## Positive solutions of multi-point boundary value problems at resonance.(English)Zbl 1203.34041

The authors discuss existence for a multi-point second order boundary value problem at resonance. A recent fixed point theorem in a cone due to O’Regan and Zima is used in the analysis.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations

### Keywords:

boundary value problem at resonance; positive solution
Full Text:

### References:

 [1] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044 [2] An, Y., Existence of solutions for a three-point boundary value problem at resonance, Nonlinear anal., 65, 1633-1643, (2006) · Zbl 1104.34007 [3] Bai, C.; Fang, J., Existence of positive solutions for three-point boundary value problems at resonance, J. math. anal. appl., 291, 538-549, (2004) · Zbl 1056.34019 [4] Bitsadze, A.V.; Samarskiĭ, A.A., Some elementary generalizations of linear elliptic boundary value problems, Dokl. akad. nauk SSSR, 185, 739-740, (1969), (in Russian) · Zbl 0187.35501 [5] Cremins, C.T., A fixed-point index and existence theorems for semilinear equations in cones, Nonlinear anal., 46, 789-806, (2001) · Zbl 1015.47041 [6] Cremins, C.T., Existence theorems for semilinear equations in cones, J. math. anal. appl., 265, 447-457, (2002) · Zbl 1015.47042 [7] Feng, W.; Webb, J.R.L., Solvability of three-point boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019 [8] 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 [9] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045 [10] Gupta, C.P., Existence theorems for a second order $$m$$-point boundary value problem at resonance, Int. J. math. math. sci., 18, 705-710, (1995) · Zbl 0839.34027 [11] Han, X., Positive solutions for a three-point boundary value problem at resonance, J. math. anal. appl., 336, 556-568, (2007) · Zbl 1125.34014 [12] He, X.; Ge, W., Triple solutions for second-order three-point boundary value problems, J. math. anal. appl., 268, 256-265, (2002) · Zbl 1043.34015 [13] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the first kind for a sturm – liouville operator in its differential and finite difference aspects, Differ. equ., 23, 803-810, (1987) · Zbl 0668.34025 [14] G. Infante, Positive solutions of some nonlinear BVPs involving singularities and integral BCs, Discrete Contin. Dyn. Syst. Ser. S (in press) · Zbl 1160.34018 [15] Infante, G.; Webb, J.R.L., Three point boundary value problems with solutions that change sign, J. integral equations appl., 15, 37-57, (2003) · Zbl 1055.34023 [16] Infante, G.; Webb, J.R.L., Loss of positivity in a nonlinear scalar heat equation, Nodea nonlinear differential equations appl., 13, 249-261, (2006) · Zbl 1112.34017 [17] Infante, G.; Webb, J.R.L., Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations, Proc. edinb. math. soc., 49, 637-656, (2006) · Zbl 1115.34026 [18] Kosmatov, N., A symmetric solution of a multipoint boundary value problem at resonance, Abstr. appl. anal., (2006), 11 pp. Art. ID 54121 · Zbl 1137.34313 [19] Kosmatov, N., Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear anal., (2007) [20] Lan, K.Q., Properties of kernels and eigenvalues for three point boundary value problems, Discrete contin. dyn. syst., Suppl., 546-555, (2005) · Zbl 1162.34019 [21] Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033 [22] Liu, B., Solvability of multi-point boundary value problem at resonance. IV, Appl. math. comput., 143, 275-299, (2003) · Zbl 1071.34014 [23] Liu, B.; Zhao, Z., A note on multi-point boundary value problems, Nonlinear anal., 67, 2680-2689, (2007) · Zbl 1127.34006 [24] Ma, R., Positive solutions for a nonlinear three-point boundary-value problem, Electron. J. differential equations, 34, (1999), 8 pp [25] Ma, R., Multiplicity results for an $$m$$-point boundary value problem at resonance, Indian J. math., 47, 15-31, (2005) · Zbl 1078.34005 [26] Mawhin, J., Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. differential equations, 12, 610-636, (1972) · Zbl 0244.47049 [27] O’Regan, D.; Zima, M., Leggett – williams norm-type theorems for coincidences, Arch. math., 87, 233-244, (2006) · Zbl 1109.47051 [28] Palamides, P.K., Multi point boundary-value problems at resonance for $$n$$-order differential equations: positive and monotone solutions, Electron. J. differential equations, 25, (2004), 14 pp · Zbl 1066.34013 [29] Petryshyn, W.V., On the solvability of $$x \in T x + \lambda F x$$ in quasinormal cones with $$T$$ and $$F$$ k-set contractive, Nonlinear anal., 5, 585-591, (1981) · Zbl 0474.47028 [30] Przeradzki, B.; Stańczy, R., Solvability of a multi-point boundary value problem at resonance, J. math. anal. appl., 264, 253-261, (2001) · Zbl 1043.34016 [31] Santanilla, J., Some coincidence theorems in wedges, cones, and convex sets, J. math. anal. appl., 105, 357-371, (1985) · Zbl 0576.34018 [32] Webb, J.R.L., (), 137-147 [33] Webb, J.R.L., Positive solutions of some three point boundary value problems via fixed point index theory, Nonlinear anal., 47, 4319-4332, (2001) · Zbl 1042.34527 [34] Webb, J.R.L., Multiple positive solutions of some nonlinear heat flow problems, Discrete contin. dyn. syst., Suppl., 895-903, (2005) · Zbl 1161.34007 [35] Webb, J.R.L.; Lan, K.Q., Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. methods nonlinear anal., 27, 91-115, (2006) · Zbl 1146.34020 [36] Zima, M., Fixed point theorem of leggett – williams type and its application, J. math. anal. appl., 299, 254-260, (2004) · Zbl 1066.47059
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.