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November  2016, 15(6): 2281-2300. doi: 10.3934/cpaa.2016037

Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities

1. 

Department of Mathematics, Texas A\&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202 , United States

2. 

Department of Mathematics, Atilim University 06836, Incek, Ankara

Received  February 2016 Revised  June 2016 Published  September 2016

In the case of oscillatory potentials, we present Lyapunov type inequalities for $n$th order forced differential equations of the form \begin{eqnarray} x^{(n)}(t)+\sum_{j=1}^{m}q_j(t)|x(t)|^{\alpha_j-1}x(t)=f(t) \end{eqnarray} satisfying the boundary conditions \begin{eqnarray} x(a_i)=x'(a_i)=x''(a_i)=\cdots=x^{(k_i)}(a_i)=0;\qquad i=1,2,\ldots,r, \end{eqnarray} where $a_1 < a_2 < \cdots < a_r$, $0\leq k_i$ and \begin{eqnarray} \sum_{j=1}^{r}k_j+r=n;\qquad r\geq 2. \end{eqnarray} No sign restriction is imposed on the forcing term and the nonlinearities satisfy \begin{eqnarray} 0 < \alpha_1 < \cdots < \alpha_j < 1 < \alpha_{j+1} < \cdots < \alpha_m < 2. \end{eqnarray} The obtained inequalities generalize and compliment the existing results in the literature.
Citation: Ravi P. Agarwal, Abdullah Özbekler. Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2281-2300. doi: 10.3934/cpaa.2016037
References:
[1]

R. P. Agarwal, Boundary value problems for higher order integro-differential equations,, \emph{Nonlinear Anal.}, 7 (1983), 259. doi: 10.1016/0362-546X(83)90070-6. Google Scholar

[2]

R. P. Agarwal, Some inequalities for a function having $n$ zeros. General inequalities, 3 (Oberwolfach, 1981), 371-378,, \emph{Internat. Schriftenreihe Numer. Math.}, (1983). Google Scholar

[3]

R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations,, Singapore: World Scientific, (1986). doi: 10.1142/0266. Google Scholar

[4]

R. P. Agarwal and P. J. Y. Wong, Lidstone polynomial and boundary value problems,, \emph{Computers Math. Applic.}, 17 (1989), 1397. doi: 10.1016/0898-1221(89)90023-0. Google Scholar

[5]

R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications,, Dordrecht, (1993). doi: 10.1007/978-94-011-2026-5. Google Scholar

[6]

R. P. Agarwal, D. O'Regan, I. Rachunková and S. Staněk, Two-point higher-order BVPs with singularities in phase variables,, \emph{Computers Math. Applic.}, 46 (2003), 1799. doi: 10.1016/S0898-1221(03)90238-0. Google Scholar

[7]

R. P. Agarwal and P. J. Y. Wong, Eigenvalues of complementary Lidstone boundary value problems,, \emph{Bound. Value Probl.}, 2012 (2012), 1. doi: 10.1186/1687-2770-2012-49. Google Scholar

[8]

R. P. Agarwal and A. Özbekler, Lyapunov type inequalities for second order sub and super-half-linear differential equations,, \emph{Dynam. Systems Appl.}, 24 (2015), 211. Google Scholar

[9]

R. P. Agarwal and A. Özbekler, Lyapunov type inequalities for even order differential equations with mixed nonlinearities,, \emph{J. Inequal. Appl.}, 2015 (2015). doi: 10.1186/s13660-015-0633-4. Google Scholar

[10]

R. P. Agarwal and A. Özbekler, Disconjugacy via Lyapunov and Vallée-Poussin type inequalities for forced differential equations,, \emph{Appl. Math. Comput.}, 265 (2015), 456. doi: 10.1016/j.amc.2015.05.038. Google Scholar

[11]

P. R. Beesack, On Green's function of an $N$-point boundary value problem,, \emph{Pasific J. Math.}, 12 (1962), 801. Google Scholar

[12]

A. Beurling, Un théoréme sur les fonctions bornées et uniformément continues sur l'axe réel,, \emph{Acta Math.}, 77 (1945), 127. Google Scholar

