Multistep collocation technique implementation for a pantograph-type second-kind Volterra integral equation

In this research, we have elaborated high-rate multistep collocation strategies in order to concern with second-type vanishing delay VIEs. Herein, characteristics of uniqueness, existence, and regularity for both numerical and analytical solutions have been shown. To explore the solvability of the system derived from the numerical method, we have defined particular operators and demonstrated that these operators are both compact and bounded. Solvability is studied by means of the innovative compact operator concepts. The concept of convergence has been examined in greater detail, revealing that the convergence of the method is influenced by the spectral radius of the matrix generated according to the collocation parameters in the difference equation resulting from the method's error. Finally, two numerical examples are given to certify our theoretically gained results. Also, since the proposed numerical method is local in nature, it can be compared to other local methods, such as those used in reference ^[1]. We will compare our method with ^[1] in the last section.