## 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.

### MSC:

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