# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Liu, B., Solvability of multi-point boundary value problem at resonance (IV), Appl. math. comput., 143, 275-299, (2003) · Zbl 1071.34014  Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (), 74-142 · Zbl 0798.34025  Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025  Kosmatov, N., A multi-point boundary value problem with two critical conditions, Nonlinear anal., 65, 622-633, (2006) · Zbl 1121.34023  Bitsadze, A.V., On the theory of nonlocal boundary value problems, Soviet. math. dock., 30, 8-10, (1964) · Zbl 0586.30036  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  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  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  Feng, W.; webb, J.R.L., Solvability of three point boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019  Liu, B., Solvability of multi-point boundary value problem at resonance (III), Appl. math. comput., 129, 119-143, (2002) · Zbl 1054.34033
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.