A connection between the Landesman-Lazer condition and the solvability of the equation Lx = N(x) in a cone with a noninvertible linear operator L is studied. The result is based on the abstract framework from [5], applied to the existence of periodic solutions of ordinary differential equations, and compared with theorems by Santanilla (see [7]).
@article{bwmeta1.element.bwnjournal-article-apmv58z1p95bwm,
author = {Bogdan Przeradzki},
title = {A note on solutions of semilinear equations at resonance in a cone},
journal = {Annales Polonici Mathematici},
volume = {58},
year = {1993},
pages = {95-103},
zbl = {0776.34035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv58z1p95bwm}
}
Bogdan Przeradzki. A note on solutions of semilinear equations at resonance in a cone. Annales Polonici Mathematici, Tome 58 (1993) pp. 95-103. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv58z1p95bwm/
[000] [1] R. E. Gaines and J. Santanilla, A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations, Rocky Mountain J. Math. 12 (1982), 669-678. | Zbl 0508.34030
[001] [2] E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609-623. | Zbl 0193.39203
[002] [3] J. L. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conf. Ser. in Math. 40, Amer. Math. Soc., Providence, R.I., 1979.
[003] [4] B. Przeradzki, An abstract version of the resonance theorem, Ann. Polon. Math. 53 (1991), 35-43. | Zbl 0746.47043
[004] [5] B. Przeradzki, Operator equations at resonance with unbounded nonlinearities, preprint.
[005] [6] B. Przeradzki, A new continuation method for the study of nonlinear equations at resonance, J. Math. Anal. Appl., to appear.
[006] [7] J. Santanilla, Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations, ibid. 126 (1987), 397-408. | Zbl 0629.34017
[007] [8] J. Santanilla, Existence of nonnegative solutions of a semilinear equation at resonance with linear growth, Proc. Amer. Math. Soc. 105 (1989), 963-971. | Zbl 0687.47045
[008] [9] S. A. Williams, A sharp sufficient condition for solution of a nonlinear elliptic boundary value problem, J. Differential Equations 8 (1970), 580-586. | Zbl 0209.13003