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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:
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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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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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