Oscillation criteria for half-linear dynamic equations on time scales

This paper is concerned with oscillation of the second-order half- linear delay dynamic equation (r(t)(x � )) � + p(t)x(τ(t)) = 0, on a time scale T, where γ ≥ 1 is the quotient of odd positive integers, p(t), and τ : T → T are positive rd-continuous functions on T, r(t) is positive and (delta) differentiable, τ(t) ≤ t, and limt→∞ τ(t) = ∞. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results in the special cases when T = R and T = N involve and improve some oscillation results for second-order differential and difference equations; and when T = hN, T = q N 0 and T = N 2 our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.

[1]  Ravi P. Agarwal,et al.  Oscillation of Second Order Delay Dynamic Equations , 2005 .

[2]  I. V. Kamenev,et al.  An integral criterion for oscillation of linear differential equations of second order , 1978 .

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

[4]  C. Philos,et al.  Oscillation theorems for linear differential equations of second order , 1989 .

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

[6]  William Benjamin Fite,et al.  Concerning the zeros of the solutions of certain differential equations , 1918 .

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

[8]  Ravi P. Agarwal,et al.  Philos-Type Oscillation Criteria for Second Order Half-Linear Dynamic Equations on Time Scales , 2007 .

[9]  Samir H. Saker,et al.  Oscillation criteria of second-order half-linear dynamic equations on time scales , 2005 .

[10]  Qi-Ru Wang.  Oscillation and asymptotics for second-order half-linear differential equations , 2001, Appl. Math. Comput..

[11]  Zeev Nehari,et al.  Oscillation criteria for second-order linear differential equations , 1957 .

[12]  John R. Graef,et al.  Oscillation and comparison theorems for half-linear second-order difference equations , 2001 .

[13]  Aurel Wintner,et al.  ON THE NON-EXISTENCE OF CONJUGATE POINTS.* , 1951 .

[14]  Ravi P. Agarwal,et al.  Best possible gronwall inequalities , 1997 .

[15]  Ján Ohriska,et al.  Oscillation of second order delay and ordinary differential equation , 1984 .

[16]  W. Leighton.  On Self-Adjoint Differential Equations of Second Order. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Lynn Erbe,et al.  Oscillation Criteria for Second Order Nonlinear Delay Equations , 1973, Canadian Mathematical Bulletin.

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

[19]  F. V. Atkinson,et al.  On second-order non-linear oscillations , 1955 .

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

[21]  O. Doslý.  Qualitative theory of half-linear second order differential equations , 2002 .

[22]  J. Wong.  SECOND ORDER OSCILLATION WITH RETARDED ARGUMENTS , 1972 .

[23]  J. Yan.  A NOTE ON AN OSCILLATION CRITERION FOR AN EQUATION WITH DAMPED TERM , 1984 .

[24]  P. Waltman.  An oscillation criterion for a nonlinear second order equation , 1965 .

[25]  Ravi P. Agarwal,et al.  Oscillation criteria for second-order retarded differential equations , 1997 .

[26]  J. V. Manojlovi,et al.  Oscillation criteria for second-order half-linear differential equations , 1999 .