The Numerical Solution of Volterra Integral Equations

Abstract

Recently we studied a collocation–quadrature method in weighted ${\mathbf{L}}^2$ spaces as well as in the space of continuous functions for a Volterra-like integral equation of the form $u\left(x\right)-{\int }_{\alpha x}^{1}h(x-\alpha y)u\left(y\right)dy=f\left(x\right),0<x<1,$ where $h\left(x\right)$ (with a possible singularity at $x=0$) and $f\left(x\right)$ are given (in general complex-valued) functions, and $\alpha \in (0,1)$ is a fixed parameter. Here, we want to investigate the same method for the case when $\alpha =1.$ More precisely, we consider (in general weakly singular) Volterra integral equations of the form $u\left(x\right)-{\int }_0^{x}h(x,y){(x-y)}^{\kappa }u\left(y\right)dy=f\left(x\right),0<x<1,$ where $\kappa >-1$, and $h:D⟶\mathbb{C}$ is a continuous function, $D=\left\{(x,y)\in {\mathbb{R}}^2:0<y<x<1\right\}.$ The passage from $0<\alpha <1$ to $\alpha =1$ and the consideration of more general kernel functions $h(x,y)$ make the studies more involved. Moreover, we enhance the family of interpolation operators defining the approximating operators, and, finally, we ask if, in comparison to collocation–quadrature methods, the application of the Nyström method together with the theory of collectively compact operator sequences is possible.

1. Introduction

In [1], a collocation–quadrature method is studied in weighted ${\mathbf{L}}^2$ spaces as well as in the space of continuous functions for a Volterra-like integral equation of the form

$$u\left(x\right)-{\int }_{\alpha x}^{1}h(x-\alpha y)u\left(y\right)dy=f\left(x\right),0<x<1,$$

where $h\left(x\right)$ (with a possible singularity at $x=0$) and $f\left(x\right)$ are given (in general complex-valued) functions, and $\alpha \in (0,1)$ is a fixed parameter (see [1], Section 3.2). Here, we are interested in the numerical solution of Volterra integral equations of the second kind,

$$u\left(x\right)-{\int }_0^{x}g(x,y)u\left(y\right)dy=f\left(x\right),0<x<1,$$

(1)

where the functions $f:(0,1)⟶\mathbb{C}$ and $g:D⟶\mathbb{C}$ with $D=\left\{(x,y)\in {\mathbb{R}}^2:0<y<x<1\right\}$ are given. In particular, we prefer global polynomials as ansatz functions for the approximating solution $u_n\left(x\right)$ and the application of Gaussian rules for the approximation of the integral operator. We do this with the aim not to restrict the convergence rate of the method by taking piecewise polynomials of fixed degree. As a consequence, the convergence rate of the method is only bounded by the smoothness of the given data $g(x,y)$ and $f\left(x\right).$

The following result (see Lemma 1) on the solvability of (1) is well known and very special for Volterra integral equations of the above type. The space of continuous functions $f:[0,1]⟶\mathbb{C}$ is denoted by $\mathbf{C}=\mathbf{C}[0,1]$ and equipped with the infinity norm ${∥f∥}_{\infty }=max\left\{\left|f\right(x\left)\right|:x\in [0,1]\right\}$, making $\left(\mathbf{C}[0,1],{∥.∥}_{\infty }\right)$ a Banach space.

Equation (1) is written in operator form as

$$(\mathcal{I}-\mathcal{K})u=f,$$

(2)

where $\mathcal{I}:\mathbf{C}⟶\mathbf{C}$ and $\mathcal{K}:\mathbf{C}⟶\mathbf{C}$ denote the identity and the integral operator

$$\left(\mathcal{K}u\right)\left(x\right)={\int }_0^{x}g(x,y)u\left(y\right)dy,0<x<1,$$

(3)

respectively. In particular, the special case of Volterra convolution equations is of interest, where the kernel function of the integral operator is of the form $g(x-y)$ with a given continuous function $g:(0,1)⟶\mathbb{C},$

$$u\left(x\right)-{\int }_0^{x}g(x-y)u\left(y\right)dy=f\left(x\right),0<x<1.$$

(4)

Often the kernel function of the integral operator is not continuous but weakly singular along the line $\left\{(x,y)\in D:x=y\right\}$ of triangle $D.$ That is why, in what follows, we consider a kernel function of the form $g_{\kappa }(x,y)=h(x,y){(x-y)}^{\kappa }$ with a continuous function $h:D⟶\mathbb{C}$ or, in case of the convolution equation, of the form $g_{\kappa }(x,y)=h_0(x-y){(x-y)}^{\kappa }$ (i.e., $h(x,y)=h_0(x-y)$) with a continuous function $h_0:(0,1)⟶\mathbb{C},$ where $\kappa >-1$ is a given real number. The respective integral operator is denoted by ${\mathcal{K}}_{\kappa },$ i.e.,

$$\left({\mathcal{K}}_{\kappa }u\right)\left(x\right)={\int }_0^x{g}_{\kappa }(x,y)u\left(y\right)dy,-1<x<1.$$

(5)

The corresponding integral Equations (1) and (4) are written as

$$\left(\mathcal{I}-{\mathcal{K}}_{\kappa }\right)u=f.$$

(6)

The remainder of this paper is organized as follows: In Section 2 we collect a series of mapping properties of the above-mentioned integral operators in weighted ${\mathbf{L}}^2$ spaces as well as in the space of continuous functions. In Section 3 we describe the collocation–quadrature method, study its stability and convergence, and prove convergence rates for the approximate solutions. In Section 4 we answer the question under which conditions it is possible to apply the famous Nyström method (see, for example, [2], Chapter 4, and [3], Section 12.2) together with the theory of collectively compact operator sequences to the Volterra integral equations under consideration. We will see that from a theoretical point of view (concerning stability and convergence rates) the collocation–quadrature method outperforms the Nyström method. The present paper concentrates on these theoretical issues. The discussion of computational aspects of the proposed collocation–quadrature method is reserved for forthcoming studies.

2. Mapping Properties of the Involved Integral Operators

From ([3], Theorems 3.10 and 10.17), we infer the following Lemma. $\overline{D}$ denotes the closure of the set $D,$ $\overline{D}=\left\{(x,y)\in {\mathbb{R}}^2:0\le y\le x\le 1\right\}$.

Lemma 1.

If $h:\overline{D}⟶\mathbb{C}$ is continuous, then the linear operator $\mathcal{I}-{\mathcal{K}}_{\kappa }:\mathbf{C}⟶\mathbf{C}$ is invertible, i.e., for every $f\in \mathbf{C}$ there exists a unique solution $u=u^{\ast }\in \mathbf{C}$ of (6).

In the following we try to extend this result to weighted spaces of continuous functions as well as weighted spaces of square integrable functions. First, we weaken the assumption on the kernel function $h(x,y)$ in Lemma 3. Let $v^{\gamma ,\delta }$ denote the Jacobi weight function $v^{\gamma ,\delta }\left(x\right)={(1-x)}^{\gamma }{x}^{\delta },$ $-1<x<1.$ For $\gamma ,\delta \ge 0,$ we define the space ${\tilde{\mathbf{C}}}_{\gamma ,\delta }={\tilde{\mathbf{C}}}_{\gamma ,\delta }(0,1)$ as the set of all continuous functions $f:(0,1)⟶\mathbb{C},$ such that $v^{\gamma ,\delta }f$ belongs to $\mathbf{C},$ i.e., for which the finite limits ${lim}_{x\to 1-0}{v}^{\gamma ,\delta }\left(x\right)f\left(x\right)$ and ${lim}_{x\to +0}{v}^{\gamma ,\delta }\left(x\right)f\left(x\right)$ exist. Equipped with the norm

$${∥f∥}_{\gamma ,\delta ,\infty }=sup\left\{v^{\gamma ,\delta }\left(x\right)\left|f\left(x\right)\right|:0<x<1\right\},$$

${\tilde{\mathbf{C}}}_{\gamma ,\delta }$ becomes a Banach space. ${\mathbf{C}}_{\gamma ,\delta }={\mathbf{C}}_{\gamma ,\delta }(0,1)$ refers to the closed subspace of ${\tilde{\mathbf{C}}}_{\gamma ,\delta }$ of all functions $f\in {\tilde{\mathbf{C}}}_{\gamma ,\delta },$ for which

$$\underset{x\to 1-0}{lim}{v}^{\gamma ,\delta }\left(x\right)f\left(x\right)=0\mathrm{if}\gamma >0\mathrm{and}\underset{x\to +0}{lim}{v}^{\gamma ,\delta }\left(x\right)f\left(x\right)=0\mathrm{if}\delta >0.$$

Obviously, the map $\mathcal{T}:{\tilde{\mathbf{C}}}_{\gamma ,\delta }⟶\mathbf{C},$ $f\mapsto v^{\gamma ,\delta }f$ is an isometric isomorphism. Thus, the operator $\mathcal{I}-{\mathcal{K}}_{\kappa }:{\tilde{\mathbf{C}}}_{\gamma ,\delta }⟶{\tilde{\mathbf{C}}}_{\gamma ,\delta }$ is invertible if and only if the operator $\mathcal{I}-\mathcal{T}{\mathcal{K}}_{\kappa }{\mathcal{T}}^{-1}:\mathbf{C}⟶\mathbf{C}$ is invertible. Note that

$$\left(\mathcal{T}{\mathcal{K}}_{\kappa }{\mathcal{T}}^{-1}f\right)\left(x\right)={\int }_0^x{v}^{\gamma ,\delta }\left(x\right)h(x,y){(x-y)}^{\kappa }{v}^{-\gamma ,-\delta }\left(y\right)f\left(y\right)dy.$$

(7)

Consequently, a simple conclusion of Lemma 1 is the following corollary:

Corollary 1.

If $\gamma ,\delta \ge 0$ and the function $\overline{D}⟶\mathbb{C},$ $(x,y)\mapsto v^{\gamma ,\delta }\left(x\right)h(x,y)v^{-\gamma ,-\delta }\left(y\right)$ is continuous, then for every $f\in {\tilde{\mathbf{C}}}_{\gamma ,\delta },$ Equation (6) has a unique solution $u\in {\tilde{\mathbf{C}}}_{\gamma ,\delta }.$

We continue to discuss the unique solvability in ${\tilde{\mathbf{C}}}_{\gamma ,\delta }.$ In comparison to the previous corollary, in the following lemma we weaken the condition on the kernel function part $h(x,y).$ In what follows we assume that there are non-negative numbers ${\gamma }_0,{\delta }_0,{\gamma }_1,{\delta }_1$ such that the following condition (A) is fulfilled:

(A)

The function $\overline{D}⟶\mathbb{C},$ $(x,y)\mapsto v^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,y)v^{{\gamma }_0,{\delta }_0}\left(y\right)$ is continuous.

Lemma 2.

Let condition (A) be satisfied, and take $\gamma ,\delta \ge 0$ such that ${\gamma }_1\le \gamma ,$ ${\delta }_1\le \delta ,$ and ${\delta }_0+\delta <1.$ Moreover, assume that

$$1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast }>0,where{\gamma }_{\ast }={\gamma }_0+{\gamma }_1,{\delta }_{\ast }={\delta }_0+{\delta }_1.$$

Then, for $f\equiv 0,$ (6) has only the trivial solution in ${\tilde{\mathbf{C}}}_{\gamma ,\delta }.$

Proof. 

With $M:=sup\left\{v^{{\gamma }_1,{\delta }_1}\left(x\right)\left|h(x,y)\right|v^{{\gamma }_0,{\delta }_0}\left(y\right):x,y\in D\right\}$ we have, for $u\in {\tilde{\mathbf{C}}}_{\gamma ,\delta },$

$$\begin{array}{cc}\hfill \left|\left({\mathcal{K}}_{\kappa }u\right)\left(x\right)\right|& =\left|{\int }_0^{x}h(x,y){(x-y)}^{\kappa }u\left(y\right)dy\right|\hfill \\ \hfill & \le M{v}^{-{\gamma }_1,-{\delta }_1}\left(x\right){\int }_0^x{(x-y)}^{\kappa }{v}^{-{\gamma }_0,-{\delta }_0}\left(y\right)\left|u\left(y\right)\right|dy,0<x<1.\hfill \end{array}$$

(8)

By induction we prove that, for $m\in \mathbb{N},$

$$\left|\left({\mathcal{K}}_{\kappa }^{m}u\right)\left(x\right)\right|\le {\mathcal{C}}_m{v}^{-{\gamma }_1,-{\delta }_1}\left(x\right){\int }_0^x{(x-y)}^{\kappa +(m-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })}{v}^{-{\gamma }_0,-{\delta }_0}\left(y\right)\left|u\left(y\right)\right|dy,$$

(9)

where the constant ${\mathcal{C}}_m$ does not depend on $x\in (0,1)$ and $u\in \mathbf{C}(0,1).$ Of course, this is true for $m=1$ with ${\mathcal{C}}_1=M$ due to (8). Let (9) be true for some $m\in \mathbb{N}.$ Then, using (8) again, we get

$$\begin{array}{cc}& \left|\left({\mathcal{K}}_{\kappa }^{m+1}u\right)\left(x\right)\right|\hfill \\ & \le M{v}^{-{\gamma }_1,-{\delta }_1}\left(x\right){\int }_0^x{(x-y)}^{\kappa }{v}^{-{\gamma }_0,-{\delta }_0}\left(y\right)\left|\left({\mathcal{K}}_{\kappa }^{m}u\right)\left(y\right)\right|dy\hfill \\ & \le M{\mathcal{C}}_m{v}^{-{\gamma }_1,-{\delta }_1}\left(x\right)·\hfill \\ & ·{\int }_0^x{(x-y)}^{\kappa }{v}^{-{\gamma }_{\ast },-{\delta }_{\ast }}\left(y\right){\int }_0^y{(y-z)}^{\kappa +(m-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })}{v}^{-{\gamma }_0,-{\delta }_0}\left(z\right)\left|u\left(z\right)\right|dzdy\hfill \\ & =M{\mathcal{C}}_m{v}^{-{\gamma }_1,-{\delta }_1}\left(x\right)·\hfill \\ & ·{\int }_0^x{\int }_z^x{(x-y)}^{\kappa }{(y-z)}^{\kappa +(m-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })}{v}^{-{\gamma }_{\ast },-{\delta }_{\ast }}\left(y\right)dy{v}^{-{\gamma }_0,-{\delta }_0}\left(z\right)\left|u\left(z\right)\right|dz,\hfill \end{array}$$

where, using the substitution $y=z+s(x-z),$ $0\le s\le 1,$

$$\begin{array}{cc}& {\int }_z^x{(x-y)}^{\kappa }{(y-z)}^{\kappa +(m-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })}{v}^{-{\gamma }_{\ast },-{\delta }_{\ast }}\left(y\right)dy\hfill \\ & ={(x-z)}^{\kappa +\kappa +(m-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })+1}·\hfill \\ & ·{\int }_0^1{(1-s)}^{\kappa }{s}^{\kappa +(m-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })}{[1-z-s(x-z)]}^{-{\gamma }_{\ast }}{[z+s(x-z)]}^{-{\delta }_{\ast }}ds\hfill \\ & \le {(x-z)}^{\kappa +\kappa +(m-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })+1-{\gamma }_{\ast }-{\delta }_{\ast }}{\int }_0^1{(1-s)}^{\kappa -{\gamma }_{\ast }}{s}^{\kappa +(m-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })-{\delta }_{\ast }}ds\hfill \\ & ={\tilde{\mathcal{C}}}_m{(x-z)}^{\kappa +m(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })}.\hfill \end{array}$$

Thus, we obtain (9) for $m:=m+1$ and ${\mathcal{C}}_{m+1}:=M{\mathcal{C}}_m{\tilde{\mathcal{C}}}_m.$ Now, we choose $m_0\in \mathbb{N}$ such that $\kappa +(m_0-1)(1+\kappa -{\gamma }_{\ast }-{\delta }_{\ast })-{\gamma }_0-\gamma \ge 0.$ Then, due to (9),

$$\begin{array}{cc}\hfill \left|\left({\mathcal{K}}_{\kappa }^{m_0}u\right)\left(x\right)\right|& \le {\mathcal{C}}_{m_0}{v}^{-{\gamma }_1,-{\delta }_1}\left(x\right){\int }_0^x{(1-y)}^{\gamma }{y}^{-{\delta }_0}\left|u\left(y\right)\right|dy\hfill \\ \hfill & \le {\mathcal{C}}_{m_0}{v}^{-{\gamma }_1,-{\delta }_1}\left(x\right){\int }_0^x{y}^{-{\delta }_0-\delta }dy{∥u∥}_{\gamma ,\delta ,\infty },0<x<1.\hfill \end{array}$$

(10)

Let $u\in {\tilde{\mathbf{C}}}_{\gamma ,\delta }$ be a solution of the homogeneous equation $u={\mathcal{K}}_{\kappa }u.$ Then, u is also a solution of $u={\mathcal{K}}_{\kappa }^{m_0}u$, and, again by induction, we can prove

$$\left|u\left(x\right)\right|\le v^{-{\gamma }_1,-{\delta }_1}\left(x\right){\left({\displaystyle \frac{{\mathcal{C}}_{m_0}}{1-{\delta }_0-\delta }}\right)}^n{∥u∥}_{\gamma ,\delta ,\infty }{\displaystyle \frac{x^{n(1-{\delta }_0-\delta )}}{n!}},n\in \mathbb{N},0<x<1.$$

(11)

