On coupled systems of Lidstone-type boundary value problems
Abstract
This research concerns the existence and location of solutions for coupled system of differential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented.
Keyword : coupled nonlinear systems, coupled lower and upper solutions, Lidstone-type boundary value problems, operator theory, suspension bridges
How to Cite
de Sousa, R., Minhós, F. ., & Fialho, J. (2021). On coupled systems of Lidstone-type boundary value problems. Mathematical Modelling and Analysis, 26(3), 358-371. https://doi.org/10.3846/mma.2021.12977
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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R.P. Agarwal and P.J.Y. Wong. Lidstone polynomials and boundary value problems. Computers & Mathematics with Applications, 17(10):1397–1421, 1989. https://doi.org/10.1016/0898-1221(89)90023-0
D.R. Anderson and F. Minhós. A discrete fourth-order Lidstone problem with parameters. Applied Mathematics and Computation, 214(2):523–533, 2009. https://doi.org/10.1016/j.amc.2009.04.034
A. Bagheri, M. Alipour, O.E. Ozbulut and D.K. Harris. Identification of flexural rigidity in bridges with limited structural information. Journal of Structural Engineering, 144(8):04018126, 2018. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002131
A. Cabada and L. López-Somoza. Lower and upper solutions for even order boundary value problems. Mathematics, 7(10):878, 2019. https://doi.org/10.3390/math7100878
J.A. Cid, D. Franco and F. Minhós. Positive fixed points and fourth-order equations. Bulletin of the London Mathematical Society, 41(1):72–78, 2009. https://doi.org/10.1112/blms/bdn105
C. de Coster and P. Habets. Two-Point Boundary Value Problems: Lower and Upper Solutions, volume 205. Elsevier Science & Technology, Oxford, United Kingdom, 2006.
R. de Sousa and F. Minhós. Coupled systems of Hammerstein-type integral equations with sign-changing kernels. Nonlinear Analysis: Real World Applications, 50:469–483, 2019. https://doi.org/10.1016/j.nonrwa.2019.05.011
J. Fialho and F. Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Discrete and Continuous Dynamical Systems, 2013(Special):217–226, 2013. https://doi.org/10.3934/proc.2013.2013.217
P. Fitzpatrick, M. Martelli, J. Mawhin and R. Nussbaum. Theory of Ordinary Differential Equations. Springer-Verlag Berlin Heidelberg, 1993. https://doi.org/10.1007/BFb0085073
C. Gao and J. Xu. Bifurcation techniques and positive solutions of discrete Lidstone boundary value problems. Applied Mathematics and Computation, 218(2):434–444, 2011. https://doi.org/10.1016/j.amc.2011.05.083
F. Gazzola, Y. Wang and R. Pavani. Variational formulation of the Melan equation. Mathematical Methods in the Applied Sciences, 41(3):943–951, 2016. https://doi.org/10.1002/mma.3962
C.P. Gupta. Existence and uniqueness results for the bending of an elastic beam equation at resonance. Journal of Mathematical Analysis and Applications, 135(1):208–225, 1988. https://doi.org/10.1016/0022-247X(88)90149-7
C.P. Gupta. Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis: An International Journal, 26(4):289–304, 2007. https://doi.org/10.1080/00036818808839715
Y. Li and Y. Gao. Existence and uniqueness results for the bending elastic beam equations. Applied Mathematics Letters, 95:72–77, 2019. https://doi.org/10.1016/j.aml.2019.03.025
J. Melan. Theory of arches and suspension bridges. Myron Clark Pul. Comp., London, 1913.
F. Minhós and H. Carrasco. Higher Order Boundary Value Problems on Unbounded Domains. World Scientific Publishing Company, Singapore, 2017. https://doi.org/10.1142/10448
F. Minhós, T. Gyulov and A.I. Santos. Existence and location result for a fourth order boundary value problem. Discrete and Continuous Dynamical Systems, 2005(0133-0189 2005 Special 662):662–671, 2005.
