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Oscillation of third order nonlinear delay dynamic equations on time scales

Authors
Journal
Mathematical and Computer Modelling
0895-7177
Publisher
Elsevier
Publication Date
Volume
49
Identifiers
DOI: 10.1016/j.mcm.2008.12.011
Keywords
  • Oscillation
  • Delay Nonlinear Dynamic Equations
  • Time Scales

Abstract

Abstract It is the purpose of this paper to give oscillation criteria for the third order nonlinear delay dynamic equation ( a ( t ) { [ r ( t ) x Δ ( t ) ] Δ } γ ) Δ + f ( t , x ( τ ( t ) ) ) = 0 , on a time scale T , where γ ≥ 1 is the quotient of odd positive integers, a and r are positive r d -continuous functions on T , and the so-called delay function τ : T → T satisfies τ ( t ) ≤ t for t ∈ T and lim t → ∞ τ ( t ) = ∞ and f ∈ C ( T × R , R ) . Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when T = R and T = N involve and improve some oscillation results for third order delay differential and difference equations; when T = h N , T = q N 0 and T = N 2 our oscillation results are essentially new. Some examples are given to illustrate the main results.

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