Existence results on the one-dimensional Dirichlet problem suggested by the piecewise linear case

Author:
M. Arias

Journal:
Proc. Amer. Math. Soc. **97** (1986), 121-127

MSC:
Primary 34B15

DOI:
https://doi.org/10.1090/S0002-9939-1986-0831399-2

MathSciNet review:
831399

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the existence of solutions of a two-point boundary value problem at resonance in which the nonlinearity grows at most linearly. Sharp results for the linear growth of the nonlinearity in each direction are obtained.

**[1]**S. Ahmad,*A resonance problem in which the nonlinearity may grow linearly*, Proc. Amer. Math. Soc.**92**(1984), 381-384. MR**759657 (86e:34031)****[2]**L. Cesari,*Functional analysis, nonlinear differential equations and the alternative method*, Nonlinear Functional Analysis and Differential Equations (L. Cesari, R. Kannan and J. D. Schuur, eds.), Dekker, New York, 1977, pp. 1-197. MR**0487630 (58:7249)****[3]**L. Cesari and R. Kannan,*Existence of solutions of a nonlinear differential equation*, Proc. Amer. Math. Soc.**88**(1983), 605-613. MR**702284 (85d:34017)****[4]**E. N. Dancer,*On the Dirichlet problem for weakly nonlinear elliptic partial differential equations*, Proc. Roy. Soc. Edinburgh Sect. A**76**(1977), 283-300. MR**0499709 (58:17506)****[5]**-,*Boundary-value problems for weakly nonlinear ordinary differential equations*, Bull. Austral. Math. Soc.**15**(1976), 321-328. MR**0430384 (55:3389)****[6]**R. W. Gaines and J. L. Mawhin,*Coincidence degree and nonlinear differential equations*, Lecture Notes in Math., vol. 568, Springer-Verlag, Berlin and New York, 1977. MR**0637067 (58:30551)****[7]**D. E. Leach,*On Poincaré's perturbation theorem and a theorem of W. S. Loud*, J. Differential Equations**7**(1970), 34-53. MR**0251308 (40:4539)****[8]**W. S. Loud,*Periodic solutions of nonlinear differential equations of Duffing type*, Proc. U.S. Japan Seminar on Differential and Functional Equations, Benjamin, New York, 1967, pp. 199-224. MR**0223656 (36:6704)****[9]**J. L. Mawhin, J. R. Ward and M. Willen,*Necessary and sufficient conditions for the solvability of nonlinear two-point boundary value problem*, preprint.**[10]**J. R. Ward,*Perturbations with some superlinear growth for a class of second order elliptic boundary value problems*, Nonlinear Anal.**6**(1982), 307-374. MR**654812 (83i:35076)**

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/S0002-9939-1986-0831399-2

Keywords:
Resonance,
a priori bounds,
Sturm comparison theory,
Dirichlet piecewise linear problem

Article copyright:
© Copyright 1986
American Mathematical Society