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Solvability of a second-order multi-point boundary value problem at resonance. (English) Zbl 1168.34310
Summary: Based on the coincidence degree theory of Mawhin, we get a general existence result for the following second-order multi-point boundary value problem at resonance
$x''(t)= f(t,x(t),x'(t))+e(t), \quad t\in(0,1),$
$x(0)= \sum_{i=1}^m \alpha_ix(\xi_i), \qquad x'(1)= \sum_{j=1}^n \beta_jx'(\eta_j),$
where $$f:[0,1]\times\mathbb R^2\to\mathbb R$$ is a Carathéodory function, $$e\in L^1[0,1]$$, $$0<\xi_1<\xi_2<\cdots< \xi_m<1$$, $$\alpha_i\in\mathbb R$$, $$i=1,2,\dots,m$$, $$m\geq 2$$ and $$0<\eta_1<\cdots<\eta_n<1$$, $$\beta_j\in\mathbb R$$, $$j=1,\dots,n$$, $$n\geq 1$$. In this paper, both of the boundary value conditions are responsible for resonance.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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##### References:
  Gupta, C.P., A second order m-point boundary value problem at resonance, Nonlinear anal., 24, 1483-1489, (1995) · Zbl 0824.34023  Feng, W.; Webb, J.R.L., Solvability of three-point boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019  Ma, R., Multiplicity results for a third order boundary value problem at resonance, Nonlinear anal., 32, 493-499, (1998) · Zbl 0932.34014  Bai, Z.; Li, W.; Ge, W., Existence and multiplicity of solutions for four-point boundary value problems at resonance, Nonlinear anal., 60, 1151-1162, (2005) · Zbl 1070.34026  Kosmatov, N., A multi-point boundary value problem with two critical conditions, Nonlinear anal., 65, 622-633, (2006) · Zbl 1121.34023  Liu, B.; Zhao, Z., A note on multi-point boundary value problems, Nonlinear anal., 67, 2680-2689, (2007) · Zbl 1127.34006  Liu, B., Solvability of multi-point boundary value problem at resonance (II), Appl. math. comput., 136, 353-377, (2003) · Zbl 1053.34016  Liu, B.; Yu, J., Solvability of multi-point boundary value problem at resonance (III), Appl. math. comput., 129, 119-143, (2002) · Zbl 1054.34033  Liu, B., Solvability of multi-point boundary value problem at resonance (IV), Appl. math. comput., 143, 275-299, (2003) · Zbl 1071.34014  Du, Z.; Lin, X.; Ge, W., On a third-order multi-point boundary value problem at resonance, J. math. anal. appl., 302, 217-229, (2005) · Zbl 1072.34012  Mawhin, J., Topological degree methods in nonlinear boundary value problems, NSFCBMS regional conference series in mathematics, (1979), American Mathematical Society Providence, RI  Ge, W., Boundary value problems for ordinary nonlinear differential equations, (2007), Science Press Beijing, (in Chinese)
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