A necessary and sufficient condition for the existence of \(C^{4n-1}[0,1]\) positive solutions of higher-order singular sublinear boundary value problems (Q2461939)

The authors consider the higher order boundary value problem \[ u^{(4n)}=f(t,u,u^{(4n-2)}),\;\;u(0)=u(1)=0, \] \[ au^{(2k)}(0)-bu^{(2k+1)}(0)=0,\;cu^{(2k)}(1)-bu^{(2k+1)}(1)=0, \quad 1\leq k\leq 2n-1,\tag{1} \] where \(a, b, c, d\) are nonnegative numbers and \(f\) is continuous and nonnegative in \(]0,1[\times]0,\infty[\). Moreover \(f\) is ``quasi-homogeneous'' in the second and third variables (a model is \(\sum_{i=1}^np_i(t)u^{\alpha_i}(-v)^{\beta_i},\) where \(p_i\) is continuous in \(]0,1[\), some of the \(\alpha_i\) take negative values, the remaining take positive values and the \(\beta_i\) are nonnegative). The authors give several conditions (necessary and sufficient, necessary, or sufficient) for (1) to have a \(C^{4n-1}[0,1]\) solution, in cases where some of the coefficients \(a,\,b,\,c,\,d\) vanish. The conditions are of the type \[ 0<\int_0^1f(t,t(1-t),h(t))\,dt<\infty \] with \(h(t)=-t\) or \(h(t)=-(1-t)\) or \(h(t)=-1\). The proof uses Green's functions and a fixed point theorem in cones. The results are extended to the case; when \(f\) has arguments \((t,\,u,\,u'',\,\cdots,\,u^{(4n-2)})\).