Positive solution for \(m\)-point boundary value problems
The authors are concerned with the existence of positive solutions of \(m\)-point (\(m\geq 3\)) boundary value problem \[ u''(t) + \lambda h (t) f (t,u)= 0,\quad 0\leq t\leq 1, \quad u(0) = 0, \quad u(1) =\sum_{i=1}^{m-2} \alpha_iu(\xi_i), \tag{*} \] where \(0<\xi_1<\xi_2<\dots<\xi_{m-2}<1\), \(f:[0,1]\times[0,\infty)\to\mathbb R\) is continuous and is allowed to change sign, \(h(t)\) is a nonnegative measurable function on \([0, 1]\) with \[ 0<\int_0^1h(t)dt<\infty\text{ and }0<\sum_{i=1}^{m-2}\alpha_i<1. \] They have combined the method of lower solution with the topological degree theory to achieve their objective. In the following, one of the results is stated: Suppose that there exist \(r>M>0\) such that \[ 0<\frac{M}{\underset {0\leq t\leq1} \min f(t,Mw(t))}=a\leq b=\frac{r}{A\underset{{0\leq t\leq 1}\atop{Mw(t)\leq u\leq r}}\max f(t,u)}\,, \] where \(A =\max_{0\leq t\leq1} w(t)\), \(w(t)=\int_0^1G(t,s)h(s)ds\), and \(G(t, s)\) is Green's function of the boundary value problem \[ -u''=0,\quad0\leq t\leq1,\;u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i), \] with \(\sum_{i=1}^{m-2}\alpha_i\xi_i\neq1\). Then, the boundary value problem (\(*\)) has at least one positive solution \(u_1(t)\) satisfying \(0<Mw(t)\leq u_1(t)\), \(0<t<1\) and \(\| u_1\|\leq r\) if \(\lambda\in[a,b]\). It seems that this is the first work on multipoint boundary value problem where the nonlinear term \(f(t, u)\) is allowed to change sign. Remark: It is a good paper.
scientific article; zbMATH DE number 2153765