Analysis of collocation methods for nonlinear Volterra integral equations of the third kind

Abstract

We study the approximation of solutions of a class of nonlinear Volterra integral equations (VIEs) of the third kind by using collocation in certain piecewise polynomial spaces. If the underlying Volterra integral operator is not compact, the solvability of the collocation equations is generally guaranteed only if special (so-called modified graded) meshes are employed. It is then shown that for sufficiently regular data the collocation solutions converge to the analytical solution with the same optimal order as for VIEs with compact operators. Numerical examples are given to verify the theoretically predicted orders of convergence.

Appendices

Some lemmas for solvability

Based on the global Lipschitz condition on G, a useful estimation of the mapping $J_n$ (cf. 4.1), $n=0, \ldots , N-1$, is presented in the following lemma.

Lemma A.1

Assume that G satisfies a global Lipschitz condition with Lipschitz constant $L>0$ (cf. Theorem 2.4). If $k\in C(D)$, $\beta \in (0,1]$ and $c_1< \cdots <c_m$, the following estimate holds for $n=0,1,\ldots , N-1$,

$$\begin{aligned} \sup _{U\ne V}\frac{\Vert J_n(U)-J_n(V)\Vert _\infty }{\Vert U-V\Vert _\infty }\le h_nt_{n,1}^{-\beta }L\varLambda (c)\sup _{t_n\le s\le t\le t_{n+1}}|k(t,s)|, \end{aligned}$$

where $\varLambda (c):=\sup \limits _{\theta \in [0,1]}\sum \limits _{i=1}^{m}|L_i(\theta )|$ is the Lebesgue constant corresponding to the points $\{c_i\}$.

Proof

Let $U,V\in \mathbb R^m$ with $U\ne V$. Then it follows from (4.1) that for $i=1,\ldots ,m$,

$$\begin{aligned} \begin{aligned}&|(J_n(U)-J_n(V))_i| \\&\quad = h_nt_{n,i}^{-\beta }\left| \int _0^{c_i} {k(t_{n,i},t_n+\theta h_n)\left( G\left( \sum \limits _{j = 1}^m {U^j } L_j (\theta )\right) -G\left( \sum \limits _{j = 1}^m {V^j } L_j (\theta )\right) \right) } \mathrm{{d}}\theta \right| \\&\quad \le h_nt_{n,i}^{-\beta }L\int _0^{c_i} |k(t_{n,i},t_n+\theta h_n)|\sum \limits _{j = 1}^m |U^j-V^j|| L_j (\theta ))| \mathrm{{d}}\theta \\&\quad \le h_nt_{n,i}^{-\beta }L\int _0^{c_i} \sum \limits _{j = 1}^m | L_j (\theta )| \mathrm{{d}}\theta \sup _{t_n\le s\le t\le t_{n+1}}|k(t,s)| \Vert U-V\Vert _\infty \\&\quad \le h_nt_{n,1}^{-\beta }L\varLambda (c)\sup _{t_n\le s\le t\le t_{n+1}}|k(t,s)| \Vert U-V\Vert _\infty . \end{aligned} \end{aligned}$$

Hence

$$\begin{aligned} \Vert J_n(U)-J_n(V)\Vert _\infty \le h_nt_{n,1}^{-\beta }L\varLambda (c)\sup _{t_n\le s\le t\le t_{n+1}}|k(t,s)|\Vert U-V\Vert _\infty \end{aligned}$$

and the proof is complete. $\square$

However, for the noncompact case, the estimate for $n=0$ in Lemma A.1 may now be very large, and hence it may be impossible to achieve contractivity in the first subinterval $[0,t_1]$. Therefore, we will resort to the implicit function theorem in this subinterval for the mapping $\bar{J}(U,h)$ (cf. 4.2).

Lemma A.2

Assume that $\beta =1$, $k(0,0)\ne 0$, $f(0)\in D^* = \{\xi -k(0,0)G(\xi ) : \xi \in \mathbb R\}$ and u(0) is the unique solution of the algebraic equation (1.5). If $k(0,0)G'(u(0))\ne l \;(l=1,\ldots ,m)$, then for any collocation parameters $0<c_1< \cdots < c_m \le 1$ there exists an $h^{*}$ and a continuous function U(h) such that, for all $0< h < h^{*}$, $U(0)=u(0)e$ and $\bar{J}(U(h),h)=0$, where $e := (\,1,\ldots ,1\,)^T\in \mathbb R^m$.

