A second order \(m\)-point boundary value problem at resonance. (English) Zbl 0824.34023

The problem of existence of solutions for the boundary value problem \(x''= f(t, x, x')+ e(t)\), \(t\in (0, 1)\), \(x(0)= 0\), \(x'(1)= \sum^{m- 2}_{i= 1} a_ i x'(\xi_ i)\) is studied. Here \(f: [0, 1]\times \mathbb{R}^ 2\to \mathbb{R}\) is a Carathéodory function of sublinear growth, \(e\in L_ 1[0, 1]\), \(a_ i\geq 0\); \(0< \xi_ 1<\cdots< \xi_{m- 2}< 1\), \(m\geq 3\). Using Mawhin’s version of the Leray-Schauder continuation theorem, the author investigates this problem in the case of resonance, when \(\sum^{m- 2}_{i= 1} a_ i= 1\).


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Ll’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Diff. Eqns, 23, 8, 979-987 (1988) · Zbl 0668.34024
[2] Gupta, C. P.; Ntouyas, S. K.; Tsamatos, P. Ch., On an \(m\)-point boundary value problem for second order ordinary differential equations, Nonlinear Analysis, 23, 11, 1427-1436 (1994) · Zbl 0815.34012
[4] Mawhin, J., Landesman-Lazer Type Problems for Nonlinear Equations, (Conference Sem. Mat. Univ. Bari, Vol. 147 (1977))
[5] Mawhin, J., Topological Degree Methods in Nonlinear Boundary Value Problems, (NSF-CBMS Regional Conference Series in Math. (1979), Am. Math. Soc: Am. Math. Soc Providence, RI), No. 40 · Zbl 0414.34025
[6] Mawhin, J., Compacité, monotonie et convexité dans l’etude de problèmes aux limite semilinéaires, Sem. Analyse. Moderne, 19 (1981) · Zbl 0497.47033
[7] Gupta, C. P., Solvability of a three-point boundary value problem for a second order ordinary differential equation, J. math. Analysis Applic., 168, 540-551 (1992) · Zbl 0763.34009
[8] Gupta, C. P., A note on a second order three-point boundary value problem, J. math. Analysis Appl., 186, 277-281 (1994) · Zbl 0805.34017
[10] Marano, S. A., A remark on a second order three-point boundary value problem, J. math. Analysis Applic., 183, 518-522 (1994) · Zbl 0801.34025
[11] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1967), Cambridge University Press: Cambridge University Press London · Zbl 0634.26008
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.