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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 $$\sum^{m- 2}_{i= 1} a_ i= 1$$.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
  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  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  GUPTA C. P., NTOUYAS S. K. & TSAMATOS P. Ch., Existence results for m-point boundary value problems, J. Diff. Eqns Dyn. Syst. (to appear). · Zbl 1185.34023  Mawhin, J., Landesman-lazer type problems for nonlinear equations, ()  Mawhin, J., Topological degree methods in nonlinear boundary value problems, (), No. 40 · Zbl 0414.34025  Mawhin, J., Compacité, monotonie et convexité dans l’etude de problèmes aux limite semilinéaires, Sem. analyse. moderne, 19, (1981) · Zbl 0497.47033  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  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  GUPTA C. P., Solvability of a multi-point boundary value problem at resonance (submitted).  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  Hardy, G.H.; Littlewood, J.E.; Polya, G., Inequalities, (1967), Cambridge University Press London · Zbl 0634.26008
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