Existence and location of solutions to fourth-order Lidstone coupled systems with dependence on odd derivatives
Tóm tắt
This paper addresses the existence and location results for coupled system with two fourth-order differential equations with dependence on all derivatives in nonlinearities and subject to Lidstone-type boundary conditions. To guarantee the existence and location of the solutions, we applied lower and upper solutions technique and degree theory. In this context, we highlight a new type of Nagumo condition to control the growth of the third derivatives and increases the number of applications, as well as a new type of definitions of upper and lower solutions for such coupled systems. Last section contains an application to a coupled system composed by two fourth order equations, which models the estimated bending of simply-supported beam with torsional solitons.
Tài liệu tham khảo
Aftabizadeh, A.R.: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 116, 415–426 (1986)
Agarwal, R.: On fourth order boundary value problems arising in beam analysis. Differ. Integral Equ. 2(1), 91–110 (1989)
Agarwal, R., Wong, P.: Lidstone polynomials and boundary value problems. Comput. Math. Appl. 17(10), 1397–1421 (1989)
Agarwal, R., Wong, P.J.Y.: Quasilinearization and approximate quasilinearization for lidstone boundary value problems. Int. J. Comput. Math. 42(1–2), 99–116 (1992)
Anderson, D., Minhós, F.: A discrete fourth-order Lidstone problem with parameters. Appl. Math. Comput. 214, 523–533 (2009)
Bai, Z., Ge, W.: Solutions of 2nth Lidstone boundary value problems and dependence on higher order derivatives. J. Math. Anal. Appl. 279, 442–450 (2003)
Benci, V., Fortunato, D., Gazzola, F.: Existence of torsional solitons in a beam model of suspension bridge. Arch. Ration. Mech. Anal. 226, 559–585 (2017)
Costabile, F., Napoli, A.: Collocation for high-order differential equations with lidstone boundary conditions. Hindawi Publ. Corp. J. Appl. Math. (2012)
Coster, C., Habets, P.: Two-Point Boundary Value Problems: Lower and Upper Solutions, vol. 205. Elsevier Science & Technology, Oxford (2006)
Davis, J., Henderson, J., Wong, P.J.Y.: General Lidstone problems: multiplicity and symmetry of solutions. J. Math. Anal. Appl. 251, 527–548 (2000)
Fialho, J., Minhós, F.: The role of lower and upper solutions in the generalization of Lidstone problems. In: Discret. Contin. Dyn. Syst. Dynamical systems, differential equations and applications, 9th AIMS Conference, suppl. 217–226 (2013)
Fitzpatrick, P., Martelli, M., Mawhin, J., Nussbaum, R.: Topological Methods for Ordinary Differential Equations. Springer, Berlin (1993)
Gao, C., Xu, J.: Bifurcation techniques and positive solutions of discrete Lidstone boundary value problems. Appl. Math. Comput. 218, 434–444 (2011)
Gazzola, F.: Mathematical Models for Suspension Bridges, vol. 15. Springer International Publishing, Geneva (2015)
Gupta, C.: Existence and uniqueness results for the bending of an elastic beam equation at resonance. J. Math. Anal. Appl. 135, 208–225 (1988)
Gupta, C.P.: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. Int. J. 26(4), 289–304 (1988)
Kang, P., Wei, Z.: Existence of positive solutions for systems of bending elastic beam equations. Electron. J. Differ. Equ. 2012, 1–9 (2012)
Lazer, A.C., Mckenna, P.J.: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990)
Li, Y., Gao, Y.: Existence and uniqueness results for the bending elastic beam equations. Appl. Math. Lett. 95, 72–77 (2019)
Meng, G., Shen, K., Yan, P., Zhang, M.: Strong continuity of the Lidstone Eigenvalues of the beam equation in potentials. Oper. Matrices 8(3), 889–899 (2014)
Minhós, F., Carrasco, H.: Higher Order Boundary Value Problems on Unbounded Domains: Types of Solutions, Functional Problems and Applications. Trends in Abstract and Applied Analysis, vol. 5, World Scientific Publishing Company, Singapore (2017)
Minhós, F., Carrasco, H.: Lidstone-type problems on the whole real line and homoclinic solutions applied to infinite beams. Neural Comput. Appl. (2020)
Minhós, F., Gyulov, T., Santos, A.: Existence and location result for a fourth order boundary value Problem. Discrete Continu. Dyn. Syst. 662–671 (2005)
Minhós. F., Coxe, I.: Solvability for nth order coupled systems with full nonlinearities. In I. Area et al. (eds.), Nonlinear Analysis and Boundary Value Problems NABVP 2018, Springer Proceedings in Mathematics & Statistics, vol. 292, pp 63–80, Springer, Cham (2019)
de Sousa, R., Minhós, F.: Coupled systems of Hammerstein-type integral equations with sign-changing kernels. Nonlinear Anal. Real World Appl. 50, 469–483 (2019)
Sun, J.-P., Wang, X.: Monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. Hindawi Publ. Corp. Math. Probl. Eng. (2011)
Talwar, J., Mohanty, R.K.: A class of numerical methods for the solution of fourth-order ordinary differential equations in polar coordinates. Hindawi Publ. Corp. Adv. Numer. Anal. (2012)
Vrabel, R.: Formation of boundary layers for singularly perturbed fourth-order ordinary differential equations with the Lidstone boundary conditions. J. Math. Anal. Appl. 440, 65–73 (2016)
Wang, Y.-M.: Higher-order Lidstone boundary value problems for elliptic partial differential equations. J. Math. Anal. Appl. 308, 314–333 (2005)
Wong, P.J.Y., Agarwal, R.: Eigenvalues of Lidstone boundary value problems. Appl. Math. Comput. 104, 15–31 (1999)
Yang, B.: Upper and lower estimates for positive solutions of the higher order Lidstone boundary value problem. J. Math. Anal. Appl. 382, 290–302 (2011)
Yao, Q.: Existence of n solutions and/or positive solutions to a semipositone elastic beam equation. Nonlinear Anal. 66, 138–150 (2007)
Zhang, X., Feng, M.: Positive solutions of singular beam equations with the bending term. Bound. Value Probl. 2015, 84 (2015)
Zhu, F., Liu, L., Wu, Y.: Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems. Appl. Math. Comput. 216, 448–457 (2010)