zbMATH — the first resource for mathematics

Solvability for second-order three-point boundary value problems on a half-line. (English) Zbl 1123.34307
Summary: This work is concerned with the existence of a solution to the second-order three-point boundary value problem on the half-line $\begin{cases} x''(t)+f(t,x(t), x'(t))=0,\;0<t<+\infty,\\ x(0)=\alpha x(\eta),\;\lim_{t\to+\infty}x'(t)=0, \end{cases}$ where $$\alpha\in \mathbb{R}$$, $$\alpha\neq 1$$ and $$\eta\in(0,+\infty)$$ are given. After the discussion of the Green function for the corresponding homogeneous system on the half-line, we establish some criteria for the existence of solutions to the system discussed with suitable conditions imposed on $$f$$. The results are obtained by the Leray-Schauder continuation theorem.

MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B40 Boundary value problems on infinite intervals for ordinary differential equations
Full Text:
References:
 [1] Gupta, C.P., A note on a second order three-point value problem, J. math. anal. appl., 186, 277-281, (1994) · Zbl 0805.34017 [2] Gupta, C.P.; Trofimchuk, S.I., A sharper condition for the solvability of a three-point second order boundary value problem, J. math. anal. appl., 205, 586-597, (1997) · Zbl 0874.34014 [3] Ma, R., Positive solutions for second order three-point boundary value problems, Appl. math. lett., 14, 1-5, (2001) · Zbl 0989.34009 [4] 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 [5] O’Regan, D., Theory of singular boundary value problems, (1994), World Scientific Singapore · Zbl 0808.34022 [6] Agarwal, R.P.; O’Regan, D., Infinite interval problems for differential, difference and integral equations, (2001), Kluwer Academic Publisher Netherlands · Zbl 1003.39017 [7] Baxley, J.V., Existence and uniqueness for nonlinear boundary value problems on infinite interval, J. math. anal. appl., 147, 127-133, (1990) · Zbl 0719.34037 [8] Guo, D., Second order impulsive integro-differential equations on unbounded domains in Banach spaces, Nonlinear anal., 35, 413-423, (1999) · Zbl 0917.45010 [9] Agarwal, R.P.; O’Regan, D., Fixed point theory for self maps between Fréchet spaces, J. math. anal. appl., 256, 498-512, (2001) · Zbl 0997.47044 [10] Jiang, D.; Agarwal, R.P., A uniqueness and existence theorem for a singular third-order boundary value problem on $$[0, \infty)$$, Appl. math. lett., 15, 445-451, (2002) · Zbl 1021.34020 [11] Frigon, M., Fixed point results for compact maps on closed subsets of Fréchet spaces and applications to differential and integral equations, Bull. belg. math. soc., 9, 23-37, (2002) · Zbl 1026.47047 [12] Frigon, M.; O’Regan, D., Fixed point of cone-compressing and cone-extending operators in Fréchet spaces, Bull. London math. soc., 35, 672-680, (2003) · Zbl 1041.47040 [13] Ma, R., Existence of positive solution for second-order boundary value problems on infinite intervals, Appl. math. lett., 16, 33-39, (2003) · Zbl 1046.34045 [14] Bai, C.; Fang, J., On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals, J. math. anal. appl., 282, 711-731, (2003) · Zbl 1036.34075 [15] Yan, B.; Liu, Y., Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line, Appl. math. comput., 147, 629-644, (2004) · Zbl 1045.34009
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.