Positive solutions of a singular boundary value problem for systems of second-order differential equations (Q1004441)

The authors consider the periodic boundary value problem \[ \begin{cases} u''-2au'+(a^2+b^2)u=\lambda\varphi_1(t)f_1(t,u,v)\\ v''-2av'+(a^2+b^2)v=\lambda\varphi_2(t)f_2(t,u,v),\;\;0<t<1\\ u(0)=u(1),\;\;u'(0)=u'(1)\\ v(0)=v(1),\;\;v'(0)=v'(1). \end{cases} \] The functions \(\varphi_i\) may be singular at \(0\) or \(1\), \(f_1\) may be singular at \(u=0\) and \(f_2\) may be singular at \(v=0\), the functions \(f_i\) are continuous and defined for positive values of their second and third variables. The authors give conditions on the ratios \(\frac{f_1(t,u,v)}{u}\), \(\frac{f_2(t,u,v)}{v }\) so that, for a certain range of values of \(\lambda\), the problem has a positive solution \((u,v)\). The proof uses a fixed point theorem in cones and approximation to overcome the singularities.