Positive solutions of singular boundary value problems for nonlinear differential systems
Consider the boundary value problem \[ \begin{gathered} -x''(t)= f(t, y)\quad\text{for }t\in (0,1),\\ -y''(t)= g(t, x)\quad\text{for }t\in (0,1),\\ \alpha_1 x(0)- \beta_1 x'(0)= \gamma_1 x(1)+ \delta_1 x'(1)= 0,\\ \alpha_2 y(0)- \beta_2 y'(0)= \gamma_2 y(1)+\delta_2 y'(1)= 0,\end{gathered}\tag{\(*\)} \] where \(\alpha_i\), \(\beta_i\), \(\gamma_i\), \(\delta_i\) are nonnegative constants and \(\rho_i= \alpha_i\gamma_i+ \alpha_i\delta_i+ \gamma_i\beta_i> 0\) for \(i= 1,2\). The functions \(f\) and \(g\) may be singular at \(t= 0\), \(t= 1\), \(x= 0\), \(y= 0\). The authors construct a special cone and apply fixed point index theory in a cone to establish the existence of at least one positive solution of \((*)\).
scientific article; zbMATH DE number 5161971