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.
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Figure 1. Comparison of the classical Lyapunov inequality, main results in [28] and our revised inequalities
S. R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Elsevier, 1974.
![]() ![]() |
|
G. Borg
, On a liapounoff criterion of stability, American Journal of Mathematics, 71 (1949)
, 67-70.
doi: 10.2307/2372093.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
S.-S. Cheng
, Lyapunov inequalities for differential and difference equations, Fasc. Math, 23 (1991)
, 25-41.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
P. Hartman, Ordinary Differential Equations, Birkhauser, Boston, 1982.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. Henderson
, Optimal interval lengths for nonlocal boundary value problems for second order lipschitz equations, Communications in Applied Analysis, 15 (2011)
, 475-482.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. Lees
, Discrete methods for nonlinear two-point boundary value problems, Numerical Solution of Partial Differential Equations, 1 (1966)
, 59-72.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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
The nontrivial solution
The nontrivial solution
The nontrivial solution
The nontrivial solution