On the convergence of collocation solutions in continuous piecewise polynomial spaces for Volterra integral equations

Abstract

This paper fills an important gap in the convergence analysis of collocation solutions in spaces of continuous piecewise polynomials for Volterra integral equations of the second kind. Our analysis is then extended to Volterra functional integral equations of the second kind with constant delays.

1 Introduction

The convergence analysis of piecewise polynomial collocation solutions for Volterra integral equations (VIEs) of the second kind,

$$\begin{aligned} u(t)=g(t)+\int _0^tK(t,s)u(s)ds, \quad t\in I:=[0,T], \end{aligned}$$

(1.1)

with continuous kernel K(t, s) is now largely well understood; see [4–6] and, especially, the surveys [1, 3]. However, there has remained an important gap in the convergence analysis of collocation solutions for the second-kind VIE (1.1): it concerns the convergence/divergence of piecewise polynomial collocation solutions for (1.1) that are globally continuous on I and correspond to collocation points that do not include the points of the underlying mesh $I_h$.

It is the aim of this paper to close this gap and to employ the gained insight to establish the analogous convergence analysis for the Volterra functional integral equation (VFIE) with constant delay $\tau > 0$,

$$\begin{aligned} \left\{ \begin{aligned}&u(t) = g(t)+\int _{t-\tau }^tK(t,s)u(s)ds, \;\; t\in I,\\&u(t) = \varphi (t),\;\; t \in [-\tau ,0]. \end{aligned}\right. \end{aligned}$$

(1.2)

The outline of the paper is as follows. In Sects. 2 and 3 we state our main results on the convergence of globally continuous piecewise polynomial collocation solutions for the second-kind VIE (1.1) and the second-kind VFIE (1.2); their proofs are given in Sects. 4 and 5. In Sect. 6, we use a number of examples to illustrate the validity of our results on the attainable order of these collocation solutions. Section 7 concludes with a concluding remark.

2 Continuous collocation solutions for second kind VIEs

2.1 Meshes and collocation spaces

Let $I_h:=\left\{ t_{n} := nh: \; n = 0,1, \ldots , N \;\, (t_{N} = T)\right\}$ be a given mesh on $I = [0,T]$, with $\sigma _n := [t_n,t_{n+1}]$ and mesh diameter $h = T/N$. We seek a collocation solution $u_h$ for (1.1) in the space

$$\begin{aligned} S_{m}^{(0)}(I_{h}) := \left\{ v \in C (I): \; v|_{\sigma _n} \in \pi _m = \pi _m(\sigma _n)\; (0 \le n \le N-1)\right\} , \end{aligned}$$

where $\pi _m$ denotes the space of all (real) polynomials of degree not exceeding m. For a prescribed set of collocation points

$$\begin{aligned} X_h:=\left\{ t=t_n+c_ih: \ 0< c_1<\cdots <c_m\le 1\ (0\le n\le N-1)\right\} , \end{aligned}$$

(2.1)

$u_h$ is defined by the collocation equation

$$\begin{aligned} u_h(t)=g(t)+\int _0^t K(t,s)u_h(s)ds,\quad t\in X_h, \end{aligned}$$

(2.2)

with $u_h(0)=g(0)$.

Consequently the collocation polynomial can be written as (see [2])

$$\begin{aligned} u'_h(t_n+sh)=\sum _{j=1}^mL_j(s)U_{n,j} ,\quad s\in (0, 1], \end{aligned}$$

(2.3)

where $U_{n,i}:=u_h'(t_{n,i}), t_{n,i}:=t_n+c_ih$ and the polynomials

$$\begin{aligned} L_j(s):=\prod _{k\ne j}^m\dfrac{s-c_k}{c_j-c_k}\; \; (j=1,\ldots , m), \end{aligned}$$

denote the Lagrange fundamental polynomials with respect to the (distinct) collocation parameters $\{c_i\}$.

Integrating (2.3), we obtain

$$\begin{aligned} u_h(t_n+sh)=u_h(t_n)+h\sum _{j=1}^m\beta _j(s)U_{n,j} ,\quad s\in [0, 1], \end{aligned}$$

(2.4)

where $\beta _j(s):=\int _0^sL_j(v)dv$.

Therefore, at $t=t_{n,i}$,

$$\begin{aligned} u_h(t_{n,i})= & {} u_h(t_n)+h\sum _{j=1}^ma_{ij}U_{n,j} =g(t_{n,i})+\int _0^{t_{n,i}}K(t_{n,i},s)u_h(s)ds\nonumber \\= & {} g(t_{n,i})+h\sum _{l=0}^{n-1}\int _0^1K(t_{n,i},t_l+sh) \left[ u_h(t_l)+h\sum _{j=1}^m\beta _j(s)U_{l,j}\right] ds\nonumber \\&+\,h\int _0^{c_i}K(t_{n,i},t_n+sh)\left[ u_h(t_n)+h\sum _{j=1}^m\beta _j(s)U_{n,j}\right] ds, \end{aligned}$$

(2.5)

where $a_{ij}:=\beta _j(c_i)$.

Denote $A:=(a_{ij})_{m\times m}$, $e:=(1,\ldots , 1)^T$, $G_n:=(g(t_{n,1}),\ldots , g(t_{n,m}))^T$, $U_n:=(U_{n,1},\ldots ,U_{n,m})^T$, $B_n^{(l)}:=\left( \int _0^{1}K(t_{n,i},t_l+sh)\beta _j(s)ds\right) \; (0\le l\le N-1)$, $B_n:=\left( \int _0^{c_i}K(t_{n,i},t_n+sh)\beta _j(s)ds\right)$, $C_n^{(l)}:=diag\left( \int _0^{1}K(t_{n,i},t_l+sh)ds\right) \; (0\le l\le N-1)$, and $C_n:=diag\left( \int _0^{c_i}K(t_{n,i},t_n+sh)ds\right)$, we have

$$\begin{aligned} \left( hA-h^2B_n\right) U_n=G_n+\left( hC_n-I_m\right) eu_h(t_n)+h\sum _{l=0}^{n-1} \left[ C_n^{(l)}eu_h(t_l)+hB_n^{(l)}U_l\right] , \end{aligned}$$

(2.6)

where $I_m$ denotes the identity in $L(\mathbb {R}^m)$.

If $g \in C(I)$ and $K \in C(D) \; (D := \{(t,s): \; 0 \le s \le t \le T\})$, (2.6) determines a unique $u_h \in S_m^{(0)}(I_h)$ for all sufficiently small mesh diameters, say $h \in (0,\bar{h})$. However, the resulting collocation solution will not converge uniformly on I to the exact solution of (1.1) for every choice of the collocation parameters $\{c_i\}$: while the convergence statement

$$\begin{aligned} \lim _{h \rightarrow 0}\Vert u-u_h\Vert _{\infty } = 0 \end{aligned}$$

(2.7)

holds whenever $c_1 > 0$ and $c_m = 1$ (cf. [3], and [7, 8]), this will in general no longer remain true when $c_m < 1$. If (2.7) holds, the order of convergence will not be the same for all $\{c_i\}$.

2.2 The main convergence results

Theorem 2.1

Assume that $g\in C^{m+2}(I), K\in C^{m+2}(D)$, and $u_h\in S_{m}^{(0)}(I_h)$ is the collocation solution for the second-kind Volterra integral equation (1.1) defined by the collocation equation (2.2) whose underlying meshes have mesh diameters $h < \bar{h}$. Then (2.7) holds if, and only if, the collocation parameters $\{c_i\}$ satisfy the condition

$$\begin{aligned} -1\le \rho _m:=(-1)^m\prod \limits _{i=1}^m\dfrac{1-c_i}{c_i}\le 1. \end{aligned}$$

The corresponding attainable global order of convergence is given by

$$\begin{aligned} \max _{t\in I}|u(t)-u_h(t)| \le C \left\{ \begin{array}{l@{\quad }l} h^{m+1}, &{} \hbox {if }\; -1\le \rho _m< 1,\\ h^{m} , &{} \hbox {if }\; \rho _m= 1, \end{array} \right. \end{aligned}$$

where the constant C depends on the collocation parameters $\{c_i\}$ but not on h.

3 Continuous collocation solutions for second kind VIEs with constant delay

3.1 Meshes and collocation spaces

It is well known (see for example [2, Ch. 4]) that the constant delay $\tau > 0$ in (1.2) induces the primary discontinuity points $\xi _{\mu } = \mu \tau \; (\mu \ge 0)$ at which the regularity of the solution u(t) is, at least for small values of $\mu$, lower than it is in $(\xi _{\mu }, \xi _{\mu +1})$. Thus, the collocation solution $u_h \in S_m^{(0)}(I_h)$ will attain an order of global convergence equal to that for VIEs (1.1) without delay only if the underlying mesh $I_h$ includes these primary discontinuity points. Assuming for ease of notation that $T = \xi _{M+1}$ for some $M \ge 1$, we choose this so-called constrained mesh to be

$$\begin{aligned} I_h:=\bigcup _{\mu =0}^M I_h^{(\mu )}, \;\; \text{ with } \;\; I_h^{(\mu )}:=\{ t_{n}^{(\mu )}:=\xi _{\mu }+nh \; (n=0,1,\ldots , N)\}, \end{aligned}$$

(3.1)

where $h = \tau /N$. We set $\sigma _n^{(\mu )}:=[t_n^{(\mu )},t_{n+1}^{(\mu )}]$. The solution u of (1.2) will be approximated by the collocation solution

$$\begin{aligned} u_h \in S_{m}^{(0)}(I_{h}) := \left\{ v \in C (I_h): \; v|_{\sigma _{n}^{(\mu )}} \in \pi _{m} \; (0 \le n \le N-1)\right\} , \end{aligned}$$

using collocation points

$$\begin{aligned} X_h:= & {} \bigcup _{\mu =0}^M X_h^{(\mu )},\nonumber \\&\text{ with } \;\; X_h^{(\mu )}:=\{t_{n,i}^{(\mu )}=t_n^{(\mu )}+c_ih: \; i = 1, \ldots , m \; (0\le n\le N-1)\} \end{aligned}$$

(3.2)

corresponding to prescribed collocation parameters $\{c_i\}$ with $0 < c_1 < \cdots < c_m \le 1$. Hence, the collocation equation for the subinterval $\sigma _n^{(\mu )}$ is

$$\begin{aligned} u_h(t)=g(t)+\int _{t-\tau }^tK(t,s)u_h(s)ds, \; t \in X_h^{(\mu )} \quad (\mu = 0,1, \ldots , M). \end{aligned}$$

(3.3)

If $\mu = 0$, the values of $u_h$ at $t\in [-\tau , 0]$ are determined by the given initial function, i.e., $u_h(t)=\varphi (t)$.

