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Positive solutions for third-order three-point nonhomogeneous boundary value problems. (English) Zbl 1163.34313
Summary: In this work, by employing the Guo-Krasnosel’skii fixed point theorem and Schauder’s fixed point theorem, we study the existence and nonexistence of positive solutions to the third-order three-point nonhomogeneous boundary value problem
\[ \begin{aligned} u^{\prime\prime\prime} (t)+ a & (t)f(u(t))=0, \quad 0<t<1 \\ u(0)=u^{\prime} & (0)=0, \qquad u^{\prime}(1) - \alpha u^{\prime}(\eta) = \lambda \end{aligned} \]
where \(\eta \in (0,1), \alpha \in [0,1/\eta)\) are constants and \(\lambda \in (0,\infty)\) is a parameter.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Anderson, D., Multiple positive solutions for a three-point boundary value problem, Math. comput. modelling, 27, 49-57, (1998) · Zbl 0906.34014
[2] Anderson, D., Green’s function for a third-order generalized right focal problem, J. math. anal. appl., 288, 1-14, (2003) · Zbl 1045.34008
[3] Anderson, D.; Davis, J.M., Multiple solutions and eigenvalues for third-order right focal boundary value problems, J. math. anal. appl., 267, 135-157, (2002) · Zbl 1003.34021
[4] Bai, Z.; Fei, X., Existence of triple positive solutions for a third order generalized right focal problem, Math. inequal. appl., 9, 437-444, (2006) · Zbl 1103.34010
[5] Boucherif, A.; Al-Malki, N., Nonlinear three-point third order boundary value problems, Appl. math. comput., 190, 1168-1177, (2007) · Zbl 1134.34007
[6] Chen, H., Positive solutions for the nonhomogeneous three-point boundary value problem of second-order differential equations, Math. comput. modelling, 45, 844-852, (2007) · Zbl 1137.34319
[7] J.R. Graef, Bo Yang, Multiple positive solutions to a three point third order boundary value problem, Discrete Contin. Dyn. Syst. 2005 (Suppl.) 1-8 · Zbl 1152.34325
[8] Grossinho, M.R.; Minhos, F.M., Existence result for some third order separated boundary value problems, Nonlinear. anal., 47, 2407-2418, (2001) · Zbl 1042.34519
[9] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045
[10] Guo, L.; Sun, J.; Zhao, Y., Existence of positive solution for nonlinear third-order three-point boundary value problem, Nonlinear. anal., 68, 10, 3151-3158, (2008) · Zbl 1141.34310
[11] Kong, L.; Kong, Q., Multi-point boundary value problems of second-order differential equations (I), Nonlinear. anal., 58, 909-931, (2004) · Zbl 1066.34012
[12] Kong, L.; Kong, Q., Multi-point boundary value problems of second-order differential equations (II), Comm. appl. nonlinear. anal., 14, 93-111, (2007) · Zbl 1140.34008
[13] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen, Netherlands
[14] Ma, R., Positive solutions for a second order three-point boundary-value problems, Appl. math. lett., 14, 1-5, (2001) · Zbl 0989.34009
[15] Sun, Y., Positive solutions of singular third-order three-point boundary value problems, J. math. anal. appl., 306, 589-603, (2005) · Zbl 1074.34028
[16] Yao, Q., The existence and multiplicity of positive solutions for a third-order three-point boundary value problem, Acta math. appl. sin., 19, 117-122, (2003) · Zbl 1048.34031
[17] Yu, H.; Lü, H.; Liu, Y., Multiple positive solutions to third-order three-point singular semipositone boundary value problem, Proc. Indian acad. sci. math. sci., 114, 409-422, (2004) · Zbl 1062.34014
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