Collocation methods for third-kind Volterra integral equations with proportional delays

In this paper, a class of Volterra delay integral equations (VDIEs) with noncompact operators is approximated by collocation methods. The properties of corresponding operators as well as existence, uniqueness and regularity of exact solution are discussed. The existence and uniqueness of collocation solutions are proved under two special graded meshes. Moreover, we present the convergence conditions and convergence order. Finally, some numerical examples are given to verify the validity of the theoretical orders of convergence.