3.2 The main convergence results

Theorem 3.1

Assume that $g\in C^{m+2}(I), K\in C^{m+2}(D), \varphi \in C^{m+1}[-\tau , 0]$, and let $u_h\in S_{m}^{(0)}(I_h)$ be the collocation solution for the second-kind VFIE (1.2) determined by the collocation equation (3.3), using constrained meshes $I_h$ of the form (3.1). Then $u_h$ converges uniformly on I to the solution u of (1.2) if, and only if, the collocation parameters in (3.2) satisfy the condition

$$\begin{aligned} -1\le \rho _m:=(-1)^m\prod \limits _{i=1}^m \dfrac{1-c_i}{c_i}\le 1. \end{aligned}$$

The resulting attainable global order of convergence is then given by

$$\begin{aligned} \max _{t\in I}|u(t)-u_h(t)|\le C \left\{ \begin{array}{l@{\quad }l} h^{m+1}, &{} \hbox {if }\; -1\le \rho _m< 1,\\ h^{m}, &{} \hbox {if }\; \rho _m= 1, \end{array} \right. \end{aligned}$$

where the constant C depends on the collocation parameters $\{c_i\}$ but not on h.

Remark 3.1

The convergence results for the case $0 < c_1 < \cdots < c_m = 1$ follow trivially from the proof of Theorem 2.1 (see also [8] and [3]). A similar conclusion holds in the case of second-kind VFIEs (Theorem 3.1 and its proof).

Remark 3.2

The case $\rho _m= 1$ can happen only when m is even, but as Theorem 2.1 and Theorem 3.1 describe, this case leads to a reduction of the order of convergence.

4 Proof of Theorem 2.1

We assume that $c_m<1$.

According to the theory of Lagrange interpolation we may write

$$\begin{aligned} u'(t_n+sh)=\sum _{j=1}^mL_j(s)u'(t_{n,j})+h^mR_{m,n}^1(s) ,\;\; s\in [0,1], \end{aligned}$$

(4.1)

where the Peano remainder term and Peano kernel (see [2]) are given by

$$\begin{aligned} R_{m,n}^{1}(v):=\int _0^1K_m(v,z)u^{(m+1)}(t_n+zh)dz \end{aligned}$$

and

$$\begin{aligned} K_m(v,z):=\frac{1}{(m-1)!}\left\{ (v-z)_{+}^{m-1}-\sum _{k=1}^mL_k(v)(c_k-z)_{+}^{m-1}\right\} ,\ v\in [0,1]. \end{aligned}$$

Here, $(v-z)_{+}^{m-1}:=0$ for $v<z$ and $(v-z)_{+}^{m-1}:=(v-z)^{m-1}$ for $v\ge z$.

Integration of (4.1) leads to

$$\begin{aligned} u(t_n+sh)=u(t_n)+h\sum _{j=1}^m\beta _j(s)u'(t_{n,j}) +h^{m+1}R_{m,n}(s), \ s\in [0,1], \end{aligned}$$

(4.2)

where $R_{m,n}(s):={\int _0^sR_{m,n}^1(v)dv}$.

We first consider the case of constant kernel $K(t, s)\equiv 1$. This case already contains all important ideas.

By (2.4) and (4.2), the collocation error $e_h:=u-u_h$ on $[t_n, t_{n+1}]$ may be written as

$$\begin{aligned} e_h(t_n+sh)=e_h(t_n)+h\sum _{j=1}^m\beta _j(s)\varepsilon _{n,j}+h^{m+1}R_{m,n}(s), \end{aligned}$$

(4.3)

where $\varepsilon _{n,i}:=u'(t_{n,i})-u_h'(t_{n,i})$. Particularly,

$$\begin{aligned} e_h(t_{n,i})=e_h(t_n)+h\sum _{j=1}^ma_{ij}\varepsilon _{n,j}+h^{m+1}R_{m,n}(c_i). \end{aligned}$$

(4.4)

By (1.1)–(2.2) and using (4.3), it can be shown that

$$\begin{aligned} e_h(t_{n,i})= & {} \int _0^{t_{n,i}}e_h(s)ds=h\sum _{l=0}^{n-1}\int _0^{1}e_h(t_l+sh)ds+h\int _0^{c_i}e_h(t_n+sh)ds\nonumber \\= & {} h\sum _{l=0}^{n-1}e_h(t_l)+h^2\sum _{l=0}^{n-1}\sum _{j=1}^m\gamma _j(1)\varepsilon _{l,j} +hc_ie_h(t_n)+h^2\sum _{j=1}^mb_{ij}\varepsilon _{n,j}+h^{m+1}\tilde{R}_{m,n}(c_i),\nonumber \\ \end{aligned}$$

(4.5)

where $\gamma _j(s):=\int _0^s\beta _j(v)dv, \;\; b_{ij}:=\int _0^{c_i}\beta _j(s)ds=\gamma _j(c_i),$ and $\tilde{R}_{m,n}(c_i):=\sum \nolimits _{l=0}^{n-1}h\int _0^{1}R_{m,l}(s)ds+h\int _0^{c_i}R_{m,n}(s)ds.$ So by (4.4) and (4.5), we have

$$\begin{aligned}&e_h(t_n)+h\sum _{j=1}^ma_{ij}\varepsilon _{n,j}+h^{m+1}R_{m,n}(c_i)\nonumber \\&\quad =h\sum _{l=0}^{n-1}e_h(t_l)+h^2\sum _{l=0}^{n-1}\sum _{j=1}^m\gamma _j(1)\varepsilon _{l,j} +hc_ie_h(t_n)+h^2\sum _{j=1}^mb_{ij}\varepsilon _{n,j}+h^{m+1}\tilde{R}_{m,n}(c_i).\nonumber \\ \end{aligned}$$

(4.6)

By the standard technique used by Brunner (see [2]), rewriting (4.6) with n replaced by $n-1$ and with $i= m$ and subtract it from (4.6), we find

$$\begin{aligned}&e_h(t_{n})-e_h(t_{n-1})+h\sum _{j=1}^ma_{ij}\varepsilon _{{n},j} -h\sum _{j=1}^ma_{mj}\varepsilon _{{n-1},j}\\&\qquad +\,h^{m+1}R_{m,{n}}(c_i)-h^{m+1}R_{m,{n-1}}(c_m)\\&\quad =he_h(t_{n-1})+h^2\sum _{j=1}^m\gamma _j(1)\varepsilon _{{n-1},j} +hc_ie_h(t_n)-hc_me_h(t_{n-1})\\&\qquad +\,h^2\sum _{j=1}^mb_{ij}\varepsilon _{n,j}- h^2\sum _{j=1}^mb_{mj}\varepsilon _{{n-1},j}+h^{m+1}\tilde{R}_{m,n}(c_i)-h^{m+1}\tilde{R}_{m,{n-1}}(c_m). \end{aligned}$$

This can be written in the more concise form

$$\begin{aligned}&\left( e_h(t_{n})-e_h(t_{n-1})\right) e+hA\varepsilon _{n} -hee_m^TA\varepsilon _{n-1}\nonumber \\&\quad =he_h(t_{n-1})e+h^2e\gamma ^T\varepsilon _{n-1} +hCee_h(t_n)-hc_mee_h(t_{n-1})\nonumber \\&\qquad +\,h^2B\varepsilon _{n}- h^2ee_m^TB\varepsilon _{n-1}+h^{m+1}R_{m,{n}}, \end{aligned}$$

(4.7)

with obvious meaning of $R_{m,{n}}$, and with $C:=diag(c_1, \ldots , c_m)$, $\varepsilon _n:=(\varepsilon _{n,1}, \ldots , \varepsilon _{n,m})^T$, $B:=(b_{ij})_{m\times m}$, $\gamma :=(\gamma _1(1),\ldots , \gamma _m(1))^T$.

Since $e_h$ is continuous in I, and hence at the mesh points, by (4.3) and $e_h(0)=0$, we also have the relation (see [2, (1.1.27)])

$$\begin{aligned} e_h(t_n)= & {} e_h(t_{n-1}+h)=e_h(t_{n-1})+h\sum _{j=1}^mb_j\varepsilon _{n-1,j}+h^{m+1}R_{m,n-1}(1)\nonumber \\= & {} h\sum _{l=0}^{n-1}\sum _{j=1}^mb_j\varepsilon _{l,j} +h^{m+1}\sum _{l=0}^{n-1}R_{m,l}(1)=h\sum _{l=0}^{n-1}b^T\varepsilon _{l} +h^{m+1}\sum _{l=0}^{n-1}R_{m,l}(1), \end{aligned}$$

(4.8)

where $b_j:=\int _0^1L_j(s)ds$ and $b^T:=(b_1,\ldots ,b_m)$.

Substituting (4.8) into (4.7), we have

$$\begin{aligned}&eb^T\varepsilon _{n-1}+h^meR_{m,n-1}(1)+A\varepsilon _{n} -ee_m^TA\varepsilon _{n-1}\\&\quad =(1-c_m) e \left[ h\sum _{l=0}^{n-2}b^T\varepsilon _{l} +h^{m}\sum _{l=0}^{n-2}hR_{m,l}(1)\right] +he\gamma ^T\varepsilon _{n-1}\\&\qquad +Ce\left[ h\sum _{l=0}^{n-1}b^T\varepsilon _{l} +h^{m}\sum _{l=0}^{n-1}hR_{m,l}(1)\right] +hB\varepsilon _{n}- hee_m^TB\varepsilon _{n-1} +h^{m}R_{m,{n}}. \end{aligned}$$

This equation can be written in the form

$$\begin{aligned} \left( A-hB\right) \varepsilon _{n}= & {} \left( ee_m^TA-eb^T+he\gamma ^T- hee_m^TB\right) \varepsilon _{n-1}+h(1-c_m)e\sum _{l=0}^{n-2}b^T\varepsilon _{l}\nonumber \\&+\,hCe\sum _{l=0}^{n-1}b^T\varepsilon _{l} +h^{m}\tilde{R}_{m,{n}}, \end{aligned}$$

(4.9)

or

$$\begin{aligned} \varepsilon _{n}= & {} \left( A^{-1}\left( ee_m^TA-eb^T\right) +O(h)\right) \varepsilon _{n-1}+h D\sum _{l=0}^{n-1}\varepsilon _{l} +h^{m}\bar{R}_{m,{n}}, \end{aligned}$$

(4.10)

with obvious meaning of $\tilde{R}_{m,{n}},\bar{R}_{m,{n}}$ and D.

