Abstract
This paper is concerned with the behaviour of solutions of nonlinear operator equations with non-invertible linear part under perturbations of the operators involved. Even in the linear case, the solutions need not change continously if the linear part is perturbed in such a way that its rank changes. We prove a continuous-dependence result for the nonlinear problem and illustrate it with an example involving the behaviour of periodic solutions of nonlinear differential equations at resonance under perturbations of the differential operator.
Keywords
- Banach Space
- Linear Operator
- Periodic Solution
- Nonlinear Problem
- Bounded Linear Operator
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References
L. CESARI: Functional analysis and periodic solutions of nonlinear differential equations, in: Contributions to Differential Equations 1, Wiley, New York 1963, 149–187.
L. CESARI: Functional analysis, nonlinear differential equations, and the alternative method, in: Nonlinear Functional Analysis and Differential Equations (L. Cesari, R. Kannan, J. Schuur, eds.), Dekker, New York 1976, 1–197.
L. CESARI, H.W. ENGL: Existence and uniqueness of solutions for nonlinear alternative problems in a Banach space, Czechoslovak Math. Jour. 31 (106), (1981), 670–678.
H.W. ENGL, R. KRESS: A singular perturbation problem for linear operators with an application to electrostatic and magnetostatic boundary and transmission problems, Math. Meth. in the Appl. Sc. 3 (1981), 249–274.
M.Z. NASHED, G.F. VOTRUBA: A unified operator theory of generalized inverses, in: Generalized Inverses and Applications (M.Z. Nashed, ed.), Academic Press, New York 1976, 1–109.
M.Z. NASHED: Perturbations and approximations for generalized inverses and linear operator equations, same volume, 325–396.
H. WACKER: A summary of the developments on imbedding methods, in: Continuation Methods (H. Wacker, ed.), Academic Press, New York 1978, 1–35.
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© 1984 Springer-Verlag
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Engl, H.W. (1984). Behaviour of solutions of nonlinear alternative problems under perturbations of the linear part with rank change. In: Vinti, C. (eds) Nonlinear Analysis and Optimization. Lecture Notes in Mathematics, vol 1107. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101494
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DOI: https://doi.org/10.1007/BFb0101494
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13903-4
Online ISBN: 978-3-540-39123-4
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