Indeed, from $u\left(x\right)=\left({\mathcal{K}}_{\kappa }^{m_0}u\right)\left(x\right)$ and (10) we see that (11) is true for $n=1.$ If we assume that (11) is true for some $n\in \mathbb{N},$ then

$$\begin{array}{cc}\hfill \left|u\right(x\left)\right|& \stackrel{\left(10\right)}{\le }{\mathcal{C}}_{m_0}{v}^{-{\gamma }_1,-{\delta }_1}\left(x\right){\int }_0^x{(1-y)}^{\gamma }{y}^{-{\delta }_0}\left|u\left(y\right)\right|dy\hfill \\ & \stackrel{\left(11\right)}{\le }{\displaystyle \frac{v^{-{\gamma }_1,-{\delta }_1}\left(x\right){\mathcal{C}}_{m_0}^{n+1}{∥u∥}_{\gamma ,\delta ,\infty }}{{(1-{\delta }_0-\delta )}^{n}n!}}{\int }_0^x{(1-y)}^{\gamma -{\gamma }_1}{y}^{-{\delta }_0-{\delta }_1+n(1-{\delta }_0-\delta )}dy\hfill \\ & \le {\displaystyle \frac{v^{-{\gamma }_1,-{\delta }_1}\left(x\right){\mathcal{C}}_{m_0}^{n+1}{∥u∥}_{\gamma ,\delta ,\infty }}{{(1-{\delta }_0-\delta )}^{n}n!}}{\int }_0^x{y}^{-{\delta }_0-\delta +n(1-{\delta }_0-\delta )}dy\hfill \\ & =v^{-{\gamma }_1,-{\delta }_1}\left(x\right){\left({\displaystyle \frac{{\mathcal{C}}_{m_0}}{1-{\delta }_0-\delta }}\right)}^{n+1}{∥u∥}_{\gamma ,\delta ,\infty }{\displaystyle \frac{x^{(n+1)(1-{\delta }_0-\delta )}}{(n+1)!}}.\hfill \end{array}$$

We send n to infinity in (11) and conclude that $u\left(x\right)=0$ for all $x\in (0,1).$ □

For $\gamma ,\delta >-1,$ ${\mathbf{L}}_{\gamma ,\delta }^2$ refers to the Hilbert space of all with respect to the Jacobi weight $v^{\gamma ,\delta }$ square integrable functions on the interval $(0,1)$ equipped with the inner product

$${⟨u,v⟩}_{\gamma ,\delta }={\int }_0^{1}u\left(x\right)\overline{v\left(x\right)}{v}^{\gamma ,\delta }\left(x\right)dx$$

and the norm ${∥u∥}_{\gamma ,\delta }=\sqrt{{⟨u,u⟩}_{\gamma ,\delta }}.$

In what follows, $\mathcal{C}$ denotes a generic positive constant, which can assume different values in different places. Moreover, $\mathcal{C}\ne \mathcal{C}(n,x,\dots )$ indicates that $\mathcal{C}$ does not depend on $n,x,\dots$ Let ${∥h∥}_{\infty }$ denote the value $sup\left\{h(x,y):x,y\in D\right\}.$

Lemma 3.

Assume that $h:\overline{D}⟶\mathbb{C}$ is continuous. Then, for $\kappa >-{\displaystyle \frac{1}{2}},$ $-1<\gamma ,\delta <1,$ and $max\left\{0,\gamma \right\}+max\left\{0,\delta \right\}<1+2\kappa ,$ the operator ${\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2⟶\mathbf{C}$ is compact.

Proof. 

Let $u\in {\mathbf{L}}_{\gamma ,\delta }^2$ and $x\in [-1,1].$ Then, using the substitution $x-y=xz,$

$$\begin{array}{cc}\hfill |\left({\mathcal{K}}_{\kappa }u\right)\left(x\right)|& \le {∥h∥}_{\infty }\sqrt{{\int }_0^x{\left|u\left(y\right)\right|}^2{v}^{\gamma ,\delta }\left(y\right)dy}\sqrt{{\int }_0^x{(x-y)}^{2\kappa }{v}^{-\gamma ,-\delta }\left(y\right)dy}\hfill \\ \hfill & \le {∥h∥}_{\infty }{∥u∥}_{\gamma ,\delta }\sqrt{x^{1+2\kappa -\delta }{\int }_0^1{z}^{2\kappa }{(1-x+xz)}^{-\gamma }{(1-z)}^{-\delta }dz}\hfill \\ \hfill & \le {∥h∥}_{\infty }{∥u∥}_{\gamma ,\delta }\sqrt{x^{1+2\kappa -\delta -max\left\{0,\gamma \right\}}{\int }_0^1{z}^{2\kappa -max\left\{0,\gamma \right\}}{(1-z)}^{-\delta }dz}\hfill \\ \hfill & \le {∥h∥}_{\infty }{∥u∥}_{\gamma ,\delta }\sqrt{{\int }_0^1{(1-z)}^{-\delta }{z}^{2\kappa -max\left\{0,\gamma \right\}}dz}=\mathcal{C}{∥u∥}_{\gamma ,\delta },\hfill \end{array}$$

(12)

where the constant $\mathcal{C}$ does not depend on $u\in {\mathbf{L}}_{\gamma ,\delta }^2.$ Now, for $0\le x_1<x_2\le 1,$ we can estimate

$$\begin{array}{cc}& |\left({\mathcal{K}}_{\kappa }u\right)\left(x_2\right)-\left({\mathcal{K}}_{\kappa }u\right)\left(x_1\right)|\hfill \\ & =\left|{\int }_0^{x_2}h(x_2,y){(x_2-y)}^{\kappa }u\left(y\right)dy-{\int }_0^{x_1}h(x_1,y){(x_1-y)}^{\kappa }u\left(y\right)dy\right|\hfill \\ & \le \left|{\int }_{x_1}^{x_2}h(x_2,y){(x_2-y)}^{\kappa }u\left(y\right)dy\right|\hfill \\ & +\left|{\int }_0^{x_1}\left[h(x_2,y)-h(x_1,y)\right]{(x_2-y)}^{\kappa }u\left(y\right)dy\right|\hfill \\ & +\left|{\int }_0^{x_1}h(x_1,y)\left[{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right]u\left(y\right)dy\right|\hfill \\ & =:I_1+I_2+I_3.\hfill \end{array}$$

We estimate these three terms separately. Firstly, we have

$$I_1\le {∥h∥}_{\infty }\sqrt{{\int }_{x_1}^{x_2}{\left|u\left(y\right)\right|}^2{v}^{\gamma ,\delta }\left(y\right)dy}\sqrt{{\int }_{x_1}^{x_2}{(x_2-y)}^{2\kappa }{v}^{-\gamma ,-\delta }\left(y\right)dy}\le {∥h∥}_{\infty }{∥u∥}_{\gamma ,\delta }\sqrt{A},$$

where, taking into account $2\kappa -max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}>-1,$

$$\begin{array}{cc}\hfill A& =(x_2-x_1){\int }_0^1{\left[(x_2-x_1)(1-z)\right]}^{2\kappa }{\left[1-x_1-(x_2-x_1)z\right]}^{-\gamma }{\left[(x_2-x_1)z+x_1\right]}^{-\delta }dz\hfill \\ & \le {(x_2-x_1)}^{1+2\kappa }{\int }_0^1{(1-z)}^{2\kappa }·\hfill \\ & ·\left\{\begin{array}{ccc}1& :& \gamma \le 0\\ {(x_2-x_1)}^{-\gamma }{(1-z)}^{-\gamma }& :& \gamma >0\end{array}\right\}\left\{\begin{array}{ccc}1& :& \delta \le 0\\ {(x_2-x_1)}^{-\delta }{z}^{-\delta }& :& \delta >0\end{array}\right\}dz\hfill \\ & \le {(x_2-x_1)}^{1+2\kappa -max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}}{\int }_0^1{(1-z)}^{2\kappa -max\left\{0,\gamma \right\}}{z}^{-max\left\{0,\delta \right\}}dz\hfill \\ & =\mathcal{C}{(x_2-x_1)}^{1+2\kappa -max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}}.\hfill \end{array}$$

Hence,

$$I_1\le \mathcal{C}{∥u∥}_{\gamma ,\delta }{(x_2-x_1)}^{{\displaystyle \frac{1}{2}}(1+2\kappa -max\left\{0,\gamma \right\}-max\left\{0,\delta \right\})},\mathcal{C}\ne \mathcal{C}(x_1,x_2,u).$$

(13)

Secondly, let $\epsilon >0.$ Then, there is an $\eta >0$ such that $|h(x_1,y)-h(x_2,y)|<\epsilon$ for all $x_1,x_2,y\in [-1,1]$ with $0\le y\le x_1\le x_2\le x_1+\eta .$ Thus, for $0\le x_1\le x_2\le x_1+\eta ,$

$$\begin{array}{cc}\hfill I_2& \le \epsilon {∥u∥}_{\gamma ,\delta }\sqrt{{\int }_0^{x_1}{(x_2-y)}^{2\kappa }{v}^{-\gamma ,-\delta }\left(y\right)dy}\hfill \\ & \le \epsilon {∥u∥}_{\gamma ,\delta }\left\{\begin{array}{ccc}{\displaystyle \sqrt{{\int }_0^1{v}^{-\gamma ,-\delta }\left(y\right)dy}}& :& \kappa \ge 0,\\ {\displaystyle \sqrt{{\int }_0^{x_1}{(x_1-y)}^{2\kappa }{(x_1-y)}^{-max\left\{0,\gamma \right\}}{y}^{-max\left\{0,\delta \right\}}dy}}& :& \kappa <0,\end{array}\right.\hfill \\ & =\epsilon {∥u∥}_{\gamma ,\delta }\left\{\begin{array}{ccc}{\displaystyle \sqrt{{\int }_0^1{v}^{-\gamma ,-\delta }\left(y\right)dy}}& :& \kappa \ge 0,\\ {\displaystyle \sqrt{x_1^{1+2\kappa -max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}}{\int }_0^1{(1-y)}^{2\kappa -max\left\{0,\gamma \right\}}{y}^{-max\left\{0,\delta \right\}}dy}}& :& \kappa <0.\end{array}\right.\hfill \end{array}$$

Consequently,

$$I_2\le \mathcal{C}\epsilon {∥u∥}_{\gamma ,\delta },\mathcal{C}\ne \mathcal{C}(x_1,x_2,u).$$

(14)

Finally, we refer to the inequality (see [4], Chapter 1 (5.2))

$$\left|{\xi }_1^{\mu }-{\xi }_2^{\mu }\right|\le {|{\xi }_1-{\xi }_2|}^{\mu },0\le \mu \le 1,{\xi }_1,{\xi }_2\ge 0,$$

(15)

and obtain

$$I_3\le {∥h∥}_{\infty }{∥u∥}_{\gamma ,\delta }\sqrt{{\int }_0^{x_1}{\left[{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right]}^2{v}^{-\gamma ,-\delta }\left(y\right)dy}=:{∥h∥}_{\infty }{∥u∥}_{\gamma ,\delta }\sqrt{B}$$

with $B\le \mathcal{C}{(x_2-x_1)}^{2min\left\{\kappa ,1\right\}}$ if $\kappa >0.$ In case of $-{\displaystyle \frac{1}{2}}<\kappa <0$, we choose $\chi >0$ such that $\kappa +\chi <0$ and $max\left\{0,\gamma \right\}+max\left\{0,\delta \right\}<1+2(\kappa -\chi ).$ Then, using the equality

$$\left|{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right|={(x_2-y)}^{\kappa }\left[{(x_2-y)}^{-\kappa }-{(x_1-y)}^{-\kappa }\right]{(x_1-y)}^{\kappa },$$

we get

$$\begin{array}{cc}\hfill B& \le {(x_2-x_1)}^{-2\kappa }{\int }_0^{x_1}{(x_2-y)}^{2\kappa }{(x_1-y)}^{2\kappa }{v}^{-\gamma ,-\delta }\left(y\right)dy\hfill \\ & ={(x_2-x_1)}^{-2\kappa }{\int }_0^{x_1}{(x_2-y)}^{2(\kappa +\chi )}{(x_2-y)}^{-2\chi }{(x_1-y)}^{2\kappa }{v}^{-\gamma ,-\delta }\left(y\right)dy\hfill \\ & \le {(x_2-x_1)}^{2\chi }{\int }_0^{x_1}{(x_1-y)}^{2(\kappa -\chi )}{v}^{-\gamma ,-\delta }\left(y\right)dy\hfill \\ & \le \mathcal{C}{(x_2-x_1)}^{2\chi }{x}_1^{1+2(\kappa -\chi )-\delta }{\int }_0^1{(1-z)}^{2(\kappa -\chi )-max\left\{0,\gamma \right\}}{z}^{-\delta }dz\hfill \\ & \le \mathcal{C}{(x_2-x_1)}^{2\chi }.\hfill \end{array}$$

Hence,

$$I_3\le \mathcal{C}{∥u∥}_{\gamma ,\delta }{(x_2-x_1)}^{{\chi }_0},{\chi }_0=\left\{\begin{array}{ccc}min\left\{\kappa ,1\right\}& :& \kappa >0,\\ \chi & :& \kappa <0,\end{array}\right.\mathcal{C}\ne \mathcal{C}(x_1,x_2,u).$$

(16)

As a conclusion of the relations (12), (13), (14), and (16), the set $\left\{{\mathcal{K}}_{\kappa }u:u\in {\mathbf{L}}_{\gamma ,\delta }^2,{∥u∥}_{\gamma ,\delta }\le 1\right\}$ is uniformly bounded and equicontinuous. It remains to refer to the Arzela–Ascoli theorem (see [2], Section 1.2.1). □

Corollary 2.

If condition (A) is valid, then the operator ${\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\tilde{\mathbf{C}}}_{{\gamma }_1,{\delta }_1}$ is compact if

$$\kappa >-{\displaystyle \frac{1}{2}},-1<2{\gamma }_0+\gamma ,2{\delta }_0+\delta <1,andmax\left\{0,2{\gamma }_0+\gamma \right\}+max\left\{0,2{\delta }_0+\delta \right\}<1+2\kappa .$$

Proof. 

We can write ${\mathcal{K}}_{\kappa }={\mathcal{J}}_1^{-1}{\mathcal{K}}_{\kappa }^{0,1}{\mathcal{J}}_0^{-1},$ where

$$\left({\mathcal{K}}_{\kappa }^{0,1}f\right)\left(x\right)={\int }_0^x{v}^{{\gamma }_1,{\delta }_1}h(x,y)v^{{\gamma }_0,{\delta }_0}\left(y\right){(x-y)}^{\kappa }f\left(y\right)dy,$$

and ${\mathcal{J}}_{0}f=v^{{\gamma }_0,{\delta }_0}f,$ ${\mathcal{J}}_{1}f=v^{{\gamma }_1,{\delta }_1}f.$ The mappings ${\mathcal{J}}_0:{\mathbf{L}}_{2{\gamma }_0+\gamma ,2{\delta }_0+\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2$ and ${\mathcal{J}}_1:{\tilde{\mathbf{C}}}_{{\gamma }_1,{\delta }_1}⟶\mathbf{C}$ are isometrical isomorphisms, and from Lemma 3, the operator ${\mathcal{K}}_{\kappa }^{0,1}:{\mathbf{L}}_{2{\gamma }_0+\gamma ,2{\delta }_0+\delta }^2⟶\mathbf{C}$ is compact. Hence,

$${\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2\stackrel{{\mathcal{J}}_0^{-1}}{⟶}{\mathbf{L}}_{2{\gamma }_0+\gamma ,2{\delta }_0+\delta }^2\stackrel{{\mathcal{K}}_{\kappa }^{0,1}}{⟶}\mathbf{C}\stackrel{{\mathcal{J}}_1^{-1}}{⟶}{\tilde{\mathbf{C}}}_{{\gamma }_1,{\delta }_1}$$

is compact. □

3. A Collocation–Quadrature Method for Equation (6)

3.1. The Method

We proceed with proposing a collocation–quadrature method for the numerical solution of the integral Equation (6). Note that the integral operator in this equation can be written in the form

$$\left({\mathcal{K}}_{\kappa }u\right)\left(x\right)=x^{1+\kappa }{\int }_0^{1}h(x,xz){(1-z)}^{\kappa }u\left(xz\right)dz,0<x<1.$$

(17)

To define an approximation of this operator, we use the Gaussian rule

$${\int }_0^{1}f\left(x\right)v^{\alpha ,\beta }\left(x\right)dx=\sum _{k=1}^n{\lambda }_{nk}^{\alpha ,\beta }f\left(x_{nk}^{\alpha ,\beta }\right)+e_n^{\alpha ,\beta }\left(f\right)$$

(18)

with the error term $e_n^{\alpha ,\beta }\left(f\right),$ where $x_{nk}^{\alpha ,\beta }$ denotes the kth zero of the nth orthonormal polynomial $p_n^{\alpha ,\beta }\left(x\right)$ with respect to the weight function $v^{\alpha ,\beta }\left(x\right),$ $0<x_{nn}^{\alpha ,\beta }<\dots <x_{n1}^{\alpha ,\beta }<1,$ and ${\lambda }_{nk}^{\alpha ,\beta }$ is the respective Christoffel number

$${\lambda }_{nk}^{\alpha ,\beta }={\int }_0^1{\ell }_{nk}^{\alpha ,\beta }\left(x\right)v^{\alpha ,\beta }\left(x\right)dx,{\ell }_{nk}^{\alpha ,\beta }\left(x\right)=\prod _{j=1,j\ne k}^n{\displaystyle \frac{x-x_{nk}^{\alpha ,\beta }}{x_{nj}^{\alpha ,\beta }-x_{nk}^{\alpha ,\beta }}}={\displaystyle \frac{p_n^{\alpha ,\beta }\left(x\right)}{{\left(p_n^{\alpha ,\beta }\right)}^{\prime }\left(x_{nk}^{\alpha ,\beta }\right)(x-x_{nk}^{\alpha ,\beta })}}.$$

