# American Institute of Mathematical Sciences

October  2017, 10(5): 1133-1148. doi: 10.3934/dcdss.2017061

## Lyapunov-type inequalities and solvability of second-order ODEs across multi-resonance

 a. College of Mathematics, Jilin University, Changchun 130012, China b. School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China c. State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, China

* Corresponding author: zhanghe14@mails.jlu.edu.cn

Received  October 2016 Revised  November 2016 Published  June 2017

Fund Project: This work was completed with the support by National Basic Research Program of China Grant 2013CB834100, NSFC Grant 11571065, NSFC Grant 11171132 and NSFC Grant 11201173.

We present some new Lyapunov-type inequalities for boundary value problems of the form $y''+u(x)y=0$, $y(0)=0=y(1)$, where $-A≤ u(x)≤ B$ and there are many resonance points lying inside the interval $[-A, B]$. The classical Lyapunov's inequality and its reverse are improved by using Pontryagin's maximum principle. As applications, we establish two readily verifiable unique solvability criteria for general $u(x)$. Some relevant examples are given to illustrate our results. Variants of Lyapunov-type inequalities for nonlinear BVPs are discussed at the end of the paper.

Citation: He Zhang, Xue Yang, Yong Li. Lyapunov-type inequalities and solvability of second-order ODEs across multi-resonance. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1133-1148. doi: 10.3934/dcdss.2017061
##### References:
 [1] S. R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Elsevier, 1974. [2] G. Borg, On a liapounoff criterion of stability, American Journal of Mathematics, 71 (1949), 67-70.  doi: 10.2307/2372093. [3] A. Cañada, J. A. Montero and S. Villegas, Lyapunov-type inequalities and neumann boundary value problems at resonance, Math. Inequal. Appl., 8 (2005), 459-475.  doi: 10.7153/mia-08-42. [4] A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, Journal of Functional Analysis, 237 (2006), 176-193.  doi: 10.1016/j.jfa.2005.12.011. [5] A. Cañada, J. A. Montero and S. Villegas, Lyapunov-type inequalities for differential equations, Mediterranean Journal of Mathematics, 3 (2006), 177-187.  doi: 10.1007/s00009-006-0071-0. [6] A. Cañada and S. Villegas, Optimal lyapunov inequalities for disfocality and neumann boundary conditions using lp norms, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 877-888.  doi: 10.3934/dcds.2008.20.877. [7] A. Cañada and S. Villegas, Lyapunov inequalities for neumann boundary conditions at higher eigenvalues, J. Eur. Math. Soc. (JEMS), 12 (2010), 163-178.  doi: 10.4171/JEMS/193. [8] A. Cañada and S. Villegas, Lyapunov inequalities for partial differential equations at radial higher eigenvalues, Discrete Contin. Dyn. Syst., 33 (2013), 111-122.  doi: 10.3934/dcds.2013.33.111. [9] X. Chang and Q. Huang, Two-point boundary value problems for duffing equations across resonance, Journal of optimization theory and applications, 140 (2009), 419-430.  doi: 10.1007/s10957-008-9461-8. [10] S.-S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math, 23 (1991), 25-41. [11] S. B. Eliason, A lyapunov inequality for a certain second order non-linear differential equation, Journal of the London Mathematical Society, 2 (1970), 461-466.  doi: 10.1112/jlms/2.Part_3.461. [12] B. Harris and Q. Kong, On the oscillation of differential equations with an oscillatory coefficient, Transactions of the American Mathematical Society, 347 (1995), 1831-1839.  doi: 10.1090/S0002-9947-1995-1283552-4. [13] P. Hartman, Ordinary Differential Equations, Birkhauser, Boston, 1982. [14] J. Henderson, Best interval lengths for boundary value problems for third order lipschitz equations, SIAM journal on mathematical analysis, 18 (1987), 293-305.  doi: 10.1137/0518023. [15] J. Henderson, Optimal interval lengths for nonlocal boundary value problems for second order lipschitz equations, Communications in Applied Analysis, 15 (2011), 475-482. [16] M. Grigor'evich Krein, On certain problems on the maximum and minimum of characteristic values and on the lyapunov zones of stability, Amer. Math. Soc. Transl., 1 (1955), 163-187.  doi: 10.1090/trans2/001/08. [17] M. Lees, Discrete methods for nonlinear two-point boundary value problems, Numerical Solution of Partial Differential Equations, 1 (1966), 59-72. [18] Y. Li and H. Wang, Neumann problems for second order ordinary differential equations across resonance, Zeitschrift für angewandte Mathematik und Physik ZAMP, 46 (1995), 393-406.  doi: 10.1007/BF01003558. [19] A. Liapounoff, Problème général de la stabilité du mouvement, In Annales de la faculté des sciences de Toulouse, 9 (1907), 203-474. Université Paul Sabatier. [20] G. López and J.-A. Montero-Sánchez, Neumann boundary value problems across resonance, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 398-408.  doi: 10.1051/cocv:2006009. [21] J. Mawhin and J. Ward, Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Analysis: Theory, Methods & Applications, 5 (1981), 677-684.  doi: 10.1016/0362-546X(81)90084-5. [22] J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-laplacian, In Abstract and Applied Analysis, Hindawi Publishing Corporation, 2004,147-153. doi: 10.1155/S108533750431002X. [23] J. Qi and S. Chen, Extremal norms of the potentials recovered from inverse dirichlet problems, Inverse Problems, 32 (2016), 035007, 13pp. doi: 10.1088/0266-5611/32/3/035007. [24] K. Shen and M. Zhang, An optimal class of non-degenerate potentials for second-order ordinary differential equations, Boundary Value Problems, 2015 (2015), 1-17.  doi: 10.1186/s13661-015-0451-0. [25] X. Tang and M. Zhang, Lyapunov inequalities and stability for linear hamiltonian systems, Journal of Differential Equations, 252 (2012), 358-381.  doi: 10.1016/j.jde.2011.08.002. [26] H. Wang and Y. Li, Two point boundary value problems for second-order ordinary differential equations across many resonant points, Journal of mathematical analysis and applications, 179 (1993), 61-75.  doi: 10.1006/jmaa.1993.1335. [27] H. Wang and Y. Li, Neumann boundary value problems for second-order ordinary differential equations across resonance, SIAM journal on control and optimization, 33 (1995), 1312-1325.  doi: 10.1137/S036301299324532X. [28] H. Wang and Y. Li, Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations, Zeitschrift für angewandte Mathematik und Physik ZAMP, 47 (1996), 373-384.  doi: 10.1007/BF00916644. [29] M. Zhang, Extremal values of smallest eigenvalues of hill's operators with potentials in $L^1$ balls, Journal of Differential Equations, 246 (2009), 4188-4220.  doi: 10.1016/j.jde.2009.03.016.

