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Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales

  • Jia Baoguo (a1), Lynn Erbe (a2) and Allan Peterson (a2)
Abstract

Consider the second order superlinear dynamic equation

where pC(, ℝ), is a time scale, ƒ : ℝ → ℝ is continuously differentiable and satisfies ƒ ′(x) > 0, and x ƒ (x) > 0 for x ≠ 0. Furthermore, f (x) also satisfies a superlinear condition, which includes the nonlinear function ƒ (x) = x α with α > 1, commonly known as the Emden–Fowler case. Here the coefficient function p(t) is allowed to be negative for arbitrarily large values of t. In addition to extending the result of Kiguradze for (∗) in the real case = ℝ, we obtain analogues in the difference equation and q-difference equation cases.

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References
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[1] Atkinson, F. V., On second order nonlinear oscillations.. Pacific J. Math. 5(1955), 643647.
[2] Baoguo, J., Erbe, L., and Peterson, A., Some new comparison results for second order linear dynamic equations.. Canad. Appl. Math. Quart. 15(2007), 349366.
[3] Baoguo, J., Erbe, L., and Peterson, A., New comparison and oscillation theorems for second order half-linear dynamic equations on time scales.. Comput. Math. Appl. 56(2008), 27442756. doi:10.1016/j.camwa.2008.05.014
[4] Bohner, M., Erbe, Lynn, and Peterson, A., Oscillation for nonlinear second order dynamic equations on a time scale.. J. Math. Anal. Appl. 301(2005), 491507. doi:10.1016/j.jmaa.2004.07.038
[5] Bohner, M. and Peterson, A., Dynamic Equation on Time Scales: An Introduction with Applications. Birkhäuser, Boston, 2001.
[6] Bohner, M. and Peterson, A., Editors, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
[7] Erbe, L., Oscillation criteria for second order linear equations on a time scale.. Canad. Appl. Math. Quart. 9(2001), 346375.
[8] Hooker, J. W. and Patula, W. T., A second order nonlinear difference equation: Oscillation and asymptotic behavior.. J. Math. Anal. Appl. 91(1983), 929. doi:10.1016/0022-247X(83)90088-4
[9] Kelley, W. and Peterson, A., Difference Equations: An Introduction with Applications. Harcourt/Academic Press, Second Edition, San Diego, 2001.
[10] Kiguradze, I. T., A note on the oscillation of solutions of the equation u″ + a(t)unsgn u = 0. Casopis Pest. Mat. 92(1967), 343350.
[11] Mingarelli, A. B., Volterra-Stieltjes Integral Equations and Generalized Differential Equations. Lecture Notes in Mathematics 989, Springer–Verlag, 989 1983.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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