${\mathcal{L}}_n^{\alpha ,\beta }$ refers to the polynomial interpolation operator associated with the nodes $x_{nk}^{\alpha ,\beta },$ i.e.,

$$\left({\mathcal{L}}_n^{\alpha ,\beta }f\right)\left(x\right)=\sum _{k=1}^{n}f\left(x_{nk}^{\alpha ,\beta }\right){\ell }_{nk}^{\alpha ,\beta }\left(x\right)$$

(19)

for every function $f:(-1,1)⟶\mathbb{C}.$ Let $\mathbf{C}(0,1)$ refer to the set of continuous functions $f:(0,1)⟶\mathbb{C}$. Writing ${(1-z)}^{\kappa }=v^{\kappa -\alpha ,-\beta }\left(z\right)v^{\alpha ,\beta }\left(z\right)$ and applying the quadrature rule (18) to (17), we define the quadrature operator ${\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }:\mathbf{C}(0,1)⟶\mathbf{C}(0,1)$ as

$$\left({\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }f\right)\left(x\right)=x^{1+\kappa }\sum _{k=1}^n{\lambda }_{nk}^{\alpha ,\beta }h(x,x{x}_{nk}^{\alpha ,\beta })v^{\kappa -\alpha ,-\beta }\left(x_{nk}^{\alpha ,\beta }\right)f\left(x{x}_{nk}^{\alpha ,\beta }\right).$$

(20)

As a possible approximation of the operator ${\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2$, we take the composed operator ${\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }{\mathcal{P}}_n^{\gamma ,\delta },$ where ${\mathcal{P}}_n^{\gamma ,\delta }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{P}}_n$ denotes the orthonormal projector

$$\left({\mathcal{P}}_n^{\gamma ,\delta }f\right)\left(x\right)=\sum _{j=0}^{n-1}{⟨f,p_j^{\gamma ,\delta }⟩}_{\gamma ,\delta }{p}_j^{\gamma ,\delta }\left(x\right),$$

and ${\mathbf{P}}_n$ refers to the linear space of all algebraic polynomials of degree less than $n.$ In other words, we are looking for an approximate solution $u_n\in {\mathbf{P}}_n$ of (6) by trying to solve the following equation:

$$u_n-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }{u}_n=f_n,$$

(21)

where $f_n\in {\mathbf{P}}_n$ is an approximation of f satisfying ${lim}_{n\to \infty }{∥f_n-f∥}_{\gamma ,\delta }=0.$ For example, if $f\in \mathbf{C}(0,1),$ then we can choose (under certain conditions, see [5] Lemma 15 as well as the beginning of Section 3.2) $f_n={\mathcal{L}}_n^{\alpha ,\beta }f.$ In that case, (21) is equivalent to

$$u_n\left(x_{nj}^{\alpha ,\beta }\right)-\left({\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }{u}_n\right)\left(x_{nj}^{\alpha ,\beta }\right)=f\left(x_{nj}^{\alpha ,\beta }\right),j=1,\dots ,n.$$

From the definition (20) of the operator ${\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }:\mathbf{C}(0,1)⟶\mathbf{C}(0,1)$, we have

$$\left({\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }f\right)\left(x\right)=x^{1+\kappa }{\int }_0^1{\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,x·)v^{\kappa -\alpha ,-\beta }f(x·)\right]\left(y\right)v^{\alpha ,\beta }\left(y\right)dy.$$

Due to the algebraic accuracy of the Gaussian rule (18), for $p_n\in {\mathbf{P}}_n,$ we get

$$\left({\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }{p}_n\right)\left(x\right)=x^{1+\kappa }{\int }_0^1{\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,x·)v^{\kappa -\alpha ,-\beta }\right]\left(y\right)p_n\left(xy\right)v^{\alpha ,\beta }\left(y\right)dy.$$

Using this relation, we can try to extend the operator ${\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }:{\mathbf{P}}_n⟶{\mathbf{L}}_{\gamma ,\delta }^2$ using linearity to a linear and bounded operator ${\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2,$

$$\left({\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }f\right)\left(x\right)=x^{1+\kappa }{\int }_0^1{\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,x·)v^{\kappa -\alpha ,-\beta }\right]\left(y\right)f\left(xy\right)v^{\alpha ,\beta }\left(y\right)dy.$$

(22)

For this we need the boundedness of the operator ${\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }:{\mathbf{P}}_n⟶{\mathbf{L}}_{\gamma ,\delta }^2,$ i.e., the existence of a constant ${\rho }_n\in \mathbb{R}$ such that

$${∥{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }{p}_n∥}_{\gamma ,\delta }\le {\rho }_n{∥p_n∥}_{\gamma ,\delta }\forall p_n\in {\mathbf{P}}_n.$$

To find out under which conditions of the involved parameters we can guarantee the existence of such a constant, we assume that $h(x,y)$ satisfies condition (A) and estimate for $p_n\in {\mathbf{P}}_n,$ $q_n(x,y):=v^{{\gamma }_1,{\delta }_1}\left(x\right){\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,x·)v^{\kappa -\alpha ,-\beta }\right]\left(y\right),$ and ${\mathcal{C}}_n:=sup\left\{|q_n(x,y)|:x,y\in [0,1]\right\},$

$$\begin{array}{cc}& {∥{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }{p}_n∥}_{\gamma ,\delta }^2\hfill \\ & ={\int }_0^1{x}^{2(1+\kappa )}{v}^{-2{\gamma }_1,-2{\delta }_1}\left(x\right){\left|{\int }_0^1{q}_n(x,y)p_n\left(xy\right)v^{\alpha ,\beta }\left(y\right)dy\right|}^2{v}^{\gamma ,\delta }\left(x\right)dx\hfill \\ & \le {\mathcal{C}}_n^2{\int }_0^1{x}^{2(1+\kappa )}{v}^{-2{\gamma }_1,-2{\delta }_1}\left(x\right)·\hfill \\ & ·{\int }_0^1{\left|p_n\left(xy\right)\right|}^2{v}^{\gamma ,\delta }\left(xy\right)dy{\int }_0^1{(1-xy)}^{-\gamma }{\left(xy\right)}^{-\delta }{(1-y)}^{2\alpha }{y}^{2\beta }dy{v}^{\gamma ,\delta }\left(x\right)dx\hfill \\ & ={\mathcal{C}}_n^2{\int }_0^1{x}^{1+2\kappa }{v}^{-2{\gamma }_1,-2{\delta }_1}\left(x\right){\int }_0^x{\left|p_n\left(z\right)\right|}^2{v}^{\gamma ,\delta }\left(z\right)dz·\hfill \\ & ·{\int }_0^1{(1-xy)}^{-\gamma }{(1-y)}^{2\alpha }{y}^{2\beta -\delta }dy{(1-x)}^{\gamma }dx\hfill \\ & \le {\mathcal{C}}_n^2{∥p_n∥}_{\gamma ,\delta }^2{\int }_0^1{(1-x)}^{\gamma -max\left\{0,\gamma \right\}-2{\gamma }_1}{x}^{1+2\kappa -2{\delta }_1}dx{\int }_0^1{(1-y)}^{2\alpha }{y}^{2\beta -\delta }dy.\hfill \end{array}$$

Thus, we can choose

$${\rho }_n={\mathcal{C}}_n\sqrt{{\int }_0^1{(1-x)}^{\gamma -max\left\{0,\gamma \right\}-2{\gamma }_1}{x}^{1+2\kappa -2{\delta }_1}dx{\int }_0^1{(1-y)}^{2\alpha }{y}^{2\beta -\delta }dy}$$

if

$${\displaystyle \frac{1+\gamma -max\left\{0,\gamma \right\}}{2}}>{\gamma }_1,1+\kappa >{\delta }_1,\alpha >-{\displaystyle \frac{1}{2}}\mathrm{and}\beta >{\displaystyle \frac{\delta -1}{2}}.$$

(23)

With definition (22) of ${\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2,$ every solution $u_n\in {\mathbf{L}}_{\gamma ,\delta }^2$ of Equation (21) belongs automatically to ${\mathbf{P}}_n$ if $f_n\in {\mathbf{P}}_n.$

3.2. Stability and Convergence

Analogously to ${\tilde{\mathbf{C}}}_{\gamma ,\delta }={\tilde{\mathbf{C}}}_{\gamma ,\delta }(0,1)$ and ${\mathbf{C}}_{\gamma ,\delta }={\mathbf{C}}_{\gamma ,\delta }(0,1),$ we define the sets ${\tilde{\mathbf{R}}}_{\gamma ,\delta }={\tilde{\mathbf{R}}}_{\gamma ,\delta }(0,1)$ and ${\mathbf{R}}_{\gamma ,\delta }={\mathbf{R}}_{\gamma ,\delta }(0,1)$ by replacing the space $\mathbf{C}(0,1)$ of continuous functions $f:(0,1)⟶\mathbb{C}$ with the set $\mathbf{R}=\mathbf{R}(0,1)$ of all functions $f:(0,1)⟶\mathbb{C}$ being bounded and Riemann-integrable on each closed subinterval of the interval $(0,1).$ Moreover, for ${\gamma }_0,{\delta }_0>0,$ we set

$${\mathbf{R}}_{{\gamma }_0,{\delta }_0}^0=\bigcup _{0\le \gamma <{\gamma }_0,0\le \delta <{\delta }_0}{\mathbf{R}}_{\gamma ,\delta }.$$

Note that

$$\begin{array}{cc}\hfill {\mathbf{R}}_{{\gamma }_0,{\delta }_0}^0& =\bigcup _{0\le \gamma <{\gamma }_0,0\le \delta <{\delta }_0}{\tilde{\mathbf{R}}}_{\gamma ,\delta }\hfill \\ & =\left\{f\in \mathbf{R}(0,1):\exists \mathcal{C}>0,\exists \epsilon >0with\left|f\left(x\right)\right|\le \mathcal{C}{v}^{\epsilon -{\gamma }_0,\epsilon -{\delta }_0}\left(x\right)\forall x\in (0,1)\right\}.\hfill \end{array}$$

Let us recall the following lemma ([5], Lemma 15):

Lemma 4.

Let $\psi ,\chi \in \mathbb{R},$ $\alpha ,\beta ,\gamma ,\delta >-1,$ and $f\in {\mathbf{R}}_{{\displaystyle \frac{1+\gamma }{2}},{\displaystyle \frac{1+\delta }{2}}}^0.$ Then we have, for $n⟶\infty ,$

$$v^{\psi ,\chi }{\mathcal{L}}_n^{\alpha ,\beta }{v}^{-\psi ,-\chi }f⟶fin{\mathbf{L}}_{\gamma ,\delta }^2$$

if

$$-{\displaystyle \frac{1}{2}}<\gamma +2\psi -\alpha <{\displaystyle \frac{3}{2}}and\gamma +2\psi >-1,$$

and if

$$-{\displaystyle \frac{1}{2}}<\delta +2\chi -\beta <{\displaystyle \frac{3}{2}}and\delta +2\chi >-1.$$

From Lemma 4 we infer that

$$\underset{n\to \infty }{lim}{∥{\mathcal{L}}_n^{\alpha ,\beta }f-f∥}_{\gamma ,\delta }=0\forall f\in {\tilde{\mathbf{R}}}_{{\gamma }^{\ast },{\delta }^{\ast }}$$

(24)

if

$$0\le {\gamma }^{\ast }<{\displaystyle \frac{1+\gamma }{2}},0\le {\delta }^{\ast }<{\displaystyle \frac{1+\delta }{2}},\mathrm{and}-{\displaystyle \frac{1}{2}}<\gamma -\alpha ,\delta -\beta <{\displaystyle \frac{3}{2}}.$$

(25)

Since pointwise convergence becomes uniform on each compact subset, the compactness of the operator ${\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\tilde{\mathbf{C}}}_{{\gamma }_1,{\delta }_1}$ (see Corollary 2) implies the norm convergence

$$\underset{n\to \infty }{lim}{∥{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa }∥}_{{\mathbf{L}}_{\gamma ,\delta }^2\to {\mathbf{L}}_{\gamma ,\delta }^2}=0$$

(26)

if condition (25) is fulfilled for ${\gamma }^{\ast }={\gamma }_1$ and ${\delta }^{\ast }={\delta }_1.$ For the proof of the next proposition, we need the following lemma (see [5], Lemma 16).

Lemma 5.

Let $-1<\alpha ,\beta ,\gamma ,\delta ,$ $0\le \tau <{\displaystyle \frac{1+\alpha }{2}},$ $0\le \chi <{\displaystyle \frac{1+\beta }{2}},$ and $-{\displaystyle \frac{1}{2}}<\gamma -\alpha ,\delta -\beta <{\displaystyle \frac{3}{2}}$ be satisfied. If the function

$$[-1,1]\times [-1,1]⟶\mathbb{C},(x,y)\mapsto g(x,y){(1-y)}^{\tau }{y}^{\chi }$$

is continuous, then

$$\underset{n\to \infty }{lim}sup\left\{\parallel {\mathcal{L}}_n^{\alpha ,\beta }g(x,·)-g(x,·){\parallel }_{\gamma ,\delta }:0\le x\le 1\right\}=0.$$

Now, let us prove a first result on the stability and the convergence of the collocation–quadrature method (21). We have the following proposition:

Proposition 1.

Assume that the function $h(x,y)$ satisfies condition (A) with ${\gamma }_0={\delta }_0=0,$

$${\delta }_1<1,1+\kappa -{\gamma }_1-{\delta }_1>0,and{\gamma }_1<{\displaystyle \frac{1+\gamma -max\left\{0,\gamma \right\}}{2}},{\delta }_1<{\displaystyle \frac{1+\delta }{2}}.$$

(27)

Let $f\in {\mathbf{L}}_{\gamma ,\delta }^2,$ and $f_n\in {\mathbf{P}}_n,$ $n\in \mathbb{N},$ with ${lim}_{n\to \infty }{∥f_n-f∥}_{\gamma ,\delta }=0.$ Furthermore, assume that

$$\kappa >-{\displaystyle \frac{1}{2}},0\le \gamma <1,-1<\delta <1,\gamma +max\left\{0,\delta \right\}<1+2\kappa ,$$

(28)

$$-{\displaystyle \frac{1}{2}}<\alpha ,and{\displaystyle \frac{\delta -1}{2}}<\beta ,$$

(29)

and

$$-{\displaystyle \frac{1}{2}}<\gamma -\alpha ,\delta -\beta \le 0,$$

(30)

$$0\le 1+2\kappa +\gamma -2\alpha +\delta -2\beta ,\alpha -\kappa <{\displaystyle \frac{1+\alpha }{2}},and\beta <{\displaystyle \frac{1+\beta }{2}},i.e.,\beta <1.$$

(31)

Then, for all sufficiently large $n,$ the collocation–quadrature Equation (21) has a unique solution $u_n^{\ast }\in {\mathbf{P}}_n$ tending to the unique solution $u^{\ast }\in {\mathbf{L}}_{\gamma ,\delta }^2$ of Equation (6) in the norm of the space ${\mathbf{L}}_{\gamma ,\delta }^2$ if n tends to infinity.

Proof. 

Due to the last two inequalities in (27), the space ${\tilde{\mathbf{C}}}_{{\gamma }_1,{\delta }_1}$ is continuously embedded in ${\mathbf{L}}_{\gamma ,\delta }^2.$ Hence, from Corollary 2 (see the conditions in (28)), we infer that the operator $\mathcal{I}-{\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2$ is compact.

