Smoothed integral equations

For a linear integral equation x ( t ) = a ( t ) − ∫ 0 t B ( t , s ) x ( s ) d s there is a resolvent equation R ( t , s ) = B ( t , s ) − ∫ s t B ( t , u ) R ( u , s ) d u and a variation of parameters formula x ( t ) = a ( t ) − ∫ 0 t R ( t , s ) a ( s ) d s . It is assumed that B is a perturbed convex function and that a ( t ) may be badly behaved in several ways. When the first two equations are treated separately by means of a Liapunov functional, restrictive conditions are required separately on a ( t ) and B ( t , s ) . Here, we treat them as a single equation f ( t ) = S ( t ) − ∫ 0 t B ( t , u ) f ( u ) d u where S is an integral combination of a ( t ) and B ( t , s ) . There are two distinct advantages. First, possibly bad behavior of a ( t ) is smoothed. Next, properties of S needed in the Liapunov functional can be obtained from an array of properties of a ( t ) and B ( t , s ) yielding considerable flexibility not seen in standard treatment. The results are used to treat nonlinear perturbation problems. Moreover, the function y ( t ) = a ( t ) − ∫ 0 t B ( t , s ) a ( s ) d s is shown to converge pointwise and in L 2 [ 0 , ∞ ) to x ( t ) .