Upper and lower solutions for \(\Phi\)-Laplacian third-order BVPs on the half line (Q2877842)

The authors study the existence of positive solutions for third-order boundary value problems of the type NEWLINE\[NEWLINE\begin{aligned} (\Phi(-x''(t)))' + f (t, x(t)) = 0, \quad t \in (0,+\infty), \\ x(0)=\mu x'(0), \;x'(+\infty)=x''(+\infty)=0, \end{aligned}NEWLINE\]NEWLINE where \(\mu\geq0\), \(f\) is a continuous map and \(\Phi:\mathbb{R}\to\mathbb{R}\) is a continuous, increasing homeomorphism such that \(\Phi(0)=0\). The map \(\Phi\), the so-called \(\Phi\)-Laplacian, is a generalization of the one-dimensional \(p\)-Laplacian. The map \(f\) verifies some monotonicity assumption and may be singular at \(0\).NEWLINENEWLINEUsing the method of upper and lower solutions and fixed point techniques, the existence of at least one positive solution is proved. The result is illustrated by a concrete example.