Let $u_H\in {\mathbf{L}}_{\gamma ,\delta }^2$ be a solution of the homogeneous equation $\left(\mathcal{I}-{\mathcal{K}}_{\kappa }\right)u=0.$ Then, due to Corollary 2, $u_H$ belongs to ${\tilde{\mathbf{C}}}_{{\gamma }_1,{\delta }_1}$ and is identically zero due to Lemma 2 (see the first two conditions in (27)). Thus, the operator $\mathcal{I}-{\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2$ has a trivial null space. This, together with the compactness of ${\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2$ and the Fredholm alternative, leads to the invertibility of the operator $\mathcal{I}-{\mathcal{K}}_{\kappa }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2.$ Thus, Equation (6) has a unique solution $u^{\ast }\in {\mathbf{L}}_{\gamma ,\delta }^2.$

Due to our assumptions we can use definition (22) of the operator ${\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }$ (see (23) with (29) as well as the second and third inequalities in (27)). For every $u\in {\mathbf{L}}_{\gamma ,\delta }^2,$ we have

$$\begin{array}{cc}\hfill \parallel \left({\mathcal{K}}_{\kappa }-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }\right)u{\parallel }_{\gamma ,\delta }& \le \parallel \left({\mathcal{K}}_{\kappa }-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa }\right)u{\parallel }_{\gamma ,\delta }+\parallel {\mathcal{L}}_n^{\alpha ,\beta }\left({\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }\right)u{\parallel }_{\gamma ,\delta }\hfill \\ & \le \parallel {\mathcal{K}}_{\kappa }-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa }{\parallel }_{{\mathbf{L}}_{\gamma ,\delta }^2\to {\mathbf{L}}_{\gamma ,\delta }^2}{\parallel u\parallel }_{\gamma ,\delta }+\mathcal{C}\parallel \left({\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }\right)u{\parallel }_{{\gamma }_1,{\delta }_1,\infty },\hfill \end{array}$$

where $\mathcal{C}=sup\left\{\parallel {\mathcal{L}}_n^{\alpha ,\beta }{\parallel }_{{\tilde{\mathbf{C}}}_{{\gamma }_1,{\delta }_1}\to {\mathbf{L}}_{\gamma ,\delta }^2}:n\in \mathbb{N}\right\}<\infty$ due to (24) (see (25) with (27), (30)), and the Banach–Steinhaus theorem. Now, we are going to prove that, for all $u\in {\mathbf{L}}_{\gamma ,\delta }^2,$

$$\parallel \left({\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }\right)u{\parallel }_{{\gamma }_1,{\delta }_1,\infty }\le {\gamma }_n{∥u∥}_{\gamma ,\delta }with{\gamma }_n⟶0ifn⟶\infty .$$

(32)

For this, we again use the definition (22) of the operator ${\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }$ and obtain, for every $u\in {\mathbf{L}}_{\gamma ,\delta }^2,$

$$\begin{array}{cc}\hfill & \parallel ({\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta })u{\parallel }_{{\gamma }_1,{\delta }_1,\infty }\hfill \\ \hfill & \le \underset{0\le x\le 1}{max}{x}^{1+\kappa }{\int }_0^1|v^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,xy)v^{\kappa -\alpha ,-\beta }\left(y\right)\hfill \\ \hfill & -{\mathcal{L}}_n^{\alpha ,\beta }\left[v^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,x·)v^{\kappa -\alpha ,-\beta }\right]\left(y\right)|·\left|u\left(xy\right)\right|v^{\alpha ,\beta }\left(y\right)dy\hfill \\ \hfill & \le \underset{0\le x\le 1}{max}{x}^{1+\kappa }\parallel v^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,x·)v^{\kappa -\alpha ,-\beta }-{\mathcal{L}}_n^{\alpha ,\beta }{v}^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,x·)v^{\kappa -\alpha ,-\beta }{\parallel }_{\gamma ,\delta }·\hfill \\ \hfill & ·{∥u(x·)v^{\alpha -\gamma ,\beta -\delta }∥}_{\gamma ,\delta }\hfill \\ \hfill & \le \underset{0\le x\le 1}{max}{x}^{{\displaystyle \frac{1}{2}}(1+2\kappa -2\alpha +\gamma -2\beta +\delta )}·\hfill \\ \hfill & ·\parallel v^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,x·)v^{\kappa -\alpha ,-\beta }-{\mathcal{L}}_n^{\alpha ,\beta }{v}^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,x·)v^{\kappa -\alpha ,-\beta }{\parallel }_{\gamma ,\delta }{∥u∥}_{\gamma ,\delta },\hfill \end{array}$$

(33)

where we have taken into account the estimate (note that $\gamma ,\alpha -\gamma ,\beta -\delta \ge 0$ due to (28) and (30))

$$\begin{array}{cc}\hfill & {∥u(x·)v^{\alpha -\gamma ,\beta -\delta }∥}_{\gamma ,\delta }^2\hfill \\ \hfill & ={\int }_0^1{\left|u\left(xy\right)\right|}^2{v}^{2\alpha -\gamma ,2\beta -\delta }\left(y\right)dy\hfill \\ \hfill & =x^{-1-2\alpha +\gamma -2\beta +\delta }{\int }_0^x{\left|u\left(z\right)\right|}^2{(x-z)}^{2(\alpha -\gamma )}{z}^{2(\beta -\delta )}{(x-z)}^{\gamma }{z}^{\delta }dz\hfill \\ \hfill & \le x^{-1-2\alpha +\gamma -2\beta +\delta }{\int }_0^x{\left|u\left(z\right)\right|}^2{(1-z)}^{\gamma }{z}^{\delta }dz=x^{-1-2\alpha +\gamma -2\beta +\delta }{∥u∥}_{\gamma ,\delta }^2.\hfill \end{array}$$

(34)

Applying Lemma 5 for $g(x,y)=v^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,xy)v^{\kappa -\alpha ,-\beta }\left(y\right),$ $\tau =max\left\{0,\alpha -\kappa \right\},$ and $\chi =max\left\{0,\beta \right\}$, we get (32) with (see (31))

$${\gamma }_n=sup\left\{\parallel v^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,x·)v^{\kappa -\alpha ,-\beta }-{\mathcal{L}}_n^{\alpha ,\beta }{v}^{{\gamma }_1,{\delta }_1}\left(x\right)h(x,x·)v^{\kappa -\alpha ,-\beta }{\parallel }_{\gamma ,\delta }:0\le x\le 1\right\}.$$

Together with (26) we obtain

$$\underset{n\to \infty }{lim}{∥{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }-{\mathcal{K}}_{\kappa }∥}_{{\mathbf{L}}_{\gamma ,\delta }^2\to {\mathbf{L}}_{\gamma ,\delta }^2}=0.$$

(Here and in what follows, ${∥\mathcal{A}∥}_{\mathbf{X}\to \mathbf{Y}}$ denotes the norm of the linear and bounded operator $\mathcal{A}:\mathbf{X}⟶\mathbf{Y}$ defined on the Banach space $\mathbf{X}$ and mapping into the Banach space $\mathbf{Y}.$) Now, a Neumann series argument shows that, for all sufficiently large n (say $n\ge n_0$), the operators $\mathcal{I}-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }:{\mathbf{L}}_{\gamma ,\delta }^2⟶{\mathbf{L}}_{\gamma ,\delta }^2$ are invertible and their inverses are uniformly bounded, i.e.,

$${\mathcal{C}}^{\ast }:=sup\left\{{∥{\left(\mathcal{I}-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }\right)}^{-1}∥}_{{\mathbf{L}}_{\gamma ,\delta }^2\to {\mathbf{L}}_{\gamma ,\delta }^2}:n\ge n_0\right\}<\infty ,$$

(35)

which means that the method is stable.

Let $u^{\ast }$ and $u_n^{\ast },$ $n\ge n_0$ be the unique solutions of Equations (6) and (21), respectively. Then,

$$\begin{array}{cc}\hfill u_n^{\ast }-u^{\ast }& ={\left(\mathcal{I}-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }\right)}^{-1}\left[f_n-\left(\mathcal{I}-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }\right)u^{\ast }\right]\hfill \\ \hfill & ={\left(\mathcal{I}-{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }\right)}^{-1}\left[f_n-f+\left({\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }-{\mathcal{K}}_{\kappa }\right)u^{\ast }\right].\hfill \end{array}$$

(36)

Consequently,

$${∥u_n^{\ast }-u^{\ast }∥}_{\gamma ,\delta }\le {\mathcal{C}}^{\ast }\left({∥f_n-f∥}_{\gamma ,\delta }+{∥{\mathcal{L}}_n^{\alpha ,\beta }{\mathcal{K}}_{\kappa ,n}^{\alpha ,\beta }-{\mathcal{K}}_{\kappa }∥}_{{\mathbf{L}}_{\gamma ,\delta }^2\to {\mathbf{L}}_{\gamma ,\delta }^2}{∥u^{\ast }∥}_{\gamma ,\delta }\right),$$

and the proof is finished. □

In the following corollary we reformulate Proposition 1 for the case of $\alpha =\gamma ,$ $\beta =\delta ,$ and ${\gamma }_0={\gamma }_1={\delta }_0={\delta }_1=0.$

Corollary 3.

Assume that the function $h:\overline{D}⟶\mathbb{C}$ is continuous. Let $f\in {\mathbf{L}}_{\gamma ,\delta }^2$ and $f_n\in {\mathbf{P}}_n,$ $n\in \mathbb{N},$ with ${lim}_{n\to \infty }{∥f_n-f∥}_{\gamma ,\delta }=0.$ Assume that $\kappa >-{\displaystyle \frac{1}{2}},$ $0\le \gamma <1,$ $-1<\delta <1,$ $\gamma +max\left\{0,\delta \right\}<1+2\kappa ,$ and $\gamma -\kappa <{\displaystyle \frac{1+\gamma }{2}}.$ Then, for all sufficiently large $n,$ the collocation–quadrature equation

$$u_n-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }{u}_n=f_n$$

(37)

has a unique solution $u_n^{\ast }\in {\mathbf{P}}_n$ tending to the unique solution $u^{\ast }\in {\mathbf{L}}_{\gamma ,\delta }^2$ of Equation (6) in the norm of the space ${\mathbf{L}}_{\gamma ,\delta }^2$ if n tends to infinity.

3.3. Convergence Rates

To prove convergence rates we recall some estimates for the interpolation error. For this, we define the space ${\mathbf{W}}^r,$ $r\in \mathbb{N},$ of all functions $f:(0,1)⟶\mathbb{C},$ which are $r-1$ times differentiable, and the $(r-1)$-th derivative is absolutely continuous on $(0,1).$ ${\mathbf{L}}^p(a,b),$ $p\in (1,\infty )$ denotes the usual ${\mathbf{L}}^p$ space on the interval $(a,b)$, ${\mathbf{L}}^p:={\mathbf{L}}^p(0,1).$ For $1<p<\infty$ and all $f\in {\mathbf{W}}^r,$ we have (see [6] (3.2.63))

$${\parallel \left(f-{\mathcal{L}}_n^{\alpha ,\beta }f\right)v^{{\displaystyle \frac{\gamma }{p}},{\displaystyle \frac{\delta }{p}}}\parallel }_{{\mathbf{L}}^p}\le {\displaystyle \frac{\mathcal{C}}{n^r}}{\parallel f^{\left(r\right)}{\phi }^r{v}^{{\displaystyle \frac{\gamma }{p}},{\displaystyle \frac{\delta }{p}}}\parallel }_{{\mathbf{L}}^p},\mathcal{C}\ne \mathcal{C}(n,f),$$

(38)

if and only if

$${\displaystyle \frac{\alpha }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}<{\displaystyle \frac{\gamma }{p}}<{\displaystyle \frac{\alpha }{2}}+{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}}\mathrm{and}{\displaystyle \frac{\beta }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}<{\displaystyle \frac{\delta }{p}}<{\displaystyle \frac{\beta }{2}}+{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}},$$

where $\phi \left(x\right)=\sqrt{(1-x)x}.$ For $r\in \mathbb{N},$ we define the space

$${\mathbf{W}}_{\gamma ,\delta }^r=\left\{f\in {\mathbf{L}}_{\gamma ,\delta }^2:f^{(r-1)}\in \mathbf{A}\mathbf{C}(0,1),f^{\left(r\right)}{\phi }^r\in {\mathbf{L}}_{\gamma ,\delta }^2\right\},$$

where $\mathbf{A}\mathbf{C}(0,1)$ is the set of all absolutely continuous functions $f:(0,1)⟶\mathbb{C}.$ In the case $p=2$ and for $r\in \mathbb{N}$ and all $f\in {\mathbf{W}}_{\gamma ,\delta }^r,$ we get

$$\parallel f-{\mathcal{L}}_n^{\alpha ,\beta }f{\parallel }_{\gamma ,\delta }\le {\displaystyle \frac{\mathcal{C}}{n^r}}\parallel f^{\left(r\right)}{\phi }^r{\parallel }_{\gamma ,\delta },\mathcal{C}\ne \mathcal{C}(n,f),$$

(39)

if and only if

$$\gamma -\alpha ,\delta -\beta \in \left(-{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}\right).$$

(40)

We will restrict ourselves to the case $\alpha =\gamma$ and $\beta =\delta ,$ such that condition (40) is satisfied. Nevertheless, with the help of (39), analogous but more involved considerations are possible if $\alpha \ne \gamma$ and/or $\beta \ne \delta .$ Thus, we assume that the conditions of Corollary 3 are in force. Then, due to (35) and (36),

$${∥u_n^{\ast }-u^{\ast }∥}_{\gamma ,\delta }\le {\mathcal{C}}^{\ast }\left({∥f_n-f∥}_{\gamma ,\delta }+\mathcal{C}\parallel \left({\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }-{\mathcal{K}}_{\kappa }\right)u^{\ast }{\parallel }_{\infty }+\parallel \left({\mathcal{L}}_n^{\gamma ,\delta }-\mathcal{I}\right){\mathcal{K}}_{\kappa }{u}^{\ast }{\parallel }_{\gamma ,\delta }\right).$$

(41)

Let us check which further conditions on the given data $f\left(x\right)$ and $h(x,y)$ are suitable to obtain convergence rates from estimate (41).

1.

Of course, the first addend in the brackets on the right-hand side in (41) can be estimated with the help of (39) by $\mathcal{C}{n}^{-r}{∥f^{\left(r\right)}{\phi }^r∥}_{\gamma ,\delta }$ if we choose $f_n={\mathcal{L}}_n^{\gamma ,\delta }f$ and assume that $f\in {\mathbf{W}}_{\gamma ,\delta }^r,$ $r\in \mathbb{N}.$

2.

Let us consider the second addend. Due to (22), we can estimate $\parallel \left({\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }-{\mathcal{K}}_{\kappa }\right)u^{\ast }{\parallel }_{\infty }$ by

$$\begin{array}{cc}& \underset{0\le x\le 1}{sup}\left|x^{1+\kappa }{\int }_0^1\left[{\mathcal{L}}_n^{\gamma ,\delta }\left[h(x,x·)v^{\kappa -\gamma ,-\delta }\right]\left(y\right)-h(x,xy)v^{\kappa -\gamma ,-\delta }\left(y\right)\right]u^{\ast }\left(xy\right)v^{\gamma ,\delta }\left(y\right)dy\right|\hfill \\ & \le \underset{0\le x\le 1}{sup}{x}^{1+\kappa }\parallel {\mathcal{L}}_n^{\gamma ,\delta }\left[h(x,x·)v^{\kappa -\gamma ,-\delta }\right]-h(x,x·)v^{\kappa -\gamma ,-\delta }{\parallel }_{\gamma ,\delta }{∥u^{\ast }(x·)∥}_{\gamma ,\delta },\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {∥u(x·)∥}_{\gamma ,\delta }^2& ={\int }_0^1{\left|u\left(xy\right)\right|}^2{(1-y)}^{\gamma }{y}^{\delta }dy=x^{-1-\gamma -\delta }{\int }_0^x{\left|u\left(z\right)\right|}^2{(x-z)}^{\gamma }{z}^{\delta }dz\hfill \\ & \stackrel{\gamma \ge 0}{\le }{x}^{-1-\gamma -\delta }{\int }_0^x{\left|u\left(z\right)\right|}^2{(1-z)}^{\gamma }{z}^{\delta }=x^{-1-\gamma -\delta }{∥u∥}_{\gamma ,\delta }^2\hfill \end{array}$$

(see (34)). Hence, since $1+2\kappa -\gamma -\delta \ge 0$ (due to the assumptions in Corollary 3),

$$\begin{array}{cc}\hfill \parallel \left({\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }-{\mathcal{K}}_{\kappa }\right)u^{\ast }{\parallel }_{\infty }& \le \underset{0\le x\le 1}{sup}\parallel {\mathcal{L}}_n^{\gamma ,\delta }\left[h(x,x·)v^{\kappa -\gamma ,-\delta }\right]-h(x,x·)v^{\kappa -\gamma ,-\delta }{\parallel }_{\gamma ,\delta }{∥u^{\ast }∥}_{\gamma ,\delta }\hfill \\ & \le \mathcal{C}{n}^{-r}{∥u^{\ast }∥}_{\gamma ,\delta },\hfill \end{array}$$

if

$$h(x,x·)v^{\kappa -\gamma ,-\delta }\in {\mathbf{W}}_{\gamma ,\delta }^r\mathrm{uniformly}\mathrm{with}\mathrm{respect}\mathrm{to}x\in [0,1].$$

(42)

This can be guaranteed if, for example, the function

$${\displaystyle \overline{D}⟶\mathbb{C},(x,y)\mapsto \frac{{\partial }^{r}h(x,y)}{\partial y^r}}$$

(43)

is continuous, and

(a)

$\gamma =\kappa ,$$\delta =0.$

Since ${\displaystyle \frac{d^r}{d{y}^r}}\left[{(1-y)}^{\alpha }{y}^{\beta }\right]={(1-y)}^{\alpha -r}{y}^{\beta -r}{g}_{\alpha ,\beta ,r}\left(y\right)$ with $g_{\alpha ,\beta ,r}\in \mathbf{C}[0,1],$ we have (42) if (43) is satisfied together with

(b)

$1+2\kappa -\gamma -r>0,$ and $1-\delta -r>0$ (i.e., $-1<\delta <0,$ and $r=1$),

or

(c)

$\gamma =\kappa ,$ $-1<\delta <0,$ and $r=1,$

or

(d)

$1+2\kappa -\gamma -r>0,$ and $\delta =0.$

3.

For the last addend on the right-hand side of (41), we can again use (39) if we know that ${\mathcal{K}}_{\kappa }{u}^{\ast }\in {\mathbf{W}}_{\gamma ,\delta }^r.$ Assume that the functions

$$\overline{D}⟶\mathbb{C},(x,y)\mapsto {\displaystyle \frac{{\partial }^k}{\partial x^k}}h(x,y),k=0,1,\dots ,r,$$

(44)

are continuous.

