## Multi-point boundary value problems on an unbounded domain at resonance.(English)Zbl 1138.34006

Summary: We consider the second-order nonlinear differential equation
$(p(t)u'(t))'=f(t,u(t),u'(t))\text{ a.e. in }(0,\infty),$
satisfying two sets of boundary conditions:
$u'(0)=0,\quad\sum^n_{i=1}\kappa_iu(T_i)=\lim_{t\to\infty}u(t)$
and
$u(0)=0,\quad \sum^n_{i=1}\kappa_iu(T_i)=\lim_{t\to\infty} u(t),$
where $$n\geq1$$, $$f:[0,\infty)\times \mathbb R^2\to\mathbb R$$ is Carathéodory with respect to $$L_1[0,\infty)$$. The parameters in the multi-point boundary conditions are such that the corresponding differential operator is non-invertible but nevertheless is a Fredholm map of index zero. As a result the coincidence degree theory can be applied to establish existence theorems.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B40 Boundary value problems on infinite intervals for ordinary differential equations
Full Text:

### References:

 [1] Agarwal, R. P.; O’Regan, D., Infinite Interval Problems for Differential, Difference and Integral Equations (2001), Kluwer Academic · Zbl 1003.39017 [2] Amster, P.; De Napoli, P.; Mariani, M. C., Periodic solutions of a resonant third-order equation, Nonlinear Anal., 60, 399-410 (2005) · Zbl 1064.34025 [3] Amster, P.; De Napoli, P.; Mariani, M. C., Periodic solutions of a resonant higher-order equation, Port. Math. (N.S.), 62, 1, 13-24 (2005) · Zbl 1084.34043 [4] Bai, Z.; Ge, W.; Wang, Y., Upper and lower solution method for a four-point boundary-value problem at resonance, Nonlinear Anal., 60, 1151-1162 (2005) · Zbl 1070.34026 [5] Cheung, W. S.; Ren, J., Periodic solutions for $$p$$-Laplacian Rayleigh equations, Nonlinear Anal., 65, 2003-2012 (2006) · Zbl 1112.34047 [6] Feng, W.; Webb, J. R.L., Solvability of three point boundary value problem at resonance, Nonlinear Anal., 30, 3227-3238 (1997) · Zbl 0891.34019 [7] Garsía-Huidobro, M.; Gupta, C. P.; Manásevich, R., A Dirichlet-Neumann $$m$$-point BVP with a $$p$$-Laplacian-like operator, Nonlinear Anal., 62, 1067-1089 (2005) · Zbl 1082.34011 [8] Gupta, C. P., A second order $$m$$-point boundary value problem at resonance, Nonlinear Anal., 24, 1483-1489 (1995) · Zbl 0824.34023 [9] Karpińska, W., On bounded solutions of nonlinear differential equations at resonance, Nonlinear Anal., 51, 723-733 (2002) · Zbl 1021.34047 [10] Kosmatov, N., Multi-point boundary value problems on time scales at resonance, J. Math. Anal. Appl., 323, 1, 253-266 (2006) · Zbl 1107.34008 [11] Kosmatov, N., A multi-point boundary value problem with two critical conditions, Nonlinear Anal., 65, 3, 622-633 (2006) · Zbl 1121.34023 [12] N. Kosmatov, Second order boundary value problems on an unbounded domain, Nonlinear Anal., in press (doi:10.1016/j.na.2006.11.043; N. Kosmatov, Second order boundary value problems on an unbounded domain, Nonlinear Anal., in press (doi:10.1016/j.na.2006.11.043 · Zbl 1135.34313 [13] Lian, H.; Ge, W., Solvability for second-order three-point boundary value problems on a half-line, Appl. Math. Lett., 16, 33-39 (2006) [14] Liu, B., Solvability of multi-point boundary value problems at resonance (IV), Appl. Math. Comput., 143, 275-299 (2003) · Zbl 1071.34014 [15] Liu, Y.; Ge, W., Solvability of a $$(p, n - p)$$-type multi-point boundary-value problems for higher-order differential equations, Electron. J. Differential Equations, 2003, 120, 1-19 (2003) · Zbl 1042.34032 [16] Mawhin, J., Topological degree methods in nonlinear boundary value problems, (NSF-CBMS Regional Conference Series in Math., vol. 40 (1979), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0414.34025
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.