Analog of the Cauchy function for a generalized equation with deviating argument
The author studies the properties of the Cauchy function for the Volterra-type integral equation with deviating argument \[ x(t)-\lambda\int_\alpha^t x(F(\cdot)) dq = f(t),\qquad \alpha, t\in D:=[a,b], \eqno{(1)} \] where \(x,f\in C(D)\) and \(q\in C(D)\cap BV(D)\), \(F : D\to D\) is a continuous function, and \(\lambda\) plays the role of a spectral parameter (actually all presented results refer to the case \(|\lambda|<<1\)). Defining the multiplication \(*\) generated by the deviation \(F\), the author introduces first the algebra of formal power series \(\sum_{k=0}^\infty \lambda^k x_k\) with function coefficients \(x_k\in C(D^n)\). This algebra is a unital associative algebra over \(\mathbb{R}\) with the Riemann-Stieltjes integral being defined in it. Then the original equation (1) is represented in an abstract form \[ x(t) - \int_\alpha^t (dQ * x) = f(t). \eqno{(2)} \] Further analysis demonstrates that the scheme of the classical theory of linear differential equations is transferred to equation (2) in full extent. In particular, there exists an analog of the Cauchy function, which allows to express every solution to equation (2) in the integral form.
scientific article; zbMATH DE number 5345603