Discrete resonance problems subject to periodic forcing
HTML articles powered by AMS MathViewer
- by Stephen B. Robinson and Klaus Schmitt PDF
- Proc. Amer. Math. Soc. 148 (2020), 471-477 Request permission
Abstract:
In this paper, we consider the following discrete nonlinear problem which is subject to a periodic nonlinear forcing term: \begin{equation*} A u = \lambda u +p(u) + h, \end{equation*} where $A$ is an $n\times n$ matrix with real components, $p: \mathbb {R}^n\to \mathbb {R}^n$ is a periodic forcing term, and $\langle h,\overline {\phi } \rangle =0$, where $\overline {\phi }$ is an eigenvector of $A^T,$ the transpose of $A$, corresponding to a simple real eigenvalue $\lambda$. Conditions on these terms will be provided such that this problem will have infinitely many distinct solutions. The results here are motivated by some recent results for discrete systems and by results obtained for analogous boundary value problems for semilinear elliptic problems at resonance.References
- A. S. Besicovitch, Almost periodic functions, Dover Publications, Inc., New York, 1955. MR 0068029
- Harald Bohr, Almost Periodic Functions, Chelsea Publishing Co., New York, N.Y., 1947. MR 0020163
- A. Cañada and F. Roca, Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions, Differential Integral Equations 10 (1997), no. 6, 1113–1122. MR 1608041
- E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl. (4) 131 (1982), 167–185. MR 681562, DOI 10.1007/BF01765151
- Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
- Stanislaus Maier-Paape and Klaus Schmitt, Asymptotic behavior of solution continua for semilinear elliptic problems, Canad. Appl. Math. Quart. 4 (1996), no. 2, 211–228. MR 1418271
- S. Parsons and S. B. Robinson, A discrete resonance problem with periodic nonlinear forcing, North Carolina J. Math. Stat. 1 (2015), 22–29.
- Sarah Parsons and Stephen B. Robinson, A discrete analog of a theorem by Schaaf and Schmitt, Dynamic systems and applications. Vol. 7, Dynamic, Atlanta, GA, 2016, pp. 300–302. MR 3645024
- H.-O. Peitgen, D. Saupe, and K. Schmitt, Nonlinear elliptic boundary value problems versus their finite difference approximations: numerically irrelevant solutions, J. Reine Angew. Math. 322 (1981), 74–117. MR 603027
- Paul H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462–475. MR 328705, DOI 10.1016/0022-0396(73)90061-2
- Renate Schaaf and Klaus Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc. 306 (1988), no. 2, 853–859. MR 933322, DOI 10.1090/S0002-9947-1988-0933322-5
- Renate Schaaf and Klaus Schmitt, Asymptotic behavior of positive solution branches of elliptic problems with linear part at resonance, Z. Angew. Math. Phys. 43 (1992), no. 4, 645–676. MR 1176305, DOI 10.1007/BF00946255
- Johann Schröder, $M$-matrices and generalizations using an operator theory approach, SIAM Rev. 20 (1978), no. 2, 213–244. MR 498642, DOI 10.1137/1020037
Additional Information
- Stephen B. Robinson
- Affiliation: Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, North Carolina 27109
- MR Author ID: 341844
- Email: sbr@wfu.edu
- Klaus Schmitt
- Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112
- MR Author ID: 192515
- Email: schmitt@math.utah.edu
- Received by editor(s): March 16, 2019
- Received by editor(s) in revised form: May 17, 2019
- Published electronically: July 30, 2019
- Communicated by: Wenxian Shen
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 471-477
- MSC (2010): Primary 15A24, 35P30, 39A45, 65H10
- DOI: https://doi.org/10.1090/proc/14713
- MathSciNet review: 4052187