Positive solutions for semipositone BVPs of second-order difference equations
The authors consider the following Sturm-Liouville discrete time problem \[ \triangle[p(t-1)\triangle u(t-1)] + \lambda f(t,u(t)) = 0 \;,\;a+1\leq t\leq b+1 \] \[ \gamma_1u(a) - \gamma_2p(a)\triangle u(a) = 0,\quad \gamma_3u(b+2)+\gamma_4p(b+1)\triangle u(b+1)=0 \] with \(\gamma_i\geq 0\), \(\gamma_1\gamma_3+\gamma_1\gamma_4+\gamma_2\gamma_3>0\), \(\lambda>0\) and \(\triangle\) the forward difference; also \(p(t)>0\), \(a\leq t\leq b+1\). Explicit conditions expressed through the properties of the nonlinearity \(f\) and the problem's coefficients are given, for the existence of at least one positive solution.
scientific article; zbMATH DE number 5305213