Partial integral equations of the third kind (Q942001)

The author investigates partial integral equations of the third kind, that is, \(Cx=(L+M+N)x+f\), where \(C,L,M,N\) are linear operators on the space of continuous functions defined on \([a,b]\times[c,d]\). Conditions under which such equations are Fredholm and Fredholm of index zero are given. Note that an equation \(Ax=f\) in a complex Banach space is Fredholm if \(A\) is a closed bounded linear operator whose range is closed, the dimension of the kernel \(n(A)\) and of the cokernel \(d(A)\) are finite. The index of the operator is \(ind\,A=n(A)-d(A)\).