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Positive solutions to a generalized second order three-point boundary value problem. (English) Zbl 1140.34313
Summary: By using Krasnoselskii’s fixed point theorem, we study the existence of at least one or two positive solutions to the nonlinear second order three-point boundary value problem $$y''(t)+a(t)f(y(t))=0,\quad 0<t<T,\quad y(0)=\beta y(\eta),\quad y(T)=\alpha y(n),$$ where $0<\eta<T$, $0<\alpha<\frac T\eta$, $0<\beta<\frac{T-\alpha\eta}{T-\eta}$ are given constants. As an application, we also present some examples to illustrate our results.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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##### References:
 [1] I1’in, V. A.; Moiseev, E. I.: Nonlocal boundary value problem of the first kind for a strum -- Liouville operator in its differential and finite difference aspects. Differ. equat. 23, 803-810 (1987) [2] I1’in, V. A.; Moiseev, E. I.: Nonlocal boundary value problem of the second kind for a strum -- Liouville operator. Differ. equat. 23, 979-987 (1987) · Zbl 0668.34024 [3] Gupta, C. P.: A sharper condition for the solvability of a three-point second order boundary value problem. J. math. Anal. appl. 205, 579-586 (1997) · Zbl 0874.34014 [4] Gupta, C. P.: A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. math. Comput. 89, 133-146 (1998) · Zbl 0910.34032 [5] Sun, J.; Li, W.: Multiple positive solutions to second order neumama boundary value problem. Appl. maht. Comput. 146, 187-194 (2003) · Zbl 1041.34013 [6] Avery, R. I.; Henderson, J.: Three symmetric positive solutions for second order boundary value problem. Appl. math. Lett. 13, 1-7 (2000) · Zbl 0961.34014 [7] Luo, H.; Ma, Q.: Positive solutions to a generalized second-order three point boundary value problem on time scales. Electron J. Differ. equat. 1, 1-14 (2005) · Zbl 1075.34014 [8] Ma, R.; Wang, H.: Positive solutions of nonlinear three-point boundary value problem. J. math. Anal. appl. 279, 216-227 (2003) · Zbl 1028.34014 [9] 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 [10] Ma, R.: Positive solutions for nonlinear three-point boundary value problem. Electron J. Differ. equat. 34, 1-8 (1999) · Zbl 0926.34009 [11] Liu, B.: Positive solutions of a nonlinear three-point boundary value problem. Comput. math. Appl. 44, 201-211 (2002) · Zbl 1008.34014 [12] Henderson, D. R.: Solutions to second order three-point problems on time scales. J. differ. Equat. appl. 8, 673-688 (2002) · Zbl 1021.34011 [13] Krasnoselskii, M. A.: Positive solutions of operator equations. (1964) [14] 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