Since

$$\begin{aligned} ee_m^TA-eb^T=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} a_{m1}-b_1 &{} a_{m2}-b_2 &{} \cdots &{} a_{mm}-b_m \\ a_{m1}-b_1 &{} a_{m2}-b_2 &{} \cdots &{} a_{mm}-b_m \\ \cdots &{} \cdots &{} \cdots &{} \cdots \\ a_{m1}-b_1 &{}a_{m2}- b_2 &{} \cdots &{} a_{mm}-b_m \\ \end{array} \right) , \end{aligned}$$

the rank of the matrix $ee_m^TA-eb^T$ is one, implying that the rank of the matrix $A^{-1}(ee_m^TA-eb^T)$ is also one. This means that this matrix has exactly one nonzero eigenvalue. Setting $A^{-1}:=(v_{ij})_{m\times m}$, we have

$$\begin{aligned}&A^{-1}\left( ee_m^TA-eb^T\right) \\&\quad =\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} (a_{m1}-b_1)\sum \limits _{j=1}^mv_{1j} &{} (a_{m2}-b_2)\sum \limits _{j=1}^mv_{1j} &{} \cdots &{} (a_{mm}-b_m)\sum \limits _{j=1}^mv_{1j} \\ (a_{m1}-b_1)\sum \limits _{j=1}^mv_{2j} &{} (a_{m2}-b_2)\sum \limits _{j=1}^mv_{2j} &{} \cdots &{} (a_{mm}-b_m)\sum \limits _{j=1}^mv_{2j} \\ \cdots &{} \cdots &{} \cdots &{} \cdots \\ (a_{m1}-b_1)\sum \limits _{j=1}^mv_{mj} &{} (a_{m2}-b_2)\sum \limits _{j=1}^mv_{mj} &{} \cdots &{} (a_{mm}-b_m)\sum \limits _{j=1}^mv_{mj} \\ \end{array} \right) , \end{aligned}$$

and the nonzero eigenvalue is

$$\begin{aligned} \lambda \left( A^{-1}(ee_m^TA-eb^T)\right) =\sum \limits _{i=1}^m(a_{mi}-b_i)\sum \limits _{j=1}^mv_{ij}=1-b^TA^{-1}e. \end{aligned}$$

By Proposition 3.8 and Theorem 3.10 of [5], we know that the stability function $R(z) = P(z)/Q(z)$ of the collocation method has the value $R(\infty )=1-b^TA^{-1}e$, where Q(z) and P(z) are the polynomials

$$\begin{aligned} Q(z)= & {} M^{(m)}(0)+M^{(m-1)}(0)z+\cdots +M(0)z^m,\\ P(z)= & {} M^{(m)}(1)+M^{(m-1)}(1)z+\cdots +M(1)z^m, \end{aligned}$$

with

$$\begin{aligned} M(z)=\frac{1}{m!}\prod _{i=1}^m(z-c_i). \end{aligned}$$

Therefore,

$$\begin{aligned} R(\infty )=\frac{M(1)}{M(0)}=(-1)^m\prod _{i=1}^m \dfrac{1-c_i}{c_i}; \end{aligned}$$

that is, the only nonzero eigenvalue of $A^{-1}(ee_m^TA-eb^T)$ is

$$\begin{aligned} 1-b^TA^{-1}e=(-1)^m\prod _{i=1}^m \dfrac{1-c_i}{c_i}=\rho _m. \end{aligned}$$

Therefore $A^{-1}\left( ee_m^TA-eb^T\right)$ is diagonalizable and there exists a nonsingular matrix T such that

$$\begin{aligned} T^{-1}A^{-1}\left( ee_m^TA-eb^T\right) T=:F=diag(\rho _m,\underbrace{0,\ldots ,0}_{m-1}). \end{aligned}$$

Multiplying (4.10) by $T^{-1}$ and setting $Z_n:=T^{-1}\varepsilon _{n}$, we obtain

$$\begin{aligned} \begin{aligned} Z_{n} =(F+O(h))Z_{n-1}+h T^{-1}DT\sum _{l=0}^{n-1}Z_{l} +h^{m}T^{-1}\bar{R}_{m,{n}}. \end{aligned} \end{aligned}$$

(4.11)

We consider the following three cases:

Case I $-1< \rho _m<1$

Using standard techniques of error estimation for collocation solutions of VIEs (see [2, 5, 7]), we know that there exists a constant $C_{1}$, such that

$$\begin{aligned} \begin{aligned} \Vert \varepsilon _n\Vert _1\le C_{1} h^m. \end{aligned} \end{aligned}$$

(4.12)

It follows from (4.6) that there exist constants $C_{2}$ and $C_{3}$ such that

$$\begin{aligned} \begin{aligned} |e_h(t_n)|\le hC_{2}\sum _{l=0}^{n-1}|e_h(t_l)|+C_{3}h^{m+1}, \end{aligned} \end{aligned}$$

(4.13)

and hence by the discrete Gronwall inequality (see [2]), there exists a constant $C_{4}$, such that

$$\begin{aligned} \begin{aligned} |e_h(t_n)|\le C_{4}h^{m+1} \quad (n = 1, \ldots , N). \end{aligned} \end{aligned}$$

(4.14)

Case II $\rho _m=-1$

Rewriting (4.11) with n replaced by $n-1$ and subtract it from (4.11), we have

$$\begin{aligned} Z_{n}-Z_{n-1}= & {} (F+O(h))(Z_{n-1}-Z_{n-2})+hT^{-1}DTZ_{n-1}\nonumber \\&+\,h^{m}T^{-1}(\bar{R}_{m,{n}}-\bar{R}_{m,{n-1}}). \end{aligned}$$

(4.15)

Therefore,

$$\begin{aligned} \left( \begin{array}{c} Z_{n} \\ Z_{n-1} \\ \end{array} \right)= & {} \left( \begin{array}{c@{\quad }c} I_m+F+O(h) &{} -F+O(h)\\ I_m &{} 0 \\ \end{array} \right) \left( \begin{array}{c} Z_{n-1} \\ Z_{n-2} \\ \end{array} \right) \\&+\left( \begin{array}{c} h^{m}T^{-1}\left( \bar{R}_{m,{n}}-\bar{R}_{m,{n-1}}\right) \\ 0 \\ \end{array} \right) . \end{aligned}$$

Since $\bar{R}_{m,{n}}-\bar{R}_{m,{n-1}}=O(h)$ for $u\in C^{m+2}$, we define $r_{m,n}:=\frac{\bar{R}_{m,{n}}-\bar{R}_{m,{n-1}}}{h}$ and set

$$\begin{aligned} X_n:=\left( \begin{array}{c} Z_{n} \\ Z_{n-1} \\ \end{array} \right) , G:=\left( \begin{array}{c@{\quad }c} I_m+F &{} -F\\ I_m &{} 0 \\ \end{array} \right) , \bar{r}_{m,{n}}:=\left( \begin{array}{c} T^{-1}r_{m,{n}} \\ 0 \\ \end{array} \right) . \end{aligned}$$

We may then write

$$\begin{aligned} X_n=GX_{n-1}+O(h)X_{n-1}+h^{m+1}\bar{r}_{m,{n}}. \end{aligned}$$

(4.16)

The eigenvalues of the matrix G are $\underbrace{1, 1,\ldots , 1}_m; -1, \underbrace{0, \ldots , 0}_{m-1}$. The eigenvalue 1 of multiplicity m has m linearly independent eigenvectors, while to the eigenvalue 0 of multiplicity $m-1$ there correspond $m-1$ linearly independent eigenvectors. Therefore, G is diagonalizable, and there exists a nonsingular matrix P such that

$$\begin{aligned} P^{-1}GP=:\varLambda =diag(\underbrace{1,\ldots ,1}_{m},-1,\underbrace{0,\ldots ,0}_{m-1}). \end{aligned}$$

Defining $Y_n:=P^{-1}X_n$ we obtain

$$\begin{aligned} Y_n=(\varLambda +O(h))Y_{n-1}+h^{m+1}P^{-1}\bar{r}_{m,{n}}. \end{aligned}$$

(4.17)

Similar to [7] we can assert that there exist constants $C_5, C_6$ so that

$$\begin{aligned} \Vert Y_n\Vert \le (1+C_5h)\Vert Y_{n-1}\Vert +C_6h^{m+1}. \end{aligned}$$

An induction argument then leads to

$$\begin{aligned} \Vert Y_n\Vert \le (1+C_5h)^{n}\Vert Y_{0}\Vert +\frac{(1+C_5h)^{n}-1}{C_5h}C_6h^{m+1}, \end{aligned}$$

and we can then show that there exists a constant $C_7$ such that,

$$\begin{aligned} \begin{aligned} \Vert \varepsilon _{n}\Vert _1\le C_7h^{m}. \end{aligned} \end{aligned}$$

(4.18)

By (4.6), and as in Case I, there exists hence a constant $C_8$ so that

$$\begin{aligned} \begin{aligned} |e_h(t_n)|\le C_8h^{m+1}. \end{aligned} \end{aligned}$$

(4.19)

Case III $\rho _m=1$

Here, the eigenvalues of G defined in Case II are $\underbrace{1, 1,\ldots , 1}_m; 1, \underbrace{0, \ldots , 0}_{m-1}$, where now the eigenvalue 1 of multiplicity $m+1$ also has m linearly independent eigenvectors. This means that G is not diagonalizable, but there exists a nonsingular matrix Q, such that

$$\begin{aligned}Q^{-1}GQ=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 &{} 1 &{} &{} &{} &{} &{} \\ &{} 1 &{} &{} &{} &{} &{} \\ &{} &{} \ddots &{} &{} &{} &{} \\ &{} &{} &{} 1 &{} &{} &{} \\ &{} &{} &{} &{} 0 &{} &{} \\ &{} &{} &{} &{} &{} \ddots &{} \\ &{} &{} &{} &{} &{} &{}0\\ \end{array} \right) . \end{aligned}$$

Defining $\bar{\varLambda }:=Q^{-1}GQ, \bar{Y}_n:=Q^{-1}X_n$ and recalling (4.16) we obtain

$$\begin{aligned} \bar{Y}_n=(\bar{\varLambda }+O(h))\bar{Y}_{n-1}+h^{m+1}Q^{-1}\bar{r}_{m,{n}}. \end{aligned}$$

(4.20)

An induction argument yields

$$\begin{aligned} \bar{Y}_n=(\bar{\varLambda }+O(h))^n\bar{Y}_{0}+h^{m+1}\sum _{l=0}^{n-1}(\bar{\varLambda }+O(h))^{l}Q^{-1}\bar{r}_{m,{n-l}}, \end{aligned}$$

and thus there exist constants $C_9, C_{10}$ such that

$$\begin{aligned} \Vert \bar{Y}_n\Vert \le C_9\Vert \bar{\varLambda }^n\Vert \Vert \bar{Y}_{0}\Vert +h^{m+1}C_{10}\sum _{l=0}^{n-1}\Vert \bar{\varLambda }^{l}\Vert . \end{aligned}$$

It is easily to check that

$$\begin{aligned} \bar{\varLambda }^n=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 &{} n &{} &{} &{} &{} &{} \\ &{} 1 &{} &{} &{} &{} &{} \\ &{} &{} \ddots &{} &{} &{} &{} \\ &{} &{} &{} 1 &{} &{} &{} \\ &{} &{} &{} &{} 0 &{} &{} \\ &{} &{} &{} &{} &{} \ddots &{} \\ &{} &{} &{} &{} &{} &{} 0\\ \end{array} \right) . \end{aligned}$$

Therefore, there exists a constant $C_{11}$ such that

$$\begin{aligned} \begin{aligned} \Vert \varepsilon _{n}\Vert _1\le C_{11}h^{m-1}, \end{aligned} \end{aligned}$$

(4.21)

and an argument analogous to the one employed in the analysis of Case I shows that there exists a constant $C_{12}$ such that

$$\begin{aligned} \begin{aligned} |e_h(t_n)|\le C_{12}h^{m}. \end{aligned} \end{aligned}$$

(4.22)

Obviously, the collocation solution $u_h$ is divergent if $|\rho _m|>1$. The proof is completed by recalling (2.3), (4.1) and (4.3).

