Irregular boundary value problems with discontinuous coefficients and the eigenvalue parameter (Q1038598)

The authors are concerned with the existence of solutions to the Birkhoff-irregular boundary value problem with discontinuous coefficients and transmission conditions \[ \begin{cases} L(\lambda)u=f(x), \quad x\in[-1,0)\cup(0,1]\\ L_1(\lambda)u=f_1, \\ L_2u=f_2\\ L_3u=f_3\\ L_4u=f_4, \end{cases}\tag{P} \] where \[ \begin{aligned} L(\lambda)u&=\lambda^2u(x)+\lambda[a_1(x)u'(x)+b_1(x)u(x)]+a_2(x)u''(x)+b_2(x)u'(x)+Bu_{\mid x},\\ L_1(\lambda)u&=\lambda(\alpha_{11}u(-1)+\beta_{11}u(1))+\alpha_{10}u'(-1)+\beta_{10}u'(1)+\gamma_{10}u(-1)+\delta_{10}u(1),\\ L_2u&=\alpha_{20}u(-1)+\beta_{20}u(1),\\ L_3u&=\eta_{10}u(-0)+\eta_{11}u(+0),\\ L_4u&=\eta_{20}u'(-0)+\eta_{21}u'(+0).\end{aligned} \] The function \(f\) and the coefficients \(a_i, b_i\) (\(i=1,2\)) are discontinuous at \(0\) while the remaining parameters are complex numbers, \(B\) is an abstract linear operator. \(\lambda\) is an eigenvalue of this irregular transmission problem. According to the given parameters, the authors presents some existence and uniqueness results for some values of the eigenvalue \(\lambda.\) The discussion concerns the homogeneous and nonhomogeneous equations as well as the case of an abstract linear operator and the biharmonic equation.