On the solution of linear and nonlinear integral equation. (Q1412574)

The author considers Fredholm-Volterra integral equations of the second kind in the space \(L_2(\Omega)\times C[0,T]\). The linear as well as the nonlinear case is under consideration. In the linear case, using separation of variables, the author obtains a Volterra integral equation of the second kind with respect to time in the space \(C[0,T]\), and a Fredholm integral equation of the second kind with respect to position in the space \(L_2(\Omega)\). A sequence convergent to the unique solution of the considered equation is constructed. In the case of a Fredholm-Volterra integral equation with nonlinear Volterra integral term, two forms of nonlinear functions are investigated. In each of these cases it is shown that the equation can be represented as a system of Fredholm integral equations of the second kind. Also in the case of a Fredholm-Volterra integral equation with nonlinear Fredholm integral term it is shown that this equation becomes a system of Hammerstein integral equations. In the last section the author considers some special shapes of the kernel of the Fredholm integral term.