Positive solutions for second-order \(m\)-point boundary value problems. (English) Zbl 1026.34016

Guo, Yanping; Shan, Wenrui; Ge, Weigao

J. Comput. Appl. Math. 151, No. 2, 415-424 (2003).

Summary: By using the Schauder fixed-point theorem and analysis method, we establish the existence of solutions to the \(m\)-point boundary value problem \[ u''(t)+ a(t)f(u)=0, \quad u(0)=0,\;u(1)-\sum^{m-2}_{i=1} k_iu(\xi_i)= b, \] where \(b,k_i>0\), \(i=1,2,\dots, m-2\), \(0<\xi_1 <\xi_2< \cdots <\xi_{m-2}<1\) and \(a(t)\) is allowed to be singular at \(t=0,1\). Under some conditions, we show that there exists a positive number \(b^*\) such that the problem has at least one positive solution for \(0<b<b^*\) and no solution for \(b>b^*\).

MSC:

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 solution; Schauder fixed-point theorem; m-point boundary value problem

References:

[1]	Feng, W.; Webb, J. R.L., Solvability of a m-point boundary value problems with nonlinear growth, J. Math. Anal. Appl., 212, 467-480 (1997) · Zbl 0883.34020
[2]	Feng, W., On a m-point nonlinear boundary value problem, Nonlinear Anal., 30, 6, 5369-5374 (1997) · Zbl 0895.34014
[3]	Gupta, C. P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168, 540-551 (1992) · Zbl 0763.34009
[4]	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
[5]	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
[6]	Il’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differential Equations, 23, 8, 979-987 (1987) · Zbl 0668.34024
[7]	Ma, R., Positive solutions of a nonlinear three-point boundary value problem, Electron. J. Differential Equations, 34, 1-8 (1999)
[8]	Ma, R., Existence theorems for a second order m-point boundary value problem, J. Math. Anal. Appl., 211, 545-555 (1997) · Zbl 0884.34024
[9]	Ma, R., Positive solutions for second order three-point boundary value problems, Appl. Math. Lett., 14, 1-5 (2001) · Zbl 0989.34009