## Solvability for second-order three-point boundary value problems at resonance on a half-line.(English)Zbl 1136.34034

Summary: This paper deals with the solvability and uniqueness of the second-order three-point boundary value problems at resonance on a half-line
$x''(t)=f(t,x(t),x'(t)),\quad 0<t<+\infty,$
$x(0)= x(\eta),\quad \lim_{t\to+\infty} x'(t)=0,$
and
$x''(t)=f(t,x(t),x'(t))+e(t),\quad 0<t<+\infty,$
$x(0)=x(\eta),\quad \lim_{t\to+\infty} x'(t)=0,$
where $$f:[0,+\infty]\times\mathbb R^2\to\mathbb R,$$ $$e:[0,+\infty]\to\mathbb R$$ are continuous and $$\eta\in (0,+\infty)$$. By using the coincidence degree theory, we establish some existence and uniqueness criteria.

### MSC:

 34B40 Boundary value problems on infinite intervals for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations

### Keywords:

coincidence degree theory; infinite intervals
Full Text:

### References:

  Agarwal, R.P.; O’Regan, D., Infinite interval problems for differential, difference and integral equations, (2001), Kluwer Academic · Zbl 1003.39017  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  Chen, Shaozhu; Zhang, Yong, Singular boundary value problems on a half-line, J. math. anal. appl., 195, 449-468, (1995) · Zbl 0852.34019  Guo, Dajun, Second-order impulsive integro-differential equations on unbounded domains in Banach spaces, Nonlinear anal., 35, 413-423, (1999) · Zbl 0917.45010  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  Yan, Baoqiang, Boundary value problems on the half-line with impulse and infinite delay, J. math. anal. appl., 259, 94-114, (2001) · Zbl 1009.34059  Jiang, Daqing; 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  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  Ma, Ruyun, Existence of positive solution for second-order boundary value problems on infinite intervals, Appl. math. lett., 16, 33-39, (2003) · Zbl 1046.34045  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, Differ. equ., 23, 803-810, (1987) · Zbl 0668.34025  Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a sturm – liuville operator, Differ. equ., 23, 979-987, (1987) · Zbl 0668.34024  Gupta, C.P., Solvability of multi-point boundary value problems at resonance, Results math., 28, 270-276, (1995) · Zbl 0843.34023  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  Feng, W.; Webb, J.R.L., Solvability of three point boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019  Feng, W.; Webb, J.R.L., Solvability of m-point boundary value problems with nonlinear growth, J. math. anal. appl., 212, 467-480, (1997) · Zbl 0883.34020  Liu, Bin, Solvability of multi-point boundary value problem at resonance (III), Appl. math. comput., 129, 119-143, (2002) · Zbl 1054.34033  Liu, Bin, Solvability of multi-point boundary value problem at resonance (II), Appl. math. comput., 136, 353-377, (2003) · Zbl 1053.34016  Du, Zengji; Lin, Xiaojie; Ge, Weigao, On a third-order multi-point boundary value problem at resonance, J. math. anal. appl., 302, 217-229, (2005) · Zbl 1072.34012  Lian, Hairong; Ge, Weigao, Solvability for second-order three-point boundary value problems on a half-line, Appl. math. lett., 19, 1000-1006, (2006) · Zbl 1123.34307  Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations in topological methods for ordinary differential equations, (), 74-142 · Zbl 0798.34025
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