Uniqueness and parameter dependence of solutions of second-order boundary value problems. (English) Zbl 1181.34021

Kong, Lingju; Kong, Qingkai

Appl. Math. Lett. 22, No. 11, 1633-1638 (2009).

Summary: We consider the boundary value problem with nonhomogeneous multi-point boundary condition
\[ \begin{aligned} & u'+a(t)f(u)=0,\quad t\in(0,1),\\ & u(0)=\sum^m_{i=1} a_iu(t_i)+\lambda,\quad u(1)=\sum^m_{i=1} b_iu(t_i)+\mu.\end{aligned} \]
A sufficient condition is obtained for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters \(\lambda\) and \(\mu\) is also studied. Our work complements some results in the literature.

MSC:

34B08	Parameter dependent boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

positive solutions; boundary value problems; uniqueness; dependence on parameters; normal solid cones

References:

[1]	Graef, J. R.; Kong, L., Solutions of second order multi-point boundary value problems, Math. Proc. Cambridge. Philos. Soc., 145, 489-510 (2008) · Zbl 1166.34007
[2]	Graef, J. R.; Yang, B., Positive solutions to a multi-point higher order boundary value problem, J. Math. Anal. Appl., 316, 409-421 (2006) · Zbl 1101.34004
[3]	Henderson, J., Solutions of multipoint boundary value problems for second order equations, Dynam. Systems Appl., 15, 111-117 (2006) · Zbl 1104.34310
[4]	Henderson, J.; Karna, B.; Tisdell, C. C., Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc., 133, 1365-1369 (2005) · Zbl 1061.34009
[5]	Sun, Y., Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem, Nonlinear Anal., 66, 1051-1063 (2007) · Zbl 1114.34022
[6]	Webb, J. R.L.; Infante, G., Positive solutions of nonlocal boundary value problems, a unified approach, J. London Math. Soc., 74, 673-693 (2006) · Zbl 1115.34028
[7]	Webb, J. R.L.; Lan, K. Q., Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal., 27, 91-115 (2006) · Zbl 1146.34020
[8]	Guo, Y.; Shan, W.; Ge, W., Positive solutions for second-order \(m\)-point boundary value problems, J. Comput. Appl. Math., 151, 415-424 (2003) · Zbl 1026.34016
[9]	Kong, L.; Kong, Q., Second-order boundary value problems with nonhomogeneous boundary conditions I, Math. Nachr., 278, 173-193 (2005) · Zbl 1060.34005
[10]	Kong, L.; Kong, Q., Second-order boundary value problems with nonhomogeneous boundary conditions II, J. Math. Anal. Appl., 330, 1393-1411 (2007) · Zbl 1119.34009
[11]	Kwong, M. K.; Wong, J. S.W., The shooting method and non-homogeneous multi-point BVPs of second-order ODE, Bound. Value Probl., 2007 (2007), Article ID 64012, 16 pages
[12]	Ma, R., Positive solutions for second-order three-point boundary value problems, Appl. Math. Lett., 14, 1-15 (2001) · Zbl 0989.34009
[13]	Ma, R., Positive solutions for nonhomogeneous \(m\)-point boundary value problems, Comput. Math. Appl., 47, 689-698 (2004) · Zbl 1062.34022
[14]	Sun, W.; Chen, S.; Zhang, Q.; Wang, C., Existence of positive solutions to \(n\)-point nonhomogeneous boundary value problems, J. Math. Anal. Appl., 330, 612-621 (2007) · Zbl 1123.34017
[15]	Zhang, Z.; Wang, J., Positive solutions to a second-order three-point boundary value problem, J. Math. Anal. Appl., 285, 237-249 (2003) · Zbl 1035.34011
[16]	Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press Orlando · Zbl 0661.47045