order differential equations with damping. J Austral Math Soc (Series A) 1990, 49: 43-54. 10.1017/S1446788700030226View ArticleGoogle Scholar
- Grace SR: Oscillation theorems for nonlinear differential equations of second order. Math Anal And Appl 1992, 171: 220-241. 10.1016/0022-247X(92)90386-RView ArticleGoogle Scholar
- Grace SR, Lalli BS: On the second order nonlinear oscillations. Bull Inst Math Acad Sinica 1987, 15(no. 3):297-309.Google Scholar
- Grace SR, Lalli BS: Oscillation theorems for second order nonlinear differential equations with a damping term, Comment. Math Univ Carolinae 1989, 30(4):691-697.Google Scholar
- Grace SR, Lalli BS: Integral averaging technique for the oscillation of second order nonlinear differential equations. J Math Anal Appl 1990, 149: 277-311. 10.1016/0022-247X(90)90301-UView ArticleGoogle Scholar
- Grace SR, Lalli BS, Yeh CC: Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term. SIAM J Math Anal 1984, 15: 1082-1093. 10.1137/0515084View ArticleGoogle Scholar
- Grace SR, Lalli BS, Yeh CC: Addendum: Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term. SIAM J Math Anal 1988, 19(5):1252-1253. 10.1137/0519089View ArticleGoogle Scholar
- Graef JR, Rankin SM, Spikes PW: Oscillation theorems for perturbed non-linear differential equation. J Math Anal Appl 1978, 65: 375-390. 10.1016/0022-247X(78)90189-0View ArticleGoogle Scholar
- Greaf JR, Spikes PW: On the oscillatory behaviour of solutions of second order nonlinear differential equation. Czech Math J 1986, 36: 275-284.Google Scholar
- Kamenev IV: Integral criterion for oscillation of linear differential equations of second order. Math Zametki 1978, 23: 249-251.Google Scholar
- Kartsatos AG: On oscillation of nonlinear equations of second order. J Math Anal Appl 1968, 24: 665-668. 10.1016/0022-247X(68)90019-XView ArticleGoogle Scholar
- Lee CF, Yeh CC: An Oscillation theorems. Appl Math Lett 2007, 20: 238-240. 10.1016/j.aml.2006.04.005View ArticleGoogle Scholar
- Li WT, Agarwal RP: Interval oscillation criteria for second order nonlinear equations with damping. Computers Math Applic 2000, 40: 217-230. 10.1016/S0898-1221(00)00155-3View ArticleGoogle Scholar
- Meng FW: An oscillation theorem for second order superlinear differential equations. Ind J Pure Appl Math 1996, 27: 651-658.Google Scholar
- Nagabuchi Y, Yamamoto M: Some oscillation criteria for second order nonlinear ordinary differential equations with damping. Proc Japan Acad 1988, 64: 282-285.View ArticleGoogle Scholar
- Ohriska J, Zulova A: Oscillation criteria for second order nonlinear differential equation. IM Preprint Series A 2004, 10: 1-11.Google Scholar
- Ouyang Z, Zhong J, Zou S: Oscillation criteria for a class of second-order nonlinear differential equations with damping term. Abstr Appl Anal 2009, 2009: 1-12.View ArticleGoogle Scholar
- Philos CHG: Oscillation of sublinear differential equations of second order. Nonlinear Anal 1983, 7(10):1071-1080. 10.1016/0362-546X(83)90016-0View ArticleGoogle Scholar
- Philos CHG: On second order sublinear oscillation. Aequations Math 1984, 27: 242-254. 10.1007/BF02192675View ArticleGoogle Scholar
- Philos CHG: Integral averages and second order superlinear oscillation. Math Nachr 1985, 120: 127-138. 10.1002/mana.19851200112View ArticleGoogle Scholar
- Remili M: Oscillation criteria for second order nonlinear perturbed differential equations. Electron J Qual Theor Differ Equat 2010, 25: 1-11.View ArticleGoogle Scholar
- Salhin AA: Oscillation criteria of second order nonlinear differential equations with variable Coefficients. Discret Dyn Nat Soc 2014, 2014: 1-9.Google Scholar
- Temtek P, Tiryaki A: Oscillation criteria for a certain second-order nonlinear perturbed differential equations. Journal of Inequalities and Applications 2013, 524: 1-12.Google Scholar
- Tiryaki A: Oscillation criteria for a certain second-order nonlinear differential equations with deviating arguments. Electron J Qual Theor Differ Equat 2009, 61: 1-11.View ArticleGoogle Scholar
- Tiryaki A, Basci Y: Oscillation theorems for certain even-order nonlinear damped differential equations. Rocky Mt J Math 2008, 38(3):1011-1035. 10.1216/RMJ-2008-38-3-1011View ArticleGoogle Scholar
- Yan J: Oscillation theorems for second order linear differential equations with damping. Proc Amer Math Soc 1986, 98(2):276-282. 10.1090/S0002-9939-1986-0854033-4View ArticleGoogle Scholar
- Yibing S, Zhenlai H, Shurong S, Chao Z: Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales. Abstr Appl Anal 2013, 2013: 1-11.Google Scholar
- Yibing S, Zhenlai H, Shurong S, Chao Z: Fite-Wintner-Leighton-Type Oscillation Criteria for Second-Order Differential Equations with Nonlinear Damping. Abstr Appl Anal 2013, 1–10: 2012.Google Scholar
- Zhang Q, Wang L: Oscillatory behavior of solutions for a class of second-order nonlinear differential equation with perturbation. Acta Appl Math 2010, 110: 885-893. 10.1007/s10440-009-9483-8View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.