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

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

Saroj Panigrahi 1,

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 $[a, \sigma(b)]_{\mathbb{T}}$.
Keywords: dynamic equations, Liapunov-type inequality, disconjugacy, time scales., oscillatory solutions.
Mathematics Subject Classification: 34C10, 34N05, 26E7.
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), 535-557.

[2]

E. Akin, Boundary value problem for a differential equation on a measure chain, Panamer. Math. J. 10 (3) (2000), 17-30.

[3]

M. Bohner, A.Peterson, " Advances in Dynamic Equations on Time Scale," Birkhaüser, 2002.

[4]

M. Bohner, S. Clark, J. Ridenhour, Liapunov inequalities for time scales, J. of Inequal. Appl.7 (1) (2002), 61-77.

[5]

R. C. Brown, D. B. Hinton, Opial's inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc. 125 (1997), 1123-1129.

[6]

S. S. Cheng, A discrete analogue of the inequality of Liapunov, Hokkaido Math J. 12 (1983), 105-112.

[7]

S. S. Cheng, Liapunov inequalities for differential and difference equations, Fasc. Math. 23 (1991), 25-41.

[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), 163-170.

[9]

S. B. Eliason, A Liapunov inequality, J. Math. Anal. Appl. 32 (1972), 461-466.

[10]

S. B. Eliason, A Liapunov inequality for a certain nonlinear differential equation, J. London Math. Soc. 2 (1970), 461-466.

[11]

S. B. Eliason, A Liapunov inequalities for certain second order functional differential equations, SIAM J. Appl. Math.27 (1) (1974), 180-199.

[12]

P. Hartman, "Ordinary Differential Equations," Wiley, New york, 1964 an Birkhaüser, Boston, 1982.

[13]

S. Hilger, Ein MaβKettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten , Ph. D. Thesis, Universität Würzburg, 1988.

[14]

A. M. Liapunov, Probleme général de la stabilité du mouvment, (French translation of Russian paper dated 1893), Ann. Fac. Sci. Univ. Toulouse, 2 (1907) 27 - 247, Reprinted as Ann. Math. Studies, 17, Prineton, 1947.

[15]

B. G. Pachpatte, On Liapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl. 195 (1995), 527-536.

[16]

N. Parhi, S. Panigrahi, On Liapunov-type inequality for third order differential equations, J. Math. Anal. Appl. 233 (2) (1999), 445-460.

show all references

References:
[1]

R. P. Agarwal, M. Bohner and Peterson, Inequalities on time scales: A Survey, Math. Ineq. Appl., 4 (2001), 535-557.

[2]

E. Akin, Boundary value problem for a differential equation on a measure chain, Panamer. Math. J. 10 (3) (2000), 17-30.

[3]

M. Bohner, A.Peterson, " Advances in Dynamic Equations on Time Scale," Birkhaüser, 2002.

[4]

M. Bohner, S. Clark, J. Ridenhour, Liapunov inequalities for time scales, J. of Inequal. Appl.7 (1) (2002), 61-77.

[5]

R. C. Brown, D. B. Hinton, Opial's inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc. 125 (1997), 1123-1129.

[6]

S. S. Cheng, A discrete analogue of the inequality of Liapunov, Hokkaido Math J. 12 (1983), 105-112.

[7]

S. S. Cheng, Liapunov inequalities for differential and difference equations, Fasc. Math. 23 (1991), 25-41.

[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), 163-170.

[9]

S. B. Eliason, A Liapunov inequality, J. Math. Anal. Appl. 32 (1972), 461-466.

[10]

S. B. Eliason, A Liapunov inequality for a certain nonlinear differential equation, J. London Math. Soc. 2 (1970), 461-466.

[11]

S. B. Eliason, A Liapunov inequalities for certain second order functional differential equations, SIAM J. Appl. Math.27 (1) (1974), 180-199.

[12]

P. Hartman, "Ordinary Differential Equations," Wiley, New york, 1964 an Birkhaüser, Boston, 1982.

[13]

S. Hilger, Ein MaβKettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten , Ph. D. Thesis, Universität Würzburg, 1988.

[14]

A. M. Liapunov, Probleme général de la stabilité du mouvment, (French translation of Russian paper dated 1893), Ann. Fac. Sci. Univ. Toulouse, 2 (1907) 27 - 247, Reprinted as Ann. Math. Studies, 17, Prineton, 1947.

[15]

B. G. Pachpatte, On Liapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl. 195 (1995), 527-536.

[16]

N. Parhi, S. Panigrahi, On Liapunov-type inequality for third order differential equations, J. Math. Anal. Appl. 233 (2) (1999), 445-460.

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