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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, Nonlinear Anal., 7 (1983), 259-270. 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, Internat. Schriftenreihe Numer. Math., 64, Birkhäuser, Basel, 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, Computers Math. Applic., 17 (1989), 1397-1421. 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, Boston, London: Kluwer Academic Publishers, 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, Computers Math. Applic., 46 (2003), 1799-1826. 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, Bound. Value Probl., 2012 (2012), 1-23. 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, Dynam. Systems Appl., 24 (2015), 211-220.  Google Scholar [9] R. P. Agarwal and A. Özbekler, Lyapunov type inequalities for even order differential equations with mixed nonlinearities, J. Inequal. Appl., 2015 (2015), 142, 10 pp. 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, Appl. Math. Comput., 265 (2015), 456-468. 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, Pasific J. Math., 12 (1962), 801-812.  Google Scholar [12] A. Beurling, Un théoréme sur les fonctions bornées et uniformément continues sur l'axe réel, Acta Math., 77 (1945), 127-136.  Google Scholar [13] G. Borg, On a Liapunoff criterion of stability, Amer. J. Math., 71 (1949), 67-70.  Google Scholar [14] R. C. Brown and D. B. Hinton, Opial's inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc., 125 (1997), 1123-1129. doi: 10.1090/S0002-9939-97-03907-5.  Google Scholar [15] D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368-373. doi: 10.1016/j.amc.2010.01.010.  Google Scholar [16] S. S. Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math., 12 (1983), 105-112. doi: 10.14492/hokmj/1381757783.  Google Scholar [17] S. S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math., 23 (1991), 25-41.  Google Scholar [18] R. S. Dahiya and B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations, J. Math. Phys. Sci., 7 (1973), 163-170.  Google Scholar [19] K. M. Das and A. S. Vatsala, On the Green's function of an $n$-point boundary value problem, Trans. Amer. Math. Soc., 182 (1973), 469-480.  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, J. Math. Anal. Appl., 51 (1975), 670-677.  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, Colloq Math Soc János Bolyai, 30 (1979), 158-180. Google Scholar [23] S. B. Eliason, A Lyapunov inequality for a certain non-linear differential equation, J. London Math. Soc., 2 (1970), 461-466.  Google Scholar [24] S. B. Eliason, Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math., 27 (1974), 180-199.  Google Scholar [25] S. B. Eliason, Lyapunov inequalities and bounds on solutions of certain second order equations, Canad. Math. Bull., 17 (1974), 499-504.  Google Scholar [26] G. G. Gustafson, A Green's function convergence principle, with applications to computation and norm estimates, Rocky Mountain J. Math., 6 (1976), 457-492.  Google Scholar [27] G. S. Guseinov and B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl., 45 (2003), 1399-1416. 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, J. Math. Anal. Appl., 35 (2007), 1195-1206. doi: 10.1016/j.jmaa.2007.01.095.  Google Scholar [29] P. Hartman, Ordinary Differential Equations, New York, 1964 and Birkhäuser, Boston: Wiley, 1982.  Google Scholar [30] X. He and X. H. Tang, Lyapunov-type inequalities for even order differential equations, Commun. Pure. Appl. Anal., 11 (2012), 465-473. doi: 10.3934/cpaa.2012.11.465.  Google Scholar [31] H. Hochstadt, A new proof of stability estimate of Lyapunov, Proc. Amer. Math. Soc., 14 (1963), 525-526.  Google Scholar [32] L. Jiang and Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl., 310 (2005), 579-593. 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, Appl. Math. Comput., 230 (2014), 680-686. doi: 10.1016/j.amc.2013.12.128.  Google Scholar [35] M. K. Kwong, On Lyapunov's inequality for disfocality, J. Math. Anal. Appl., 83 (1981), 486-494. 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, Appl. Math. Lett., 17 (2004), 847-853. 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), Ann Fac Sci Univ Toulouse 2 (1907), 27-247, Reprinted as Ann Math Studies, No. 17, Princeton, 1947. 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, J. Differential Equations, 227 (2006), 102-115. doi: 10.1016/j.jde.2006.01.004.  