In the following, we prove the results for general, non-constant kernels. Now, by (4.4), (1.1)–(2.2) and using (4.3), we obtain

$$\begin{aligned} e_h(t_{n,i})= & {} e_h(t_n)+h\sum _{j=1}^ma_{ij}\varepsilon _{n,j}+h^{m+1}R_{m,n}(c_i)\nonumber \\= & {} \int _0^{t_{n,i}}K(t_{n,i}, s)e_h(s)ds\nonumber \\= & {} h\sum _{l=0}^{n-1}\int _0^{1}K(t_{n,i}, t_l+sh)e_h(t_l+sh)ds+h\int _0^{c_i}K(t_{n,i}, t_n+sh)e_h(t_n+sh)ds\nonumber \\= & {} h\sum _{l=0}^{n-1}\int _0^{1}K(t_{n,i}, t_l+sh) \left[ e_h(t_l)+h\sum _{j=1}^m\beta _j(s)\varepsilon _{l,j}\right] ds\nonumber \\&+\,h\int _0^{c_i}K(t_{n,i}, t_n+sh)\left[ e_h(t_n)+h\sum _{j=1}^m\beta _j(s)\varepsilon _{n,j}\right] ds\nonumber \\&+\, h^{m+1}\sum _{l=0}^{n-1}h\int _0^{1}K(t_{n,i}, t_l+sh)R_{m,l}(s)ds\nonumber \\&+\,h^{m+2}\int _0^{c_i}K(t_{n,i}, t_n+sh)R_{m,n}(s)ds\nonumber \\= & {} h\sum _{l=0}^{n-1}\int _0^{1}K(t_{n,i}, t_l+sh)ds e_h(t_l)+h^2\sum _{l=0}^{n-1}\sum _{j=1}^m\int _0^{1}K(t_{n,i}, t_l+sh)\beta _j(s)ds\varepsilon _{l,j}\nonumber \\&+\,h\int _0^{c_i}K(t_{n,i}, t_n+sh)dse_h(t_n)+h^2\sum _{j=1}^m\int _0^{c_i}K(t_{n,i}, t_n+sh)\beta _j(s)ds\varepsilon _{n,j}\nonumber \\&+\, h^{m+1}\sum _{l=0}^{n-1}h\int _0^{1}K(t_{n,i}, t_l +sh)R_{m,l}(s)ds+h^{m+2}\int _0^{c_i}K(t_{n,i}, t_n+sh)R_{m,n}(s)ds. \end{aligned}$$

(4.23)

By the standard technique used by Brunner (see [2]), rewriting (4.23) with n replaced by $n-1$ and with $i= m$ and subtract it from (4.23), we find

$$\begin{aligned}&e_h(t_{n})-e_h(t_{n-1})+h\sum _{j=1}^ma_{ij}\varepsilon _{{n},j} -h\sum _{j=1}^ma_{mj}\varepsilon _{{n-1},j}\\&\quad =h\int _0^{1}K(t_{n,i}, t_{n-1}+sh)ds e_h(t_{n-1})\\&\qquad +\,h^2(c_i+1-c_m)\sum _{l=0}^{n-2}\int _0^{1}K'_1(\xi _{n,i}, t_l+sh)ds e_h(t_l)\nonumber \\&\qquad +\, h^2\sum _{j=1}^m\int _0^{1}K(t_{n,i}, t_{n-1}+sh)\beta _j(s)ds\varepsilon _{{n-1},j}\nonumber \\&\qquad +\, h^3(c_i+1-c_m)\sum _{l=0}^{n-2}\sum _{j=1}^m\int _0^{1}K'_1(\xi _{n,i}, t_l+sh)\beta _j(s)ds\varepsilon _{l,j}\nonumber \\&\qquad +\, h\int _0^{c_i}K(t_{n,i}, t_n+sh)dse_h(t_n)-h\int _0^{c_m}K(t_{n-1,m}, t_{n-1}+sh)ds e_h(t_{n-1})\nonumber \\&\qquad +\, h^2\sum _{j=1}^m\int _0^{c_i}K(t_{n,i}, t_n+sh)\beta _j(s)ds\varepsilon _{n,j}\\&\qquad -\,h^2\sum _{j=1}^m\int _0^{c_m}K(t_{n-1,m}, t_{n-1}+sh)\beta _j(s)ds\varepsilon _{n-1,j}\nonumber \\&\qquad +\, h^{m+1}\tilde{R}_{m,n}(c_i), \end{aligned}$$

where $\tilde{R}_{m,n}(c_i):=-R_{m,{n}}(c_i)+R_{m,{n-1}}(c_m)+h\int _0^{1}K(t_{n,i}, t_{n-1}+sh)R_{m,{n-1}}(s)ds +\sum \nolimits _{l=0}^{n-2}h\int _0^{1}[K(t_{n,i}, t_l+sh)-K(t_{n-1,m}, t_l+sh)]R_{m,l}(s)ds +h\int _0^{c_i}K(t_{n,i}, t_n+sh)R_{m,n}(s)ds-h\int _0^{c_m}K(t_{n-1,m}, t_{n-1}+sh)R_{m,n-1}(s)ds$, $\xi _{{n,i}}\in (t_{n-1,m}, t_{n,i})$.

This can be written in the more concise form

$$\begin{aligned}&\left( e_h(t_{n})-e_h(t_{n-1})\right) e+hA\varepsilon _{n} -hee_m^TA\varepsilon _{n-1}\nonumber \\&\quad =hC_n^{(n-1)}ee_h(t_{n-1})+h^2\left( C+(1-c_m)I_m\right) \sum _{l=0}^{n-2}\bar{C}_n^{(l)}ee_h(t_l)+h^2B_n^{(n-1)}\varepsilon _{n-1} \nonumber \\&\qquad +\,h^3\left( C+(1-c_m)I_m\right) \sum _{l=0}^{n-2}\bar{B}_n^{l}\varepsilon _{l}+hC_nee_h(t_n) -hee_m^TC_{n-1}ee_h(t_{n-1})\nonumber \\&\qquad +\,h^2B_n\varepsilon _{n}- h^2ee_m^TB_{n-1}\varepsilon _{n-1} +h^{m+1}R^{(1)}_{m,{n}}, \end{aligned}$$

(4.24)

with obvious meaning of $R^{(1)}_{m,{n}}$, and with $\bar{C}_n^{(l)}:=diag (\int _0^{1}K'_1(\xi _{n,i}, t_l+sh)ds)\; (0\le l\le N-1)$ and $\bar{B}_n^{(l)}:= (\int _0^{1}K'_1(\xi _{n,i}, t_l+sh)\beta _j(s)ds)\; (0\le l\le N-1)$.

Substituting (4.8) into (4.24), we have

$$\begin{aligned}\begin{aligned}&eb^T\varepsilon _{n-1}+A\varepsilon _{n} -ee_m^TA\varepsilon _{n-1}\\&\quad =hC_n^{n-1}eb^T\sum _{l=0}^{n-2}\varepsilon _{l} +h^2\left( C+(1-c_m)I_m\right) \sum _{l=0}^{n-2}\bar{C}_n^{(l)}e\sum _{k=0}^{l-1}\varepsilon _{k}+hB_n^{(n-1)}\varepsilon _{n-1}\\&\qquad +h^2\left( C+(1-c_m)I_m\right) \sum _{l=0}^{n-2}\bar{B}_n^{l}\varepsilon _{l}+ hC_neb^T\sum _{l=0}^{n-1}\varepsilon _{l} -hee_m^TC_{n-1}eeb^T\sum _{l=0}^{n-2}\varepsilon _{l}\\&\qquad +hB_n\varepsilon _{n}- hee_m^TB_{n-1}\varepsilon _{n-1} +h^{m}\bar{\tilde{R}}_{m,{n}}, \end{aligned} \end{aligned}$$

with obvious meaning of $\bar{\tilde{R}}_{m,{n}}$. This equation can be written in the form

$$\begin{aligned}&\left( A-hB_n\right) \varepsilon _{n}\nonumber \\&\quad =\left( ee_m^TA-eb^T+hB_n^{(n-1)}- hee_m^TB_{n-1}\right) \varepsilon _{n-1}+hC_n^{n-1}eb^T\sum _{l=0}^{n-2}\varepsilon _{l}\nonumber \\&\quad \quad +\,h^2\left( C+(1-c_m)I_m\right) \sum _{l=0}^{n-2}\bar{C}_n^{(l)}e\sum _{k=0}^{l-1}\varepsilon _{k} +h^2\left( C+(1-c_m)I_m\right) \sum _{l=0}^{n-2}\bar{B}_n^{l}\varepsilon _{l}\nonumber \\&\quad \quad +\,hC_neb^T\sum _{l=0}^{n-1}\varepsilon _{l} -hee_m^TC_{n-1}eeb^T\sum _{l=0}^{n-2}\varepsilon _{l} +h^{m}\bar{\tilde{R}}_{m,{n}}, \end{aligned}$$

(4.25)

or

$$\begin{aligned} \begin{aligned} \varepsilon _{n} =&\left( A^{-1}\left( ee_m^TA-eb^T\right) +O(h)\right) \varepsilon _{n-1}+h \tilde{D}_n\sum _{l=0}^{n-1}\varepsilon _{l} +h^{m}\bar{\bar{R}}_{m,{n}}, \end{aligned} \end{aligned}$$

(4.26)

with obvious meaning of $\bar{\bar{R}}_{m,{n}}$ and $\tilde{D}_n$.

