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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)

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:
[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
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