Proof

It follows from (1.5) that $\bar{J}(u(0) e,0)=0$ and the Jacobian matrix at (u(0)e, 0) is given by

$$\begin{aligned} \left. \frac{\partial \bar{J}}{\partial U}\right| _{(U,0)=(u(0)e,0)} = \mathbf I_m - k(0,0)G'(u(0))\mathbf D, \end{aligned}$$

where $\mathbf I_m\in \mathbb R^{m\times m}$ is the identity matrix and $\mathbf D = \begin{pmatrix} c_i^{-1}{\displaystyle \int _0^{c_i} L_j(\theta ) \mathrm{{d}}\theta } \end{pmatrix}_{i,j=1,\ldots ,m}$.

Let $\mathbf S := \mathrm {diag}\big (\,1,\ldots ,1/m\,\big )$; and $\mathbf V =(c_i^{j-1})_{i,j=1,\ldots ,m}$ is the Vandermonde matrix. It follows from [18] that $\mathbf D = \mathbf V\mathbf S\mathbf V^{-1}$ and that

$$\begin{aligned} \left. \frac{\partial \bar{J}}{\partial U}\right| _{(U,0)=(u(0)e,0)} = \mathbf V(\mathbf I_m - k(0,0)G'(u(0))\mathbf S)\mathbf V^{-1} \end{aligned}$$

is invertible if and only if $k(0,0)G'(u(0))\ne l \;(l=1,\ldots ,m)$. Hence the implicit function theorem (see [14]) implies that there exists a continuous function U(h) defined for $0<h<h^*$ such that $U(0)=u(0)e$ and $\bar{J}(U(h),h) = 0$ for $0<h<h^*$. $\square$

Remark A.1

For the solution u(0) to the algebraic equation (1.5), the value $k(0,0)G'(u(0))$ is available and hence the conditions in Lemma A.2 can be easily verified. It follows from Theorem 2.1 (see (iii))

$$\begin{aligned} 1\not \in k(0,0)G'(u(0))\sigma _0(\mathscr {V}_{1}), \end{aligned}$$

and thus (1.1) possesses a unique solution.

Some lemmas for convergence

It follows from [20] that the operator $\mathscr {I}-\mathscr {V}_{\varphi ,a}$ is invertible in $\mathfrak L(L^\infty (I))$ whenever $\varphi \in L^1(0,1)$, $a\in C(D)$ and $|a(0,0)|\Vert \varphi \Vert _1<1$. In the followings, we estimate the uniform bound of the inverse operator by $\lambda$-norms.

Lemma B.1

Assume that $\varphi \in L^1(0,1)$ is a core and $a\in C(D)$ with $|a(0,0)|\Vert \varphi \Vert _1<1$. Then there exists a constant $\lambda = \lambda (\varphi ,a)$ such that

$$\begin{aligned} \sup _{\Vert u\Vert _\lambda = 1}\Vert \mathscr {V}_{\varphi ,a}u\Vert _\lambda \le \frac{1+|a(0,0)|\Vert \varphi \Vert _1}{2}. \end{aligned}$$

Moreover, the operator $\mathscr {I} - \mathscr {V}_{\varphi ,a}$ in $\mathfrak L(L^\infty (I))$ is invertible and

$$\begin{aligned} \Vert (\mathscr {I} - \mathscr {V}_{\varphi ,a})^{-1}\Vert _{\infty , \mathfrak L(L^\infty (I))}= & {} \Vert (\mathscr {I} - \mathscr {V}_{\varphi ,a})^{-1}\Vert _{\infty , \mathfrak L(C(I))}\\\le & {} \frac{2}{1-|a(0,0)|\Vert \varphi \Vert _1}\exp (\lambda T). \end{aligned}$$

Proof

Let $\varphi \in L^1(0,1)$ and $a\in C(D)$. Then there exist $\delta >0$ and $\lambda >0$ such that

$$\begin{aligned} \begin{aligned}&\max _{t\in [0,\delta ], r\in (0,1)}|a(t,tr)-a(0,0)|\le \frac{1-|a(0,0)|\Vert \varphi \Vert _1}{2\Vert \varphi \Vert _1^2},\\&\int _{1-\delta }^1\left| \varphi (r)\right| \mathrm{d} r\le \frac{1-|a(0,0)|\Vert \varphi \Vert _1}{8\Vert \varphi \Vert _1\Vert a\Vert _\infty },\\&e^{-\lambda \delta ^2}\Vert a\Vert _\infty \le \frac{1-|a(0,0)|\Vert \varphi \Vert _1}{8\Vert \varphi \Vert _1}. \end{aligned} \end{aligned}$$