Google Scholar [40] Z. Nehari, Some eigenvalue estimates, J. Anal. Math., 7 (1959), 79-88.  Google Scholar [41] Z. Nehari, On an inequality of Lyapunov, in: Studies in Mathematical Analysis and Related Topics, Stanford, CA: Stanford University Press, 1962.  Google Scholar [42] B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Anal. Math., 195 (1995), 527-536. doi: 10.1006/jmaa.1995.1372.  Google Scholar [43] B. G. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J., 4 (1997), 139-148. doi: 10.1023/A:1022930116838.  Google Scholar [44] B. G. Pachpatte, Inequalities related to the zeros of solutions of certain second order differential equations, Facta. Univ. Ser. Math. Inform., 16 (2001), 35-44.  Google Scholar [45] S. Panigrahi, Lyapunov-type integral inequalities for certain higher order differential equations, Electron J Differential Equations, 2009 (2009), 1-14.  Google Scholar [46] N. Parhi and S. Panigrahi, On Liapunov-type inequality for third-order differential equations, J. Math. Anal. Appl., 233 (1999), 445-460. doi: 10.1006/jmaa.1999.6265.  Google Scholar [47] N. Parhi and S. Panigrahi, Liapunov-type inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31-46.  Google Scholar [48] T. W. Reid, A matrix equation related to an non-oscillation criterion and Lyapunov stability, Quart. Appl. Math. Soc., 23 (1965), 83-87.  Google Scholar [49] T. W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl., 32 (1970), 424-434.  Google Scholar [50] B. Singh, Forced oscillation in general ordinary differential equations, Tamkang J. Math., 6 (1975), 5-11.  Google Scholar [51] A. Tiryaki, M. Unal and D. Cakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl., 332 (2007), 497-511. doi: 10.1016/j.jmaa.2006.10.010.  Google Scholar [52] A. Tiryaki, Recent developments of Lyapunov-type inequalities, Advances in Dynam. Sys. Appl., 5 (2010), 231-248.  Google Scholar [53] M. Unal, D. Cakmak and A. Tiryaki, A discrete analogue of Lyapunov-type inequalities for nonlinear systems, Comput. Math. Appl., 55 (2008), 2631-2642. 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, Turkish J. Math., 32 (2008), 255-275.  Google Scholar [55] X. Yang, On Liapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134 (2003), 307-317. doi: 10.1016/S0096-3003(01)00285-5.  Google Scholar [56] X. Yang, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884-3890. doi: 10.1016/j.amc.2009.11.032.  Google Scholar [57] A. Wintner, On the nonexistence of conjugate points, Amer. J. Math., 73 (1951), 368-380.  Google Scholar [58] Q. M. Zhang and X. He, Lyapunov-type inequalities for a class of even-order differential equations, J. Inequal. Appl., 2012 (2012), 1-7. 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, Nonlinear Anal., 7 (1983), 259-270. 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, Internat. Schriftenreihe Numer. Math., 64, Birkhäuser, Basel, 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, Computers Math. Applic., 17 (1989), 1397-1421. 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, Boston, London: Kluwer Academic Publishers, 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, Computers Math. Applic., 46 (2003), 1799-1826. 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, Bound. Value Probl., 2012 (2012), 1-23. 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, Dynam. Systems Appl., 24 (2015), 211-220.  Google Scholar [9] R. P. Agarwal and A. Özbekler, Lyapunov type inequalities for even order differential equations with mixed nonlinearities, J. Inequal. Appl., 2015 (2015), 142, 10 pp. 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, Appl. Math. Comput., 265 (2015), 456-468. 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, Pasific J. Math., 12 (1962), 801-812.  Google Scholar [12] A. Beurling, Un théoréme sur les fonctions bornées et uniformément continues sur l'axe réel, Acta Math., 77 (1945), 127-136.  Google Scholar [13] G. Borg, On a Liapunoff criterion of stability, Amer. J. Math., 71 (1949), 67-70.  Google Scholar [14] R. C. Brown and D. B. Hinton, Opial's inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc., 125 (1997), 1123-1129. doi: 10.1090/S0002-9939-97-03907-5.  Google Scholar [15] D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368-373. doi: 10.1016/j.amc.2010.01.010.  Google Scholar [16] S. S. Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math., 12 (1983), 105-112. doi: 10.14492/hokmj/1381757783.  Google Scholar [17] S. S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math., 23 (1991), 25-41.  