Abstract Volterra equations of the second kind (Q1275266)

The author studies the equation \(x = Vx+f\) where \(V\) is a nonlinear Volterra operator. Given a family \(P_i\) of projections indexed by a linearly ordered set an operator \(V\) is said to be a \(0\)-Volterra operator provided \(P_ix=P_iy\) implies that \(P_iVx= P_iVy\), and \(V\) is said to be a Volterra operator provided one in addition has \(V(P_ix + (I-P_i)y) +V((I-P_i)x + P_iy) = Vx + Vy\). The author then studies the existence of local solutions and the connection between local and global solutions and shows in a quite general setting that compact linear Volterra operators have spectral radius zero. A number of additional results is established as well and this supports the claim that this more general definition of a Volterra operator gives useful new results.