×

zbMATH — the first resource for mathematics

On the existence of positive solution for a kind of multi-point boundary value problem at resonance. (English) Zbl 1200.34018
The paper deals with the existence of positive solutions for a second-order multi-point boundary value problem at resonance
\[ x''(t)+f(t,x(t))=0, \quad t\in (0,1), \]
\[ x(0)=\sum_{i=1}^{m-2}\alpha_{i}x(\xi_{i}), \quad x(1)=\sum_{i=1}^{m-2}\beta_{i}x(\xi_{i}). \]
The key tool is the Leggett-Williams norm-type theorem for coincidence equations due to D. O’Regan and M. Zima [Arch. Math. 87, No. 3, 233–244 (2006; Zbl 1109.47051)].
Reviewer’s remark: There is a gap in the proof of one of the main results of the paper. Namely, the assumptions of Theorem 2 do not imply the condition (C2) of Lemma 2.1.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bitsadze, A.V.; Samarskii˘, A.A., Some elementary generalizations of linear elliptic boundary value problems, Dokl. akad. nauk SSSR, 185, 739-740, (1969), (in Russian) · Zbl 0187.35501
[2] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the first kind for a sturm – liuville operator in its differential and finite difference aspects, Differential equations, 23, 803-810, (1987) · Zbl 0668.34025
[3] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a sturm – liuville operator, Differential equations, 23, 979-987, (1987) · Zbl 0668.34024
[4] Gupta, C.P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. math. anal. appl., 168, 540-551, (1992) · Zbl 0763.34009
[5] Gupta, C.P., A sharper condition for the solvability of a three-point second order boundary value problem, J. math. anal. appl., 205, (1997), 586-579 · Zbl 0874.34014
[6] Ma, R., Positive solutions of a nonlinear three-point boundary valve problem, Electron. J. differential equations, 34, 1-8, (1999)
[7] Yang, L., Multiplicity results for second-order \(m\)-point boundary value problem, J. math. anal. appl., 324, 532-542, (2006) · Zbl 1112.34011
[8] Avery, R.I., Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. math. appl., 42, 695-704, (2001) · Zbl 1006.34022
[9] 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
[10] Guo, Y.; Ge, W., Positive solutions for three-point boundary-value problems with dependence on the first order derivative, J. math. anal. appl., 290, 291-301, (2004) · Zbl 1054.34025
[11] Xian, X.; Regan, D.O., Multiplicity of sign-changing solutions for some four-point boundary value problem, Nonlinear anal., 69, 434-447, (2008) · Zbl 1152.34006
[12] Agarwal, R.P.; O’regan, D., Some new existence results for differential and integral equations, Nonlinear anal., 29, 679-692, (1997) · Zbl 0878.45001
[13] Henderson, J., Double solutions of three-point boundary-value problems for second-order differential equations, Electron. J. diff. eqs., 115, 1-7, (2004) · Zbl 1075.34013
[14] Agarwal, R.P.; O’regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Publishers Boston · Zbl 0923.39002
[15] Ma, R., Positive solutions of a nonlinear \(m\)-point boundary value problem, Comput. math. appl., 42, 755-765, (2001) · Zbl 0987.34018
[16] Zhang, G., Positive solutions of \(m\)-point boundary value problem, J. math. anal. appl., 291, 406-418, (2004) · Zbl 1069.34037
[17] Yang, L., Existence of three positive solutions for some second-order \(m\)-point boundary value problems, Acta. math. appl. sinica, 24, 253-264, (2008) · Zbl 1160.34020
[18] Ji, D., Multiple positive solutions for some \(p\)-Laplacian boundary value problems, Appl. math. comput., 187, 1315-1325, (2007) · Zbl 1135.34012
[19] Feng, H., Multiplicity of symmetric positive solutions for a multi-point boundary value problem with a one-dimensional \(p\)-Laplacian, Nonlinear anal., 69, 3050-3059, (2008) · Zbl 1148.34006
[20] Wang, Y., Existence of triple positive solutions for multi-point boundary value problems with a one dimensional \(p\)-Laplacian, Comput. math. appl., 54, 793-807, (2007) · Zbl 1134.34017
[21] Avery, R.I.; Peterson, A.C., Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. math. appl., 42, 313-322, (2001) · Zbl 1005.47051
[22] Przeradzki, Bogdan, Solvability of a multi-point boundary value problem at resonance, J. math. anal. appl., 264, 253-261, (2001) · Zbl 1043.34016
[23] Gupta, C.P., On a \(m\)-point boundary value problem for second-order ordinary differential equations, Nonlinear anal., 23, 1427-1436, (1994) · Zbl 0815.34012
[24] Feng, W.; Webb, J.R., Solvability of three point boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019
[25] Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025
[26] Liu, B., Solvability of multi-point boundary value problem at resonance (IV), Appl. math. comput., 143, 275-299, (2003) · Zbl 1071.34014
[27] 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
[28] Infante, G.; Zima, M., Positive solutions of multi-point boundary value problems at resonance, Nonlinear anal., 69, 2458-2465, (2008) · Zbl 1203.34041
[29] Palamides, P.K., Multi-point boundary-value problems at resonance for \(n\)-order differential equations: positive and monotone solutions, Electron J. differential equations, 25, 1-14, (2004) · Zbl 1066.34013
[30] Zima, M., Existence of positive solutions for a second-order three-point boundary value proble at resonance, Dynam. systems appl., 5, 527-532, (2008) · Zbl 1203.34035
[31] Webb, J.R.L.; Zima, M., Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems, Nonlinear anal., 71, 1369-1378, (2009) · Zbl 1179.34023
[32] Cremins, C.T., A fixed-point index and existence theorems for semi-linear equations in cones, Nonlinear anal., 46, 789-806, (2001) · Zbl 1015.47041
[33] O’Regan, D.; Zima, M., Leggett – williams norm-type theorems for coincidences, Arch. math., 87, 233-244, (2006) · Zbl 1109.47051
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.