Article

Full entry | PDF   (0.2 MB)
Keywords:
multi-point boundary value problem; coincidence degree theory; resonance; higher-order ODE; degree arguments
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.
References:
[1] 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. DOI 10.1016/j.na.2004.10.013 | MR 2115118 | Zbl 1070.34026
[2] Du, Z., Lin, X., Ge, W.: Some higher order multi-point boundary value problem at resonance. J. Comput. Appl. Math. 177 (2005), 55-65. DOI 10.1016/j.cam.2004.08.003 | MR 2118659 | Zbl 1059.34010
[3] Feng, W., Webb, J. R. L.: Solvability of three-point boundary value problems at resonance. Nonlinear Anal., Theory Methods Appl. 30 (1997), 3227-3238. DOI 10.1016/S0362-546X(96)00118-6 | MR 1603039 | Zbl 0891.34019
[4] Gupta, C. P.: A second order $m$-point boundary value problem at resonance. Nonlinear Anal., Theory Methods Appl. 24 (1995), 1483-1489. DOI 10.1016/0362-546X(94)00204-U | MR 1327929 | Zbl 0839.34027
[5] Kosmatov, N.: A multi-point boundary value problem with two critical conditions. Nonlinear Anal., Theory Methods Appl. 65 (2006), 622-633. DOI 10.1016/j.na.2005.09.042 | MR 2231078 | Zbl 1121.34023
[6] Liu, B., Yu, J.: Solvability of multi-point boundary value problem at resonance. III. Appl. Math. Comput. 129 (2002), 119-143. DOI 10.1016/S0096-3003(01)00036-4 | MR 1897323 | Zbl 1054.34033
[7] Liu, B., Zhao, Z.: A note on multi-point boundary value problems. Nonlinear Anal., Theory Methods Appl. 67 (2007), 2680-2689. DOI 10.1016/j.na.2006.09.032 | MR 2345756 | Zbl 1127.34006
[8] 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. DOI 10.1016/S0022-247X(03)00567-5 | MR 2024338 | Zbl 1046.34029
[9] 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
[10] Meng, F., Du, Z.: Solvability of a second-order multi-point boundary value problem at resonance. Appl. Math. Comput. 208 (2009), 23-30. DOI 10.1016/j.amc.2008.11.026 | MR 2490766 | Zbl 1168.34310
[11] 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. DOI 10.1007/BF02936051 | MR 2108802 | Zbl 1070.34031
[12] 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. DOI 10.1016/j.jmaa.2008.11.082 | MR 2508869 | Zbl 1180.34016

Partner of