We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.
@article{bwmeta1.element.doi-10_2478_s11533-011-0052-9,
author = {Michael Gil'},
title = {Positivity conditions and bounds for Green's functions for higher order two-point BVP},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {1156-1163},
zbl = {1315.34038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0052-9}
}
Michael Gil’. Positivity conditions and bounds for Green’s functions for higher order two-point BVP. Open Mathematics, Tome 9 (2011) pp. 1156-1163. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0052-9/
[1] Agarwal R.P., O’Regan D., Ordinary and Partial Differential Equations, Universitext, Springer, New York, 2009
[2] Agarwal R.P., O’Regan D., Wong P.J.Y., Positive Solutions of Differential, Difference and Integral Equations, Kluwer, Dordrecht, 1999
[3] Bai Z., W. Ge, Solutions of 2nth Lidstone boundary value problems and dependence on higher order derivatives, J. Math. Anal. Appl., 2003, 279(2), 442–450 http://dx.doi.org/10.1016/S0022-247X(03)00011-8 | Zbl 1029.34019
[4] Chen Y.S., The singular perturbation of two-point boundary value problem for nonlinear systems, Ann. Differential Equations, 1995, 10(5), 18–21
[5] Eloe P.W., Henderson J., Positive solutions for higher order ordinary differential equations, Electron. J. Differential Equations, 1995, #3 | Zbl 0814.34017
[6] Gil’ M.I., Positive solutions of equations with nonlinear causal mappings, Positivity, 2007, 11(3), 523–535 http://dx.doi.org/10.1007/s11117-007-2076-8 | Zbl 1136.34064
[7] Gil’ M.I., Positivity of the Green functions for higher order ordinary differential equations, Electron. J. Differential Equations, 2008, #97 | Zbl 1172.34321
[8] Gil’ M.I., Two sided bounds and positivity conditions for the Green function to periodic problem for a higher order ODE, Int. J. Dyn. Syst. Differ. Equ., 2009, 2(3–4), 253–261 | Zbl 1204.34033
[9] Guo Y., Gao Y., The method of upper and lower solutions for a Lidstone boundary value problem, Czechoslovak Math. J., 2005, 55(130) (3), 639–652 http://dx.doi.org/10.1007/s10587-005-0051-8 | Zbl 1081.34019
[10] Jódar L., Villanueva R.J., Navarro E., Solving nonmonic higher order two-point boundary value matrix problems using quasi-Green’s matrix functions, Analysis, 1992, 12(1–2), 139–157 | Zbl 0757.34020
[11] Krasnosel’skii M.A., Burd V.Sh., Kolesov Yu.S., Nonlinear Almost Periodic Oscillations, Nauka, Moscow, 1970 (in Russian)
[12] Krasnosel’skii M.A., Zabreiko P.P., Geometric Methods of Nonlinear Analysis, Nauka, Moscow, 1975 (in Russian)
[13] Kusano T., Naito M., Unbounded nonoscillatory solutions of nonlinear ordinary differential equations of arbitrary order, Hiroshima Math. J., 1988, 18(2), 361–372 | Zbl 0654.34030
[14] Laptinsky V.N., Makovetsky I.I., On the two-point boundary-value problem for the Riccati matrix differential equation, Cent. Eur. J. Math., 2005, 3(1), 143–154 http://dx.doi.org/10.2478/BF02475661 | Zbl 1069.34023
[15] Murty K.N., Sarma G.V.R.L., Theory of differential inequalities for two-point boundary value problems and their applications to three-point B.V.Ps associated with nth order non-linear system of differential equations, Appl. Anal., 2002, 81(1), 39–49 http://dx.doi.org/10.1080/0003681021000021051 | Zbl 1031.34015
[16] Murty M.S.N., Apparao B.V., Two point boundary value problems for matrix differential equations, J. Indian Math. Soc., 2006, 73(1–2), 1–7 | Zbl 1126.34314
[17] Nagle R.K., Saff E.B., Snider A.D., Fundamentals of Differential Equations and Boundary Value Problems, Addison-Wesley, New York, 1996 | Zbl 0949.34001
[18] Naito M., Yano K., Positive solutions of higher order ordinary differential equations with general nonlinearities, J. Math. Anal. Appl., 2000, 250(1), 27–48 http://dx.doi.org/10.1006/jmaa.2000.6953 | Zbl 0967.34004
[19] Wang Y.-M., On 2nth-order Lidstone boundary value problems, J. Math. Anal. Appl., 2005, 312(2), 383–400 http://dx.doi.org/10.1016/j.jmaa.2005.03.039 | Zbl 1090.34015
[20] Yao Q., Positive solutions to a semilinear system of second-order two-point boundary value problems, Ann. Differential Equations, 2006, 22(1), 87–94 | Zbl 1113.34012