(a)

If $f\in \mathbf{C}[0,1]$, then, due to Lemma 3, the solution $u^{\ast }$ also belongs to $\mathbf{C}[0,1].$ Consequently, in view of

$$\left({\mathcal{K}}_{\kappa }{u}^{\ast }\right)\left(x\right)={\int }_0^{x}h(x,y){(x-y)}^{\kappa }{u}^{\ast }\left(x\right)dx,$$

we have ${\mathcal{K}}_{\kappa }{u}^{\ast }\in {\mathbf{W}}_{\gamma ,\delta }^r$ if $\kappa \in {\mathbb{N}}_0$ (a non-negative integer) and $r\le 1+\kappa$ (see (54)).

(b)

If $f\in \mathbf{C}[0,1],$ $r\in \mathbb{N}$ (a positive integer), and $r-1<\kappa <r,$ then we can also conclude ${\mathcal{K}}_{\kappa }{u}^{\ast }\in {\mathbf{W}}_{\gamma ,\delta }^r,$ since the operator ${\mathcal{K}}_{\beta }$ for $\beta >-1$ maps $\mathbf{C}$ into $\mathbf{C}$ (see Lemma 1 and (54)).

Let us summarize the considerations in items 1 to 3 in the following proposition.

Proposition 2.

Assume that the conditions of Corollary 3 are valid. Let $u^{\ast }\in {\mathbf{L}}_{\gamma ,\delta }^2$ and $u_n^{\ast }\in {\mathbf{L}}_{\gamma ,\delta }^2$ denote the solutions of (6) and (37), respectively. Moreover, let $r\in \mathbb{N},$ $f\in {\mathbf{W}}_{\gamma ,\delta }^r,$ and $f_n={\mathcal{L}}_n^{\gamma ,\delta }f$, and let the conditions (43) and (44) be fulfilled. Then, there is a constant $\mathcal{C}\ne \mathcal{C}\left(n\right)$ such that

$${∥u_n^{\ast }-u^{\ast }∥}_{\gamma ,\delta }\le \mathcal{C}{n}^{-r},$$

if one of conditions 2.(a)–2.(d) and one of conditions 3.(a)–3.(b) are additionally satisfied by the given data.

We see that the case where $-{\displaystyle \frac{1}{2}}<\kappa <0$ (not prohibited in Corollary 3) does not fit the conditions of Proposition 2. That is why we consider that case separately. We assume $f\in \mathbf{C}[0,1],$ which implies $u^{\ast }\in \mathbf{C}[0,1]$ (see Lemma 1). Let $0\le x_1<x_2\le 1$ and estimate

$$\begin{array}{cc}\hfill \left|\left({\mathcal{K}}_{\kappa }{u}^{\ast }\right)\left(x_2\right)-\left({\mathcal{K}}_{\kappa }{u}^{\ast }\right)\left(x_1\right)\right|& \le \left|{\int }_{x_1}^{x_2}h(x_2,y){(x_2-y)}^{\kappa }{u}^{\ast }\left(y\right)dy\right|\hfill \\ & +\left|{\int }_0^{x-1}\left[h(x_2,y)-h(x_1,y)\right]{(x_2-y)}^{\kappa }{u}^{\ast }\left(y\right)dy\right|\hfill \\ & +\left|{\int }_0^{x_1}h(x_1,y)\left[{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right]u^{\ast }\left(y\right)dy\right|\hfill \\ & =:J_1+J_2+J_3,\hfill \end{array}$$

where, with the notation ${∥u^{\ast }∥}_{\infty }={∥u^{\ast }∥}_{0,0,\infty },$

$$\begin{array}{cc}\hfill J_1& \le {∥h∥}_{\infty }{∥u^{\ast }∥}_{\infty }{\int }_{x_1}^{x_2}{(x_2-y)}^{\kappa }dy={∥h∥}_{\infty }{∥u^{\ast }∥}_{\infty }{\displaystyle \frac{1}{1+\kappa }}{(x_2-x_1)}^{1+\kappa },\hfill \\ \hfill J_2& \le {∥u^{\ast }∥}_{\infty }\underset{0\le y\le x_1}{sup}|h(x_2,y)-h(x_1,y)|{\displaystyle \frac{1}{1+\kappa }}\left[x_2^{1+\kappa }-{(x_2-x_1)}^{1+\kappa }\right|,\hfill \\ \hfill J_3& \le {∥h∥}_{\infty }{∥u^{\ast }∥}_{\infty }{\displaystyle \frac{1}{1+\kappa }}\left[x_2^{1+\kappa }-x_1^{1+\kappa }+{(x_2-x_1)}^{1+\kappa }\right]\le {\displaystyle \frac{2{∥h∥}_{\infty }{∥u^{\ast }∥}_{\infty }}{1+\kappa }}{(x_2-x_1)}^{1+\kappa }.\hfill \end{array}$$

Hence, if we assume that the function $h(x,y)$ is uniformly Hölder-continuous in x with exponent $1+\kappa$ with respect to $y,$ i.e.,

$$\underset{0\le y\le x_1}{sup}|h(x_2,y)-h(x_1,y)|\le \mathcal{C}{(x_2-x_1)}^{1+\kappa }$$

(45)

$\forall x_1,x_2,0\le x_1<x_2\le 1,\mathcal{C}\ne \mathcal{C}(x_1,x_2),$ then we have

$${\mathcal{K}}_{\kappa }{u}^{\ast }\in {\mathbf{C}}^{1+\kappa }[0,1],$$

(46)

where ${\mathbf{C}}^{\chi }[0,1]$ denotes the space of Hölder-continuous functions on $[0,1]$ (with Hölder exponent $\chi \in (0,1]$).

For a function g satisfying $gu\in {\mathbf{L}}^p(0,1)$ for a Jacobi weight function $w\in {\mathbf{L}}^p(0,1)$ and some $p\in (1,\infty ),$ we introduce the main part of the weighted modulus of smoothness:

$${\Omega }_{\phi }^r{(g,t)}_{w,p}=\underset{0<h\le t}{sup}{∥\left({\Delta }_{h\phi }^{r}g\right)w∥}_{{\mathbf{L}}^p(4h^2,1-4h^2)},$$

where, for $r\in \mathbb{N},$

$$\left({\Delta }_{h\phi }^{r}g\right)\left(x\right)=\sum _{k=0}^r{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{r}{k}}\right)g\left(x+\left({\displaystyle \frac{r}{2}}-k\right)h\phi \left(x\right)\right).$$

Let us recall the relation (see [7], Theorem 3.1)

$${∥\left(g-{\mathcal{L}}_n^{\gamma ,\delta }g\right)w∥}_{{\mathbf{L}}^p}\le \mathcal{C}{n}^{-{\displaystyle \frac{1}{p}}}{\int }_0^{{\displaystyle \frac{1}{n}}}{\displaystyle \frac{{\Omega }_{\phi }^r{(g,t)}_{w,p}dt}{t^{1+\frac{1}{p}}}},\mathcal{C}\ne \mathcal{C}(n,g),$$

(47)

which is true if and only if

$${\displaystyle \frac{w}{\sqrt{v^{\gamma ,\delta }\phi }}}\in {\mathbf{L}}^p(0,1)\mathrm{and}{\displaystyle \frac{\sqrt{v^{\gamma ,\delta }\phi }}{w}}\in {\mathbf{L}}^q(0,1),{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.$$

(48)

Note that, when $w=v^{{\gamma }_1,{\delta }_1},$ condition (48) is equivalent to

$${\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}<{\gamma }_1<{\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}}\mathrm{and}{\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}<{\delta }_1<{\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}}.$$

(49)

Corollary 4.

Let $-{\displaystyle \frac{1}{2}}<\kappa <0$ and $-{\displaystyle \frac{1}{2}}\le \delta <1.$ In additional to the conditions of Corollary 3, assume that $h(·,y)\in {\mathbf{C}}^{1+\kappa }[0,1]$ holds uniformly with respect to $y\in [0,1]$(see (45)), the function $\overline{D}⟶\mathbb{C},(x,y)\mapsto {\displaystyle \frac{\partial }{\partial y}}h(x,y)$ is continuous, $f\in {\mathbf{C}}^{1+\kappa }[0,1],$ and $f_n={\mathcal{L}}_n^{\gamma ,\delta }f.$ Then, there is a constant $\mathcal{C}\ne \mathcal{C}\left(n\right)$ such that, for all sufficiently large $n,$

$${∥u_n^{\ast }-u^{\ast }∥}_{\gamma ,\delta }\le \mathcal{C}{n}^{-(1+\kappa )},$$

where $u_n^{\ast }$ and $u^{\ast }$ are the unique solutions in ${\mathbf{L}}_{\gamma ,\delta }^2$ of (37) and (6), respectively.

Proof. 

Due to the above considerations and our assumptions, we have ${\mathcal{K}}_{\kappa }{u}^{\ast }\in {\mathbf{C}}^{1+\kappa }[0,1].$ Consequently, ${\Omega }_{\phi }^1{({\mathcal{K}}_{\kappa }{u}^{\ast },t)}_{u,2}\le \mathcal{C}{t}^{1+\kappa },$ which follows directly from the definition of ${\Omega }_{\phi }^1{(g,t)}_{u,p}$ using $\left|\left({\mathcal{K}}_{\kappa }{u}^{\ast }\right)(x+{\displaystyle \frac{h}{2}})-\left({\mathcal{K}}_{\kappa }{u}^{\ast }\right)(x-{\displaystyle \frac{h}{2}})\right|\le \mathcal{C}{h}^{1+\kappa }.$ When $u=v^{{\displaystyle \frac{\gamma }{2}},{\displaystyle \frac{\delta }{2}}}$ and $p=q=2$, condition (49) is satisfied, so that from (47) we can infer

$${∥({\mathcal{L}}_n^{\gamma ,\delta }-\mathcal{I}){\mathcal{K}}_{\kappa }{u}^{\ast }∥}_{\gamma ,\delta }\le \mathcal{C}{n}^{-(1+\kappa )}.$$

The same convergence rate is obtained for the first addend on the right-hand side of (41) since $f\in {\mathbf{C}}^{1+\kappa }[0,1]$ and $f_n={\mathcal{L}}_n^{\gamma ,\delta }f$ are assumed.

It remains to estimate the second addend. For this, set $g_x\left(y\right)=h(x,xy)v^{\kappa -\gamma ,-\delta }\left(y\right).$ Moreover, let ${\gamma }_0+{\gamma }_1=\gamma$ and ${\delta }_0+{\delta }_1=\delta .$ Then, from (22),

$$\begin{array}{cc}\hfill {∥({\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }-{\mathcal{K}}_{\kappa })u^{\ast }∥}_{\infty }& =\underset{0\le x\le 1}{sup}\left|x^{1+\kappa }{\int }_0^1\left[\left({\mathcal{L}}_n^{\gamma ,\delta }{g}_x\right)\left(y\right)-g_x\left(y\right)\right]u^{\ast }\left(xy\right)v^{\gamma ,\delta }\left(y\right)dy\right|\hfill \\ \hfill & \le \underset{0\le x\le 1}{sup}\parallel \left({\mathcal{L}}_n^{\gamma ,\delta }{g}_x-g_x\right)v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^2}\parallel u^{\ast }(x·)v^{{\gamma }_1,{\delta }_1}{\parallel }_{{\mathbf{L}}^2}.\hfill \end{array}$$

(50)

Since the solution $u^{\ast }$ is a continuous function, we have

$$\parallel u(x·)v^{{\gamma }_1,{\delta }_1}{\parallel }_{{\mathbf{L}}^2}^2={\int }_0^1{\left|u^{\ast }\left(xy\right)\right|}^2{v}^{2{\gamma }_1,2{\delta }_1}\left(y\right)dy\le \mathcal{C}\ne \mathcal{C}\left(x\right)<\infty ,$$

if ${\gamma }_1,{\delta }_1>-{\displaystyle \frac{1}{2}}.$ From (38) we conclude

$$\parallel \left({\mathcal{L}}_n^{\gamma ,\delta }{g}_x-g_x\right)v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^2}\le \mathcal{C}{n}^{-1}\parallel g_x^{\prime }\phi v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^2}$$

if ${\displaystyle \frac{\gamma }{2}}-{\displaystyle \frac{1}{4}}<{\gamma }_0<{\displaystyle \frac{\gamma }{2}}-{\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{1}{4}}<{\delta }_0<{\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{3}{4}}.$ Thus, we have to show that ${\gamma }_0,{\delta }_0$ and ${\gamma }_1,{\delta }_1$ can be chosen in such a way that the norm $\parallel g_x^{\prime }\phi v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^2}$ can be bounded by a constant $\mathcal{C}\ne \mathcal{C}\left(x\right).$

We choose ${\gamma }_0,{\delta }_0$ and ${\gamma }_1,{\delta }_1$ so that ${\gamma }_0+{\gamma }_1=\gamma ,$ ${\delta }_0+{\delta }_1=\delta ,$ ${\gamma }_0>\gamma -\kappa ,$ ${\delta }_0>\delta ,$ and ${\gamma }_1,{\delta }_1>-{\displaystyle \frac{1}{2}}.$ This is possible, since $\kappa >-{\displaystyle \frac{1}{2}}$ and $\delta -{\displaystyle \frac{1}{2}}<\delta .$ The inequalities $\gamma <1+2\kappa$ and $\delta <1$ ensure that ${\gamma }_0$ and ${\delta }_0$ can be chosen in such a way that ${\gamma }_0<{\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{3}{4}}$ and ${\delta }_0<{\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{3}{4}}.$ Furthermore, we have

$${\displaystyle \frac{\gamma }{2}}-{\displaystyle \frac{1}{4}}<{\displaystyle \frac{1}{2}}\left(\gamma +{\displaystyle \frac{1}{2}}\right)-\kappa -{\displaystyle \frac{1}{2}}<\gamma +{\displaystyle \frac{1}{2}}-\kappa -{\displaystyle \frac{1}{2}}=\gamma -\kappa <{\gamma }_0$$

and

$${\displaystyle \frac{\delta }{2}}-{\displaystyle \frac{1}{4}}={\displaystyle \frac{1}{2}}\left(\delta +{\displaystyle \frac{1}{2}}\right)-{\displaystyle \frac{1}{2}}<\delta <{\delta }_0.$$

Finally, since

$$\left|g_x^{\prime }\left(y\right)\right|\phi \left(y\right)v^{{\gamma }_0,{\delta }_0}\left(y\right)\le \mathcal{C}{(1-y)}^{{\gamma }_0+\kappa -\gamma -{\displaystyle \frac{1}{2}}}{y}^{{\delta }_0-\delta -{\displaystyle \frac{1}{2}}},\mathcal{C}\ne \mathcal{C}(x,y),$$

and ${\gamma }_0+\kappa -\gamma >0,$ ${\delta }_0-\delta >0,$ we have $\parallel g_x^{\prime }\phi v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^2}\le \mathcal{C}\ne \mathcal{C}\left(x\right),$ which finishes the proof of the corollary. □

3.4. Uniform Convergence

In the previous sections, the studies were restricted to the case where $\kappa >-{\displaystyle \frac{1}{2}}.$ The reason for this is the respective condition in Lemma 3, which is important for the proof of the stability of the collocation–quadrature method in the ${\mathbf{L}}_{\gamma ,\delta }^2$ space (see the proof of Proposition 1). As one can see from Corollary 5, under certain conditions the operator ${\mathcal{K}}_{\kappa }:\mathbf{C}⟶\mathbf{C}$ is compact for $\kappa >-1.$ That is why we studied the collocation–quadrature method (37) in the $\mathbf{C}$ space, which allowed us to prove convergence rates in the infinity norm.

If $A=A(x,n,\dots )$ and $B=B(x,n,\dots )$ are two positive functions depending on certain parameters $n,x,\dots ,$ then we use the notion $A{\sim }_{x,n,\dots }B$ if there is a positive constant $\mathcal{C}\ne \mathcal{C}(x,n,\dots )$ such that ${\mathcal{C}}^{-1}B\le A\le \mathcal{C}B.$ Let us recall the following classical result on the Lebesgue constant $\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}},$ being the norm of the interpolation operator ${\mathcal{L}}_n^{\gamma ,\delta }$ in the $\mathbf{C}$ space.

Lemma 6

([8], Theorem 14.4, p. 335). For all $n\in \mathbb{N},$ we have

$$\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}{\sim }_n\left\{\begin{array}{ccc}1+logn& :& -1<\gamma ,\delta \le -{\displaystyle \frac{1}{2}},\\ n^{max\left\{\gamma ,\delta \right\}+{\displaystyle \frac{1}{2}}}& :& \mathrm{otherwise}.\end{array}\right.$$

Note that, for $f\in \mathbf{C}$ and every $p\in {\mathbf{P}}_n$,

$$\parallel f-{\mathcal{L}}_n^{\gamma ,\delta }f{\parallel }_{\infty }=\parallel f-p+{\mathcal{L}}_n^{\gamma ,\delta }(p-f){\parallel }_{\infty }\le \left(1+\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\right){∥p-f∥}_{\infty }.$$

Hence,

$$\parallel f-{\mathcal{L}}_n^{\gamma ,\delta }f{\parallel }_{\infty }\le \left(1+\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\right)E_n{\left(f\right)}_{\infty },$$

(51)

where $E_n{\left(f\right)}_{\infty }$ denotes the best approximation of f by polynomials $p\in {\mathbf{P}}_n$ in the infinity norm,

$$E_n{\left(f\right)}_{\infty }=\underset{p\in {\mathbf{P}}_n}{inf}{∥p-f∥}_{\infty }.$$

We remember the following inequality (see [9], (2.5.22)):

$$E_n{\left(f\right)}_{\infty }\le {\displaystyle \frac{\mathcal{C}}{n^r}}\parallel f^{\left(r\right)}{\phi }^r{\parallel }_{\infty },N,r\in \mathbb{N},\mathcal{C}\ne \mathcal{C}(n,f),$$

(52)

which is true for all $f\in \mathbf{C}$ satisfying $f^{\left(r\right)}{\phi }^r\in \mathbf{C}$ and is a consequence of the iterated Favard inequality

$$E_n{\left(f\right)}_{\infty }\le {\displaystyle \frac{\mathcal{C}}{n^r}}{E}_{N-r}{\left(f^{\left(r\right)}\right)}_{{\phi }^r,\infty },f^{\left(r\right)}{\phi }^r\in \mathcal{C},n>r\in \mathbb{N},\mathcal{C}\ne \mathcal{C}(n,f),$$

(53)

with $E_n{\left(f\right)}_{{\phi }^r,\infty }={inf}_{p\in {\mathbf{P}}_n}\parallel (p-\psi ){\phi }^r{\parallel }_{\infty }.$ In what follows we will assume that, for some $r\in {\mathbb{N}}_0,$

(D)

the functions ${\displaystyle \overline{D}⟶\mathbb{C},(x,y)\mapsto \frac{{\partial }^{k}h(x,y)}{\partial x^k},}$ $k=0,1,\dots ,r,$ are continuous.