[13]

G. Borg, On a Liapunoff criterion of stability,, \emph{Amer. J. Math.}, 71 (1949), 67. Google Scholar

[14]

R. C. Brown and D. B. Hinton, Opial's inequality and oscillation of 2nd order equations,, \emph{Proc. Amer. Math. Soc.}, 125 (1997), 1123. doi: 10.1090/S0002-9939-97-03907-5. Google Scholar

[15]

D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations,, \emph{Appl. Math. Comput.}, 216 (2010), 368. doi: 10.1016/j.amc.2010.01.010. Google Scholar

[16]

S. S. Cheng, A discrete analogue of the inequality of Lyapunov,, \emph{Hokkaido Math.}, 12 (1983), 105. doi: 10.14492/hokmj/1381757783. Google Scholar

[17]

S. S. Cheng, Lyapunov inequalities for differential and difference equations,, \emph{Fasc. Math.}, 23 (1991), 25. Google Scholar

[18]

R. S. Dahiya and B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations,, \emph{J. Math. Phys. Sci.}, 7 (1973), 163. Google Scholar

[19]

K. M. Das and A. S. Vatsala, On the Green's function of an $n$-point boundary value problem,, \emph{Trans. Amer. Math. Soc.}, 182 (1973), 469. Google Scholar

[20]

K. M. Das and A. S. Vatsala, Green function for $n-n$ boundary value problem and an analogue of Hartman's result,, \emph{J. Math. Anal. Appl.}, 51 (1975), 670. Google Scholar

[21]

O. Došlý and P. Řehák, Half-Linear Differential Equations,, Heidelberg: Elsevier Ltd, (2005). Google Scholar

[22]

A. Elbert, A half-linear second order differential equation,, \emph{Colloq Math Soc J\'anos Bolyai}, 30 (1979), 158. Google Scholar

[23]

S. B. Eliason, A Lyapunov inequality for a certain non-linear differential equation,, \emph{J. London Math. Soc.}, 2 (1970), 461. Google Scholar

[24]

S. B. Eliason, Lyapunov type inequalities for certain second order functional differential equations,, \emph{SIAM J. Appl. Math.}, 27 (1974), 180. Google Scholar

[25]

S. B. Eliason, Lyapunov inequalities and bounds on solutions of certain second order equations,, \emph{Canad. Math. Bull.}, 17 (1974), 499. Google Scholar

[26]

G. G. Gustafson, A Green's function convergence principle, with applications to computation and norm estimates,, \emph{Rocky Mountain J. Math.}, 6 (1976), 457. Google Scholar

[27]

G. S. Guseinov and B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems,, \emph{Comput. Math. Appl.}, 45 (2003), 1399. doi: 10.1016/S0898-1221(03)00095-6. Google Scholar

[28]

G. S. Guseinov and A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems,, \emph{J. Math. Anal. Appl.}, 35 (2007), 1195. doi: 10.1016/j.jmaa.2007.01.095. Google Scholar

[29]

P. Hartman, Ordinary Differential Equations,, New York, (1964). Google Scholar

[30]

X. He and X. H. Tang, Lyapunov-type inequalities for even order differential equations,, \emph{Commun. Pure. Appl. Anal.}, 11 (2012), 465. doi: 10.3934/cpaa.2012.11.465. Google Scholar

[31]

H. Hochstadt, A new proof of stability estimate of Lyapunov,, \emph{Proc. Amer. Math. Soc.}, 14 (1963), 525. Google Scholar

[32]

L. Jiang and Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales,, \emph{J. Math. Anal. Appl.}, 310 (2005), 579. doi: 10.1016/j.jmaa.2005.02.026. Google Scholar

[33]

S. Karlin, Total Positivity, Vol. I,, Stanford California: Stanford University Press, (1968). Google Scholar

[34]

Z. Kayar and A. Zafer, Stability criteria for linear Hamiltonian systems under impulsive perturbations,, \emph{Appl. Math. Comput.}, 230 (2014), 680. doi: 10.1016/j.amc.2013.12.128. Google Scholar

[35]