Google Scholar [18] R. S. Dahiya and B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations, J. Math. Phys. Sci., 7 (1973), 163-170.  Google Scholar [19] K. M. Das and A. S. Vatsala, On the Green's function of an $n$-point boundary value problem, Trans. Amer. Math. Soc., 182 (1973), 469-480.  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, J. Math. Anal. Appl., 51 (1975), 670-677.  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, Colloq Math Soc János Bolyai, 30 (1979), 158-180. Google Scholar [23] S. B. Eliason, A Lyapunov inequality for a certain non-linear differential equation, J. London Math. Soc., 2 (1970), 461-466.  Google Scholar [24] S. B. Eliason, Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math., 27 (1974), 180-199.  Google Scholar [25] S. B. Eliason, Lyapunov inequalities and bounds on solutions of certain second order equations, Canad. Math. Bull., 17 (1974), 499-504.  Google Scholar [26] G. G. Gustafson, A Green's function convergence principle, with applications to computation and norm estimates, Rocky Mountain J. Math., 6 (1976), 457-492.  Google Scholar [27] G. S. Guseinov and B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl., 45 (2003), 1399-1416. 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, J. Math. Anal. Appl., 35 (2007), 1195-1206. doi: 10.1016/j.jmaa.2007.01.095.  Google Scholar [29] P. Hartman, Ordinary Differential Equations, New York, 1964 and Birkhäuser, Boston: Wiley, 1982.  Google Scholar [30] X. He and X. H. Tang, Lyapunov-type inequalities for even order differential equations, Commun. Pure. Appl. Anal., 11 (2012), 465-473. doi: 10.3934/cpaa.2012.11.465.  Google Scholar [31] H. Hochstadt, A new proof of stability estimate of Lyapunov, Proc. Amer. Math. Soc., 14 (1963), 525-526.  Google Scholar [32] L. Jiang and Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl., 310 (2005), 579-593. 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, Appl. Math. Comput., 230 (2014), 680-686. doi: 10.1016/j.amc.2013.12.128.  Google Scholar [35] M. K. Kwong, On Lyapunov's inequality for disfocality, J. Math. Anal. Appl., 83 (1981), 486-494. 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, Appl. Math. Lett., 17 (2004), 847-853. 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), Ann Fac Sci Univ Toulouse 2 (1907), 27-247, Reprinted as Ann Math Studies, No. 17, Princeton, 1947. 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, J. Differential Equations, 227 (2006), 102-115. doi: 10.1016/j.jde.2006.01.004.  Google Scholar [40] Z. Nehari, Some eigenvalue estimates, J. Anal. Math., 7 (1959), 79-88.  Google Scholar [41] Z. Nehari, On an inequality of Lyapunov, in: Studies in Mathematical Analysis and Related Topics, Stanford, CA: Stanford University Press, 1962.  Google Scholar [42] B. G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Anal. Math., 195 (1995), 527-536. doi: 10.1006/jmaa.1995.1372.  Google Scholar [43] B. G. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J., 4 (1997), 139-148. doi: 10.1023/A:1022930116838.  Google Scholar [44] B. G. Pachpatte, Inequalities related to the zeros of solutions of certain second order differential equations, Facta. Univ. Ser. Math. Inform., 16 (2001), 35-44.  Google Scholar [45] S. Panigrahi, Lyapunov-type integral inequalities for certain higher order differential equations, Electron J Differential Equations, 2009 (2009), 1-14.  Google Scholar [46] N. Parhi and S. Panigrahi, On Liapunov-type inequality for third-order differential equations, J. Math. Anal. Appl., 233 (1999), 445-460. doi: 10.1006/jmaa.1999.6265.  Google Scholar [47] N. Parhi and S. Panigrahi, Liapunov-type inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31-46.  Google Scholar [48] T. W. Reid, A matrix equation related to an non-oscillation criterion and Lyapunov stability, Quart. Appl. Math. Soc., 23 (1965), 83-87.  Google Scholar [49] T. W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl., 32 (1970), 424-434.  Google Scholar [50] B. Singh, Forced oscillation in general ordinary differential equations, Tamkang J. Math., 6 (1975), 5-11.  Google Scholar [51] A. Tiryaki, M. Unal and D. Cakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl., 332 (2007), 497-511. doi: 10.1016/j.jmaa.2006.10.010.  Google Scholar [52] A. Tiryaki, Recent developments of Lyapunov-type inequalities, Advances in Dynam. Sys. 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