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Positive solutions for multi-point boundary value problem on the half-line. (English) Zbl 1110.34018

Summary: This paper presents a variety of existence results for nonlinear multipoint boundary value problems on the half-line. In particular, we consider the problem \[ x''(t)-px'(t)-qx(t)+f(t,x(t))=0,\quad t\in[0, \infty), \]
\[ \alpha x(0)-\beta x'(0)-\sum^n_{i=1}k_i(\xi_i)=a, \quad \lim_{t\to\infty}\frac{x(t)}{e^{rt}}=b,\;r \in\left[0,\frac{p+\sqrt {p^2+4q}}{2}\right]. \] Existence results are established using fixed-point theorems on a cone.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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[1] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Dordrecht · Zbl 0923.39002
[2] Agarwal, R.P.; O’Regan, D., Infinite interval problems for differential, difference and integral equations, (2001), Kluwer Academic Dordrecht · Zbl 1003.39017
[3] Aronson, D.; Crandall, M.G.; Peletier, L.A., Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear anal., 6, 1001-1022, (1982) · Zbl 0518.35050
[4] Bai, Chuanzhi; Fang, Jinxuan, 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
[5] Baxley, J.V., Existence and uniqueness of nonlinear boundary value problems on infinite intervals, J. math. anal. appl., 147, 127-133, (1990) · Zbl 0719.34037
[6] Corduneanu, C., Integral equations and stability of feedback systems, (1973), Academic Press New York · Zbl 0268.34070
[7] Dajun, G.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045
[8] Guo, Yanping; Shan, Wenrui; Ge, Weigao, Positive solutions for second-order m-point boundary value problems, J. comput. appl. math., 151, 415-424, (2003) · Zbl 1026.34016
[9] Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., Solvablity of an m-point boundary value problem for second-order ordinary differential equations, J. math. anal. appl., 189, 575-584, (1995) · Zbl 0819.34012
[10] 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
[11] Iffland, G., Positive solutions of a problem emden – fowler type with a type free boundary, SIAM J. math. anal., 18, 283-292, (1987) · Zbl 0637.34013
[12] Ilin, V.A.; Moiseev, E.I., Non-local boundary value problem of the first kind for a sturm – liouville operator in its differential and finite difference aspects, Differ. equ., 23, 803-810, (1987)
[13] Kawano, N.; Yanagida, E.; Yotsutani, S., Structure theorems for positive radial solutions to \(\operatorname{\Delta} u + K(| x |) u^p = 0\) in \(R^n\), Funkcial. ekvac., 36, 557-579, (1993) · Zbl 0793.34024
[14] Meehan, M.; O’Regan, D., Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals, Nonlinear anal., 35, 355-387, (1999) · Zbl 0920.45006
[15] Yan, Baoqiang, Multiple unbounded solutions of boundary value problems for second-order differential equations on the half-line, Nonlinear anal., 51, 1031-1044, (2002) · Zbl 1021.34021
[16] Yan, Baoqiang; Liu, Yansheng, 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
[17] Zima, M., On positive solution of boundary value problems on the half-line, J. math. anal. appl., 259, 127-136, (2001) · Zbl 1003.34024
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