A singular Sturm-Liouville equation under non-homogeneous boundary conditions. (Q434507)

The authors study the non-homogeneous weighted Neumann problem NEWLINE\[NEWLINE -(x^{2\alpha }u'(x))'+u(x)=0,\,\,x\in (0,1), NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(1)=0,\; {\lim }_{x\to 0^{+}}\psi _{\alpha }(x) u'(x)=1, NEWLINE\]NEWLINE where \( \psi _{\alpha }(x)= x^{2\alpha }\) if \(0<\alpha <1\), \( \psi _{\alpha }(x)= x^{(3+\sqrt 5)/2}\) if \(\alpha =1\), and \(\psi _{\alpha }(x)= x^{(3\alpha )/2}e^{x^{(1-\alpha )}/(1-\alpha )}\) if \(\alpha >1\), and the non-homogeneous weighted Dirichlet problem NEWLINE\[NEWLINE -(x^{2\alpha }u'(x))'+u(x)=0,\,\,x\in (0,1), NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(1)=0,\; {\lim }_{x\to 0^{+}}\psi _{\alpha }(x) u(x)=1, NEWLINE\]NEWLINE where \(\psi _{\alpha }(x)= 1\) if \(0<\alpha <1/2\), \(\psi _{\alpha }(x)= (1-\ln x)^{-1}\) if \(\alpha =1/2\), \(\psi _{\alpha }(x)= x^{2\alpha -1}\) if \(1/2<\alpha <1\), \(\psi _{\alpha }(x)= x^{(1+\sqrt 5)/2}\) if \(\alpha =1\), and \(\psi _{\alpha }(x)= x^{\alpha /2}e^{x^{(1-\alpha )}/(1-\alpha )}\) if \(\alpha >1\). Using methods of functional analysis they prove the solvability of both problems.