Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

and Applied Analysis 3 The class of Runge-Kutta methods with CQ formula has been applied to delay-integro-differential equations by many authors (c.f. [18, 19]). For the CQ formula (9), we usually adopt the repeated trapezoidal rule, the repeated Simpson’s rule, or the repeated Newton-cotes rule, and so forth, denote η = max{?̃? 0 , ?̃? 1 , . . . , ?̃? m }. It should be pointed out that the adopted quadrature formula (9) is only a class of quadrature formula for ?̃?(n) i , there also exist some other types of quadrature formula, such as Pouzet quadrature (PQ) formula and the quadrature formula based on Laguerre-Radau interpolations [20, 21]. It is the aim of our future research to investigate the adaptation of PQ formula and the quadrature formula based on Laguerre-Radau interpolations to VDIDEs. 3. The Nonlinear Stability Analysis In this section, we will investigate the stability of the twostep Runge-Kutta methods for VDIDEs. In order to consider the stability property, we also need to consider the perturbed problem of (1): z 󸀠 (t) = f(t, z (t) , z (t − τ) , ∫ t t−τ g (t, ξ, z (ξ)) dξ) , t ∈ [0, Τ] , z (t) = ψ (t) , t ∈ [−τ, 0] , (10) where ψ : [−τ, 0] → CN is a given continuous function. The unique exact solution of the problem (10) is denoted as z(t). Applying the two-step Runge-Kutta method (7a)–(7c) to (10) leads to Z (n) i = z n + h s ∑ j=1 a ij f(t n + c j h, Z (n) j , Z (n)