show all references

##### References:
 [1] S. R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Elsevier, 1974. [2] G. Borg, On a liapounoff criterion of stability, American Journal of Mathematics, 71 (1949), 67-70.  doi: 10.2307/2372093. [3] A. Cañada, J. A. Montero and S. Villegas, Lyapunov-type inequalities and neumann boundary value problems at resonance, Math. Inequal. Appl., 8 (2005), 459-475.  doi: 10.7153/mia-08-42. [4] A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, Journal of Functional Analysis, 237 (2006), 176-193.  doi: 10.1016/j.jfa.2005.12.011. [5] A. Cañada, J. A. Montero and S. Villegas, Lyapunov-type inequalities for differential equations, Mediterranean Journal of Mathematics, 3 (2006), 177-187.  doi: 10.1007/s00009-006-0071-0. [6] A. Cañada and S. Villegas, Optimal lyapunov inequalities for disfocality and neumann boundary conditions using lp norms, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 877-888.  doi: 10.3934/dcds.2008.20.877. [7] A. Cañada and S. Villegas, Lyapunov inequalities for neumann boundary conditions at higher eigenvalues, J. Eur. Math. Soc. (JEMS), 12 (2010), 163-178.  doi: 10.4171/JEMS/193. [8] A. Cañada and S. Villegas, Lyapunov inequalities for partial differential equations at radial higher eigenvalues, Discrete Contin. Dyn. Syst., 33 (2013), 111-122.  doi: 10.3934/dcds.2013.33.111. [9] X. Chang and Q. Huang, Two-point boundary value problems for duffing equations across resonance, Journal of optimization theory and applications, 140 (2009), 419-430.  doi: 10.1007/s10957-008-9461-8. [10] S.-S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math, 23 (1991), 25-41. [11] S. B. Eliason, A lyapunov inequality for a certain second order non-linear differential equation, Journal of the London Mathematical Society, 2 (1970), 461-466.  doi: 10.1112/jlms/2.Part_3.461. [12] B. Harris and Q. Kong, On the oscillation of differential equations with an oscillatory coefficient, Transactions of the American Mathematical Society, 347 (1995), 1831-1839.  doi: 10.1090/S0002-9947-1995-1283552-4. [13] P. Hartman, Ordinary Differential Equations, Birkhauser, Boston, 1982. [14] J. Henderson, Best interval lengths for boundary value problems for third order lipschitz equations, SIAM journal on mathematical analysis, 18 (1987), 293-305.  doi: 10.1137/0518023. [15] J. Henderson, Optimal interval lengths for nonlocal boundary value problems for second order lipschitz equations, Communications in Applied Analysis, 15 (2011), 475-482. [16] M. Grigor'evich Krein, On certain problems on the maximum and minimum of characteristic values and on the lyapunov zones of stability, Amer. Math. Soc. Transl., 1 (1955), 163-187.  doi: 10.1090/trans2/001/08. [17] M. Lees, Discrete methods for nonlinear two-point boundary value problems, Numerical Solution of Partial Differential Equations, 1 (1966), 59-72. [18] Y. Li and H. Wang, Neumann problems for second order ordinary differential equations across resonance, Zeitschrift für angewandte Mathematik und Physik ZAMP, 46 (1995), 393-406.  doi: 10.1007/BF01003558. [19] A. Liapounoff, Problème général de la stabilité du mouvement, In Annales de la faculté des sciences de Toulouse, 9 (1907), 203-474. Université Paul Sabatier. [20] G. López and J.-A. Montero-Sánchez, Neumann boundary value problems across resonance, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 398-408.  doi: 10.1051/cocv:2006009. [21] J. Mawhin and J. Ward, Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Analysis: Theory, Methods & Applications, 5 (1981), 677-684.  