Comparison (4.10) of the case $K(t,s)\equiv 1$ with (4.26), and similar to the proof of the case $K(t,s)\equiv 1$, we can obtain now (4.11) becomes

$$\begin{aligned} \begin{aligned} Z_{n} =&(F+O(h))Z_{n-1}+h T^{-1}\tilde{D}_nT\sum _{l=0}^{n-1}Z_{l} +h^{m}T^{-1}\bar{\bar{R}}_{m,{n}}. \end{aligned} \end{aligned}$$

(4.27)

We also consider the following three cases:

Case I $-1< \rho _m<1$

By the same technique of the case $K(t,s)\equiv 1$, we can prove that there exists a constant $\tilde{C}_{4}$, such that

$$\begin{aligned} \begin{aligned} |e_h(t_n)|\le \tilde{C}_{4}h^{m+1} \quad (n = 1, \ldots , N). \end{aligned} \end{aligned}$$

(4.28)

Case II $\rho _m=-1$

Rewriting (4.27) with n replaced by $n-1$ and subtract it from (4.27), we find

$$\begin{aligned} Z_{n}-Z_{n-1}= & {} (F+O(h))(Z_{n-1}-Z_{n-2})+hT^{-1}\tilde{D}_nTZ_{n-1}\nonumber \\&+\,hT^{-1}\left( \tilde{D}_{n}-\tilde{D}_{n-1}\right) T\sum _{l=0}^{n-2}Z_{l}\nonumber \\&+\,h^{m}T^{-1}(\bar{\bar{R}}_{m,{n}}-\bar{\bar{R}}_{m,{n-1}}). \end{aligned}$$

(4.29)

Notice that $\tilde{D}_{n}-\tilde{D}_{n-1}=O(h)$, therefore,

$$\begin{aligned} \begin{aligned} \left( \begin{array}{c} Z_{n} \\ Z_{n-1} \\ \end{array} \right)&=\left( \begin{array}{c@{\quad }c} I_m+F+O(h) &{} -F+O(h)\\ I_m &{} 0 \\ \end{array} \right) \left( \begin{array}{c} Z_{n-1} \\ Z_{n-2} \\ \end{array} \right) \\&\quad +\sum _{l=1}^{n-2}\left( \begin{array}{c@{\quad }c} O(h^2) &{} 0\\ 0 &{} 0 \\ \end{array} \right) \left( \begin{array}{c} Z_{l} \\ Z_{l-1} \\ \end{array} \right) +\left( \begin{array}{c} h^{m}T^{-1}\left( \bar{\bar{R}}_{m,{n}}-\bar{\bar{R}}_{m,{n-1}}\right) \\ 0 \\ \end{array} \right) . \end{aligned} \end{aligned}$$

Now (4.17) becomes

$$\begin{aligned} Y_n=(\varLambda +O(h))Y_{n-1}+O(h^2)\sum _{l=1}^{n-2}Y_l+O(h^{m+1}). \end{aligned}$$

(4.30)

Similar to the case of $K(t,s)\equiv 1$, we can assert that there exist constants $\tilde{C}_5, \tilde{C}'_5, \tilde{C}_6$ so that

$$\begin{aligned} \Vert Y_n\Vert \le (1+\tilde{C}_5h)\Vert Y_{n-1}\Vert +\tilde{C}'_5h^2\sum _{l=1}^{n-2}\Vert Y_l\Vert +\tilde{C}_6h^{m+1}. \end{aligned}$$

An induction argument then leads to

$$\begin{aligned} \Vert Y_n\Vert \le (1+\tilde{C}_5h)^n\Vert Y_{0}\Vert +\tilde{C}'_5h^2\dfrac{(1+\tilde{C}_5h)^n-1}{\tilde{C}_5h}\sum _{l=1}^{n-2}\Vert Y_l\Vert +\,\tilde{C}_6\dfrac{(1+\tilde{C}_5h)^n-1}{\tilde{C}_5h}h^{m+1}. \end{aligned}$$

Therefore, by the discrete Gronwall inequality (see [2]), we can get that there exists a constant $\tilde{C}_7$ such that,

$$\begin{aligned} \begin{aligned} \Vert \varepsilon _{n}\Vert _1\le \tilde{C}_7h^{m}, \end{aligned} \end{aligned}$$

(4.31)

and similar to the case of $K(t,s)\equiv 1$, we can then show that there exists a constant $\tilde{C}_8$ so that

$$\begin{aligned} \begin{aligned} |e_h(t_n)|\le \tilde{C}_8h^{m+1}. \end{aligned} \end{aligned}$$

(4.32)

Case III $\rho _m=1$

Using the technique of [7], we write the collocation approximation $u_h$ and the exact solution in the form

$$\begin{aligned} u_h(t_n+sh)=\sum _{j=1}^mL_j(s)u_h(t_{n,j})+h^m\frac{u_h^{(m)}(\eta _n)}{m!}\prod _{i=1}^m(s-c_i), \end{aligned}$$

(4.33)

and

$$\begin{aligned} u(t_n+sh)=\sum _{j=1}^mL_j(s)u(t_{n,j})+h^m\frac{u^{(m)}(\eta '_n)}{m!}\prod _{i=1}^m(s-c_i), \end{aligned}$$

(4.34)

where $\eta _n, \eta '_n\in (t_n, t_{n+1})$.

So (4.34)–(4.33) yields

$$\begin{aligned} e_h((t_n+sh)=\sum _{j=1}^mL_j(s)e_h(t_{n,j})+h^m\hat{R}_{n}(s), \end{aligned}$$

(4.35)

where $\hat{R}_{n}(s):=\frac{u^{(m)}(\eta '_n)-u_h^{(m)}(\eta _n)}{m!}\prod \limits _{i=1}^m(s-c_i)$.

Now, by (1.1)–(2.2) and using (4.35), we obtain

$$\begin{aligned} e_h(t_{n,i})= & {} h\sum _{l=0}^{n-1}\int _0^{1}K(t_{n,i}, t_l+sh)e_h(t_l+sh)ds\nonumber \\&+\,h\int _0^{c_i}K(t_{n,i}, t_n+sh)e_h(t_n+sh)ds\nonumber \\= & {} h\sum _{l=0}^{n-1}\int _0^{1}K(t_{n,i}, t_l+sh)\sum _{j=1}^mL_j(s)ds e_h(t_{l,j})\nonumber \\&+\,h\int _0^{c_i}K(t_{n,i}, t_n+sh)\sum _{j=1}^mL_j(s)ds e_h(t_{n,j})\nonumber \\&+\,h^m\bar{\hat{R}}_{n}(s), \end{aligned}$$

(4.36)

with obvious meanings of $\bar{\hat{R}}_{n}(s)$.

Rewriting (4.36) with n replaced by $n-1$ and $i=m$ and subtract it from (4.36), we can get

$$\begin{aligned}&e_h(t_{n,i})-e_h(t_{n-1,m})\nonumber \\&\quad =h\int _0^{1}K(t_{n,i}, t_{n-1}+sh)\sum _{j=1}^mL_j(s)ds e_h(t_{n-1,j})\nonumber \\&\quad \quad +\, h^2(c_i+1-c_m)\sum _{l=0}^{n-2}\int _0^{1}K_1'(\xi _{n,i}, t_l+sh)\sum _{j=1}^mL_j(s)ds e_h(t_{l,j})\nonumber \\&\quad \quad +\, h\int _0^{c_i}K(t_{n,i}, t_n+sh)\sum _{j=1}^mL_j(s)ds e_h(t_{n,j})\nonumber \\&\quad \quad -\, h\int _0^{c_m}K(t_{n-1,m}, t_{n-1}+sh)\sum _{j=1}^mL_j(s)ds e_h(t_{n-1,j})\nonumber \\&\quad \quad + \,h^m\bar{\hat{R}}_{n}(s)-h^m\bar{\hat{R}}_{n-1}(s). \end{aligned}$$

(4.37)

Denoting $E_n:=(e_h(t_{n,1}), \ldots , e_h(t_{n,m}))^T$ and noticing that $\bar{\hat{R}}_{n}(s)-\bar{\hat{R}}_{n-1}(s)=O(h)$, we can rewrite (4.37) as the more concise form

$$\begin{aligned} \begin{aligned} E_n-ee_m^TE_{n-1} =O(h)E_{n-1}+O(h^2)\sum _{l=0}^{n-2}E_l +O(h)E_n+O(h^{m+1}). \end{aligned} \end{aligned}$$

(4.38)

Since

$$\begin{aligned} ee_m^T=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} \cdots &{} 0 &{} 1 \\ 0 &{} \cdots &{} 0 &{} 1 \\ \cdots &{} \cdots &{} \cdots &{} \cdots \\ 0 &{} \cdots &{} 0 &{} 1 \\ \end{array} \right) , \end{aligned}$$

the rank of $ee_m^T$ is 1, and the unique nonzero eigenvalue is 1, so $ee_m^T$ is diagonalizable and there exists a nonsingular matrix $\tilde{T}$, such that

$$\begin{aligned} \tilde{T}^{-1}ee_m^T\tilde{T}=:\tilde{F}=diag(1,\underbrace{0,\ldots ,0}_{m-1}). \end{aligned}$$

Denote $\tilde{E}_n=\tilde{T}^{-1}E_n$. Then (4.38) becomes

$$\begin{aligned} \begin{aligned} \tilde{E}_n =&(\tilde{F}+O(h))\tilde{E}_{n-1}+O(h^2)\sum _{l=0}^{n-2}\tilde{E}_l+O(h^{m+1}). \end{aligned} \end{aligned}$$

Therefore, there exist constants $\bar{C}_5, \bar{C}'_5$ and $\bar{C}_6$, such that

$$\begin{aligned} \begin{aligned} \Vert \tilde{E}_n\Vert \le (1+\bar{C}_5h)\Vert \tilde{E}_{n-1}\Vert +\bar{C}'_5h^2\sum _{l=0}^{n-2}\Vert \tilde{E}_l\Vert +\bar{C}_6h^{m+1}. \end{aligned} \end{aligned}$$

(4.39)

Similar to Case II, we can get that there exists a constant $\bar{C}_7$ such that

$$\begin{aligned} \Vert E_n\Vert \le \bar{C}_7 h^m. \end{aligned}$$

By (4.35), we can then get there exists a constant $\bar{C}_8$ such that

$$\begin{aligned} |e_h(t_n+sh)|\le \bar{C}_8h^m. \end{aligned}$$

Obviously, the collocation solution $u_h$ is divergent also if $|\rho _m|>1$. The proof is completed by recalling (2.3), (4.1) and (4.3).