If $\kappa \ge 0$ and $u\in \mathbf{C},$ then one can see by induction and

$${\left({\mathcal{K}}_{\kappa }u\right)}^{\prime }\left(x\right)=\left\{\begin{array}{ccc}{\displaystyle {\int }_0^x\left[\frac{\partial h(x,y)}{\partial x}{(x-y)}^{\kappa }+\kappa h(x,y){(x-y)}^{\kappa -1}\right]u\left(y\right)dy}& :& \kappa >0,\\ {\displaystyle {\int }_0^x\frac{\partial h(x,y)}{\partial x}u\left(y\right)dy+h(x,x)u\left(x\right)}& :& \kappa =0,\end{array}\right.$$

(54)

that ${\mathcal{K}}_{\kappa }u$ belongs to ${\mathbf{C}}^{\left(r\right)}$ if $r\le 1+\kappa .$ Moreover, there is a constant $\mathcal{C}\ne \mathcal{C}(x,u)$ such that, for all $u\in \mathbf{C}$ and all $x\in [0,1],$

$$\left|{\left({\mathcal{K}}_{\kappa }u\right)}^{\left(r\right)}\left(x\right)\right|\le \mathcal{C}{∥u∥}_{\infty }.$$

(55)

For example, for $r=1$ and $\kappa >0$ we can take

$$\mathcal{C}=\underset{0\le x\le 1}{sup}{\int }_0^x\left[\left|{\displaystyle \frac{\partial h(x,y)}{\partial x}}\right|{(x-y)}^{\kappa }+\kappa \left|h(x,y)\right|{(x-y)}^{\kappa -1}\right]dy.$$

If we define the weighted modulus of smoothness ${\omega }_{\phi }{(f,t)}_{\infty }$ as

$${\omega }_{\phi }{(f,t)}_{\infty }=\underset{0<h\le t}{sup}{∥f\left(·+{\displaystyle \frac{h\phi }{2}}\right)-f\left(·-{\displaystyle \frac{h\phi }{2}}\right)∥}_{\infty },$$

where $f\left(x\pm {\displaystyle \frac{h\phi \left(x\right)}{2}}\right):=0$ if $x\pm {\displaystyle \frac{h\phi \left(x\right)}{2}}\notin [0,1],$ then the following Lemma is true (see [10], Theorem 7.2.19):

Lemma 7.

For $f\in \mathbf{C}$ and $n\in \mathbb{N},$

$$E_n{\left(f\right)}_{\infty }\le \mathcal{C}{\omega }_{\phi }{(f,n^{-1})}_{\infty },\mathcal{C}\ne \mathcal{C}(n,f).$$

In the following lemma we estimate the norm of the linear operator ${\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }:\mathbf{C}⟶\mathbf{C}$ for certain constellations of the parameters $\gamma ,\delta$ and $\kappa .$

Lemma 8.

Let $h(x,y)$ satisfy condition (D) and $\gamma ,\delta ,\kappa >-1.$ Then, for all $n\in \mathbb{N}$ and $f\in \mathbf{C},$

$$\parallel \left({\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }\right)f{\parallel }_{\infty }\le \mathcal{C}{n}^{-\rho }{∥f∥}_{\infty },\mathcal{C}\ne \mathcal{C}(n,f)),$$

where

(a)

$\rho =r,$ if $\kappa -\gamma \in {\mathbb{N}}_0,$ $\delta =0,$ and $r\in \mathbb{N};$

(b)

$\rho =1,$ if $\kappa -\gamma \in {\mathbb{N}}_0,$ $-{\displaystyle \frac{1}{2}}\le \delta <0$ or $0<\delta <{\displaystyle \frac{3}{2}},$ and $r=1;$

(c)

$\rho =\kappa -\gamma ,$ if $r-1<\kappa -\gamma <r,$ $\delta =0,$ and $r\in \mathbb{N};$

(d)

$\rho =min\left\{\kappa -\delta ,-\delta \right\},$ if $0<\kappa -\gamma <1,$ $-1<\delta <0,$ and $r=1.$

Proof. 

Note that we can use definition (22) for the operator ${\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }:\mathbf{C}⟶\mathbf{C}$ without further conditions like (23) in case of the ${\mathbf{L}}_{\gamma ,\delta }^2$ space. As in (50) we have, for $1<p<\infty$ and $f\in \mathbf{C},$

$$\begin{array}{cc}\hfill \parallel \left({\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }\right)f{\parallel }_{\infty }& \le \underset{0\le x\le 1}{sup}\parallel \left(g_x-{\mathcal{L}}_n^{\gamma ,\delta }{g}_x\right)v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^p}\parallel f(x·)v^{{\gamma }_1,{\delta }_1}{\parallel }_{{\mathbf{L}}^q}\hfill \\ \hfill & \le \underset{0\le x\le 1}{sup}\parallel \left(g_x-{\mathcal{L}}_n^{\gamma ,\delta }{g}_x\right)v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^p}\parallel v^{{\gamma }_1,{\delta }_1}{\parallel }_{{\mathbf{L}}^q}{∥f∥}_{\infty },\hfill \end{array}$$

(56)

where $g_x\left(y\right)=h(x,xy)v^{\kappa -\gamma ,-\delta }\left(y\right),$ ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,$ ${\gamma }_1,{\delta }_1>-{\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{p}}-1,$ and ${\gamma }_0+{\gamma }_1=\gamma ,$ ${\delta }_0+{\delta }_1=\delta .$

If $\kappa -\gamma \in {\mathbb{N}}_0$ and $\delta =0,$ then $g_x\in {\mathbf{C}}^{\left(r\right)}[0,1]$ holds uniformly with respect to $x\in [0,1].$ With ${\gamma }_0={\gamma }_1={\displaystyle \frac{\gamma }{2}}$ and ${\delta }_0={\delta }_1={\displaystyle \frac{\delta }{2}}$ in (56), we obtain, due to (39),

$$\underset{0\le x\le 1}{sup}\parallel \left(g_x-{\mathcal{L}}_n^{\gamma ,\delta }{g}_x\right)v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^2}=\underset{0\le x\le 1}{sup}\parallel g_x-{\mathcal{L}}_n^{\gamma ,\delta }{g}_x{\parallel }_{\gamma ,\delta }\le \mathcal{C}{n}^{-r},$$

and (a) is proven.

If $\kappa -\gamma$ is a non-negative integer, $-{\displaystyle \frac{1}{2}}\le \delta <{\displaystyle \frac{3}{2}},$ and $r=1,$ then $g_x^{\prime }\left(y\right)=g_1(x,y)y^{-\delta -1}$ with a continuous function $g_1:{[0,1]}^2⟶\mathbb{C}.$ We choose ${\gamma }_0,{\gamma }_1$ such that ${\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}<{\gamma }_0<{\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}}$ and ${\delta }_1>{\displaystyle \frac{1}{p}}-1,$ which is possible because of $\left({\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}\right)+\left({\displaystyle \frac{1}{p}}-1\right)={\displaystyle \frac{\gamma }{2}}-{\displaystyle \frac{3}{4}}<\gamma .$ Furthermore, we can choose ${\delta }_0,{\delta }_1$ such that ${\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}}>{\delta }_0>\delta +{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}$ (i.e., $g_x^{\prime }\phi v^{{\gamma }_0,{\delta }_0}\in {\mathbf{L}}^p$) and ${\delta }_1>{\displaystyle \frac{1}{p}}-1,$ which is possible since $\left(\delta +{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}\right)+\left({\displaystyle \frac{1}{p}}-1\right)=\delta -{\displaystyle \frac{1}{2}}<\delta .$ Finally, we have ${\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}\le \delta +{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}<{\displaystyle \frac{\delta }{2}}+{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}}$ due to $-{\displaystyle \frac{1}{2}}\le \delta <{\displaystyle \frac{3}{2}}.$ Consequently, from (38) we can infer

$$\parallel \left(g_x-{\mathcal{L}}_n^{\gamma ,\delta }{g}_x\right)v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^p}\le \mathcal{C}{n}^{-1}\parallel g_x^{\prime }\phi v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^p}\le \mathcal{C}{n}^{-1},\mathcal{C}\ne \mathcal{C}\left(x\right)$$

to obtain assertion (b).

If $r-1<\kappa -\gamma <r$ for some $r\in \mathbb{N}$ and $\delta =0,$ then $g_x^{(r-1)}\in {\mathbf{C}}^{\kappa -\gamma -r+1}[0,1],$ where

$$\left|g_x^{(r-1)}\left(y_1\right)-g_x^{(r-1)}\left(y_2\right)\right|\le \mathcal{C}{|y_1-y_2|}^{\kappa -\gamma -r+1},y_1,y_2\in [0,1],\mathcal{C}\ne \mathcal{C}\left(x\right).$$

From this we obtain

$${\Omega }_{\phi }^r{(g_x,t)}_{v^{{\gamma }_0,{\delta }_0}}\le \mathcal{C}{t}^{\kappa -\gamma }.$$

(57)

Indeed, from the mean value theorem, we have

$$\begin{array}{cc}\hfill \left({\Delta }_{h\phi }^{2}f\right)\left(x\right)& =f(x+h\phi (x\left)\right)-2f\left(x\right)+f(x-h\phi (x\left)\right)\hfill \\ & =\left[f(x+h\phi (x\left)\right)-f\left(x\right)\right]-\left[f\left(x\right)-f(x-h\phi (x\left)\right)\right]\hfill \\ & =\left[f^{\prime }\left({\xi }_2\right)-f^{\prime }\left({\xi }_1\right)\right]h\phi \left(x\right),\hfill \end{array}$$

where $x-h\phi \left(x\right)\le {\xi }_1\le x\le {\xi }_2\le x+h\phi \left(x\right).$ Hence,

$$\begin{array}{cc}\hfill \left({\Delta }_{h\phi }^{3}f\right)\left(x\right)& =\left({\Delta }_{h\phi }^1\left({\Delta }_{h\phi }^{2}f\right)\right)\left(x\right)=\left({\Delta }_{h\phi }^{2}f\right)(x+{\displaystyle \frac{h}{2}}\phi \left(x\right))-\left({\Delta }_{h\phi }^{2}f\right)(x-{\displaystyle \frac{h}{2}}\phi \left(x\right))\hfill \\ & =\left[f^{\prime }\left({\eta }_4\right)-f^{\prime }\left({\eta }_3\right)\right]h\phi \left(x\right)-\left[f^{\prime }\left({\eta }_2\right)-f^{\prime }\left({\eta }_1\right)\right]h\phi \left(x\right)\hfill \\ & =\left[f^{\prime \prime }\left({\zeta }_2\right)({\eta }_4-{\eta }_3)-f^{\prime \prime }\left({\zeta }_1\right)({\eta }_2-{\eta }_1)\right]h\phi \left(x\right)\hfill \end{array}$$

with $x-{\displaystyle \frac{3}{2}}h\phi \left(x\right)\le {\eta }_1\le x-{\displaystyle \frac{h}{2}}\phi \left(x\right)\le {\eta }_2,{\eta }_3\le x+{\displaystyle \frac{h}{2}}\phi \left(x\right)\le {\eta }_4\le x+{\displaystyle \frac{3}{2}}h\phi \left(x\right)$ as well as $x-{\displaystyle \frac{3}{2}}h\phi \left(x\right)\le {\zeta }_1,{\zeta }_2\le x+{\displaystyle \frac{3}{2}}h\phi \left(x\right).$ Consequently, $\left|\left({\Delta }_{h\phi }^{3}f\right)\left(x\right)\right|\le \mathcal{C}\left|f^{\prime \prime }\left({\zeta }_2\right)-f^{\prime \prime }\left({\zeta }_1\right)\right|{\left[h\phi \left(x\right)\right]}^2.$ Since $\left({\Delta }_{h\phi }^{r}f\right)\left(x\right)=\left({\Delta }_{h\phi }^1\left({\Delta }_{h\phi }^{r-1}f\right)\right)\left(x\right)$ we obtain by induction

$$\left({\Delta }_{h\phi }^{r}f\right)\left(x\right)\le \mathcal{C}\left|f^{(r-1)}\left({\omega }_2\right)-f^{(r-1)}\left({\omega }_1\right)\right|{\left[h\phi \left(x\right)\right]}^{r-1}$$

for some ${\omega }_1,{\omega }_2\in \left[x-{\displaystyle \frac{r}{2}}h\phi \left(x\right),x+{\displaystyle \frac{r}{2}}h\phi \left(x\right)\right].$ This leads to

$$\begin{array}{cc}\hfill \left|\left({\Delta }_{h\phi }^r{g}_x\right)\left(x\right)\right|& \le \mathcal{C}\left|g_x^{(r-1)}\left({\omega }_2\right)-g_x^{(r-1)}\left({\omega }_1\right)\right|{\left[h\phi \left(x\right)\right]}^{r-1}\hfill \\ & \le \mathcal{C}{\left|{\omega }_2-{\omega }_1\right|}^{\kappa -\gamma -r+1}{\left[h\phi \left(x\right)\right]}^{r-1}\le \mathcal{C}{h}^{\kappa -\gamma },\hfill \end{array}$$

from which (57) follows.

Now, choose a large enough $p\in (1,\infty )$ such that $\kappa -\gamma >{\displaystyle \frac{1}{p}},$ ${\gamma }_1,{\delta }_1>{\displaystyle \frac{1}{p}}-1,$ ${\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}<{\gamma }_0<{\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}},$ ${\gamma }_0+{\gamma }_1=\gamma ,$ ${\delta }_0=-{\delta }_1,$ and ${\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}<{\delta }_0<{\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}}.$ This is possible, since $\left({\displaystyle \frac{\gamma }{2}}+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}\right)+\left({\displaystyle \frac{1}{p}}-1\right)={\displaystyle \frac{\gamma }{2}}-{\displaystyle \frac{3}{4}}<\gamma$ and $\left({\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}\right)+\left({\displaystyle \frac{1}{p}}-1\right)=-{\displaystyle \frac{3}{4}}<0={\delta }_0+{\delta }_1<{\displaystyle \frac{9}{4}}-{\displaystyle \frac{2}{p}}=\left({\displaystyle \frac{5}{4}}-{\displaystyle \frac{1}{p}}\right)+\left(1-{\displaystyle \frac{1}{p}}\right).$ Applying (47) yields

$$\parallel \left(g_x-{\mathcal{L}}_n^{\gamma ,\delta }{g}_x\right)v^{{\gamma }_0,{\delta }_0}{\parallel }_{{\mathbf{L}}^p}\le \mathcal{C}{n}^{-{\displaystyle \frac{1}{p}}}{\int }_0^{{\displaystyle \frac{1}{n}}}{t}^{\kappa -\gamma -1-{\displaystyle \frac{1}{p}}}dt=\mathcal{C}{n}^{-(\kappa -\gamma )},$$

and the proof of (c) is finished.

Assertion (d) can be proved analogously to (c). □

Proposition 3.

Assume that $\kappa \in {\mathbb{N}}_0$ and that, for some $r\in \mathbb{N},$ the kernel function $h(x,y)$ fulfills condition (D). Furthermore, assume that the right-hand side of (6) belongs to ${\mathbf{C}}^{\left(r\right)}.$ Moreover, let $\gamma ,\delta >-1$ be chosen in such a way that one of the conditions (a)–(d) of Lemma 8 and the condition

$$max\left\{\gamma ,\delta \right\}+{\displaystyle \frac{1}{2}}<min\left\{1+\kappa ,\rho \right\}$$

(58)

with ρ defined in Lemma 8 are satisfied. Then, for all sufficiently large $n,$ the collocation–quadrature Equation (37) has a unique solution $u_n^{\ast }\in \mathbf{C},$ which converges in the infinity norm to the unique solution $u^{\ast }\in \mathbf{C}$ of (6), where, for $f_n={\mathcal{L}}_n^{\gamma ,\delta }f,$

$${∥u_n^{\ast }-u^{\ast }∥}_{\infty }\le \mathcal{C}{n}^{-\rho }\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}$$

(59)

with a constant $\mathcal{C}\ne \mathcal{C}\left(n\right).$

Proof. 

