Some oscillation results for second order linear delay dynamic equations

In this paper, we extend the oscillation criteria that have been established by Hille [E. Hille, Nonoscillation theorems, Trans. Amer. Math. Soc. 64 (1948) 234-252] and Nehari [Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957) 428-445] for second-order differential equations to third-order dynamic equations on an arbitrary time scale T, which is unbounded above. Our results are essentially new even for third-order differential and difference equations, i.e., when T = R and T = N. We consider several examples to illustrate our results.

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“…Before stating our main results, we begin with the following lemma which is extracted from [10] (also see [11]). …”

Section: Resultsmentioning

confidence: 99%

“…We refer the reader to the papers [1][2][3]6,7,[9][10][11][12][22][23][24][25][26][27][28] and references cited therein. In this paper, we are concerned with the oscillatory behavior of solutions of the third-order linear dynamic equation …”

Section: Definitionmentioning

confidence: 99%

Hille and Nehari type criteria for third-order dynamic equations

Erbe

Peterson

Saker

2007

Journal of Mathematical Analysis and Applications

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“…We refer the reader to the papers [2][3][4][5][6][8][9][10][11][12][13][14][15][18][19][20][21][22][23][24] and the references cited therein. For oscillation of nonlinear delay dynamic equations, Zhang and Shanliang [24] considered the equation…”

Section: Introductionmentioning

confidence: 99%

Oscillation criteria for second-order nonlinear delay dynamic equations

Erbe

Peterson

Saker

2007

Journal of Mathematical Analysis and Applications

117

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In this paper, we consider the second-order nonlinear delay dynamic equationon a time scale T. By employing the generalized Riccati technique we will establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. The obtained results improve the well-known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations. Some applications and examples are considered to illustrate the main results.

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“…1, criterion (15) reduces to the oscillation criterion of Atkinson [2] for the superlinear Emden -Fowler differential equation (8) and that of Hooker and Patula [15] for the superlinear EmdenFowler difference equation (9).…”

Section: Superlinear Casementioning

confidence: 98%

“…Of our particular interest in this study are the papers [14] and [21]. To be more specific, in [14], Graef and Thandapani have generalized the results of Hooker and Patula [15] for the superlinear and sublinear difference equation (9) to system (4). They have also established the discrete generalization of the well-known theorem of Waltman [25] for the superlinear differential equation (8).…”

Section: Introductionmentioning

confidence: 97%