A unified approach to Darboux transformations. (English) Zbl 1185.45006

The authors study the one-parameter family of integral equations \[ \beta(x,y) +\zeta(x,y) +\int_x^\infty dz\beta(x,z)w(z,y) =0,\;y>x, \] where \(\beta(x,y) \) is the unknown term and \(w(z,y)\) is a given \(N\times N\) matrix kernel.
Under some conjectures to \(w\), the authors find the resolvent kernel \(r(x,y,z)\). This kernel is written in term of \(\alpha (x,y)\) that is the solution of the equation
\[ \alpha(x,y) +w(x,y) +\int_x^\infty dz \alpha(x,z) w(z,y) =0. \tag{1} \]
The unique solvability to (1) is a priori condition to \(w\).
There are many other results. The proved results allow to sharpen some earlier propositions. The authors give many applications.


45F15 Systems of singular linear integral equations
45D05 Volterra integral equations
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