On Oscillation of a Certain Class of Third-Order Nonlinear Functional Dynamic Equations on Time Scales

In this paper, we establish some new sufficient conditions for oscillati on of the third order nonlinear functional dynamic equation h p(t) h (r(t)x ¢ (t)) ¢ i ° i¢ + q(t)f(x(?(t))) = 0, for t ∈ [t0, ∞)T, on a time scale T, where ° > 0 is the quotient of odd positive integers, p, q, r and ? are positive rd-continuous functions defined on T and f ∈ C(R, R), uf(u) > 0 and f(u)/u ° > K > 0, for u 6 0. The results provided substantial improvement over those obtained by Yu and Wang [J. Comp. Appl. Math. 225 (2009), 531-540] and Hassan [Oscillation of third order nonlinear delay dynamic equations on time scales, Math. Comp. Modelling 49 (2009), 1573-1586], in the sense that our results can be applied when 0 < ° < 1, (? ◦ ¾)(t) 6 (¾ ◦ ?)(t), and do not require that R 1 t0 q(t)¢t = ∞. Some examples illustrating the main results are given.

[1]  Martin Bohner,et al.  Oscillation criteria for perturbed nonlinear dynamic equations , 2004, Math. Comput. Model..

[2]  Qi-Ru Wang,et al.  Asymptotic behavior of solutions of third-order nonlinear dynamic equations on time scales , 2009 .

[3]  Higgins Raegan.  Some oscillation criteria for second-order delay dynamic equations , 2010 .

[4]  S. Hilger.  Analysis on Measure Chains — A Unified Approach to Continuous and Discrete Calculus , 1990 .

[5]  Samir H. Saker,et al.  Oscillation Criteria for Second‐Order Nonlinear Dynamic Equations On Time Scales , 2003 .

[6]  A. Peterson,et al.  Advances in Dynamic Equations on Time Scales , 2012 .

[7]  Samir H. Saker,et al.  Oscillation criteria for second-order nonlinear delay dynamic equations , 2007 .

[8]  Taher S. Hassan,et al.  Oscillation of third order nonlinear delay dynamic equations on time scales , 2009, Math. Comput. Model..

[9]  Samir H. Saker,et al.  Oscillation of nonlinear dynamic equations on time scales , 2004, Appl. Math. Comput..

[10]  Samir H. Saker,et al.  Oscillation of second-order nonlinear neutral delay dynamic equations on time scales , 2006 .

[11]  Vanessa Speeding,et al.  Taming nature's numbers , 2003 .

[12]  Martin Bohner,et al.  Oscillation of Second Order Nonlinear Dynamic Equations on Time Scales , 2004 .

[13]  P. Waltman,et al.  Oscillation criteria for third order nonlinear differential equations. , 1966 .

[14]  Samir H. Saker,et al.  Oscillation Criteria for Half-Linear Delay Dynamic Equations on Time Scales , 2009 .

[15]  E. Hille,et al.  Non-oscillation theorems , 1948 .

[16]  B. Baculíková,et al.  Oscillation criteria for third-order nonlinear differential equations , 2008 .

[17]  Martin Bohner,et al.  Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations. , 2007 .

[18]  Peng Shi,et al.  Oscillation Criteria for Second Order Linear Differential Equations , 1999 .

[19]  Samir H. Saker,et al.  Hille and Nehari type criteria for third-order dynamic equations , 2007 .

[20]  Samir H. Saker,et al.  Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales , 2005 .

[21]  Samir H. Saker,et al.  Oscillation Criteria for Sublinear Half-Linear Delay Dynamic Equations , 2009 .

[22]  Ravi P. Agarwal,et al.  Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales , 2008 .

[23]  A. Peterson,et al.  OSCILLATION AND ASYMPTOTIC BEHAVIOR OF A THIRD-ORDER NONLINEAR DYNAMIC EQUATION , 2007 .

[24]  Anton Škerlík.  An integral condition of oscillation for equation $y^{\prime \prime \prime }+p(t)y^{\prime }+q(t)y=0$ with nonnegative coefficients , 1995 .

[25]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .