On the Solvability of Second Kind Integral Equations on the Real Line

This paper considers general second kind integral equations of the form OE(s) \Gamma Z R k(s; t)OE(t)dt = /(s) (in operator form OE \Gamma K k OE = /), where the functions k and / are assumed known, with / 2 Y; the space of bounded continuous functions on R, and k such that the mapping s ! k(s; \Delta); from R to L 1 (R); is bounded and continuous. The function OE 2 Y is the solution to be determined. Conditions on a set W ae BC(R;L 1 (R)) are obtained such that a generalised Fredholm alternative holds: if W satisfies these conditions and I \Gamma K k is injective for all k 2 W then I \Gamma K k is also surjective for all k 2 W and, moreover, the inverse operators (I \Gamma K k ) \Gamma1 on Y are uniformly bounded for k 2 W: The aproximation of the kernel in the integral equation by a sequence (k n ) converging in a weak sense to k is also considered and results on stability and convergence are obtained. These general theorems are used to establish results for two special classe..