For $u\in C(I)$ with $\Vert u\Vert _\lambda =1$, and for $t\in [0,\delta ]$ we may write

$$\begin{aligned} \begin{aligned}&e^{-\lambda t}\left| \int _0^tt^{-1}\varphi (s/t)a(t,s)u(s)\mathrm{d} s\right| \\&\quad \le \int _0^tt^{-1}e^{-\lambda (t-s)}\left| \varphi (s/t)a(t,s)e^{-\lambda s}u(s)\right| \mathrm{d} s\\&\quad \le \int _0^1e^{-\lambda t(1-r)}\left| \varphi (r)a(t,tr)\right| \mathrm{d} r\\&\quad \le \int _0^1 \left| \varphi (r)a(0,0)\right| \mathrm{d} r+\int _0^1 e^{-\lambda t(1-r)}\left| \varphi (r)(a(t,tr)-a(0,0))\right| \mathrm{d} r\\&\quad \le \left( |a(0,0)|+\max _{t\in [0,\delta ], r\in (0,1)}|a(t,tr)-a(0,0)|\right) \Vert \varphi \Vert _1 \,. \end{aligned} \end{aligned}$$

Moreover, for $t\in [\delta ,T]$ there holds

$$\begin{aligned} \begin{aligned}&e^{-\lambda t}\left| \int _0^tt^{-1}\varphi (s/t)a(t,s)u(s)\mathrm{d} s\right| \\&\quad \le |a(0,0)|\Vert \varphi \Vert _1+\int _0^1 e^{-\lambda t(1-r)}\left| \varphi (r)(a(t,tr)-a(0,0))\right| \mathrm{d} r\\&\quad \le |a(0,0)|\Vert \varphi \Vert _1+\int _0^{1-\delta } e^{-\lambda t(1-r)}\left| \varphi (r)(a(t,tr)-a(0,0))\right| \mathrm{d} r\\&\qquad +\int _{1-\delta }^1 e^{-\lambda t(1-r)}\left| \varphi (r)(a(t,tr)-a(0,0))\right| \mathrm{d} r\\&\quad \le |a(0,0)|\Vert \varphi \Vert _1+2e^{-\lambda \delta ^2}\Vert a\Vert _\infty \Vert \varphi \Vert _1+2\Vert a\Vert _\infty \int _{1-\delta }^1\left| \varphi (r)\right| \mathrm{d} r. \end{aligned} \end{aligned}$$

Hence the proof is complete. $\square$

With the same discussion, the uniform bounded result is extended to the inverse operators $\mathscr {I} - \mathscr {V}_{\varphi , a_h}$ for $a_h(t,s) = t^{1-\beta }k(t,s)r_h(s)$ with $r_h\in L^\infty (I)$ and $a_h(0,0)=\lim \limits _{t\rightarrow 0}a_h(t,tr)$ uniformly with respect to $r\in [0,1]$.

Lemma B.2

Assume that $\beta \in (0,1]$, $k\in C(D)$, $r_h\in L^\infty (I)$ with $\Vert r_h\Vert _\infty \le L$ and $a_h(0,0)=\lim \limits _{t\rightarrow 0}a_h(t,tr)$ uniformly with respect to $r\in [0,1]$. Let $|c(\beta ,k)|\Vert \varphi \Vert _1L<1$. Then there exists a $\lambda = \lambda (\beta ,k,L)$ such that

$$\begin{aligned} \sup _{\Vert u\Vert _\lambda = 1}\Vert \mathscr {V}_{\varphi ,a_h}u\Vert _\lambda \le \frac{1+|c(\beta ,k)|\Vert \varphi \Vert _1L}{2}. \end{aligned}$$

In addition, the operator $\mathscr {I} - \mathscr {V}_{\varphi ,a_h}$ is invertible in $\mathfrak L(L^\infty (I))$, and

$$\begin{aligned} \Vert (\mathscr {I} - \mathscr {V}_{\varphi ,a_h})^{-1}\Vert _{\infty ,\mathfrak L(L^\infty (I))}\le \frac{2}{1-|c(\beta ,k)|\Vert \varphi \Vert _1L}\exp (\lambda T) \end{aligned}$$

is uniformly bounded for all $r_h\in L^\infty (I)$ with $\Vert r_h\Vert _\infty \le L$.

