On a generalized integro-differential equation of Barbashin type
The main goal of this paper is to consider the following integro-differential equation: \[ \begin{multlined} {{\partial u (t, \tau, s)} \over {\partial \tau}}= C(t, \tau, s) u(t, \tau, s) + \int^b_a C(t, \tau, s, \sigma) u(t, \tau, \sigma)d\sigma\\+ \int^b_a m(t, \tau, s, r)u (r, \tau, s)dr+ \int^b_a \int^b_a n(t, \tau, s, \sigma, r) u(r, \tau, \sigma) d\sigma dr+ f(t, \tau, s) \end{multlined} \tag{1} \] and the operator differential equation \[ {{du (\tau)} \over {d\tau}}= A(\tau) u(\tau)+ f(\tau) \qquad (\tau\in J), \tag{2} \] where \(A(\tau)= C(\tau) K(\tau)\), \(K(\tau)= L(\tau)+ M(\tau)+ N(\tau)\), with \(C(\tau) u(t, s)= C(t, \tau, s) u(t,s)\), \(L(\tau) u(t,s)= \int^b_a L(t, \tau, s, \sigma) u(t, \sigma) d\sigma\), \(M(\tau) u(t,s)= \int^b_a m(t, \tau, s, r) u(r, s)dr\), \(N(\tau) u(t,s)= \int^b_a \int^b_a n(t, \tau, s, \sigma, r) u(r, \sigma) d\sigma dr\). The author studies the solution of (1) by means of the investigation of equation (2) and obtains some results. Moreover, he discusses a corresponding generalized boundary value problem.
generalized integro-differential equation of Barbashin type
scientific article; zbMATH DE number 856799