A priori estimate and existence of solutions with symmetric derivatives for a third-order two-point boundary value problem
Abstract
We study a priori estimate, existence, and uniqueness of solutions with symmetric derivatives for a third-order boundary value problem. The main tool in the proof of our existence result is Leray-Schauder continuation principle. Two examples are included to illustrate the applicability of the results.
Keyword : nonlinear boundary value problems, a priori estimate of solutions, existence of solutions, uniqueness of solution, Leray-Schauder continuation principle
How to Cite
Smirnov, S. (2025). A priori estimate and existence of solutions with symmetric derivatives for a third-order two-point boundary value problem. Mathematical Modelling and Analysis, 30(1), 159–168. https://doi.org/10.3846/mma.2025.21412
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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K. Bingelė, A. Bankauskienė and A. Štikonas. Investigation of spectrum curves for a Sturm-Liouville problem with two-point nonlocal boundary conditions. Math. Model. Anal., 25(1):37–52, 2020. https://doi.org/10.3846/mma.2020.10787
K. Bingelė and A. Štikonas. Investigation of a discrete Sturm-Liouville problem with two-point nonlocal boundary condition and natural approximation of a derivative in boundary condition. Math. Model. Anal., 29(2):309–330, 2024. https://doi.org/10.3846/mma.2024.19829
M. Feng. Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett., 24(8):1419–1427, 2011. https://doi.org/10.1016/j.aml.2011.03.023
J.R. Graef, L. Kong and Q. Kong. Symmetric positive solutions of nonlinear boundary value problems. J. Math. Anal. Appl., 326(2):1310–1327, 2007. https://doi.org/10.1016/j.jmaa.2006.03.064
C.P. Gupta and V. Lakshmikantham. Existence and uniqueness theorems for a third-order three-point boundary value problem. Nonlinear Anal., 16(11):949– 957, 1991. https://doi.org/10.1016/0362-546X(91)90099-M
B. Hopkins and N. Kosmatov. Third-order boundary value problems with sign-changing solutions. Nonlinear Anal., 67(1):126–137, 2007. https://doi.org/10.1016/j.na.2006.05.003
N. Kosmatov. Second order boundary value problems on an unbounded domain. Nonlinear Anal., 68(4):875–882, 2008. https://doi.org/10.1016/j.na.2006.11.043
X. Lin and Z. Zhao. Existence and uniqueness of symmetric positive solutions of 2n-order nonlinear singular boundary value problems. Appl. Math. Lett., 26(7):692–698, 2013. https://doi.org/10.1016/j.aml.2013.01.007
Y. Luo and Z. Luo. Symmetric positive solutions for nonlinear boundary value problems with ϕ-Laplacian operator. Appl. Math. Lett., 23(6):657–664, 2010. https://doi.org/10.1016/j.aml.2010.01.027
Y. Luo and Z. Luo. A necessary and sufficient condition for the existence of symmetric positive solutions of higher-order boundary value problems. Appl. Math. Lett., 25(5):862–868, 2012. https://doi.org/10.1016/j.aml.2011.10.033
K. Pupalaigė, M. Sapagovas and R. Čiupaila. Nonlinear elliptic equation with nonlocal integral boundary condition depending on two parameters. Math. Model. Anal., 27(4):610–628, 2022. https://doi.org/10.3846/mma.2022.16209
S. Roman and A. Štikonas. Third-order linear differential equation with three ad-ditional conditions and formula for Green’s function. Lith. Math. J., 50(4):426– 446, 2010. https://doi.org/10.1007/s10986-010-9097-x
Y. Sun. Existence and multiplicity of symmetric positive solutions for threepoint boundary value problem. J. Math. Anal. Appl., 329(2):998–1009, 2007. https://doi.org/10.1016/j.jmaa.2006.07.001
A. Štikonas. A survey on stationary problems, Green’s functions and spectrum of Sturm-Liouville problem with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 19(3):301–334, 2014. https://doi.org/10.15388/NA.2014.3.1
A. Štikonas and E. Şen. Asymptotic analysis of Sturm-Liouville problem with Dirichlet and nonlocal two-point boundary conditions. Math. Model. Anal., 28(2):308–329, 2023. https://doi.org/10.3846/mma.2023.17617
E. Zeidler. Nonlinear functional analysis and its applications I. Fixed-point theorems. Springer-Verlag, New York, 1986.
X. Zhang and W. Ge. Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput., 219(8):3553–3564, 2012. https://doi.org/10.1016/j.amc.2012.09.037