The unique solvability of (6) in $\mathbf{C}$ is guaranteed by Lemma 1. Now, our aim is to estimate

$$\parallel {\mathcal{K}}_{\kappa }-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\le \parallel {\mathcal{K}}_{\kappa }-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa }{\parallel }_{\mathbf{C}\to \mathbf{C}}+\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\parallel {\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}.$$

(60)

For the second addend on the right-hand side, we can use Lemma 6 and Lemma 8, while for the first addend we take into account ${\mathcal{K}}_{\kappa }g\in {\mathbf{C}}^{\left(s\right)}$ and $\left|{\left({\mathcal{K}}_{\kappa }g\right)}^{\left(s\right)}\left(x\right)\right|\le \mathcal{C}{∥g∥}_{\infty }$ for all $g\in \mathbf{C}$ and $s=1+\kappa$ (see (55)). Hence, due to (51) and (52),

$${∥{\mathcal{K}}_{\kappa }g-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa }g∥}_{\infty }\le \mathcal{C}{n}^{-(1+\kappa )}\left\{\begin{array}{ccc}1+logn& :& -1<\gamma ,\delta \le -{\displaystyle \frac{1}{2}}\\ n^{max\left\{\gamma ,\delta \right\}+{\displaystyle \frac{1}{2}}}& :& \mathrm{otherwise}\end{array}\right\}{∥g∥}_{\infty }.$$

Thus, from (59) together with (58), we infer

$$\parallel {\mathcal{K}}_{\kappa }-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\le \mathcal{C}\left(n^{-(1+\kappa )}+n^{-\rho }\right)\left\{\begin{array}{ccc}1+logn& :& -1<\gamma ,\delta \le -{\displaystyle \frac{1}{2}}\\ n^{max\left\{\gamma ,\delta \right\}+{\displaystyle \frac{1}{2}}}& :& \mathrm{otherwise}\end{array}\right\}⟶0,$$

which, in view of the invertibility of the operator $\mathcal{I}-{\mathcal{K}}_{\kappa }:\mathbf{C}⟶\mathbf{C}$ and a standard Neumann series argument, implies the invertibility of $\mathcal{I}-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }:\mathbf{C}⟶\mathbf{C}$ for all sufficiently large $n,$ say $n>n_0,$ and

$${\mathcal{C}}_{\ast }:=sup\left\{{∥{\left(\mathcal{I}-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }\right)}^{-1}∥}_{\mathbf{C}\to \mathbf{C}}\right\}<\infty .$$

Since $\kappa \ge 0,$ from (54) and $u^{\ast }=f+{\mathcal{K}}_{\kappa }{u}^{\ast }$, we conclude $u^{\ast }\in {\mathbf{C}}^{\left(1\right)}.$ By induction we obtain ${\mathcal{K}}_{\kappa }{u}^{\ast }\in {\mathbf{C}}^{\left(r\right)}$ and, for $f_n={\mathcal{L}}_n^{\gamma ,\delta }f,$

$$\begin{array}{cc}& {∥u_n^{\ast }-u^{\ast }∥}_{\infty }\hfill \\ & \le {\mathcal{C}}_{\ast }\parallel \left(\mathcal{I}-{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }\right)(u_n^{\ast }-u^{\ast }){\parallel }_{\infty }\hfill \\ & \le {\mathcal{C}}_{\ast }\left(\parallel {\mathcal{L}}_n^{\gamma ,\delta }f-f{\parallel }_{\infty }+\parallel {\mathcal{K}}_{\kappa }{u}^{\ast }-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa }{u}^{\ast }{\parallel }_{\infty }+\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\parallel {\mathcal{K}}_{\kappa }-{\mathcal{K}}_{\kappa ,n}^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}{∥u^{\ast }∥}_{\infty }\right)\hfill \\ & \le \mathcal{C}\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}{n}^{-\rho },\hfill \end{array}$$

taking into account (51) and (52) as well as Lemma 8, together with $\rho \le r.$ This completes the proof. □

Proposition 4.

Let $r\in {\mathbb{N}}_0,$ $r-1<\kappa <r,$ and condition (D) be satisfied, where

$$\underset{0\le y\le x_1}{sup}\left|{\displaystyle \frac{{\partial }^{r}h(x_2,y)}{\partial y^r}}-{\displaystyle \frac{{\partial }^{r}h(x_1,y)}{\partial y^r}}\right|\le \mathcal{C}{(x_2-x_1)}^{{\chi }_0}\forall x_1,x_2,0\le x_1<x_2\le 1,\mathcal{C}\ne \mathcal{C}(x_1,x_2),$$

for a certain number ${\chi }_0\in (0,1].$ Moreover, let $f\in {\mathbf{C}}^{\left(r\right)}$ with $f^{\left(r\right)}\in {\mathbf{C}}^{min\left\{{\chi }_0,1+\kappa -r\right\}}[0,1].$ Assume that the parameters $\gamma ,\delta >-1$ fulfill one of the conditions (a)–(d) as well as condition (58). Then, for all sufficiently large $n,$ the collocation–quadrature Equation (37) has a unique solution $u_n^{\ast }\in \mathbf{C},$ which converges in the infinity norm to the unique solution $u^{\ast }\in \mathbf{C}$ of (6), where, for $f_n={\mathcal{L}}_n^{\gamma ,\delta }f,$

$${∥u_n^{\ast }-u^{\ast }∥}_{\infty }\le \mathcal{C}{n}^{-\chi }\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}$$

(61)

with $\chi =min\left\{{\chi }_0+r,1+\kappa ,\rho \right\}$ and a constant $\mathcal{C}\ne \mathcal{C}\left(n\right).$

Proof. 

We can proceed as in the proof for the previous proposition. We only have to check the estimate of $\parallel {\mathcal{K}}_{\kappa }{u}^{\ast }-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa }{u}^{\ast }{\parallel }_{\infty }.$ Since $r-1<\kappa <r,$ from the proof of (46), we can infer ${\left({\mathcal{K}}_{\kappa }u\right)}^{\left(r\right)}\in {\mathbf{C}}^{min\left\{{\chi }_0,1+\kappa \right\}-r}[0,1],$ where

$$\left|{\left({\mathcal{K}}_{\kappa }u\right)}^{\left(r\right)}\left(x_1\right)-{\left({\mathcal{K}}_{\kappa }u\right)}^{\left(r\right)}\left(x_2\right)\right|\le \mathcal{C}{∥u∥}_{\infty }{|x_1-x_2|}^{min\left\{{\chi }_0,1+\kappa -r\right\}},$$

(62)

$x_1,x_2\in [0,1],$ $u\in \mathbf{C},$ $\mathcal{C}\ne \mathcal{C}(u,x_1,x_2).$ This leads to

$$\begin{array}{cc}\hfill \parallel {\mathcal{K}}_{\kappa }{u}^{\ast }-{\mathcal{L}}_n^{\gamma ,\delta }{\mathcal{K}}_{\kappa }{u}^{\ast }{\parallel }_{\infty }& \stackrel{\left(51\right)}{\le }\left(1+\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\right)E_n{\left({\mathcal{K}}_{\kappa }{u}^{\ast }\right)}_{\infty }\hfill \\ & \stackrel{\left(53\right)}{\le }\mathcal{C}{n}^{-r}\left(1+\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\right)E_{n-r}{\left({\left({\mathcal{K}}_{\kappa }{u}^{\ast }\right)}^{\left(r\right)}\right)}_{\infty }\hfill \\ & \stackrel{\mathrm{Lemma}7}{\le }\mathcal{C}{n}^{-r}\left(1+\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\right){\omega }_{\phi }{\left({\left({\mathcal{K}}_{\kappa }{u}^{\ast }\right)}^{\left(r\right)},n^{-1}\right)}_{\infty }\hfill \\ & \stackrel{\left(62\right)}{\le }\mathcal{C}\left(1+\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\right)n^{-r-min\left\{{\chi }_0,1+\kappa -r\right\}},\hfill \end{array}$$

where we took into account that, for $g\in {\mathbf{C}}^{\psi }[0,1],$

$${\omega }_{\phi }{(g,t)}_{\infty }\le \mathcal{C}\underset{0<h\le t}{sup}\underset{0\le x\le 1}{sup}{h}^{\psi }{\left[\phi \left(x\right)\right]}^{\psi }=\mathcal{C}{t}^{\psi }.$$

Analogously, we get $\parallel {\mathcal{L}}_n^{\gamma ,\delta }f-f{\parallel }_{\infty }\le \mathcal{C}{n}^{-r-{\chi }_0}.$ Thus, as at the end of the proof of Proposition 3,

$$\begin{array}{cc}\hfill {∥u_n^{\ast }-u^{\ast }∥}_{\infty }& \le \mathcal{C}\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}\left(n^{-r-{\chi }_0}+n^{-r-min\left\{{\chi }_0,1+\kappa -r\right\}}+n^{-\rho }\right)\hfill \\ & \le \mathcal{C}\parallel {\mathcal{L}}_n^{\gamma ,\delta }{\parallel }_{\mathbf{C}\to \mathbf{C}}{n}^{-min\left\{{\chi }_0+r,1+\kappa ,\rho \right\}},\hfill \end{array}$$

and the proposition is proved. □

4. What About the Nyström Method?

In different papers the Nyström method was studied for Volterra integral equations (see, for example, [11,12,13,14,15,16] for linear equations and [17,18] for nonlinear equations).

Let us use the interpolation operator ${\mathcal{L}}_n^{\alpha ,\beta },$ $\alpha ,\beta >-1$ to construct a Nyström approximation for the solution of (6) based on a product integration rule, namely, (cf. [13], (6),(7))

$$\begin{array}{cc}\hfill \left({\mathcal{K}}_{\kappa ,n}^{N,\alpha ,\beta }u\right)\left(x\right)& ={\int }_0^x{(x-y)}^{\kappa }{\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,·)v^{-\alpha ,-\beta }u\right]\left(y\right)v^{\alpha ,\beta }\left(y\right)dy\hfill \\ \hfill & =\sum _{k=1}^n{q}_{nk}^{\alpha ,\beta }\left(x\right)h(x,x_{nk}^{\alpha ,\beta })v^{-\alpha ,-\beta }\left(x_{nk}^{\alpha ,\beta }\right)u\left(x_{nk}^{\alpha ,\beta }\right)=:\sum _{k=1}^n{\Lambda }_{nk}^{\alpha ,\beta }\left(x\right)u\left(x_{nk}^{\alpha ,\beta }\right)\hfill \end{array}$$

(63)

with

$$q_{nk}^{\alpha ,\beta }\left(x\right)={\int }_0^x{(x-y)}^{\kappa }{v}^{\alpha ,\beta }\left(y\right){\ell }_{nk}^{\alpha ,\beta }\left(y\right)dy={\lambda }_{nk}^{\alpha ,\beta }\sum _{j=0}^{n-1}{p}_j^{\alpha ,\beta }\left(x_{nk}^{\alpha ,\beta }\right){\int }_0^x{p}_j^{\alpha ,\beta }\left(y\right){(x-y)}^{\kappa }{v}^{\alpha ,\beta }\left(y\right)dy.$$

With the aim of studying the Nyström method for Equation (6) in the weighted space ${\tilde{\mathbf{C}}}_{\gamma ,\delta }$ of continuous functions, we multiply equation

$$\left(\mathcal{I}+{\mathcal{K}}_{\kappa ,n}^{N,\alpha ,\beta }\right)u_n=f$$

(64)

by $v^{\gamma ,\delta }\left(x\right),$ collocate at the points $x_{nj}^{\alpha ,\beta },$ $j=1,\dots ,n,$ and take $a_k:=v^{\gamma ,\delta }\left(x_{nk}^{\alpha ,\beta }\right)u\left(x_{nk}^{\alpha ,\beta }\right),$ $k=1,\dots ,n,$ as the unknowns. This results in the following system:

$$\sum _{k=1}^n\left[{\delta }_{jk}+q_k\left(x_{nj}^{\alpha ,\beta }\right)v^{\gamma ,\delta }\left(x_{nj}^{\alpha ,\beta }\right)h(x_{nj}^{\alpha ,\beta },x_{nk}^{\alpha ,\beta })v^{-\alpha -\gamma ,-\beta -\delta }\left(x_{nk}^{\alpha ,\beta }\right)\right]a_k=\tilde{f}\left(x_{nj}^{\alpha ,\beta }\right),$$

(65)

$j=1,\dots ,n,$ where $\tilde{f}\left(x\right):=v^{\gamma ,\delta }\left(x\right)f\left(x\right).$ Note that we obtain the same system (65) if we consider equation

$$\left(\mathcal{I}-{\tilde{K}}_{\kappa }\right)\tilde{u}:=\left(\mathcal{I}-\mathcal{T}{\mathcal{K}}_{\kappa }{\mathcal{T}}^{-1}\right)\tilde{u}=\tilde{f}:=\mathcal{T}f$$

(66)

(see (7)) in the $\mathbf{C}$ space of continuous functions on $[0,1]$ and approximate the operator ${\tilde{\mathcal{K}}}_{\kappa },$

$$\left({\tilde{\mathcal{K}}}_{\kappa }\tilde{u}\right)\left(x\right)={\int }_0^x{(x-y)}^{\kappa }\tilde{h}(x,y)\tilde{u}\left(y\right)dy,\tilde{h}(x,y)=v^{\gamma ,\delta }\left(x\right)h(x,y)v^{-\gamma ,-\delta }\left(y\right),$$

by

$$\begin{array}{cc}\hfill \left({\tilde{\mathcal{K}}}_{\kappa ,n}^{N,\alpha ,\beta }\tilde{u}\right)\left(x\right)& ={\int }_0^x{(x-y)}^{\kappa }{\mathcal{L}}_n^{\alpha ,\beta }\left[\tilde{h}(x,·)v^{-\alpha ,-\beta }\tilde{u}\right]\left(y\right)v^{\alpha ,\beta }\left(y\right)dy\hfill \\ & =\sum _{k=1}^n{q}_k\left(x\right)\tilde{h}(x,x_{nk}^{\alpha ,\beta })v^{-\alpha ,-\beta }\left(x_{nk}^{\alpha ,\beta }\right)\tilde{u}\left(x_{nk}^{\alpha ,\beta }\right).\hfill \end{array}$$

That is why we will study (63) in the $\mathbf{C}$ space assuming that $h:D⟶\mathbb{C}$ is a continuous function.

We refer to the notation in ([6], Chapter 6) and consider an operator

$${\mathcal{K}}_{\kappa }:\mathbf{C}[0,1]⟶\mathbf{C}[0,1],\left({\mathcal{K}}_{\kappa }f\right)\left(x\right)={\int }_0^{1}K(x,y)f\left(y\right)dy$$

with

$$K(x,y)=\left\{\begin{array}{ccc}h(x,y){(x-y)}^{\kappa }& :& 0\le y\le x,\\ 0& :& x<y\le 1,\end{array}\right.$$

where $\kappa >-1$ is a fixed real number. For a function $f:[0,1]⟶\mathbb{C},$ $\gamma ,\delta \in \mathbb{R},$ and $0\le a\le b\le 1,$ we use the following notations (cf. the beginning of Section 2):

• ${∥f∥}_{\gamma ,\delta ,\infty ,[a,b]}=sup\left\{\left|f\left(y\right)\right|v^{\gamma ,\delta }\left(y\right):a\le y\le b\right\},$${∥f∥}_{\gamma ,\delta ,\infty }={∥f∥}_{{\tilde{\mathbf{C}}}_{\gamma ,\delta }}={∥f∥}_{\gamma ,\delta ,\infty ,[0,1]},$
• ${\displaystyle {∥f∥}_{\gamma ,\delta ,p,[a,b]}={\left({\int }_a^b\left|f\left(y\right)\right|v^{\gamma ,\delta }\left(y\right)dy\right)}^{\frac{1}{p}},}$${∥f∥}_{\gamma ,\delta ,p}={∥f∥}_{{\mathbf{L}}_{\gamma ,\delta }^p}={∥f∥}_{\gamma ,\delta ,p,[0,1]}.$

Let us consider conditions (K1)–(K4) in ([6], p. 356) (cf. also [19]). The first condition is

$$\left(\mathrm{K}1\right):{\int }_0^1\left|K(x,y)\right|dy<\infty \forall x\in [0,1].$$

(67)

We estimate

$${\int }_0^x\left|h(x,y)\right|{(x-y)}^{\kappa }dy\le \left\{\begin{array}{c}{\displaystyle {∥h(x,·)∥}_{\gamma ,\delta ,\infty ,[0,x]}{\int }_0^x{(x-y)}^{\kappa }{v}^{-\gamma ,-\delta }\left(y\right)dy,}\hfill \\ {\displaystyle {∥h(x,·)∥}_{\gamma ,\delta ,p,[0,x]}{\left({\int }_0^x{(x-y)}^{\kappa q}{v}^{-\gamma ,-\delta }\left(y\right)dy\right)}^{\frac{1}{q}},}\hfill \end{array}\right.$$

(68)

where (also hereafter) $1<p<\infty$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.$ Note that, for $q\in [1,\infty )$ and all $x\in [0,1],$ the integral

$${\int }_0^x{(x-y)}^{\kappa q}{v}^{-\gamma ,-\delta }\left(y\right)dy=x^{1+\kappa q-\delta }{\int }_0^1{(1-z)}^{\kappa q}{(1-xz)}^{-\gamma }{z}^{-\delta }dz$$

is finite if and only if

$$1+\kappa q-\delta \ge 0,1+\kappa q-\gamma >0,1+\kappa q>0,\mathrm{and}1>\delta .$$

(69)

