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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 |
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.
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. |





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