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A note on multi-point boundary value problems. (English) Zbl 1127.34006
This work is concerned with solvability of the multipoint boundary value problem \[ x''(t) = f(t,x(t),x'(t)), \quad t \in (0,1), \]
\[ x(0) = \sum_{j=1}^n \alpha_j x(\xi_j), \quad x(1) = \sum_{j=1}^n \beta_j x(\eta_j), \] where \(0 < \eta_j, \xi_j < 1\), \(\eta_j, \xi_j \in \mathbb{R}\), \(j = 1,2,\dots,n\), \(n \geq 2\), \(e \in L^1[0,1]\), and \(f:[0,1] \times \mathbb{R}^2 \to \mathbb{R}\) is a Carathéodory function. The additional assumptions
\[ \sum_{j=1}^n \alpha_j = 1 = \sum_{j=1}^n \beta_j, \quad \sum_{j=1}^n \alpha_j \xi_j = 0 = \sum_{j=1}^n \beta_j \eta_j \] are “critical”, that is, responsible for resonance.
This note complements the result by N. Kosmatov [Nonlinear Anal., Theory Methods Appl. 65, 622–633 (2006; Zbl 1121.34023)] and extends the result by the first author [Appl. Math. Comput. 143, 275–299 (2003; Zbl 1071.34014)]. The existence result follows from the celebrated coincidence degree theorem due to J. Mawhin [Topological degree methods in nonlinear boundary value problems (Regional Conference Series in Mathematics, No. 40, Providence, R.I., The American Mathematical Society) (1979; Zbl 0414.34025)].

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
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[1] Liu, B., Solvability of multi-point boundary value problem at resonance (IV), Appl. math. comput., 143, 275-299, (2003) · Zbl 1071.34014
[2] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (), 74-142 · Zbl 0798.34025
[3] Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025
[4] Kosmatov, N., A multi-point boundary value problem with two critical conditions, Nonlinear anal., 65, 622-633, (2006) · Zbl 1121.34023
[5] Bitsadze, A.V., On the theory of nonlocal boundary value problems, Soviet. math. dock., 30, 8-10, (1964) · Zbl 0586.30036
[6] Bitsadze, A.V.; Samarskii, A.A., Some elementary generalizations of linear elliptic boundary value problems, Dokil. akad. nauk SSSR, 185, 739-740, (1969) · Zbl 0187.35501
[7] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problems of the second kind for a Sturm-Liouville operator, Differ. equ., 23, 979-987, (1987) · Zbl 0668.34024
[8] Feng, W.; Webb, J.R.L., Solvability of m-point boundary value problems with nonlinear growth, J. math. anal. appl., 212, 467-480, (1997) · Zbl 0883.34020
[9] Feng, W.; webb, J.R.L., Solvability of three point boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019
[10] Liu, B., Solvability of multi-point boundary value problem at resonance (III), Appl. math. comput., 129, 119-143, (2002) · Zbl 1054.34033
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