5 Proof of Theorems 3.1

On $\sigma _n^{(\mu )}:=(t_n^{(\mu )}, t_{n+1}^{(\mu )}]$, the derivative $u_h'$ of the collocation solution has the local Lagrange representation,

$$\begin{aligned} u'_h(t_n^{(\mu )}+sh)=\sum _{j=1}^mL_j(s)U_{n,j}^{(\mu )} ,\ \ s\in (0, 1], \end{aligned}$$

(5.1)

where $U_{n,i}^{(\mu )}:=u_h'(t_{n,i}^{(\mu )})$. Upon integration of (5.1) we obtain

$$\begin{aligned} u_h(t_n^{(\mu )}+sh)=u_h(t_n^{(\mu )})+h\sum _{j=1}^m\beta _j(s)U_{n,j}^{(\mu )} ,\ \ s\in [0, 1]. \end{aligned}$$

(5.2)

Since on each subinterval $\sigma _{n}^{(\mu )}$ the exact solution of the delay VIE (1.2) is in $C^{m+2}$, we may write

$$\begin{aligned} u'(t_n^{(\mu )}+sh)=\sum _{j=1}^mL_j(s)u'(t_{n,j}^{(\mu )})+h^mR_{m,n}^{(1,\mu )}(s) ,\quad s\in (0, 1], \end{aligned}$$

(5.3)

where the Peano remainder term is given by

$$\begin{aligned} R_{m,n}^{(1,\mu )}(v):=\int _0^1K_m(v,z)u^{(m+1)}(t_n^{(\mu )}+zh)dz. \end{aligned}$$

Integration of (5.3) leads to

$$\begin{aligned} u(t_n^{(\mu )}+sh_n)=u(t_n^{(\mu )})+h\sum _{j=1}^m\beta _j(s)u'(t_{n,j}^{(\mu )}) +h^{m+1}R^{(\mu )}_{m,n}(s), \quad s\in [0,1], \end{aligned}$$

(5.4)

with $R^{(\mu )}_{m,n}(s):=\int _0^sR_{m,n}^{(1, \mu )}(v)dv$.

For ease of notation we will again assume that $K(t,s)\equiv 1$, and we can extend to the proof to the non-constant kernel by the same technique as the proof of Theorem 2.1.

By (5.2) and (5.4), the collocation error $e_h:=u-u_h$ on $\bar{\sigma _n}^{(\mu )}:=[t_n^{(\mu )}, t_{n+1}^{(\mu )}]$ can be written as

$$\begin{aligned} e_h(t_n^{(\mu )}+sh)=e_h(t_n^{(\mu )})+h\sum _{j=1}^m\beta _j(s)\varepsilon _{n,j}^{(\mu )}+h^{m+1}R_{m,n}^{(\mu )}(s), \end{aligned}$$

(5.5)

where $\varepsilon _{n,i}^{(\mu )}=u'(t_{n,i}^{(\mu )})-u_h'(t_{n,i}^{(\mu )})$. Particularly,

$$\begin{aligned} e_h(t_{n,i}^{(\mu )})=e_h(t_n^{(\mu )})+h\sum _{j=1}^ma_{ij}\varepsilon _{n,j}^{(\mu )}+h^{m+1}R_{m,n}^{(\mu )}(c_i). \end{aligned}$$

(5.6)

By (1.2)–(3.3) and using (5.5), we have

$$\begin{aligned} e_h(t_{n,i}^{(\mu )})= & {} \int _{t_{n,i}^{(\mu -1)}}^{t_{n,i}^{(\mu )}}e_h(s)ds =h\int _{c_i}^1e_h(t_n^{(\mu -1)}+sh)ds +h\sum _{l=n+1}^{N-1}\int _0^1e_h(t_l^{(\mu -1)}+sh)ds\nonumber \\&+\,h\sum _{l=0}^{n-1}\int _0^1e_h(t_l^{(\mu )}+sh)ds +h\int _0^{c_i}e_h(t_n^{(\mu )}+sh)ds\nonumber \\= & {} h\int _{c_i}^1\left[ e_h(t_n^{(\mu -1)}) +h\sum _{j=1}^m\beta _j(s)\varepsilon _{n,j}^{(\mu -1)}\right] ds\nonumber \\&+\,h\sum _{l=n+1}^{N-1}\int _0^1\left[ e_h(t_l^{(\mu -1)}) +h\sum _{j=1}^m\beta _j(s)\varepsilon _{l,j}^{(\mu -1)}\right] ds\nonumber \\&+\,h\sum _{l=0}^{n-1}\int _0^{1} \left[ e_h(t_l^{(\mu )})+h\sum _{j=1}^m\beta _j(s)\varepsilon _{l,j}^{(\mu )}\right] ds\nonumber \\&+\,h\int _0^{c_i}\left[ e_h(t_n^{(\mu )})+h\sum _{j=1}^m\beta _j(s)\varepsilon _{n,j}^{(\mu )}\right] ds\nonumber \\&+\,h^{m+1}\sum _{l=n+1}^{N-1}h\int _0^{1}R^{(\mu -1)}_{m,l}(s)ds+h^{m+2}\int _{c_i}^1R^{(\mu -1)}_{m,n}(s)ds\nonumber \\&+\, h^{m+1}\sum _{l=0}^{n-1}h\int _0^{1}R^{(\mu )}_{m,l}(s)ds+h^{m+2}\int _0^{c_i}R^{(\mu )}_{m,n}(s)ds\nonumber \\ \end{aligned}$$

$$\begin{aligned}= & {} h(1-c_i)e_h(t_n^{(\mu -1)})+h^2\sum _{j=1}^m(\gamma _j(1)-b_{ij})\varepsilon _{n,j}^{(\mu -1)} +h\sum _{l=n+1}^{N-1}e_h(t_l^{(\mu -1)})\nonumber \\&+\,h^2\sum _{l=n+1}^{N-1}\sum _{j=1}^m\gamma _j(1)\varepsilon _{l,j}^{(\mu -1)} +h\sum _{l=0}^{n-1}e_h(t_l^{(\mu )})\nonumber \\&+\,h^2\sum _{l=0}^{n-1}\sum _{j=1}^m\gamma _j(1)\varepsilon _{l,j}^{(\mu )} +hc_ie_h(t_n^{(\mu )})\nonumber \\&+\,h^2\sum _{j=1}^mb_{ij}\varepsilon _{n,j}^{(\mu )} +h^{m+1}\tilde{R}_{m,n}^{(\mu )}(c_i), \end{aligned}$$

(5.7)

where $\tilde{R}_{m,n}^{(\mu )}(c_i) = \sum \nolimits _{l=n+1}^{N-1}h\int _0^{1}R^{(\mu -1)}_{m,l}(s)ds +h\int _{c_i}^1R^{(\mu -1)}_{m,n}(s)ds +\sum \nolimits _{l=0}^{n-1}h\int _0^{1} R_{m,l}^{(\mu )}(s)ds +h\int _0^{c_i}R_{m,n}^{(\mu )}(s)ds.$ Thus, by (5.5) and (5.7), we have

$$\begin{aligned}&e_h(t_n^{(\mu )})+h\sum _{j=1}^ma_{ij}\varepsilon _{n,j}^{(\mu )} +h^{m+1}R^{(\mu )}_{m,n}(c_i)\nonumber \\&\quad =h(1-c_i)e_h(t_n^{(\mu -1)})+h^2\sum _{j=1}^m(\gamma _j(1)-b_{ij})\varepsilon _{n,j}^{(\mu -1)} +h\sum _{l=n+1}^{N-1}e_h(t_l^{(\mu -1)})\nonumber \\&\qquad +\,h^2\sum _{l=n+1}^{N-1}\sum _{j=1}^m\gamma _j(1)\varepsilon _{l,j}^{(\mu -1)} +h\sum _{l=0}^{n-1}e_h(t_l^{(\mu )}) +h^2\sum _{l=0}^{n-1}\sum _{j=1}^m\gamma _j(1)\varepsilon _{l,j}^{(\mu )} \nonumber \\&\qquad +\,hc_ie_h(t_n^{(\mu )})+h^2\sum _{j=1}^mb_{ij}\varepsilon _{n,j}^{(\mu )} +h^{m+1}\tilde{R}_{m,n}^{(\mu )}(c_i). \end{aligned}$$

(5.8)

Rewriting (5.8) with n replaced by $n-1$ and with $i=m$ and subtract it from (5.8), we can obtain

$$\begin{aligned}&e_h(t_n^{(\mu )})-e_h(t_{n-1}^{(\mu )})+h\sum _{j=1}^ma_{ij}\varepsilon _{n,j}^{(\mu )}- h\sum _{j=1}^ma_{mj}\varepsilon _{n-1,j}^{(\mu )}\nonumber \\&\qquad +\,h^{m+1}R^{(\mu )}_{m,n}(c_i)-h^{m+1}R^{(\mu )}_{m,n-1}(c_m)\nonumber \\&\quad =h(1-c_i)e_h(t_n^{(\mu -1)})-h(1-c_m)e_h(t_{n-1}^{(\mu -1)}) +h^2\sum _{j=1}^m(\gamma _j(1)-b_{ij})\varepsilon _{n,j}^{(\mu -1)}\nonumber \\&\qquad -\,h^2\sum _{j=1}^m(\gamma _j(1)-b_{mj})\varepsilon _{n-1,j}^{(\mu -1)} -he_h(t_n^{(\mu -1)})-h^2\sum _{j=1}^m\gamma _j(1)\varepsilon _{n,j}^{(\mu -1)}\nonumber \\&\qquad +\,he_h(t_{n-1}^{(\mu )}) +h^2\sum _{j=1}^m\gamma _j(1)\varepsilon _{{n-1},j}^{(\mu )} +hc_ie_h(t_n^{(\mu )})-hc_me_h(t_{n-1}^{(\mu )}) \nonumber \\&\qquad +\,h^2\sum _{j=1}^mb_{ij}\varepsilon _{n,j}^{(\mu )}-h^2\sum _{j=1}^mb_{mj}\varepsilon _{n-1,j}^{(\mu )} +h^{m+1}\tilde{R}_{m,n}^{(\mu )}(c_i) -h^{m+1}\tilde{R}_{m,n-1}^{(\mu )}(c_m). \end{aligned}$$

(5.9)

By (5.5), the continuity of $e_h$ on I, and $e_h(0)=0$, we find by induction

$$\begin{aligned} e_h(t_n^{(\mu )})= & {} e_h(t_{n-1}^{(\mu )}+h) =e_h(t_{n-1}^{(\mu )})+h\sum _{j=1}^mb_j\varepsilon _{n-1,j}^{(\mu )}+h^{m+1}R^{(\mu )}_{m,n-1}(1)\nonumber \\= & {} h\sum _{\nu =1}^{\mu }\sum _{l=0}^{N-1}b^T\varepsilon _{l}^{(\nu -1)} +h^{m+1}\sum _{\nu =1}^{\mu }\sum _{l=0}^{N-1}R_{m,l}^{(\nu -1)}(1) +h\sum _{l=0}^{n-1}b^T\varepsilon _{l}^{(\mu )}\nonumber \\&+\,h^{m+1}\sum _{l=0}^{n-1}R_{m,l}^{(\mu )}(1). \end{aligned}$$

