Positive solutions to a three point fourth order focal boundary value problem (Q457952)

The authors considers the three point fourth-order nonlinear boundary value problem NEWLINE\[NEWLINE\begin{gathered} u''''(t)+ g(t)\cdot f(u(t))= 0,\quad 0<t< 1,\\ u(0)= u'(p)= u''(p)= u'''(1)= 0,\end{gathered}\tag{1}NEWLINE\]NEWLINE where \(f:[0,\infty)\to [0,\infty)\); \(g:[0,1]\to [0,\infty]\) are continuous functions, \(g(t)\not\equiv 0\) on \([0,1]\) and \(p\in(0,1)\) is a constant.NEWLINENEWLINE Krasnoselskij's methods of the theory of positive solutions of nonlinear operator equations in half-ordered spaces with a cone is used by the authors. They prove the existence and non-existence of positive solutions of the boundary value problems (1) under some assumption according the functions \(f\) and \(g\).NEWLINENEWLINE The authors derive lower and upper bounds for the solution to the problem (1).