M. K. Kwong, On Lyapunov's inequality for disfocality,, \emph{J. Math. Anal. Appl.}, 83 (1981), 486. doi: 10.1016/0022-247X(81)90137-2. Google Scholar

[36]

C. Lee, C. Yeh, C. Hong and R. P. Agarwal, Lyapunov and Wirtinger inequalities,, \emph{Appl. Math. Lett.}, 17 (2004), 847. doi: 10.1016/j.aml.2004.06.016. Google Scholar

[37]

A. M. Liapunov, Probleme général de la stabilité du mouvement, (French Translation of a Russian paper dated 1893),, \emph{Ann Fac Sci Univ Toulouse 2 (1907), (1907), 27. Google Scholar

[38]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and its Applications (East European Series),, Dordrecht: 53 Kluwer Academic Publishers Group, (1991). doi: 10.1007/978-94-011-3562-7. Google Scholar

[39]

P. L. Napoli and J. P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems,, \emph{J. Differential Equations}, 227 (2006), 102. doi: 10.1016/j.jde.2006.01.004. Google Scholar

[40]

Z. Nehari, Some eigenvalue estimates,, \emph{J. Anal. Math.}, 7 (1959), 79. Google Scholar

[41]

Z. Nehari, On an inequality of Lyapunov, in: Studies in Mathematical Analysis and Related Topics,, Stanford, (1962). Google Scholar

[42]

B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations,, \emph{J. Anal. Math.}, 195 (1995), 527. doi: 10.1006/jmaa.1995.1372. Google Scholar

[43]

B. G. Pachpatte, Lyapunov type integral inequalities for certain differential equations,, \emph{Georgian Math. J.}, 4 (1997), 139. doi: 10.1023/A:1022930116838. Google Scholar

[44]

B. G. Pachpatte, Inequalities related to the zeros of solutions of certain second order differential equations,, \emph{Facta. Univ. Ser. Math. Inform.}, 16 (2001), 35. Google Scholar

[45]

S. Panigrahi, Lyapunov-type integral inequalities for certain higher order differential equations,, \emph{Electron J Differential Equations}, 2009 (2009), 1. Google Scholar

[46]

N. Parhi and S. Panigrahi, On Liapunov-type inequality for third-order differential equations,, \emph{J. Math. Anal. Appl.}, 233 (1999), 445. doi: 10.1006/jmaa.1999.6265. Google Scholar

[47]

N. Parhi and S. Panigrahi, Liapunov-type inequality for higher order differential equations,, \emph{Math. Slovaca}, 52 (2002), 31. Google Scholar

[48]

T. W. Reid, A matrix equation related to an non-oscillation criterion and Lyapunov stability,, \emph{Quart. Appl. Math. Soc.}, 23 (1965), 83. Google Scholar

[49]

T. W. Reid, A matrix Lyapunov inequality,, \emph{J. Math. Anal. Appl.}, 32 (1970), 424. Google Scholar

[50]

B. Singh, Forced oscillation in general ordinary differential equations,, \emph{Tamkang J. Math.}, 6 (1975), 5. Google Scholar

[51]

A. Tiryaki, M. Unal and D. Cakmak, Lyapunov-type inequalities for nonlinear systems,, \emph{J. Math. Anal. Appl.}, 332 (2007), 497. doi: 10.1016/j.jmaa.2006.10.010. Google Scholar

[52]

A. Tiryaki, Recent developments of Lyapunov-type inequalities,, \emph{Advances in Dynam. Sys. Appl.}, 5 (2010), 231. Google Scholar

[53]

M. Unal, D. Cakmak and A. Tiryaki, A discrete analogue of Lyapunov-type inequalities for nonlinear systems,, \emph{Comput. Math. Appl.}, 55 (2008), 2631. doi: 10.1016/j.camwa.2007.10.014. Google Scholar

[54]

M. Unal and D. Cakmak, Lyapunov-type inequalities for certain nonlinear systems on time scales,, \emph{Turkish J. Math.}, 32 (2008), 255. Google Scholar

[55]