(5.10)

For $\mu =0$, it follows from (5.9) and (5.10) that we can write the error equation in the form

$$\begin{aligned}&e\left( b^T\varepsilon _{n-1}^{(0)}+h^{m}R^{(0)}_{m,n-1}(1)\right) +A\varepsilon _{n}^{(0)}- ee_m^TA\varepsilon _{n-1}^{(0)}\\&\quad =(1-c_m)e\left( h\sum _{l=0}^{n-2}b^T\varepsilon _{l}^{(0)} +h^{m+1}\sum _{l=0}^{n-2}R_{m,l}^{(0)}(1)\right) +he\gamma ^T\varepsilon _{n-1}^{(0)}\\&\qquad +\,Ce\left( h\sum _{l=0}^{n-1}b^T\varepsilon _{l}^{(0)} +h^{m}\sum _{l=0}^{n-1}R_{m,l}^{(0)}(1)\right) +hB\varepsilon _{n}^{(0)} -hee_m^TB\varepsilon _{n-1}^{(0)} +h^{m}R_{m,n}^{(0)}, \end{aligned}$$

or

$$\begin{aligned} \begin{aligned} \left( A-hB\right) \varepsilon _{n}^{(0)}&=\left( ee_m^TA-eb^T+he\gamma ^T-hee_m^TB\right) \varepsilon _{n-1}^{(0)}\\&\quad +(1-c_m)eh\sum _{l=0}^{n-2}b^T\varepsilon _{l}^{(0)} +Ceh\sum _{l=0}^{n-1}b^T\varepsilon _{l}^{(0)} +h^{m}\tilde{R}_{m,n}^{(0)}, \end{aligned} \end{aligned}$$

with obvious meaning of $\tilde{R}_{m,n}^{(0)}$.

The proof of Theorem 2.1 reveals that $\varepsilon _{n}^{(0)}$ converges if, and only if

$$\begin{aligned} -1\le \rho _m=(-1)^m\prod _{i=1}^m \dfrac{1-c_i}{c_i}\le 1, \end{aligned}$$

and that there exist constant $C_{1}^{(0)}$, such that

$$\begin{aligned} \begin{aligned} \Vert \varepsilon _{n}^{(0)}\Vert _1\le C_{1}^{(0)}\left\{ \begin{array}{l@{\quad }l} h^m, &{}\; \hbox {if }\; -1\le \rho _m< 1;\\ h^{m-1}, &{} \; \hbox {if }\; \rho _m= 1. \end{array} \right. \end{aligned} \end{aligned}$$

(5.11)

Equation (5.8) implies that there exist constants $C_{2}^{(0)}$ and $C_{3}^{(0)}$ such that

$$\begin{aligned} \begin{aligned} |e_h(t_n^{(0)})|\le hC_{2}^{(0)}\sum _{l=0}^{n-1}|e_h(t_l^{(0)})|+C_{3}^{(0)}\left\{ \begin{array}{l@{\quad }l} h^{m+1}, &{}\; \hbox {if }\; -1\le \rho _m< 1,\\ h^{m}, &{} \; \hbox {if }\; \rho _m= 1, \end{array} \right. \end{aligned} \end{aligned}$$

(5.12)

and thus the discrete Gronwall inequality (see [2]) guarantees the existence of a constant $C_{4}^{(0)}$ for which

$$\begin{aligned} \begin{aligned} |e_h(t_n^{(0)})|\le C_{4}^{(0)}\left\{ \begin{array}{l@{\quad }l} h^{m+1}, &{}\; \hbox {if }\; -1\le \rho _m< 1,\\ h^{m}, &{} \; \hbox {if }\; \rho _m= 1, \end{array} \right. \end{aligned} \end{aligned}$$

(5.13)

holds. Assume that for $\nu =1,\ldots , \mu -1$, $\varepsilon _{n}^{(\nu )}$ converges if, and only if

$$\begin{aligned} -1\le \rho _m=(-1)^m\prod _{i=1}^m \dfrac{1-c_i}{c_i}\le 1, \end{aligned}$$

and that there exist constants $C_{1}^{(\nu )}$ and $C_{4}^{(\nu )}$ such that

$$\begin{aligned} \begin{aligned} \Vert \varepsilon _{n}^{(\nu )}\Vert _1\le C_{1}^{(\nu )}\left\{ \begin{array}{l@{\quad }l} h^{m}, &{}\; \hbox {if }\; -1\le \rho _m < 1,\\ h^{m-1}, &{} \; \hbox {if }\; \rho _m= 1, \end{array} \right. \end{aligned} \end{aligned}$$

(5.14)

and

$$\begin{aligned} \begin{aligned} |e_h(t_n^{(\nu )})|\le C_{4}^{(\nu )}\left\{ \begin{array}{l@{\quad }l} h^{m+1}, &{}\; \hbox {if }\; -1\le \rho _m< 1;\\ h^{m}, &{} \; \hbox {if }\; \rho _m= 1. \end{array} \right. \end{aligned} \end{aligned}$$

(5.15)

By (5.9) and (5.10), we have

$$\begin{aligned}&eb^T\varepsilon _{n-1}^{(\mu )} +A\varepsilon _{n}^{(\mu )}- ee_m^TA\varepsilon _{n-1}^{(\mu )}\nonumber \\&\quad =(I_m-C)ee_h(t_n^{(\mu -1)})-(1-c_m)ee_h(t_{n-1}^{(\mu -1)}) +h(e\gamma ^T-B)\varepsilon _{n}^{(\mu -1)}\nonumber \\&\qquad -\,h e(\gamma ^T-e_m^TB)\varepsilon _{n-1}^{(\mu -1)} -ee_h(t_n^{(\mu -1)})-he\gamma ^T\varepsilon _{n}^{(\mu -1)}\nonumber \\&\qquad +\,(1-c_m)e\left( h\sum _{\nu =1}^{\mu }\sum _{l=0}^{N-1}b^T\varepsilon _{l}^{(\nu -1)} +h\sum _{l=0}^{n-2}b^T\varepsilon _{l}^{(\mu )}\right) +he\gamma ^T\varepsilon _{n-1}^{(\mu )}\nonumber \\&\qquad +\,Ce\left( h\sum _{\nu =1}^{\mu }\sum _{l=0}^{N-1}b^T\varepsilon _{l}^{(\nu -1)} +h\sum _{l=0}^{n-1}b^T\varepsilon _{l}^{(\mu )}\right) +hB\varepsilon _{n}^{(\mu )}\nonumber \\&\qquad -\,hee_m^TB\varepsilon _{n-1}^{(\mu )} +h^{m}R_{m,n}^{(\mu )}, \end{aligned}$$

(5.16)

with obvious meaning of $R_{m,n}^{(\mu )}$.

In the remaining part of the proof we consider the following three cases.

Case I $-1 < \rho _m <1$

By assumption (5.14) and (5.15), we obtain from (5.16)

$$\begin{aligned} \begin{aligned} \left( A-hB\right) \varepsilon _{n}^{(\mu )}&=\left( ee_m^TA-eb^T+he\gamma ^T-hee_m^TB+h Ce b^T\right) \varepsilon _{n-1}^{(\mu )}\\&\quad +h\left( (1-c_m)e+Ce\right) \sum _{l=0}^{n-2}b^T\varepsilon _{l}^{(\mu )} +O(h^{m}). \end{aligned} \end{aligned}$$

Proceeding as in the proof of Theorem 2.1 we see that there exists a constant $C_{1}^{(\mu )}$ such that

$$\begin{aligned} \begin{aligned} \Vert \varepsilon _{n}^{(\mu )}\Vert _1\le C_{1}^{(\mu )}h^{m}, \end{aligned} \end{aligned}$$

(5.17)

and hence, by (5.8) and the discrete Gronwall lemma (see [2]), there exist constants $C_{2}^{(\mu )},\, C_{3}^{(\mu )}$ and $C_{4}^{(\mu )}$ for which

$$\begin{aligned} |e_h(t_n^{(\mu )})|\le hC_{2}^{(\mu )}\sum _{l=0}^{n-1}|e_h(t_l^{(\mu )})|+C_{3}^{(\mu )}h^{m+1} \end{aligned}$$

and

$$\begin{aligned} |e_h(t_n^{(\mu )})|\le C_{4}^{(\mu )}h^{m+1} \end{aligned}$$

are true.

Case II $\rho _m=-1$

Rewriting (5.16) with n replaced by $n-1$ and with $i=m$ and subtract it from (5.16), and by (5.10), we can obtain

$$\begin{aligned}&eb^T\varepsilon _{n-1}^{(\mu )}-eb^T\varepsilon _{n-2}^{(\mu )} +A\varepsilon _{n}^{(\mu )}-A\varepsilon _{n-1}^{(\mu )}- ee_m^TA\varepsilon _{n-1}^{(\mu )} + ee_m^TA\varepsilon _{n-2}^{(\mu )}\nonumber \\&\quad =(I_m-C)e\left( e_h(t_n^{(\mu -1)})-e_h(t_{n-1}^{(\mu -1)})\right) -(1-c_m)e\left( e_h(t_{n-1}^{(\mu -1)})-e_h(t_{n-2}^{(\mu -1)})\right) \nonumber \\&\qquad +\, h(e\gamma ^T-B)\left( \varepsilon _{n}^{(\mu -1)}-\varepsilon _{n-1}^{(\mu -1)}\right) -h e(\gamma ^T-e_m^TB)\left( \varepsilon _{n-1}^{(\mu -1)}-\varepsilon _{n-2}^{(\mu -1)}\right) \nonumber \\&\qquad -\,e\left( e_h(t_n^{(\mu -1)})-e_h(t_{n-1}^{(\mu -1)})\right) -he\gamma ^T\left( \varepsilon _{n}^{(\mu -1)}-\varepsilon _{n-1}^{(\mu -1)}\right) \nonumber \\&\qquad +\,h(1-c_m)eb^T\varepsilon _{n-2}^{(\mu )} +he\gamma ^T\left( \varepsilon _{n-1}^{(\mu )}-\varepsilon _{n-2}^{(\mu )}\right) +hCeb^T\varepsilon _{n-1}^{(\mu )}\nonumber \\&\qquad +\,hB\left( \varepsilon _{n}^{(\mu )}-\varepsilon _{n-1}^{(\mu )}\right) -hee_m^TB\left( \varepsilon _{n-1}^{(\mu )}-\varepsilon _{n-2}^{(\mu )}\right) +h^{m}\left( R_{m,n}^{(\mu )}-R_{m,n-1}^{(\mu )}\right) \nonumber \\&\quad =-Ce\left( e_h(t_n^{(\mu -1)})-e_h(t_{n-1}^{(\mu -1)})\right) -(1-c_m)e\left( e_h(t_{n-1}^{(\mu -1)})-e_h(t_{n-2}^{(\mu -1)})\right) \nonumber \\&\qquad -\,hB\left( \varepsilon _{n}^{(\mu -1)}-\varepsilon _{n-1}^{(\mu -1)}\right) -he(\gamma ^T-e_m^TB)\left( \varepsilon _{n-1}^{(\mu -1)}-\varepsilon _{n-2}^{(\mu -1)}\right) \nonumber \\&\qquad +\,h(1-c_m)eb^T\varepsilon _{n-2}^{(\mu )} +he\left( \gamma ^T-e_m^TB\right) \left( \varepsilon _{n-1}^{(\mu )}-\varepsilon _{n-2}^{(\mu )}\right) \nonumber \\&\qquad +\,hCeb^T\varepsilon _{n-1}^{(\mu )} +hB\left( \varepsilon _{n}^{(\mu )}-\varepsilon _{n-1}^{(\mu )}\right) +h^{m}\left( R_{m,n}^{(\mu )}-R_{m,n-1}^{(\mu )}\right) \nonumber \\&\quad =-hCeb^T\varepsilon _{n-1}^{(\mu -1)}-h(1-c_m)eb^T\varepsilon _{n-2}^{(\mu -1)} -hB\left( \varepsilon _{n}^{(\mu -1)}-\varepsilon _{n-1}^{(\mu -1)}\right) \nonumber \\&\qquad -\,he(\gamma ^T-e_m^TB)\left( \varepsilon _{n-1}^{(\mu -1)}-\varepsilon _{n-2}^{(\mu -1)}\right) +h(1-c_m)eb^T\varepsilon _{n-2}^{(\mu )}\nonumber \\&\qquad +\,he\left( \gamma ^T-e_m^TB\right) \left( \varepsilon _{n-1}^{(\mu )}-\varepsilon _{n-2}^{(\mu )}\right) +hCeb^T\varepsilon _{n-1}^{(\mu )}\nonumber \\&\qquad +\,hB\left( \varepsilon _{n}^{(\mu )}-\varepsilon _{n-1}^{(\mu )}\right) +h^{m}\bar{R}_{m,n}^{(\mu )}, \end{aligned}$$