Proof

Followed by Lemma B.1 and $|a_h(0,0)|\le |c(\beta ,k)|\Vert \varphi \Vert _1L$, the desired results are obtained. $\square$

For the applications of the above results to $\mathscr {I} - \mathscr {V}_{N}$, we only need to discuss the noncompact case, i.e., $\beta =1$ and $k(0,0)\ne 0$.

Lemma B.3

Assume that $\beta =1$, $G\in C^1(\mathbb R)$ satisfies a global Lipschitz condition and $|k(0,0)|L<1$. Then

$$\begin{aligned} \lim _{N\rightarrow \infty }\sup _{t\in [0,t_1]}|e_h^N(t)|=0. \end{aligned}$$

Proof

The proof is an immediate consequence of Lemma A.2, since

$$\begin{aligned} \begin{aligned} |e_h^N(t)| =&|\mathscr {P}_N(u-u(0))(t) - (u_h(t)-u(0))|\\ \le&\varLambda (c)\sup _{t\in [0,t_1]}|u(t)-u(0)| + \varLambda (c)\max _{i=1,\ldots ,m}|U(h)_i-u(0)|. \end{aligned} \end{aligned}$$

$\square$

Lemma B.4

Under the conditions in Lemma B.3, it holds that $\Vert r_N\Vert _\infty \le L$ and

$$\begin{aligned} \lim _{N\rightarrow \infty }\sup _{t\in [0,t_1]}|r_N(t)-r_N(0)|=0. \end{aligned}$$

Proof

Because $G \in C^1(\mathbb {R})$, there exists a bounded function $\xi _N(t)\in [0,1]$ such that

$$\begin{aligned} r_N(t) = G'((\mathscr {P}_Nu)(t) + \xi _N(t)e_h^N (t)), \;\, t \in [0,T]. \end{aligned}$$

Thus, $\Vert r_N\Vert _\infty \le L$, and the uniform convergence is implied by Lemma B.3 and

$$\begin{aligned} \begin{aligned} \sup _{t\in [0,t_1]}|\mathscr {P}_N(u-u(0))(t)|\le \varLambda (c)\sup _{t\in [0,t_1]}|u(t)-u(0)|. \end{aligned} \end{aligned}$$

This completes the proof. $\square$

In the rest of this section, we consider the uniform convergence of discrete cordial Volterra integral operators $\mathscr {W}_{\varphi , a_N} := \mathscr {P}_N\mathscr {V}_{\varphi , a_N}$ to a general cordial Volterra integral operator $\mathscr {V}_{\varphi , a_N}$, where $\varphi \in L^1(0,1)$ is a core (may be not a constant) and $a_N = t^{1-\beta }k(t,s)r_N(s)$ defined in Sect. 5.1. Different from the compact operators, the uniform convergence of $\mathscr {W}_{\varphi , a_N}$ to $\mathscr {V}_{\varphi , a_N}$ in $\mathfrak L(L^\infty (I))$ is not true even for a modified graded mesh $I_h^\gamma$. Whereas, for the modified graded mesh, a uniform convergence is established in $S_{m-1}^{(-1)}(I_h^\gamma )$ during the following process.

Lemma B.5

Assume that $\varphi \in L^1(0,1)$ and that $I_h^\gamma$ is a modified graded mesh with $\gamma >1$. Let $\mathscr {W}_{\varphi } := \mathscr {P}_N\mathscr {V}_{\varphi }$ be a discrete cordial Volterra integral operator. Then

$$\begin{aligned} \lim _{N\rightarrow \infty }\Vert \mathscr {W}_{\varphi }-\mathscr {V}_{\varphi }\Vert _{\infty , S_{m-1}^{(-1)}(I_h^\gamma )\rightarrow L^\infty (I)}=0. \end{aligned}$$

Proof

We first assume that $\varphi \in C[0,1]$. Then for any given $\epsilon >0$, there exists a $\delta =\delta (\epsilon )>0$ such that the estimate

$$\begin{aligned} |\varphi (x_1)-\varphi (x_2)|\le \frac{\epsilon }{2(1+\varLambda (c))} \end{aligned}$$

(B.1)

holds for all $|x_1-x_2|\le \delta$. It follows from [18] that there exists an $N_0>0$ such that for all $N>N_0$,

$$\begin{aligned} \frac{h_n}{t_n}\le \min \left\{ \delta , \frac{\epsilon }{4(1+\Vert \varphi \Vert _\infty )(1+\varLambda (c))}\right\} \text { for } n = 1,\ldots ,N-1. \end{aligned}$$