X. Yang, On Liapunov-type inequality for certain higher-order differential equations,, \emph{Appl. Math. Comput.}, 134 (2003), 307. doi: 10.1016/S0096-3003(01)00285-5. Google Scholar

[56]

X. Yang, Lyapunov-type inequality for a class of even-order differential equations,, \emph{Appl. Math. Comput.}, 215 (2010), 3884. doi: 10.1016/j.amc.2009.11.032. Google Scholar

[57]

A. Wintner, On the nonexistence of conjugate points,, \emph{Amer. J. Math.}, 73 (1951), 368. Google Scholar

[58]

Q. M. Zhang and X. He, Lyapunov-type inequalities for a class of even-order differential equations,, \emph{J. Inequal. Appl.}, 2012 (2012), 1. doi: 10.1186/1029-242X-2012-5. Google Scholar

show all references

References:
[1]

R. P. Agarwal, Boundary value problems for higher order integro-differential equations,, \emph{Nonlinear Anal.}, 7 (1983), 259. doi: 10.1016/0362-546X(83)90070-6. Google Scholar

[2]

R. P. Agarwal, Some inequalities for a function having $n$ zeros. General inequalities, 3 (Oberwolfach, 1981), 371-378,, \emph{Internat. Schriftenreihe Numer. Math.}, (1983). Google Scholar

[3]

R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations,, Singapore: World Scientific, (1986). doi: 10.1142/0266. Google Scholar

[4]

R. P. Agarwal and P. J. Y. Wong, Lidstone polynomial and boundary value problems,, \emph{Computers Math. Applic.}, 17 (1989), 1397. doi: 10.1016/0898-1221(89)90023-0. Google Scholar

[5]

R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications,, Dordrecht, (1993). doi: 10.1007/978-94-011-2026-5. Google Scholar

[6]

R. P. Agarwal, D. O'Regan, I. Rachunková and S. Staněk, Two-point higher-order BVPs with singularities in phase variables,, \emph{Computers Math. Applic.}, 46 (2003), 1799. doi: 10.1016/S0898-1221(03)90238-0. Google Scholar

[7]

R. P. Agarwal and P. J. Y. Wong, Eigenvalues of complementary Lidstone boundary value problems,, \emph{Bound. Value Probl.}, 2012 (2012), 1. doi: 10.1186/1687-2770-2012-49. Google Scholar

[8]

R. P. Agarwal and A. Özbekler, Lyapunov type inequalities for second order sub and super-half-linear differential equations,, \emph{Dynam. Systems Appl.}, 24 (2015), 211. Google Scholar

[9]

R. P. Agarwal and A. Özbekler, Lyapunov type inequalities for even order differential equations with mixed nonlinearities,, \emph{J. Inequal. Appl.}, 2015 (2015). doi: 10.1186/s13660-015-0633-4. Google Scholar

[10]

R. P. Agarwal and A. Özbekler, Disconjugacy via Lyapunov and Vallée-Poussin type inequalities for forced differential equations,, \emph{Appl. Math. Comput.}, 265 (2015), 456. doi: 10.1016/j.amc.2015.05.038. Google Scholar

[11]

P. R. Beesack, On Green's function of an $N$-point boundary value problem,, \emph{Pasific J. Math.}, 12 (1962), 801. Google Scholar

[12]

A. Beurling, Un théoréme sur les fonctions bornées et uniformément continues sur l'axe réel,, \emph{Acta Math.}, 77 (1945), 127. Google Scholar

[13]

G. Borg, On a Liapunoff criterion of stability,, \emph{Amer. J. Math.}, 71 (1949), 67. Google Scholar

[14]

R. C. Brown and D. B. Hinton, Opial's inequality and oscillation of 2nd order equations,, \emph{Proc. Amer. Math. Soc.}, 125 (1997), 1123. doi: 10.1090/S0002-9939-97-03907-5. Google Scholar

[15]

D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations,, \emph{Appl. Math. Comput.}, 216 (2010), 368. doi: 10.1016/j.amc.2010.01.010. Google Scholar

[16]

S. S. Cheng, A discrete analogue of the inequality of Lyapunov,, \emph{Hokkaido Math.}, 12 (1983), 105. doi: 10.14492/hokmj/1381757783. Google Scholar

