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Multiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition. (English) Zbl 1140.34009

The paper deals with the second-order three-point boundary value problem \[ y''(t)+f(t,y)=0,\quad 0\leq t\leq 1, \tag{1} \]
\[ y(0)=0,\quad y(1)-\alpha y(\eta)=0, \tag{2} \] where \(f\in C([0,1]\times\mathbb R,\mathbb R)\) and \(0<\eta< 1\), \(0<\alpha <1\). The purpose of this paper is to prove some multiplicity results for solutions of problem (1), (2) under conditions of non-well-ordered upper and lower solutions. The authors prove the existence of at least two, three or six solutions of the problem. The proofs are based on the fixed point index of some increasing operator with respect to some closed convex sets, which are translations of some special cones. The main idea is associating a fixed point index with pairs of lower and upper solutions. In the proofs the athors employ monotonicity conditions on the nonlinear term \(f\) of (1).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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[1] Webb, J.R.L., Positive solutions of some three point boundary value problems via fixed point index, Nonlinear anal., 47, 4319-4332, (2001) · Zbl 1042.34527
[2] Liu, B., Positive solutions of second-order three-point boundary value problems with change of sign, Comput. math. appl., 47, 1351-1361, (2004) · Zbl 1060.34015
[3] Gupta, C.P.; Trofimchuk, S.I., Existence of a solution of a three-point boundary value problem and spectral radius of a related linear operator, Nonlinear anal., 34, 489-507, (1998) · Zbl 0944.34009
[4] Zhang, Z.; Wang, J., The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems, J. comput. appl., 147, 41-52, (2002) · Zbl 1019.34021
[5] Xu, X., Three solutions for three-point boundary value problems, Nonlinear anal., 62, 1053-1066, (2005) · Zbl 1076.34011
[6] Khan, A.R.; Webb, J.R.L., Existence of at least three solutions of a second-order three-point boundary value problem, Nonlinear anal., 64, 1356-1366, (2006) · Zbl 1101.34005
[7] Picard, E., Sur l’application des méthodes d’approximations successives à l’étude de certaines équations différentielles ordinaries, J. de math., 9, 217-271, (1893) · JFM 25.0507.02
[8] Scorza Dragoni, G., Intorno a un criterio di esistenza per un problema di valori ai limiti, Rend. sem. R. accad. naz. lincei, 28, (1938) · JFM 65.0381.05
[9] Amann, H.; Ambrisetti, A.; Mancini, G., Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z., 158, 179-194, (1978) · Zbl 0368.35032
[10] De Coster, C.; Henrard, M., Existence and localization of solution for elliptic problem in presence of lower and upper solutions without any order, J. differential equations, 145, 420-452, (1998) · Zbl 0908.35042
[11] De Coster, C.; Habets, P., An overview of the method of lower and upper solutions for odes, () · Zbl 1132.34309
[12] Rachůnková, I.; Tvrdy, M., Periodic problems with p-Laplacian involving non-ordered lower and upper solutions, Fixed point theory, 6, 99-112, (2005) · Zbl 1086.34026
[13] Rachůnková, I.; Tvrdy, M., Non-ordered lower and upper functions in second order impulsive periodic problems, Dyn. contin. discrete impuls. syst., 12, 397-415, (2005) · Zbl 1086.34026
[14] Rachůnková, I., Upper and lower solutions and topological degree, J. math. anal. appl., 234, 311-327, (1999) · Zbl 1086.34017
[15] Rachůnková, I., Upper and lower solutions and multiplicity results, J. math. anal. appl., 246, 446-464, (2000) · Zbl 0961.34004
[16] Habets, P.; Omari, P., Existence and localization of second order elliptic problems using lower and upper solutions in the reversed order, Topol. methods nonlinear anal., 8, 25-56, (1996) · Zbl 0897.35030
[17] Omari, P., Non-ordered lower and upper solution and solvability of the periodic problem for Liénard and Rayleigh equations, Rend. instit. mat. univ. treiste, 20, 54-64, (1988)
[18] Sun, J., Fixed point theorems of two-point extension type and applications, J. syst. sci. math., 12, 284-286, (1992), (in Chinese) · Zbl 0795.47035
[19] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press, Inc. New York · Zbl 0661.47045
[20] Dugundji, J., An extension of tietze’s theorem, Pacific J. math., 1, 353-367, (1951) · Zbl 0043.38105
[21] Xu, X., Multiplicity results for positive solutions of some semi-positone three-point boundary value problems, J. math anal. appl., 291, 673-689, (2004) · Zbl 1056.34035
[22] Xu, X., Positive solutions for singular m-point boundary value problems with positive parameter, J. math anal. appl., 291, 352-367, (2004) · Zbl 1047.34016
[23] Sun, J.; Zhang, K., On the number of fixed points of nonlinear operators and applications, J. syst. sci. complex., 16, 2, (2003)
[24] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044
[25] Ma, R., Existence of solutions of nonlinear m-point boundary-value problems, J. math. anal. appl., 256, 556-567, (2001) · Zbl 0988.34009
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