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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.

[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).

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[41]

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

[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.

[43]

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

[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.

[45]

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

[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.

[47]

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

[48]

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

[49]

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

[50]

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

[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.

[52]

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

[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.

[54]

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

[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.

[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.

[57]

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

[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.

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.

[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).

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[41]

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

[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.

[43]

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

[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.

[45]

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

[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.

[47]

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

[48]

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

[49]

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

[50]

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

[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.

[52]

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

[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.

[54]

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

[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.

[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.

[57]

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

[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.

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