doi: 10.1016/0362-546X(81)90084-5. [22] J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-laplacian, In Abstract and Applied Analysis, Hindawi Publishing Corporation, 2004,147-153. doi: 10.1155/S108533750431002X. [23] J. Qi and S. Chen, Extremal norms of the potentials recovered from inverse dirichlet problems, Inverse Problems, 32 (2016), 035007, 13pp. doi: 10.1088/0266-5611/32/3/035007. [24] K. Shen and M. Zhang, An optimal class of non-degenerate potentials for second-order ordinary differential equations, Boundary Value Problems, 2015 (2015), 1-17.  doi: 10.1186/s13661-015-0451-0. [25] X. Tang and M. Zhang, Lyapunov inequalities and stability for linear hamiltonian systems, Journal of Differential Equations, 252 (2012), 358-381.  doi: 10.1016/j.jde.2011.08.002. [26] H. Wang and Y. Li, Two point boundary value problems for second-order ordinary differential equations across many resonant points, Journal of mathematical analysis and applications, 179 (1993), 61-75.  doi: 10.1006/jmaa.1993.1335. [27] H. Wang and Y. Li, Neumann boundary value problems for second-order ordinary differential equations across resonance, SIAM journal on control and optimization, 33 (1995), 1312-1325.  doi: 10.1137/S036301299324532X. [28] H. Wang and Y. Li, Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations, Zeitschrift für angewandte Mathematik und Physik ZAMP, 47 (1996), 373-384.  doi: 10.1007/BF00916644. [29] M. Zhang, Extremal values of smallest eigenvalues of hill's operators with potentials in $L^1$ balls, Journal of Differential Equations, 246 (2009), 4188-4220.  doi: 10.1016/j.jde.2009.03.016.
Comparison of the classical Lyapunov inequality, main results in [28] and our revised inequalities
The corresponding nontrivial solution $y(x)$
The nontrivial solution $y(x)$
The nontrivial solution $y(x)$
The nontrivial solution $y(x)$
The nontrivial solution $y(x)$
 [1] Xiaofei He, X. H. Tang. Lyapunov-type inequalities for even order differential equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 465-473. doi: 10.3934/cpaa.2012.11.465 [2] Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021059 [3] Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control and Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021 [4] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110 [5] Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571 [6] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [7] Tomasz Komorowski, Adam Bobrowski. A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3495-3502. doi: 10.3934/dcdss.2020248 [8] Takayoshi Ogawa, Kento Seraku. Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1651-1669. doi: 10.3934/cpaa.2018079 [9] H. O. Fattorini. The maximum principle in infinite dimension. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557 [10] Kunquan Lan, Wei Lin. Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1943-1960. doi: 10.3934/dcdsb.2018256 [11] Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161 [12] Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043 [13] Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499 [14] Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035 [15] Ravi P. Agarwal, Abdullah Özbekler. Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2281-2300. doi: 10.3934/cpaa.2016037 [16] Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004 [17] Leszek Gasiński. Existence results for quasilinear hemivariational inequalities at resonance. Conference Publications, 2007, 2007 (Special) : 409-418. doi: 10.3934/proc.2007.2007.409 [18] Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7 [19] Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451 [20] Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139

2020 Impact Factor: 2.425