Solutions to systems of linear Fredholm integral equations of the third kind
The authors simultaneously consider the systems of linear integral equations \[ Fu=f(t),\quad \varepsilon v(t,\varepsilon)+Fv=f(t),\quad t\in [t_0,T], \] where \(Fu=A(t)u(t)+ \int_{t_0}^TK(t,s)u(s)ds\), \(A(t)\) and \(K(t,s)\) are given \(n\times n\) matrix functions, \(f(t)\) is a given \(n\)-dimensional vector function, \(u(t)\) and \(v(t,\varepsilon)\) are the desired \(n\)-dimensional vector functions and \(0<\varepsilon \) is a small parameter. Under adequate conditions on \(A\) and \(K\) they demonstrate the uniqueness of the solutions \(u(t)\) and \(v(t,\varepsilon)\) and the convergence of \(v(t,\varepsilon)\) to \(u(t)\) as \(\varepsilon\to 0\). They also demonstrate the continuous dependence of the solutions \(u(t)\) and \(v(t,\varepsilon)\) on the right-hand side term \(f(t)\).
Fredholm integral equations of the third kind
scientific article; zbMATH DE number 5816876