A time scales version of a Wirtinger-type inequality and applications

In this paper, we derive a time scales version of a Wirtinger-type inequality. The result is applied to derive nonoscillation of a certain second order Euler-type dynamic equation on time scales.

[1]  John Todd,et al.  Discrete analogs of inequalities of Wirtinger , 1955 .

[2]  C. Ahlbrandt,et al.  Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators , 1981, Canadian Journal of Mathematics.

[3]  D. Hinton,et al.  Oscillation theory for generalized second-order differential equations , 1980 .

[4]  C. A. Swanson.  Wirtinger’s Inequality , 1978 .

[5]  O. Doslý.  Oscillation criteria for higher order sturm–liouville difference equations * , 1998 .

[6]  P. Beesack.  Integral inequalities of the Wirtinger type , 1958 .

[7]  Discrete quadratic Wirtinger's inequalities , 1987 .

[8]  C. Swanson,et al.  Higher order Wirtinger inequalities , 1980, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[9]  Ondřej Došý.  Oscillation and Spectral Properties of a Class of Singular Self‐Adjoint Differential Operators , 1997 .

[10]  Positivity of quadratic functionals on time scales: necessity , 2001 .

[11]  D. Hinton,et al.  Discrete spectra criteria for singular differential operators with middle terms , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  R. Agarwal,et al.  Quadratic functionals for second order matrix equations on time scales , 1998 .

[13]  Martin Bohner,et al.  Asymptotic Behavior of Dynamic Equations on Time Scales , 2001 .

[14]  J. M. Davis,et al.  Eventual disconjugacy on time scales , 2000, Appl. Math. Lett..

[15]  Ravi P. Agarwal,et al.  Opial Inequalities with Applications in Differential and Difference Equations , 1995 .

[16]  R. Hilscher.  Reid Roundabout Theorem for Symplectic Dynamic Systems on Time Scales , 2001 .

[17]  On disconjugacy for vector linear Hamiltonian systems on time scales , 2000 .

[18]  J. Osička,et al.  Oscillation and Nonoscillation of Higher Order Self-Adjoint Differential Equations , 2002 .

[19]  Ondrej Doslý,et al.  A necessary and sufficient condition for oscillation of the Sturm--Liouville dynamic equation on time scales , 2002 .

[20]  R. Agarwal,et al.  On discrete and continuous wirtinger inequalities , 1998 .

[21]  Richard C. Brown.  Wirtinger Inequalities, Dirichlet Functional Inequalities, and the Spectral Theory of Linear Operators and Relations , 1981 .

[22]  Simón Peňa.  DISCRETE SPECTRA CRITERIA FOR SINGULAR DIFFERENCE OPERATORS , 2002 .

[23]  R. Hilscher,et al.  A class of Sturm-Liouville difference equations:(non)oscillation constants and property BD , 2003 .

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

[25]  O. Dosly.  Oscillation and spectral properties of self-adjoint differential operators , 1997 .

[26]  A. Peterson,et al.  Asymptotic properties of solutions of a 2nth-order differential equation on a time scale , 2000 .

[27]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[28]  R. Hilscher,et al.  Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales , 2001 .