[17]

S. S. Cheng, Lyapunov inequalities for differential and difference equations,, \emph{Fasc. Math.}, 23 (1991), 25. Google Scholar

[18]

R. S. Dahiya and B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations,, \emph{J. Math. Phys. Sci.}, 7 (1973), 163. Google Scholar

[19]

K. M. Das and A. S. Vatsala, On the Green's function of an $n$-point boundary value problem,, \emph{Trans. Amer. Math. Soc.}, 182 (1973), 469. Google Scholar

[20]

K. M. Das and A. S. Vatsala, Green function for $n-n$ boundary value problem and an analogue of Hartman's result,, \emph{J. Math. Anal. Appl.}, 51 (1975), 670. Google Scholar

[21]

O. Došlý and P. Řehák, Half-Linear Differential Equations,, Heidelberg: Elsevier Ltd, (2005). Google Scholar

[22]

A. Elbert, A half-linear second order differential equation,, \emph{Colloq Math Soc J\'anos Bolyai}, 30 (1979), 158. Google Scholar

[23]

S. B. Eliason, A Lyapunov inequality for a certain non-linear differential equation,, \emph{J. London Math. Soc.}, 2 (1970), 461. Google Scholar

[24]

S. B. Eliason, Lyapunov type inequalities for certain second order functional differential equations,, \emph{SIAM J. Appl. Math.}, 27 (1974), 180. Google Scholar

[25]

S. B. Eliason, Lyapunov inequalities and bounds on solutions of certain second order equations,, \emph{Canad. Math. Bull.}, 17 (1974), 499. Google Scholar

[26]

G. G. Gustafson, A Green's function convergence principle, with applications to computation and norm estimates,, \emph{Rocky Mountain J. Math.}, 6 (1976), 457. Google Scholar

[27]

G. S. Guseinov and B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems,, \emph{Comput. Math. Appl.}, 45 (2003), 1399. doi: 10.1016/S0898-1221(03)00095-6. Google Scholar

[28]

G. S. Guseinov and A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems,, \emph{J. Math. Anal. Appl.}, 35 (2007), 1195. doi: 10.1016/j.jmaa.2007.01.095. Google Scholar

[29]

P. Hartman, Ordinary Differential Equations,, New York, (1964). Google Scholar

[30]

X. He and X. H. Tang, Lyapunov-type inequalities for even order differential equations,, \emph{Commun. Pure. Appl. Anal.}, 11 (2012), 465. doi: 10.3934/cpaa.2012.11.465. Google Scholar

[31]

H. Hochstadt, A new proof of stability estimate of Lyapunov,, \emph{Proc. Amer. Math. Soc.}, 14 (1963), 525. Google Scholar

[32]

L. Jiang and Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales,, \emph{J. Math. Anal. Appl.}, 310 (2005), 579. doi: 10.1016/j.jmaa.2005.02.026. Google Scholar

[33]

S. Karlin, Total Positivity, Vol. I,, Stanford California: Stanford University Press, (1968). Google Scholar

[34]

Z. Kayar and A. Zafer, Stability criteria for linear Hamiltonian systems under impulsive perturbations,, \emph{Appl. Math. Comput.}, 230 (2014), 680. doi: 10.1016/j.amc.2013.12.128. Google Scholar

[35]

M. K. Kwong, On Lyapunov's inequality for disfocality,, \emph{J. Math. Anal. Appl.}, 83 (1981), 486. doi: 10.1016/0022-247X(81)90137-2. Google Scholar

[36]

C. Lee, C. Yeh, C. Hong and R. P. Agarwal, Lyapunov and Wirtinger inequalities,, \emph{Appl. Math. Lett.}, 17 (2004), 847. doi: 10.1016/j.aml.2004.06.016. Google Scholar

[37]

A. M. Liapunov, Probleme général de la stabilité du mouvement, (French Translation of a Russian paper dated 1893),, \emph{Ann Fac Sci Univ Toulouse 2 (1907), (1907), 27. Google Scholar