Hence for $n = 1,\ldots ,N-1$ and $0\le s\le t_n\le t\le t_{n+1}$,

$$\begin{aligned} \left| \frac{s}{t}-\frac{s}{t_n}\right| = \frac{s(t-t_n)}{tt_n}\le \frac{h_n}{t_n}\le \delta , \end{aligned}$$

which implies, by (B.1), that

$$\begin{aligned} \max _{n=1,\ldots ,N-1}\max _{t\in (t_n,t_{n+1})}\frac{1}{t_n}\int _0^{t_n}|(\varphi (s/t)-\varphi (s/t_n))|\mathrm{d}s \le \frac{\epsilon }{2(1+\varLambda (c))}. \end{aligned}$$

Assume now that $u_h\in S_{m-1}^{(-1)}(I_h^\gamma )$, with $\Vert u_h\Vert _\infty =1$. Then

$$\begin{aligned} v_h=\mathscr {V}_\varphi u_h-\mathscr {W}_\varphi u_h\in L^\infty (I) . \end{aligned}$$

(i) A easy calculation shows that $\left. \mathscr {V}_\varphi u_h\right| _{(t_0,t_1)}\in \pi _{m-1}$, since $\left. u_h\right| _{(t_0,t_1)}\in \pi _{m-1}$. Hence $\left. \mathscr {W}_\varphi u_h\right| _{(t_0,t_1)}= \left. \mathscr {V}_\varphi u_h\right| _{(t_0,t_1)}$, i.e., $v_h(t)\equiv 0$ for $t\in [t_0,t_1]$.

(ii) For $n=1,\ldots , N-1$ and $t\in (t_n,t_{n+1})$, let

$$\begin{aligned} \delta _n(t):=\frac{1}{t}\int _0^t\varphi (s/t)u_h(s)\mathrm{d}s - \frac{1}{t_n}\int _0^{t_n}\varphi (s/t_n)u_h(s)\mathrm{d}s. \end{aligned}$$

Then for $n=1,\ldots ,N-1$,

$$\begin{aligned} \begin{aligned}&\max _{t\in (t_n,t_{n+1})}|\delta _n(t)|\\&\quad \le \max _{t\in (t_n,t_{n+1})}\frac{t-t_n}{t_n}\frac{1}{t}\int _0^t|\varphi (s/t)u_h(s)|\mathrm{d}s + \frac{1}{t_n}\int _{t_n}^t|\varphi (s/t)u_h(s)|\mathrm{d}s\\&\qquad +\frac{1}{t_n}\int _0^{t_n}|(\varphi (s/t)-\varphi (s/t_n))u_h(s)|\mathrm{d}s\\&\quad \le \frac{\epsilon }{1+\varLambda (c)}, \end{aligned} \end{aligned}$$

which implies that for $t\in (t_n,t_{n+1}]$,

$$\begin{aligned} \max _{t\in (t_n,t_{n+1})}|\mathscr {P}_N(\delta _n)(t)|\le \varLambda (c)\max _{t\in (t_n,t_{n+1})}|\delta _n(t)|\le \frac{\varLambda (c)\epsilon }{1+\varLambda (c)}. \end{aligned}$$

Therefore, for all $n=0,\ldots ,N-1$,

$$\begin{aligned} \max _{t\in (t_n,t_{n+1})}|v_h(t)|\le \max _{t\in (t_n,t_{n+1})}\left( |\delta _n(t)|+|\mathscr {P}_N(\delta _n)(t)|)\right) \le \epsilon . \end{aligned}$$

By the denseness of C(I) in $L^1(0,1)$, the conclusion for $\varphi \in L^1(0,1)$ is obtained in the following way. For any $\varepsilon >0$, there exists a continuous core $\varphi _{\varepsilon }\in C([0,1])$ such that

$$\begin{aligned} \Vert \varphi -\varphi _\epsilon \Vert _1<\epsilon , \end{aligned}$$

and this implies that

$$\begin{aligned} \Vert \mathscr {V}_\varphi -\mathscr {V}_{\varphi _\epsilon }\Vert _{\infty , L^\infty (I)\rightarrow L^\infty (I)}<\epsilon . \end{aligned}$$

