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)

 
 

 

Existence of solutions of a nonlinear differential equation


Authors: L. Cesari and R. Kannan
Journal: Proc. Amer. Math. Soc. 88 (1983), 605-613
MSC: Primary 34B15
DOI: https://doi.org/10.1090/S0002-9939-1983-0702284-9
MathSciNet review: 702284
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A criterion is proved for the existence of at least one solution to the equation $ u'' + u = g(u) + h$ with $ u(0) = u(\pi ) = 0$, where $ h \in {L_2}[0,\pi ]$ and $ g$ is continuous monotone nonincreasing.


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

  • [1] L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, Wiley, New York, 1963, pp. 149-187. MR 0151678 (27:1662)
  • [2] -, Functional analysis and a reduction method, Michigan Math. J. 11 (1964), 385-414. MR 0173839 (30:4047)
  • [3] -, Functional analysis, nonlinear differential equations, and the alternative method, Nonlinear Functional Analysis and Differential Equations (L. Cesari, R. Kannan, J. D. Schuur, eds.), Dekker, New York, 1976, pp. 1-197. MR 0487630 (58:7249)
  • [4] -, A nonlinear problem in potential theory, Michigan Math. J. 16 (1969), 3-20. MR 0280730 (43:6449)
  • [5] L. Cesari and R. Kannan, Periodic solutions in the large of nonlinear differential equations, Rend. Mat. Univ. Roma (2) 8 (1975), 633-654. MR 0379991 (52:895)
  • [6] -, Solutions in the large of Liénard systems with forcing terms, Ann. Mat. Pura Appl. (4) 111 (1976), 101-124. MR 0427751 (55:781)
  • [7] -, Periodic solutions in the large of Liénard systems with forcing terms, Boll. Un. Mat. Ital. A (6) 2 (1982), 217-223. MR 663284 (83g:34034)
  • [8] J. K. Hale, Applications of alternative problems, Lecture Notes, Brown Univ., Providence, R.I., 1971.
  • [9] A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillations at resonance, Ann. Mat. Pura Appl. 72 (1969), 49-68. MR 0249731 (40:2972)
  • [10] M. Schechter, J. Shapiro and M. Snow, Solutions of the nonlinear problem $ Au = Nu$ in a Banach space, Trans. Amer. Math. Soc. 241 (1978), 69-78. MR 492290 (81g:47069)
  • [11] P. J. McKenna and J. Rauch, Strongly nonlinear perturbations of nonnegative boundary value problems with kernel, J. Differential Equations 28 (1978), 251-265. MR 491053 (81m:35032)
  • [12] L. Cesari and R. Kannan, Functional analysis and nonlinear differential equations, Bull. Amer. Math. Soc. 79 (1973), 1216-1219. MR 0333861 (48:12183)
  • [13] -, Solutions of nonlinear hyperbolic equations at resonance, Nonlinear Anal. 6 (1982), 751-805. MR 671720 (84a:35176)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34B15

Retrieve articles in all journals with MSC: 34B15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0702284-9
Keywords: Resonance, alternative method, auxiliary and determining equations, Leray-Schauder topological argument
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society