[38]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and its Applications (East European Series),, Dordrecht: 53 Kluwer Academic Publishers Group, (1991). doi: 10.1007/978-94-011-3562-7. Google Scholar

[39]

P. L. Napoli and J. P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems,, \emph{J. Differential Equations}, 227 (2006), 102. doi: 10.1016/j.jde.2006.01.004. Google Scholar

[40]

Z. Nehari, Some eigenvalue estimates,, \emph{J. Anal. Math.}, 7 (1959), 79. Google Scholar

[41]

Z. Nehari, On an inequality of Lyapunov, in: Studies in Mathematical Analysis and Related Topics,, Stanford, (1962). Google Scholar

[42]

B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations,, \emph{J. Anal. Math.}, 195 (1995), 527. doi: 10.1006/jmaa.1995.1372. Google Scholar

[43]

B. G. Pachpatte, Lyapunov type integral inequalities for certain differential equations,, \emph{Georgian Math. J.}, 4 (1997), 139. doi: 10.1023/A:1022930116838. Google Scholar

[44]

B. G. Pachpatte, Inequalities related to the zeros of solutions of certain second order differential equations,, \emph{Facta. Univ. Ser. Math. Inform.}, 16 (2001), 35. Google Scholar

[45]

S. Panigrahi, Lyapunov-type integral inequalities for certain higher order differential equations,, \emph{Electron J Differential Equations}, 2009 (2009), 1. Google Scholar

[46]

N. Parhi and S. Panigrahi, On Liapunov-type inequality for third-order differential equations,, \emph{J. Math. Anal. Appl.}, 233 (1999), 445. doi: 10.1006/jmaa.1999.6265. Google Scholar

[47]

N. Parhi and S. Panigrahi, Liapunov-type inequality for higher order differential equations,, \emph{Math. Slovaca}, 52 (2002), 31. Google Scholar

[48]

T. W. Reid, A matrix equation related to an non-oscillation criterion and Lyapunov stability,, \emph{Quart. Appl. Math. Soc.}, 23 (1965), 83. Google Scholar

[49]

T. W. Reid, A matrix Lyapunov inequality,, \emph{J. Math. Anal. Appl.}, 32 (1970), 424. Google Scholar

[50]

B. Singh, Forced oscillation in general ordinary differential equations,, \emph{Tamkang J. Math.}, 6 (1975), 5. Google Scholar

[51]

A. Tiryaki, M. Unal and D. Cakmak, Lyapunov-type inequalities for nonlinear systems,, \emph{J. Math. Anal. Appl.}, 332 (2007), 497. doi: 10.1016/j.jmaa.2006.10.010. Google Scholar

[52]

A. Tiryaki, Recent developments of Lyapunov-type inequalities,, \emph{Advances in Dynam. Sys. Appl.}, 5 (2010), 231. Google Scholar

[53]

M. Unal, D. Cakmak and A. Tiryaki, A discrete analogue of Lyapunov-type inequalities for nonlinear systems,, \emph{Comput. Math. Appl.}, 55 (2008), 2631. doi: 10.1016/j.camwa.2007.10.014. Google Scholar

[54]

M. Unal and D. Cakmak, Lyapunov-type inequalities for certain nonlinear systems on time scales,, \emph{Turkish J. Math.}, 32 (2008), 255. Google Scholar

[55]

X. Yang, On Liapunov-type inequality for certain higher-order differential equations,, \emph{Appl. Math. Comput.}, 134 (2003), 307. doi: 10.1016/S0096-3003(01)00285-5. Google Scholar

[56]

X. Yang, Lyapunov-type inequality for a class of even-order differential equations,, \emph{Appl. Math. Comput.}, 215 (2010), 3884. doi: 10.1016/j.amc.2009.11.032. Google Scholar

[57]

A. Wintner, On the nonexistence of conjugate points,, \emph{Amer. J. Math.}, 73 (1951), 368. Google Scholar

[58]

Q. M. Zhang and X. He, Lyapunov-type inequalities for a class of even-order differential equations,, \emph{J. Inequal. Appl.}, 2012 (2012), 1. doi: 10.1186/1029-242X-2012-5. Google Scholar

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