Positive solutions for one-dimensional singular \(p\)-Laplacian boundary value problems (Q2893043)

The authors of this very interesting paper consider the one-dimensional \(p\)-Laplacian equation NEWLINE\[NEWLINE \lambda^{-1}(t)(\lambda (t)\varphi_p(x^{\prime}(t)))^{\prime }+ \mu f(t,x(t),x^{\prime }(t))=0, \;\;0<t<+\infty , NEWLINE\]NEWLINE where \(\varphi_p(s)=|s|^{p-2}s\) (\(p>1\)). This equation satisfies some singular Sturm-Liouville boundary conditions, and \(\lambda,f\) are continuous functions. For instance, a pair of these boundary conditions is NEWLINE\[NEWLINE \alpha x(0)-\beta \lim\limits_{t\to 0^{+}}\lambda (t)^{1/(p-1)}x^{\prime }(t)=0, NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \gamma \lim\limits_{t\to +\infty }x(t)+\delta \lim\limits_{t\to +\infty }\lambda (t)^{1/(p-1)}x^{\prime }(t)=0; NEWLINE\]NEWLINE here \(\alpha, \beta, \gamma, \delta, \mu >0\); there is a case when the function \(f\) can be singular at \(t=0\).NEWLINENEWLINEThe main result is the existence of positive solutions to the above stated problem. The main approach used here is based on the Krasnosel'skii fixed point theorem. The authors introduce a special Banach space \(E\) and a suitable cones in \(E\) as well as the corresponding integral operators defined on cones. Some illustrating examples are given as well.