Positive solutions of singular Dirichlet and periodic boundary value problems
Consider the equation \[ (r(x)x')'= \mu q(t) f(t,x(t)) \] where \(t \in J:=[0,T]\), \(q(t)>0\) in \(J\), \(r(x)>0\) on \((0,A]\) may be singular at \(x=0\), \(f(t,x)\leq 0\) on \(J\times (0,A)\) may be singular at \(x=0\) and \(x=A\). Sufficient conditions for the existence of solutions \(x\) with \(0<x<A\) in \((0,T)\), satisfying the Dirichlet boundary conditions \(x(0)=x(T)=0\), as well as for the existence of solutions satisfying also the periodic conditions \(x(0)=x'(0)=x(T)=x'(T)=0\), are established.
scientific article; zbMATH DE number 1781529