# Article

 Title: Solvability of a higher-order multi-point boundary value problem at resonance (English) Author: Lin, Xiaojie Author: Zhang, Qin Author: Du, Zengji Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 56 Issue: 6 Year: 2011 Pages: 557-575 Summary lang: English . Category: math . Summary: Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance $\displaylines { x^{(n)}(t)=f(t, x(t), x'(t),\cdots , x^{(n-1)}(t)),\quad t\in (0,1),\cr x(0)=\sum _{i=1}^{m}\alpha _{i}x(\xi _{i}),\quad x'(0)=\cdots =x^{(n-2)}(0)=0,\quad x^{(n-1)}(1)=\sum _{j=1}^{l}\beta _{j}x^{(n-1)}(\eta _{j}),\cr }$ where $f\colon [0, 1]\times \mathbb R^n\rightarrow \mathbb R$ is a Carathéodory function, $0<\xi _{1}<\xi _{2}<\cdots <\xi _{m}<1$, $\alpha _{i}\in \mathbb R$, $i=1,2,\cdots , m$, $m\geq 2$ and $0<\eta _{1}<\cdots <\eta _{l}<1$, $\beta _{j}\in \mathbb R$, $j=1,\cdots , l$, $l\geq 1$. In this paper, two of the boundary value conditions are responsible for resonance. (English) Keyword: multi-point boundary value problem Keyword: coincidence degree theory Keyword: resonance Keyword: higher-order ODE Keyword: degree arguments MSC: 34B10 MSC: 34B15 MSC: 47N20 idZBL: Zbl 1249.34056 idMR: MR2886238 DOI: 10.1007/s10492-011-0033-0 . Date available: 2011-12-16T15:06:31Z Last updated: 2020-07-02 Stable URL: http://hdl.handle.net/10338.dmlcz/141767 . Reference:  Bai, Z., Li, W., Ge, W.: Existence and multiplicity of solutions for four-point boundary value problems at resonance.Nonlinear Anal., Theory Methods Appl. 60 (2005), 1151-1162. Zbl 1070.34026, MR 2115118, 10.1016/j.na.2004.10.013 Reference:  Du, Z., Lin, X., Ge, W.: Some higher order multi-point boundary value problem at resonance.J. Comput. Appl. Math. 177 (2005), 55-65. Zbl 1059.34010, MR 2118659, 10.1016/j.cam.2004.08.003 Reference:  Feng, W., Webb, J. R. L.: Solvability of three-point boundary value problems at resonance.Nonlinear Anal., Theory Methods Appl. 30 (1997), 3227-3238. Zbl 0891.34019, MR 1603039, 10.1016/S0362-546X(96)00118-6 Reference:  Gupta, C. P.: A second order $m$-point boundary value problem at resonance.Nonlinear Anal., Theory Methods Appl. 24 (1995), 1483-1489. Zbl 0839.34027, MR 1327929, 10.1016/0362-546X(94)00204-U Reference:  Kosmatov, N.: A multi-point boundary value problem with two critical conditions.Nonlinear Anal., Theory Methods Appl. 65 (2006), 622-633. Zbl 1121.34023, MR 2231078, 10.1016/j.na.2005.09.042 Reference:  Liu, B., Yu, J.: Solvability of multi-point boundary value problem at resonance. III.Appl. Math. Comput. 129 (2002), 119-143. Zbl 1054.34033, MR 1897323, 10.1016/S0096-3003(01)00036-4 Reference:  Liu, B., Zhao, Z.: A note on multi-point boundary value problems.Nonlinear Anal., Theory Methods Appl. 67 (2007), 2680-2689. Zbl 1127.34006, MR 2345756, 10.1016/j.na.2006.09.032 Reference:  Lu, S., Ge, W.: On the existence of $m$-point boundary value problem at resonance for higher order differential equation.J. Math. Anal. Appl. 287 (2003), 522-539. Zbl 1046.34029, MR 2024338, 10.1016/S0022-247X(03)00567-5 Reference:  Mawhin, J.: Topological degree methods in nonlinear boundary value problems. Regional Conference Series in Mathematics, No. 40.American Mathematical Society (AMS) Providence (1979). MR 0525202 Reference:  Meng, F., Du, Z.: Solvability of a second-order multi-point boundary value problem at resonance.Appl. Math. Comput. 208 (2009), 23-30. Zbl 1168.34310, MR 2490766, 10.1016/j.amc.2008.11.026 Reference:  Xue, C., Du, Z., Ge, W.: Solutions to $m$-point boundary value problems of third-order ordinary differential equations at resonance.J. Appl. Math. Comput. 17 (2005), 229-244. Zbl 1070.34031, MR 2108802, 10.1007/BF02936051 Reference:  Zhang, X., Feng, M., Ge, W.: Existence result of second order differential equations with integral boundary conditions at resonance.J. Math. Anal. Appl. 353 (2009), 311-319. Zbl 1180.34016, MR 2508869, 10.1016/j.jmaa.2008.11.082 .

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