Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Reduction theorems for a class of semilinear equations at resonance

Author: Peter W. Bates
Journal: Proc. Amer. Math. Soc. 84 (1982), 73-78
MSC: Primary 47H15; Secondary 34C25
MathSciNet review: 633281
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In solving equations of the form $ Lu - Nu = p$ in a Hilbert space, where $ L$ is linear and $ N$ is nonlinear, the alternative method can sometimes be used to reduce the problem to one in a subspace. In this note previous reduction results are extended and at the same time the proofs are simplified. The approach is to use simple fixed point theorems in place of the traditional variational methods which are often quite delicate.

References [Enhancements On Off] (What's this?)

  • [1] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 (1979), 127-166. MR 550724 (80j:47078)
  • [2] P. W. Bates, A variational approach to solving semilinear equations at resonance, Proc. Internat. Conf. Nonlinear Phenomena in Math. Sci. (to appear). MR 727971 (85a:49031)
  • [3] P. W. Bates and A. Castro, Necessary and sufficient conditions for existence of solutions to equations with noninvertible linear part, preprint, 1979. MR 674931 (84b:47080)
  • [4] H. Brezis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 225-326. MR 0513090 (58:23813)
  • [5] F. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1100-1103. MR 0177295 (31:1558)
  • [6] A. Castro, Periodic solutions of the forced pendulum equation, Differential Equations (S. Ahmad, M. Keener, A. Lazer, Eds.), Academic Press, New York, 1980, 149-159. MR 580791 (81h:34042)
  • [7] J. Mawhin, Contractive mappings and periodically perturbed conservative systems, Arch. Math. (Brno) 12 (1976), 67-73. MR 0437858 (55:10779)
  • [8] -, Semilinear equations of gradient type in Hilbert spaces and applications to differential equations, Rep. 139, Inst. Math. Pure Appl. Univ. Catholique de Louvain, October 1980.
  • [9] M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, San Francisco, Calif., 1964. MR 0176364 (31:638)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H15, 34C25

Retrieve articles in all journals with MSC: 47H15, 34C25

Additional Information

Keywords: Hilbert space, spectrum, nonexpansive map, periodic solutions of nonlinear ordinary differential equations
Article copyright: © Copyright 1982 American Mathematical Society