zbMATH — the first resource for mathematics

Boundary value problems with nonlinear conditions. (English) Zbl 0854.34025
The paper is devoted to the boundary value problem \(x''= f(t, x, x')\), \(g_1(x(a), x'(a))= 0\), \(g_2(x(b), x'(b))= 0\), where \(-\infty< a< b< \infty\), \(f: [a, b]\times \mathbb{R}^2\to \mathbb{R}\) satisfies the Carathéodory conditions and the mappings \(g_i: \mathbb{R}^2\to \mathbb{R}\), \(i= 1,2\), are continuous and in general nonlinear. Making use of the topological degree method and the upper and lower solutions method, the author proves several theorems on the existence of solutions for the cases when neither monotonicity of \(g_1\), \(g_2\) nor growth conditions for \(f\) are required.
Reviewer: M.Tvrdý (Praha)

34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX Cite
Full Text: EuDML
[1] Gaines R. E., Mawhin J. L.: Coincidence Degree and Nonlinear Differential Equations. Springer Verlag, Berlin - Heidelberg - New York, 1977. · Zbl 0339.47031
[2] Granas A., Guenther R. B., Lee J. W.: Some General Existence Principles in the Carathéodory Theory of Nonlinear Differential Systems. J. Math. pures et appl. 70 (1991), 153-196. · Zbl 0687.34009
[3] Lepin A. Ja, Lepin L. A.: Boundary Value Problem for the Ordinary Differential Equations of the Second Order. (Russian), Zinatne, Riga, 1988. · Zbl 0661.34014
[4] Mawhin J. L.: Topological Degree Methods in Nonlinear Boundary Value Problems. Providence, R. I., 1979. · Zbl 0414.34025
[5] Mawhin J. L., Rouche N.: Equations Différentielles Ordinaires II. Masson et Cre, Paris, 1973. · Zbl 0289.34001
[6] Rachůnková L.: Sign conditions in nonlinear boundary value problems. to appear. · Zbl 0845.34036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.