# American Institute of Mathematical Sciences

2013, 2013(special): 629-641. doi: 10.3934/proc.2013.2013.629

## Liapunov-type integral inequalities for higher order dynamic equations on time scales

 1 Department of Mathematics and Statistics, University of Hyderabad, Hyderabad-500 046, India

Received  August 2012 Revised  November 2012 Published  November 2013

In this paper, we obtain Liapunov-type integral inequalities for certain nonlinear, nonhomogeneous dynamic equations of higher order without any restriction on the zeros of their higher-order delta derivatives of solutions by using time scale analysis. As an applications of our results, we show that oscillatory solutions of the equation converge to zero as $t\to \infty$. Using these inequalities, it is also shown that $(t_{m+ k} - t_{m}) \to \infty$ as $m \to \infty$, where $1 \le k \le n-1$ and $\langle t_m \rangle$ is an increasing sequence of generalized zeros of an oscillatory solution of $D^n y + y f(t, y(\sigma(t)))|y(\sigma(t))|^{p-2} = 0$, $t \ge 0$, provided that $W(., \lambda) \in L^{\mu}([0, \infty)_{\mathbb{T}}, \mathbb{R}^{+})$, $1 \le \mu \le \infty$ and for all $\lambda > 0$. A criterion for disconjugacy of nonlinear homogeneous dynamic equation is obtained in an interval $[a, \sigma(b)]_{\mathbb{T}}$.
Citation: Saroj Panigrahi. Liapunov-type integral inequalities for higher order dynamic equations on time scales. Conference Publications, 2013, 2013 (special) : 629-641. doi: 10.3934/proc.2013.2013.629
##### References:
 [1] R. P. Agarwal, M. Bohner and Peterson, Inequalities on time scales: A Survey,, Math. Ineq. Appl., 4 (2001), 535.   Google Scholar [2] E. Akin, Boundary value problem for a differential equation on a measure chain,, Panamer. Math. J. 10 (3) (2000), 10 (2000), 17.   Google Scholar [3] M. Bohner, A.Peterson, " Advances in Dynamic Equations on Time Scale,", Birkhaüser, (2002).   Google Scholar [4] M. Bohner, S. Clark, J. Ridenhour, Liapunov inequalities for time scales,, J. of Inequal. Appl.7 (1) (2002), 7 (2002), 61.   Google Scholar [5] R. C. Brown, D. B. Hinton, Opial's inequality and oscillation of 2nd order equations,, Proc. Amer. Math. Soc. 125 (1997), 125 (1997), 1123.   Google Scholar [6] S. S. Cheng, A discrete analogue of the inequality of Liapunov,, Hokkaido Math J. 12 (1983), 12 (1983), 105.   Google Scholar [7] S. S. Cheng, Liapunov inequalities for differential and difference equations,, Fasc. Math. 23 (1991), 23 (1991), 25.   Google Scholar [8] R. S. Dahiya, B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations,, J. Math. Phys. Sci. 7 (1973), 7 (1973), 163.   Google Scholar [9] S. B. Eliason, A Liapunov inequality,, J. Math. Anal. Appl. 32 (1972), 32 (1972), 461.   Google Scholar [10] S. B. Eliason, A Liapunov inequality for a certain nonlinear differential equation,, J. London Math. Soc. 2 (1970), 2 (1970), 461.   Google Scholar [11] S. B. Eliason, A Liapunov inequalities for certain second order functional differential equations,, SIAM J. Appl. Math.27 (1) (1974), 27 (1974), 180.   Google Scholar [12] P. Hartman, "Ordinary Differential Equations,", Wiley, (1964).   Google Scholar [13] S. Hilger, Ein MaβKettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten ,, Ph. D. Thesis, (1988).   Google Scholar [14] A. M. Liapunov, Probleme général de la stabilité du mouvment,, (French translation of Russian paper dated 1893), 2 (1907).   Google Scholar [15] B. G. Pachpatte, On Liapunov-type inequalities for certain higher order differential equations,, J. Math. Anal. Appl. 195 (1995), 195 (1995), 527.   Google Scholar [16] N. Parhi, S. Panigrahi, On Liapunov-type inequality for third order differential equations,, J. Math. Anal. Appl. 233 (2) (1999), 233 (1999), 445.   Google Scholar

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##### References:
 [1] R. P. Agarwal, M. Bohner and Peterson, Inequalities on time scales: A Survey,, Math. Ineq. Appl., 4 (2001), 535.   Google Scholar [2] E. Akin, Boundary value problem for a differential equation on a measure chain,, Panamer. Math. J. 10 (3) (2000), 10 (2000), 17.   Google Scholar [3] M. Bohner, A.Peterson, " Advances in Dynamic Equations on Time Scale,", Birkhaüser, (2002).   Google Scholar [4] M. Bohner, S. Clark, J. Ridenhour, Liapunov inequalities for time scales,, J. of Inequal. Appl.7 (1) (2002), 7 (2002), 61.   Google Scholar [5] R. C. Brown, D. B. Hinton, Opial's inequality and oscillation of 2nd order equations,, Proc. Amer. Math. Soc. 125 (1997), 125 (1997), 1123.   Google Scholar [6] S. S. Cheng, A discrete analogue of the inequality of Liapunov,, Hokkaido Math J. 12 (1983), 12 (1983), 105.   Google Scholar [7] S. S. Cheng, Liapunov inequalities for differential and difference equations,, Fasc. Math. 23 (1991), 23 (1991), 25.   Google Scholar [8] R. S. Dahiya, B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations,, J. Math. Phys. Sci. 7 (1973), 7 (1973), 163.   Google Scholar [9] S. B. Eliason, A Liapunov inequality,, J. Math. Anal. Appl. 32 (1972), 32 (1972), 461.   Google Scholar [10] S. B. Eliason, A Liapunov inequality for a certain nonlinear differential equation,, J. London Math. Soc. 2 (1970), 2 (1970), 461.   Google Scholar [11] S. B. Eliason, A Liapunov inequalities for certain second order functional differential equations,, SIAM J. Appl. Math.27 (1) (1974), 27 (1974), 180.   Google Scholar [12] P. Hartman, "Ordinary Differential Equations,", Wiley, (1964).   Google Scholar [13] S. Hilger, Ein MaβKettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten ,, Ph. D. Thesis, (1988).   Google Scholar [14] A. M. Liapunov, Probleme général de la stabilité du mouvment,, (French translation of Russian paper dated 1893), 2 (1907).   Google Scholar [15] B. G. Pachpatte, On Liapunov-type inequalities for certain higher order differential equations,, J. Math. Anal. Appl. 195 (1995), 195 (1995), 527.   Google Scholar [16] N. Parhi, S. Panigrahi, On Liapunov-type inequality for third order differential equations,, J. Math. Anal. Appl. 233 (2) (1999), 233 (1999), 445.   Google Scholar
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