Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-14T06:58:41.642Z Has data issue: false hasContentIssue false
  • English
  • Français

Solvability of semilinear abstract equations at resonance

Part of: Boundary value problems

Published online by Cambridge University Press:  26 February 2010

Zachariah Sinkala
Zachariah Sinkala
Affiliation:
Department of Mathematics & Statistics, Middle Tennessee State University, Murfreesboro, TN 37132, U.S.A.
Get access

Abstract

We establish a generalization of the Cesari-Kannan existence result for problems of the type Lx = N(x), x∈X where X is a separable Hilbert functional space, L is a selfadjoint linear differential operator with nontrivial finite dimensional kernel and N:X→X is a bounded continuous nonlinear operator. This generalization leads to new results when the dimension of the kernel of L is greater than one. Applications to systems of second order ordinary differential equations are given.

MSC classification

Secondary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
Mathematika , Volume 41 , Issue 2 , December 1994 , pp. 301 - 311
Copyright
Copyright © University College London 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Cesari, L.. Functional analysis, nonlinear differential equations, and the alternative method. In Nonlinear Functional Analysis and Differential Equations. Editors Cesari, L., Kannan, R. and Schuur, J. D. (Marcel Dekker, New York, 1976), 1–97.Google Scholar
2.Cesari, L. and Kannan, R.. An abstract existence theorem at resonance. Proc. Amer. Math. Soc., 63 (1977), 221–225.CrossRefGoogle Scholar
3.Cesari, L. and Kannan, R.. Solutions of nonlinear hyperbolic equations at resonance. Nonlinear Analysis TMA, 6 (1982), 751–805.CrossRefGoogle Scholar
4.Gaines, R. E. and Mawhin, J. L.. Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics, 568 (Springer-Verlag, Berlin, Heidelberg, 1977).Google Scholar
5.Goldberg, S.. Unbounded linear operators (Wiley-Interscience, New York, 1969).Google Scholar
6.Kannan, R. and Ortega, R.. Periodic solutions of pendulum-type equations. J. Diff. Equations, 59 (1985), 123–144.CrossRefGoogle Scholar
7.Lazer, A. C. and Leach, D. E.. Bounded perturbations of forced harmonic oscillators at resonance. Ann. Mat. Pura Applic, 72 (1969), 49–68.CrossRefGoogle Scholar
8.Nagle, R. K. and Sinkala, Z.. Semilinear equations at resonance where the kernel has dimension two. In Differential Equations: Stability and Control. Editor Elaydi, Saber (Marcel Dekker, 1991).Google Scholar
9.Nagle, R. K. and Sinkala, Z.. Existence of 2π-periodic solutions for nonlinear systems of first order ordinary differential equations at resonance. To appear in J. Nonlinear Analysis: Theory, Method and Applications.Google Scholar
10.Sinkala, Z.. Existence of T-periodic to second nonlinear systems at resonance. In Differential Equations: Stability and Control. Editor Elaydi, Saber (Marcel Dekker, 1991).Google Scholar
11.Zeidler, E.. Nonlinear functional analysis and its application I (Springer-Verlag, New York, 1988).CrossRefGoogle Scholar