Abstract
We study the existence, localization and multiplicity of positive solutions for a nonlinear fourth-order two-point boundary value problem. The approach is based on critical point theorems in conical shells, Krasnoselskii’s compression-expansion theorem, and unilateral Harnack type inequalities.
Authors
Alberto Cabada
Instituto de Matematicas, Facultade de Matem aticas, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia, Spain
Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Lorena Saavedra
Instituto de Matematicas, Facultade de Matematicas, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia, Spain
Stepan A. Tersian
Department of Mathematics, University of Ruse, 7017 Ruse, Bulgaria.
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 – Sofia, Bulgaria
Keywords
Fourth-order differential equation; boundary-value problem; positive solution; critical point; fixed point.
Paper coordinates
A. Cabada, R. Precup, L. Saavedra, S. Tersian, Multiple positive solutions to a fourth order boundary value problem, Electron. J. Differential Equations 2016 (2016), No. 254, 1-18.
About this paper
Journal
Electronic Journal Differential Equations
Publisher Name
Print ISSN
Online ISSN
1072-6691
google scholar link
[1] A. Cabada, J. A. Cid, B. M´aquez-Villamar´ın; Computation of Green’s functions for boundary value problems with Mathematica, Appl. Math. Comput. 219 (2012), 1919–1936.
[2] A. Cabada, S.Tersian; Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput. 219 (2013), 5261–5267.
[3] X. Cai, Z. Guo, Existence of solutions of nonlinear fourth order discrete boundary value problem, J. Difference Equ. Appl. 12 (2006), 459–466.
[4] J. A. Cid, D. Franco, F. Minh´os; Positive fixed points and fourth-order equations, Bull. Lond. Math. Soc. 41 (2009), 72–78.
[5] R. Enguica, L. Sanchez; Existence and localization of solutions for fourth-order boundary value problems, Electron. J. Differential Equations 2007 (2007), No. 127, 1–10.
[6] J. R. Graef, L. Kong, Q. Kong, B. Yang; Positive solutions to a fourth order boundary value problem, Results Math. 59 (2011), 141–155.
[7] D. Guo, V. Lakshmikantham; Nonlinear Problems in Abstract Cones, Academic Press, 1988.
[8] D.-R. Herlea; Positive solutions for second-order boundary-value problems with phiLaplacian, Electron. J. Differential Equations 2016 (2016), No. 51, 1–8.
[9] M. A. Krasnosel’ski˘ı, P. P. Zabre˘ıko; Geometrical Methods of Nonlinear Analysis, Springer, 1984.
[10] F. Li, Q. Zhang, Z. Liang; Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear Anal. 62 (2005), 803–816.
[11] S. Li and X. Zhang; Existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions, Comput. Math. Appl. 63 (2012), 1355–1360.
[12] T. F. Ma; Positive solutions for a beam equation on a nonlinear elastic foundation, Math. Comput. Modelling 39 (2004), 1195–1201.
[13] R. Precup; Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal. 75 (2012), 834–851.
[14] R. Precup; A compression type mountain pass theorem in conical shells, J. Math. Anal. Appl. 338 (2008), 1116–1130.
[15] M. I. Sonalla; Vibrations of cantevier beams with various initial conditions, Thesis, Ohio State University, 1989.
[16] S. A. Tersian, J. V. Chaparova; Periodic and homoclinic solutions of extended FisherKolmogorov equation, J. Math. Anal. Appl. 266 (2001), 490–506.
[17] L. Yang, H. Chen, X. Yang; The multiplicity of solutions for fourth-order equations generated from a boundary condition, Appl. Math. Letters 24 (2011), 1599–1603.
[18] J. R. L. Webb, G. Infante, D. Franco; Positive solutions of nonlinear fourth-order boundaryvalue problems with local and non-local boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 427–446.