On strong stability for linear integral equations

The notion of strong or adjoint stability for linear ordinary differential equations is generalized to the theory of Volterra integral equations. It is found that this generalization is not unique in that equivalent definitions for differential equations lead to different stabilities for integral equations in general. Three types of stabilities arising naturally are introduced: strong stability, adjoint stability, and uniform adjoint stability. Necessary and sufficient conditions relative to the fundamental matrix for these stabilities are proved. Some lemmas dealing with non-oscillation of solutions and a semi-group property of the fundamental matrix are also given.