Existence of positive solutions for a class of singular sublinear boundary value problems (Q2720148)

The boundary value problem NEWLINE\[NEWLINEx''+ f(t,x)= 0,\quad t\in (0,1),\quad ax(0)- bx'(0)= 0,\quad cx(1)+ dx'(1)= 0,NEWLINE\]NEWLINE with \(a\geq 0\), \(b\geq 0\), \(c\geq 0\), \(d\geq 0\), \(ac+ ad+ bc> 0\), and \(f\) can be unbounded at \(t= 0\) and \(t=1\), is considered. A necessary and sufficient condition for the existence of positive solutions in \(C[0,1]\) and also another one for the existence of positive solutions in \(C^1[0,1]\) are obtained under the hypotheses NEWLINE\[NEWLINEf(t,x)\in C((0,1)\times (0,+\infty), [0,+\infty)),\quad f(t,1)\not\equiv 0\quad\text{for }t\in (0,1),NEWLINE\]NEWLINE and there exist constants \(\lambda\), \(\mu\), \(N\) and \(M\) such that NEWLINE\[NEWLINE-\infty< \lambda< 0< \mu< 1,\qquad 0< N\leq 1\leq M,NEWLINE\]NEWLINE and for \((t,x)\in (0,1)\times (0,+\infty)\) NEWLINE\[NEWLINE\begin{aligned} k^\mu f(t,x) &\leq f(t,kx)\leq k^\lambda f(t, x)\quad\text{if }0\leq k< N,\\ k^\lambda f(t,x) &\leq f(t,kx)\leq k^\mu f(t,x)\quad\text{if }k\geq M.\end{aligned}NEWLINE\]NEWLINE Lower and upper solutions and fixed-point theorems are used.