Positive solutions for second-order functional-differential equations (Q2753274)

The author proves the following result by applying a fixed-point theorem in cones.NEWLINENEWLINENEWLINELet \(f\in C(\langle 0,\infty),\langle 0,\infty)), \langle\alpha,\beta \rangle\subset(0,2)\) and \(h\in C(\langle \alpha,\beta\rangle, \langle 0,\infty))\) such that \(\int^\beta_\alpha th(t)dt>0\), \(\rho\beta <1\), with \(\rho= \int^\beta_\alpha h(t)dt\).NEWLINENEWLINENEWLINELet \(a\in C(\langle 0,1\rangle,\langle 0, \infty))\) and let \(x_0\in\langle \beta,1\rangle\) such that \(a(x_0)> 0\). Let \(f_0=\lim_{u\to 0+} {f(u)\over u}\), \(f_\infty= \lim_{u\to\infty} {f(u) \over u}\). Then the problem NEWLINE\[NEWLINEu''+a(t)f (u)=0,\;t\in(0,1),\quad u(0)=0,\;\int^\beta_\alpha h(t)u(t)dt =u(1),NEWLINE\]NEWLINE has at least one positive solution in the caseNEWLINENEWLINENEWLINE(i) \(f_0=0\) and \(f_\infty=\infty\) (superlinear), (ii) \(f_0= \infty\) and \(f_\infty =0\) (sublinear).