L. Sun, M. Zhou and G. Wang. Existence and location results for fully nonlinear boundary value problem of nth-order nonlinear system. Boundary Value Problems, 2009(791548):17, 2009. https://doi.org/10.1155/2009/791548
R. Vrabel. Formation of boundary layers for singularly perturbed fourthorder ordinary differential equations with the Lidstone boundary conditions. Journal of Mathematical Analysis and Applications, 440(1):65–73, 2016. https://doi.org/10.1016/j.jmaa.2016.03.017
Y.-M. Wang. Higher-order Lidstone boundary value problems for elliptic partial differential equations. Journal of Mathematical Analysis and Applications, 308(1):314–333, 2005. https://doi.org/10.1016/j.jmaa.2005.01.019
P.J.Y. Wong. Triple solutions of complementary Lidstone boundary value problems via fixed point theorems. Boundary Value Problems, 125(2014), 2014. https://doi.org/10.1186/1687-2770-2014-125
F. Zhu, L. Liu and Y. Wu. Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems. Applied Mathematics and Computation, 216(2):448–457, 2010. https://doi.org/10.1016/j.amc.2010.01.038
R.P. Agarwal and P.J.Y. Wong. Lidstone polynomials and boundary value problems. Computers & Mathematics with Applications, 17(10):1397–1421, 1989. https://doi.org/10.1016/0898-1221(89)90023-0
D.R. Anderson and F. Minhós. A discrete fourth-order Lidstone problem with parameters. Applied Mathematics and Computation, 214(2):523–533, 2009. https://doi.org/10.1016/j.amc.2009.04.034
A. Bagheri, M. Alipour, O.E. Ozbulut and D.K. Harris. Identification of flexural rigidity in bridges with limited structural information. Journal of Structural Engineering, 144(8):04018126, 2018. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002131
A. Cabada and L. López-Somoza. Lower and upper solutions for even order boundary value problems. Mathematics, 7(10):878, 2019. https://doi.org/10.3390/math7100878
J.A. Cid, D. Franco and F. Minhós. Positive fixed points and fourth-order equations. Bulletin of the London Mathematical Society, 41(1):72–78, 2009. https://doi.org/10.1112/blms/bdn105
C. de Coster and P. Habets. Two-Point Boundary Value Problems: Lower and Upper Solutions, volume 205. Elsevier Science & Technology, Oxford, United Kingdom, 2006.
R. de Sousa and F. Minhós. Coupled systems of Hammerstein-type integral equations with sign-changing kernels. Nonlinear Analysis: Real World Applications, 50:469–483, 2019. https://doi.org/10.1016/j.nonrwa.2019.05.011
J. Fialho and F. Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Discrete and Continuous Dynamical Systems, 2013(Special):217–226, 2013. https://doi.org/10.3934/proc.2013.2013.217
P. Fitzpatrick, M. Martelli, J. Mawhin and R. Nussbaum. Theory of Ordinary Differential Equations. Springer-Verlag Berlin Heidelberg, 1993. https://doi.org/10.1007/BFb0085073
C. Gao and J. Xu. Bifurcation techniques and positive solutions of discrete Lidstone boundary value problems. Applied Mathematics and Computation, 218(2):434–444, 2011. https://doi.org/10.1016/j.amc.2011.05.083
F. Gazzola, Y. Wang and R. Pavani. Variational formulation of the Melan equation. Mathematical Methods in the Applied Sciences, 41(3):943–951, 2016. https://doi.org/10.1002/mma.3962
C.P. Gupta. Existence and uniqueness results for the bending of an elastic beam equation at resonance. Journal of Mathematical Analysis and Applications, 135(1):208–225, 1988. https://doi.org/10.1016/0022-247X(88)90149-7
C.P. Gupta. Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis: An International Journal, 26(4):289–304, 2007. https://doi.org/10.1080/00036818808839715
Y. Li and Y. Gao. Existence and uniqueness results for the bending elastic beam equations. Applied Mathematics Letters, 95:72–77, 2019. https://doi.org/10.1016/j.aml.2019.03.025
J. Melan. Theory of arches and suspension bridges. Myron Clark Pul. Comp., London, 1913.
F. Minhós and H. Carrasco. Higher Order Boundary Value Problems on Unbounded Domains. World Scientific Publishing Company, Singapore, 2017. https://doi.org/10.1142/10448
F. Minhós, T. Gyulov and A.I. Santos. Existence and location result for a fourth order boundary value problem. Discrete and Continuous Dynamical Systems, 2005(0133-0189 2005 Special 662):662–671, 2005.
L. Sun, M. Zhou and G. Wang. Existence and location results for fully nonlinear boundary value problem of nth-order nonlinear system. Boundary Value Problems, 2009(791548):17, 2009. https://doi.org/10.1155/2009/791548
R. Vrabel. Formation of boundary layers for singularly perturbed fourthorder ordinary differential equations with the Lidstone boundary conditions. Journal of Mathematical Analysis and Applications, 440(1):65–73, 2016. https://doi.org/10.1016/j.jmaa.2016.03.017
Y.-M. Wang. Higher-order Lidstone boundary value problems for elliptic partial differential equations. Journal of Mathematical Analysis and Applications, 308(1):314–333, 2005. https://doi.org/10.1016/j.jmaa.2005.01.019
P.J.Y. Wong. Triple solutions of complementary Lidstone boundary value problems via fixed point theorems. Boundary Value Problems, 125(2014), 2014. https://doi.org/10.1186/1687-2770-2014-125
F. Zhu, L. Liu and Y. Wu. Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems. Applied Mathematics and Computation, 216(2):448–457, 2010. https://doi.org/10.1016/j.amc.2010.01.038