Therefore, for sufficiently large N,

$$\begin{aligned} \begin{aligned}&\Vert \mathscr {V}_{\varphi }-\mathscr {W}_{\varphi }\Vert _{\infty , S_{m-1}^{(-1)}(I_h^\gamma )\rightarrow L^\infty (I)}\\&\quad \le \Vert \mathscr {V}_{\varphi }-\mathscr {V}_{\varphi _{\varepsilon }}\Vert _{\infty , S_{m-1}^{(-1)}(I_h^\gamma )\rightarrow L^\infty (I)} +\Vert \mathscr {V}_{\varphi _{\varepsilon }}-\mathscr {W}_{\varphi _{\varepsilon }}\Vert _{\infty , S_{m-1}^{(-1)}(I_h^\gamma )\rightarrow L^\infty (I)}<2\varepsilon . \end{aligned} \end{aligned}$$

Hence the proof is complete. $\square$

Lemma B.6

Assume that $\varphi \in L^1(0,1)$ and that $I_h^\gamma$ is a modified graded mesh with $\gamma >1$. Let $r_N\in L^\infty (I)$ be defined by (5.3), and set $\bar{a}_N(t,s) := r_N(s)-r_N(0)$ for $(t,s)\in D$. Then

$$\begin{aligned} \lim _{N\rightarrow \infty }\Vert \mathscr {W}_{\varphi ,\bar{a}_N}-\mathscr {V}_{\varphi ,\bar{a}_N}\Vert _{\infty , S_{m-1}^{(-1)}(I_h^\gamma )\rightarrow L^\infty (I)}=0. \end{aligned}$$

Proof

We will again first assume that $\varphi \in C(I)$. In view of Lemma B.4, for any given $\epsilon >0$ there exists an $N_0>0$ such that

$$\begin{aligned} \sup _{0<s<t<t_1}|\bar{a}_N(t,s)|\le \frac{\epsilon }{(1+\Vert \varphi \Vert _\infty )(1+\varLambda (c))} \end{aligned}$$

for all $N\ge N_0$. Hence for any given $u_h\in S_{m-1}^{(-1)}(I^\gamma )$ with $\Vert u_h\Vert _\infty \le 1$,

$$\begin{aligned} \Vert \mathscr {V}_{\varphi ,\bar{a}_N}u_h\Vert _{\infty ,[0,t_1]}\le \frac{\epsilon }{1+\varLambda (c)}, \end{aligned}$$

which implies that

$$\begin{aligned} \begin{aligned}&\Vert \mathscr {W}_{\varphi ,\bar{a}_N}-\mathscr {V}_{\varphi ,\bar{a}_N}\Vert _{\infty ,[0,t_1]}\le \Vert \mathscr {W}_{\varphi ,\bar{a}_N} \Vert _{\infty ,[0,t_1]}+\Vert \mathscr {V}_{\varphi ,\bar{a}_N}\Vert _{\infty ,[0,t_1]}\\&\quad \le \frac{\varLambda (c)\epsilon }{1+\varLambda (c)}+\frac{\epsilon }{1+\varLambda (c)}=\epsilon . \end{aligned} \end{aligned}$$

The derivation of the estimate on the interval $[t_1,T]$ is similar to one in the proof of Lemma B.5. We hence omit the details. $\square$

Lemma B.7

Assume that $\varphi \in L^1(0,1)$, $\beta =1$, $k\in C(D)$, $r_N\in L^\infty (I)$ is as defined in (5.3) and $|k(0,0)|\Vert \varphi \Vert _1L<1$. Then

$$\begin{aligned} \lim _{N\rightarrow \infty }\Vert \mathscr {W}_{\varphi ,a_N}-\mathscr {V}_{\varphi ,a_N}\Vert _{\infty , S_{m-1}^{(-1)}(I_h^\gamma )\rightarrow L^\infty (I)}=0 \end{aligned}$$

(B.2)

holds for any modified graded mesh $I^{\gamma }$ with $\gamma > 1$.

Proof

Let $\bar{k}(t,s) = k(t,s) - k(0,0)$, $\bar{r}_N(s) = r_N(s) - r_N(0)$ and $\bar{a}_N(t,s)=\bar{k}(t,s)\bar{r}_N(s)$. Thus, the uniform convergence is yielded from the results on the noncompact operator $\mathscr {V}_{\varphi }$ and the compact operator $\mathscr {V}_{\varphi ,\bar{a}_N}$ in Lemmas B.5 and B.6, respectively. $\square$