Dirichlet BVP for the second order nonlinear ordinary differential equations at resonance
Abstract
Landesman-Lazer’s type efficient sufficient conditions are established for the solvability of the Dirichlet problem 
, for 
where 
;R) and f is the L([a,b]; R) Caratheodory function, in the case where the linear problem 
has nontrivial solutions. The results obtained in the paper are optimal in the sense that if 
, i.e., when nonlinear equation turns to the linear equation, from our results follows the first part of Fredholm’s theorem.
Keyword : nonlinear ordinary differential equation, Dirichlet problem at resonance
How to Cite
Mukhigulashvili, S., & Manjikashvili, M. (2019). Dirichlet BVP for the second order nonlinear ordinary differential equations at resonance. Mathematical Modelling and Analysis, 24(4), 585-597. https://doi.org/10.3846/mma.2019.035
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S. Mukhigulashvili. The Dirichlet BVP the second order nonlinear ordinary differential equation at resonance. Italian J. Of Pure and Appl. Math., 2011(28):177–204, 2011.
S. Mukhigulashvili. The mixed BVP for second order nonlinear ordinary differential equation at resonance. Math. Nachr., 290(2):393–400, 2017. https://doi.org/10.1002/mana.201500247
N. Azbelevand, V. Maksimov and L.F. Rakhmatullina. Introduction to the Theory of Functional Differential Equations. Nauka, Moscow, 1991.
C. De Coster and P. Habets. Upper and Lower Solutions in the theory of ODE boundary value problems. Sprringer Wien, NewYork, 1996. https://doi.org/10.1007/978-3-7091-2680-6_1
P. Drabek. On the resonance problem with nonlinearity which has arbitrary linear growth. J. Math. Anal. Appl., 127(2):435–442, 1987. https://doi.org/10.1016/0022-247X(87)90121-1
R. Gaines and J Mawhin. Coincidence Degree and Nonlinear Differential Equations. Springer-Verlag. Berlin. Heidelberg. New York, 1977. https://doi.org/10.1007/BFb0089537
C. Ha and C. Kuo. On the solvability of a two point boundary value problem at resonance. Topol. Methods Nonlinear Anal., 1(3):295–302, 1993. https://doi.org/10.12775/TMNA.1993.021
C. Ha and C. Kuo. On the solvability of a two point boundary value problem at resonance II. Topol. Methods Nonlinear Anal., 11(3):159–168, 1998. https://doi.org/10.12775/TMNA.1998.010
R. Iannacci and M. Nkashama. Nonlinear boundary value problems at resonance. Proc. Amer. Math. Soc., 11(4):455–473, 1987. https://doi.org/10.1016/0362546X(87)90064-2
R. Iannacci and M. Nkashama. Nonlinear two point boundary value problems at resonance without Landesman-Lazer condition. Proc. Amer. Math. Soc., 106(4):943–952, 1989. https://doi.org/10.2307/2047278
R. Kannan, J. Nieto and M. Ray. A class of nonlinear boundary value problems without Landesman-Lazer condition. J. Math. Anal. Appl., 105(1):1–11, 1985. https://doi.org/10.1016/0022-247X(85)90093-9
I. Kiguradze and B. Shekhter. Singular boundary value problems for second order ordinary differential equations. J. Sov. Math., 43(2):2340–2417, 1988. https://doi.org/10.1007/BF01100361
E. Landesman and A. Lazer. Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech., 19(7):609–623, 1970. https://doi.org/10.1512/iumj.1970.19.19052
S. Mukhigulashvili. The Dirichlet BVP the second order nonlinear ordinary differential equation at resonance. Italian J. Of Pure and Appl. Math., 2011(28):177–204, 2011.
S. Mukhigulashvili. The mixed BVP for second order nonlinear ordinary differential equation at resonance. Math. Nachr., 290(2):393–400, 2017. https://doi.org/10.1002/mana.201500247