(5.18)

with obvious meaning of $\bar{R}_{m,n}^{(\mu )}$. Therefore,

$$\begin{aligned} \begin{aligned}&\left( \begin{array}{c@{\quad }c} A-hB &{} 0 \\ 0 &{} I_m \\ \end{array} \right) \left( \begin{array}{c} \varepsilon _{n}^{(\mu )} \\ \varepsilon _{n-1}^{(\mu )} \\ \end{array} \right) \\&\quad =\left( \begin{array}{l@{\quad }l} A+e\left( e_m^TA-b^T\right) &{}e\left( b^T-e_m^TA\right) \\ \,\, +h\left( e\gamma ^T+Ceb^T-B-ee_m^TB\right) &{} \,\,+he\left( (1-c_m)b^T-\gamma ^T+e_m^TB\right) \\ I_m &{} 0 \\ \end{array} \right) \left( \begin{array}{c} \varepsilon _{n-1}^{(\mu )} \\ \varepsilon _{n-2}^{(\mu )} \\ \end{array} \right) \\&\qquad + \left( \begin{array}{l} -hCeb^T\varepsilon _{n-1}^{(\mu -1)}-h(1-c_m)eb^T\varepsilon _{n-2}^{(\mu -1)} -hB\left( \varepsilon _{n}^{(\mu -1)}-\varepsilon _{n-1}^{(\mu -1)}\right) \\ \,\,-he(\gamma ^T-e_m^TB)\left( \varepsilon _{n-1}^{(\mu -1)}-\varepsilon _{n-2}^{(\mu -1)}\right) +h^{m}\bar{R}_{m,n}^{(\mu )} \\ 0 \\ \end{array} \right) . \end{aligned} \end{aligned}$$

Obviously, the inverse of the coefficient matrix is $,$ so that by assumption (5.14) we obtain

$$\begin{aligned} \left( \begin{array}{c} \varepsilon _{n}^{(\mu )} \\ \varepsilon _{n-1}^{(\mu )} \\ \end{array} \right)= & {} \left( \begin{array}{c@{\quad }c} I_m+A^{-1}e(e_m^T A-b^T)+O(h) &{} -A^{-1}e(e_m^TA-b^T)+O(h) \\ I_m &{} 0 \\ \end{array} \right) \left( \begin{array}{c} \varepsilon _{n-1}^{(\mu )} \\ \varepsilon _{n-2}^{(\mu )} \\ \end{array} \right) \nonumber \\&+\left( \begin{array}{c} h^{m}\tilde{\bar{R}}_{m,n}^{(\mu )} \\ 0 \\ \end{array} \right) , \end{aligned}$$

(5.19)

where the meaning of $\tilde{\bar{R}}_{m,n}^{(\mu )}$ is clear. Since the eigenvalues of the matrix on the right-hand side are $\underbrace{1, 1,\ldots , 1}_m; -1, \underbrace{0, \ldots , 0}_{m-1}$, we may use an argument similar to the one in Case II for Theorem 2.1 to establish the existence of constants $C_{5}^{(\mu )}$ and $C_{6}^{(\mu )}$ such that

$$\begin{aligned} \begin{aligned} \Vert \varepsilon _{n}^{(\mu )}\Vert \le C_{5}^{(\mu )}h^{m},\ \ \ |e_h(t_n^{(\mu )})|\le C_{6}^{(\mu )}h^{m+1}. \end{aligned} \end{aligned}$$

(5.20)

Case III $\rho _m=1$

Table 1 The absolute errors for Example 6.1 with $m=1$ at $t=1$

Using the technique of [7], on $\sigma _n^{(\mu )}:=(t_n^{(\mu )}, t_{n+1}^{(\mu )}]$, we write the collocation approximation $u_h$ and the exact solution in the form

$$\begin{aligned} u_h(t^{(\mu )}_n+sh)=\sum _{j=1}^mL_j(s)u_h(t^{(\mu )}_{n,j})+h^m\frac{u_h^{(m)}(\eta _n^{(\mu )})}{m!}\prod _{i=1}^m(s-c_i), \end{aligned}$$

(5.21)

and

$$\begin{aligned} u((t_n^{(\mu )}+sh)=\sum _{j=1}^mL_j(s)u(t^{(\mu )}_{n,j})+h^m\frac{u^{(m)}((\eta ^{(\mu )}_n)')}{m!}\prod _{i=1}^m(s-c_i), \end{aligned}$$

(5.22)

where $\eta ^{(\mu )}_n, (\eta ^{(\mu )}_n)'\in (t^{(\mu )}_n, t^{(\mu )}_{n+1})$.

So (5.22)–(5.21) yields

$$\begin{aligned} e_h((t^{(\mu )}_n+sh)=\sum _{j=1}^mL_j(s)e_h(t^{(\mu )}_{n,j})+h^m\hat{R}^{(\mu )}_{n}(s), \end{aligned}$$

(5.23)

where $\hat{R}^{(\mu )}_{n}(s):=\frac{u^{(m)}((\eta ^{(\mu )}_n)')-u_h^{(m)}(\eta ^{(\mu )}_n)}{m!}\prod \nolimits _{i=1}^m(s-c_i)$. Therefore, the proof is completed by resorting to the finial argument in the proof of Theorem 2.1 and the mathematical induction.

6 Numerical examples

In this section, we present two examples to illustrate the foregoing convergence results. For the numerical solution of (1.1) and (1.2), we choose $m=1$, $m=2$ and $m=3$. For $m=1$ we use $c_1=\frac{1}{3}, 0.49, 0.5, 0.8, 1$ respectively, and $\rho _m=-2, -\frac{51}{49}, -1, -\frac{1}{4}, 0$ respectively. For $m=2$ we use the Gauss collocation parameters, $c_1=\frac{3-\sqrt{3}}{6}, c_2=\frac{3+\sqrt{3}}{6}$; the Radau IIA collocation parameters, $c_1=\frac{1}{3}, c_2=1$; and three sets of arbitrary collocation parameters, $c_1=\frac{1}{2}, c_2=1; c_1=\frac{1}{3}, c_2=\frac{2}{3}; c_1=\frac{1}{6}, c_2=0.82$ respectively, and $\rho _m=1, 0, 0, 1, \frac{45}{41}$ respectively. For $m=3$ we use the Gauss collocation parameters, $c_1=\frac{5-\sqrt{15}}{10}, c_2=\frac{1}{2}, c_3=\frac{5+\sqrt{15}}{10}$; the Radau IIA collocation parameters, $c_1=\frac{4-\sqrt{6}}{10}, c_2=\frac{4+\sqrt{6}}{10}, c_3=1$; and three sets of arbitrary collocation parameters, $c_1=\frac{1}{2}, c_2=\frac{2}{3}, c_3=1; c_1=\frac{1}{3}, c_2=\frac{1}{2}, c_3=\frac{2}{3}; c_1=\frac{1}{4}, c_2=\frac{1}{2}, c_3=0.7$ respectively, and $\rho _m=-1, 0, 0, -1, \frac{9}{7}$ respectively. In Tables 1, 2, 3, 4, 5 and 6 we list the absolute errors for the five collocation parameters and for $m=1$, $m=2$ or $m=3$.

Example 6.1

In (1.1) let $K(t, s)=e^{t-s}$ and with g(t) such that the exact solution is $u(t)=e^{-t}$.

Example 6.2

Consider (1.2) with $K(t, s)=e^{t-s}, \;\tau =1$ and $\phi (t)=1$, and with g(t) such that the exact solution is $u(t)=\cos t$ for $t\ge 0$.

Table 2 The absolute errors for Example 6.1 with $m=2$ at $t=1$

Table 3 The absolute errors for Example 6.1 with $m=3$ at $t=1$

Table 4 The absolute errors for Example 6.2 with $m=1$ at $t=2$

Table 5 The absolute errors for Example 6.2 with $m=2$ at $t=2$

Table 6 The absolute errors for Example 6.2 with $m=3$ at $t=2$

From Tables 1, 2, 3, 4, 5 and 6, we can see that the numerical results are consistent with our theoretical analysis.

In practical applications one will rarely use collocation space $S_m^{(0)}(I_h)$ with $m > 3$, since $m = 3$ yields the global convergence order $p = m+1 = 4$ and very small absolute errors already for moderately small stepsizes.

7 Concluding remark

As we mentioned in Sect. 1, the main purpose of this paper was to close a gap in previous convergence analyses of continuous piecewise polynomial collocation solutions for second-kind Volterra integral equations. While such globally continuous collocation approximations may occasionally be desirable (for example in VFIEs with non-vanishing delays), their accuracy is in general inferior to the one obtained by using discontinuous piecewise polynomials (at essentially the same computational cost).