Let us turn to the second condition:

$$\left(\mathrm{K}2\right):\underset{x\to x_0}{lim}{\int }_0^1|K(x,y)-K(x_0,y)|dy=0\forall x_0\in [0,1].$$

(70)

Let $0\le x_1<x_2\le 1.$ Then,

$$\begin{array}{cc}& {\int }_0^1|K(x_2,y)-K(x_1,y)|dy\hfill \\ & ={\int }_{x_1}^{x_2}\left|h(x_2,y)\right|{(x_2-y)}^{\kappa }dy+{\int }_0^{x_1}\left|h(x_2,y){(x_2-y)}^{\kappa }-h(x_1,y){(x_1-y)}^{\kappa }\right|dy\hfill \\ & \le {\int }_{x_1}^{x_2}\left|h(x_2,y)\right|{(x_2-y)}^{\kappa }dy+{\int }_0^{x_1}\left|h(x_2,y)-h(x_1,y)\right|{(x_2-y)}^{\kappa }dy\hfill \\ & +{\int }_0^{x_1}\left|h(x_1,y)\right|\left|{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right|dy\hfill \\ & =:I_1+I_2+I_3.\hfill \end{array}$$

Let us estimate these three terms separately. Analogously to (68) we get

$$I_1\le \left\{\begin{array}{c}{\displaystyle {∥h(x_2,·)∥}_{\gamma ,\delta ,\infty ,[x_1,x_2]}{\int }_{x_1}^{x_2}{(x_2-y)}^{\kappa }{v}^{-\gamma ,-\delta }\left(y\right)dy,}\hfill \\ {\displaystyle {∥h(x_2,·)∥}_{\gamma ,\delta ,p,[x_1,x_2]}{\left({\int }_{x_1}^{x_2}{(x_2-y)}^{\kappa q}{v}^{-\gamma ,-\delta }\left(y\right)dy\right)}^{\frac{1}{q}},}\hfill \end{array}\right.$$

where, for $q\in [1,\infty ),$

$$\begin{array}{cc}& {\int }_{x_1}^{x_2}{(x_2-y)}^{\kappa q}{v}^{-\gamma ,-\delta }\left(y\right)dy\hfill \\ & ={(x_2-x_1)}^{1+\kappa q}{\int }_0^1{(1-z)}^{\kappa q}{\left[1-x_1-(x_2-x_1)z\right]}^{-\gamma }{\left[(x_2-x_1)z+x_1\right]}^{-\delta }dz\hfill \\ & \le \mathcal{C}{(x_2-x_1)}^{1+\kappa q-max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}}{\int }_0^1{(1-z)}^{\kappa q-max\left\{0,\gamma \right\}}{z}^{-max\left\{0,\delta \right\}}dz\hfill \\ & \le \mathcal{C}{(x_2-x_1)}^{1+\kappa q-max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}}⟶0if{x}_2-x_1⟶0,\hfill \end{array}$$

if

$$1+\kappa q-max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}>0and1>\delta .$$

(71)

For the second term we get

$$I_2\le \left\{\begin{array}{c}{\displaystyle {∥h(x_2,·)-h(x_1,·)∥}_{\gamma ,\delta ,\infty ,[0,x_1]}{\int }_0^{x_1}{(x_2-y)}^{\kappa }{v}^{-\gamma ,-\delta }\left(y\right)dy}\hfill \\ {\displaystyle {∥h(x_2,·)-h(x_1,·)∥}_{\gamma ,\delta ,p,[0,x_1]}{\left({\int }_0^{x_1}{(x_2-y)}^{\kappa q}{v}^{-\gamma ,-\delta }\left(y\right)dy\right)}^{\frac{1}{q}},}\hfill \end{array}\right.$$

where, for $q\in [1,\infty ),$

$$\begin{array}{cc}\hfill {\int }_0^{x_1}{(x_2-y)}^{\kappa q}{v}^{-\gamma ,-\delta }\left(y\right)dy& \le {\int }_0^{x_1}{(x_2-y)}^{\kappa q-max\left\{0,\gamma \right\}}{y}^{-\delta }dy\hfill \\ & \le \mathcal{C}{x}_2^{1+\kappa q-max\left\{0,\gamma \right\}-\delta }{\int }_0^1{(1-z)}^{\kappa q-max\left\{0,\gamma \right\}}{z}^{-\delta }dz\hfill \\ & \le \mathcal{C}\ne \mathcal{C}\left(x_2\right),\hfill \end{array}$$

if and only if

$$1+\kappa q-max\left\{0,\gamma \right\}-\delta \ge 0,1+\kappa q-max\left\{0,\gamma \right\}>0,and1>\delta .$$

(72)

Finally, we estimate the third term and obtain

$$I_3\le \left\{\begin{array}{c}{\displaystyle {∥h(x_1,·)∥}_{\gamma ,\delta ,\infty ,[0,x_1]}{\int }_0^{x_1}\left|{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right|v^{-\gamma ,-\delta }\left(y\right)dy,}\hfill \\ {\displaystyle {∥h(x_1,·)∥}_{\gamma ,\delta ,p,[0,x_1]}{\left({\int }_0^{x_1}{\left|{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right|}^q{v}^{-\gamma ,-\delta }\left(y\right)dy\right)}^{\frac{1}{q}},}\hfill \end{array}\right.$$

where, for $q\in [1,\infty )$ and $\kappa >0,$

$${\int }_0^{x_1}{\left|{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right|}^q{v}^{-\gamma ,-\delta }\left(y\right)dy\le \mathcal{C}{(x_2-x_1)}^{qmin\left\{1,\kappa \right\}}{\int }_0^1{v}^{-\gamma ,-\delta }\left(y\right)dy⟶0$$

(see also (15)) for $x_2-x_1⟶0,$ if $1>\gamma ,\delta .$ For $-1<\kappa <0,$ we choose $\chi >0$ such that $\kappa +\chi <0$ and again use the relation (15) to obtain

$$\begin{array}{cc}& {\int }_0^{x_1}{\left|{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right|}^q{v}^{-\gamma ,-\delta }\left(y\right)dy\hfill \\ & ={\int }_0^{x_1}{(x_1-y)}^{\kappa q}{\left[{(x_2-y)}^{-\kappa }-{(x_1-y)}^{-\kappa }\right]}^q{(x_2-y)}^{\kappa q}{v}^{-\gamma ,-\delta }\left(y\right)dy\hfill \\ & \le {(x_2-x_1)}^{-\kappa q}{\int }_0^{x_1}{(x_1-y)}^{\kappa q}{(x_2-y)}^{\kappa q}{v}^{-\gamma ,-\delta }\left(y\right)dy\hfill \\ & ={(x_2-x_1)}^{-\kappa q}{\int }_0^{x_1}{(x_2-y)}^{\kappa q}{(x_2-y)}^{(\kappa +\chi )q}{(x_2-y)}^{-\chi q}{v}^{-\gamma ,-\delta }\left(y\right)dy\hfill \\ & \le {(x_2-x_1)}^{\chi q}{x}_1^{1+(\kappa -\chi )q-\delta }{\int }_0^1{(1-z)}^{(\kappa -\chi )q-max\left\{0,\gamma \right\}}{z}^{-\delta }dz\hfill \\ & \le \mathcal{C}{(x_2-x_1)}^{\chi q},\hfill \end{array}$$

if

$$1+\kappa q-max\left\{0,\gamma \right\}>0,1+\kappa q-\delta >0,and1>\delta ,$$

(73)

and $\chi >0$ is small enough.

Note that (K1) and (K2) are necessary and sufficient for ${\mathcal{K}}_{\kappa }:\mathbf{C}⟶\mathbf{C}$ to be a compact operator (see [6], Proposition 5.3.2). Thus, by using (69), (71), (72), and (73), we obtain the following corollary:

Corollary 5.

The operator ${\mathcal{K}}_{\kappa }:\mathbf{C}⟶\mathbf{C}$ is compact if one of the following conditions is satisfied:

(A)

There exist numbers $\gamma ,\delta \in (-1,1),$ such that

$$1+\kappa -max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}>0,$$

$$sup\left\{{∥h(x,·)∥}_{\gamma ,\delta ,\infty ,[0,x]}:x\in [0,1]\right\}<\infty$$

and, for all $x_0\in [0,1),$

$$\underset{x\to x_0+0}{lim}{∥h(x,·)-h(x_0,·)∥}_{\gamma ,\delta ,\infty ,[0,x_0]}=0.$$

(B)

There exist numbers $\gamma ,\delta \in (-1,1)$ and $p>1,$ such that, for ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,$

$$1+\kappa q-max\left\{0,\gamma \right\}-max\left\{0,\delta \right\}>0,$$

$$sup\left\{{∥h(x,·)∥}_{\gamma ,\delta ,p,[0,x]}:x\in [0,1]\right\}<\infty$$

and, for all $x_0\in [0,1),$

$$\underset{x\to x_0+0}{lim}{∥h(x,·)-h(x_0,·)∥}_{\gamma ,\delta ,p,[0,x_0]}=0.$$

The following two conditions (K3) and (K4) concern the question as to whether the operators ${\mathcal{K}}_{\kappa ,n}^{N,\alpha ,\beta }:\mathbf{C}⟶\mathbf{C}$ form a collectively compact and strongly converging operator sequence. In particular, the following lemma is true (see [6], Lemma 6.1.1, cf. also [19], Section 2, Lemma and [20], Section 3, Theorem 1).

Lemma 9.

Suppose that the conditions (K1) and (K2) are in force. Then, the operators ${\mathcal{K}}_{\kappa ,n}^{N,\alpha ,\beta }:\mathbf{C}⟶\mathbf{C},$ $n\in \mathbb{N},$ are collectively compact and strongly convergent to ${\mathcal{K}}_{\kappa }:\mathbf{C}⟶\mathbf{C}$ if and only if the conditions (K3) and (K4) below are satisfied.

Remembering the definition of ${\mathcal{K}}_{\kappa ,n}^{N,\alpha ,\beta }$ (see (63)), we consider the third condition:

$$\left(\mathrm{K}3\right):\underset{n\to \infty }{lim}\sum _{k=1}^n{\Lambda }_{nk}^{\alpha ,\beta }\left(x\right)f\left(x_{nk}^{\alpha ,\beta }\right)={\int }_0^{1}K(x,y)f\left(y\right)dy\forall x\in [0,1],\forall f\in \mathbf{C}[0,1].$$

(74)

To obtain conditions under which (74) is fulfilled, we estimate

$$\begin{array}{cc}& {\int }_0^x{(x-y)}^{\kappa }\left|{\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,·)v^{-\alpha ,-\beta }f\right]\left(y\right)-h(x,y)v^{-\alpha ,-\beta }\left(y\right)f\right|v^{\alpha ,\beta }\left(y\right)dy\hfill \\ & \le \left\{\begin{array}{c}{\displaystyle {∥{\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,·)v^{-\alpha ,-\beta }f\right]-h(x,·)v^{-\alpha ,-\beta }f∥}_{\gamma ,\delta ,\infty ,[0,x]}{\int }_0^x{(x-y)}^{\kappa }{v}^{-\alpha -\gamma ,-\beta -\delta }\left(y\right)dy,}\hfill \\ {\displaystyle {∥{\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,·)v^{-\alpha ,-\beta }f\right]-h(x,·)v^{-\alpha ,-\beta }f∥}_{\gamma ,\delta ,p,[0,x]}{\int }_0^x{(x-y)}^{\kappa q}{v}^{-\alpha q-\gamma ,-\beta q-\delta }\left(y\right)dy,}\hfill \end{array}\right.\hfill \end{array}$$

where, for some $q\in [1,\infty )$ and all $x\in [0,1],$ the integral

$${\int }_0^x{(x-y)}^{\kappa q}{v}^{-\alpha q-\gamma ,-\beta q-\delta }\left(y\right)dy$$

is finite if and only if (compare with (69))

$$1+\kappa q-\beta q-\delta \ge 0,1+\kappa q-\alpha q-\gamma >0,1+\kappa q>0,and1>\beta q+\delta .$$

(75)

Moreover, we have to find conditions under which

$$\underset{n\to \infty }{lim}{∥{\mathcal{L}}_n^{\alpha ,\beta }\left[h(x,·)v^{-\alpha ,-\beta }f\right]-h(x,·)v^{-\alpha ,-\beta }∥}_{\gamma ,\delta ,p,[0,x]}=0$$

(76)

is true for all $x\in [0,1]$ and for some $p\in (1,\infty ].$ Let us restrict to $p=2$ and try to apply Lemma 4. If we set

$$g_x\left(y\right)=\left\{\begin{array}{ccc}{\displaystyle h(x,y)v^{-\alpha ,-\beta }\left(y\right)f\left(y\right)}& :& 0\le y\le x,\\ {\displaystyle 0}& :& x<y\le 1,\end{array}\right.$$

we can replace condition (76) with

$$\underset{n\to \infty }{lim}{∥{\mathcal{L}}_n^{\alpha ,\beta }{g}_x-g_x∥}_{\gamma ,\delta }=0\forall x\in [0,1].$$

(77)

Applying Lemma 4 when $\psi =\chi =0$, we obtain the conditions

$$-{\displaystyle \frac{1}{2}}<\gamma -\alpha ,\delta -\beta <{\displaystyle \frac{3}{2}}$$

(78)

and, for some $\epsilon >0$ and all $x\in [0,1],$

$$h_x\in \mathbf{R}(0,x),\left|h(x,y)\right|\le \mathcal{C}{(1-y)}^{\epsilon +\alpha -{\displaystyle \frac{1+\gamma }{2}}}{y}^{\epsilon +\beta -{\displaystyle \frac{1+\delta }{2}}}\forall y\in (0,x).$$

(79)

Let us turn to the last condition:

$$\left(\mathrm{K}4\right):\underset{x\to x_0}{lim}sup\left\{\sum _{k=1}^n\left|{\Lambda }_{nk}^{\alpha ,\beta }\left(x\right)-{\Lambda }_{nk}^{\alpha ,\beta }\left(x_0\right)\right|:n\in \mathbb{N}\right\}=0\forall x_0\in [0,1].$$

(80)

We represent ${\Lambda }_{nk}^{\alpha ,\beta }\left(x\right)$ in the form

$${\Lambda }_{nk}^{\alpha ,\beta }\left(x\right)={\lambda }_{nk}^F\left(H(x,·)\right)S(x,x_{nk}^{\alpha ,\beta }),$$

where $K(x,y)=H(x,y)S(x,y)$ with

$$H(x,y)=\left\{\begin{array}{ccc}{(x-y)}^{\kappa }{v}^{\alpha ,\beta }\left(y\right)& :& 0\le y\le x,\\ 0& :& x<y\le 1,\end{array}\right.$$

and

$$S(x,y)=\left\{\begin{array}{ccc}h(x,y)v^{-\alpha ,-\beta }\left(y\right)& :& 0\le y\le x,\\ 0& :& x<y\le 1,\end{array}\right.$$

as well as

$${\lambda }_{nk}^{F}f={\int }_0^{1}f\left(y\right){\ell }_{nk}^{\alpha ,\beta }\left(y\right)dy.$$

Let $\alpha ,\beta >-1,$ $1>\gamma ,\delta >-1,$ and $-{\displaystyle \frac{1}{2}}<\gamma -\alpha ,\delta -\beta <{\displaystyle \frac{3}{2}}.$ With the help of Lemma 4, we have

$$\begin{array}{cc}\hfill \left|\sum _{k=1}^n{\lambda }_{nk}^F\left(f\right)g\left(x_{nk}^{\alpha ,\beta }\right)-{\int }_0^{1}f\left(y\right)g\left(y\right)dy\right|& =\left|{\int }_0^{1}f\left(y\right)\left[\left({\mathcal{L}}_n^{\alpha ,\beta }g\right)\left(y\right)-g\left(y\right)\right]dy\right|\hfill \\ \hfill & \le {∥f∥}_{-\gamma ,-\delta }{∥{\mathcal{L}}_n^{\alpha ,\beta }g-g∥}_{\gamma ,\delta }⟶0\hfill \end{array}$$

(81)

for all $f\in {\mathbf{L}}_{-\gamma ,-\delta }^2$ and $g\in {\mathbf{R}}_{{\displaystyle \frac{1+\gamma }{2}},{\displaystyle \frac{1+\delta }{2}}}^0.$ For fixed ${\gamma }_0,{\delta }_0$ with $0\le {\gamma }_0<{\displaystyle \frac{1+\gamma }{2}}$ and $0\le {\delta }_0<{\displaystyle \frac{1+\delta }{2}},$ define

$${\mathcal{G}}_{n}f:{\tilde{\mathbf{R}}}_{{\gamma }_0,{\delta }_0}⟶\mathbb{C},g\mapsto \sum _{k=1}^n{\lambda }_{nk}^F\left(f\right)g\left(x_{nk}^{\alpha ,\beta }\right),$$

and equip the space ${\tilde{\mathbf{R}}}_{{\gamma }_0,{\delta }_0}$ with the norm ${∥.∥}_{{\gamma }_0,{\delta }_0,\infty }.$ It is easy to see that

$${∥{\mathcal{G}}_{n}f∥}_{{\tilde{\mathbf{R}}}_{{\gamma }_0,{\delta }_0}\to \mathbb{C}}=\sum _{k=1}^n{\displaystyle \frac{\left|{\lambda }_{nk}^F\left(f\right)\right|}{v^{{\gamma }_0,{\delta }_0}\left(x_{nk}^{\alpha ,\beta }\right)}}$$

such that the Banach–Steinhaus theorem (principle of uniform boundedness), together with (81) implies, for all $f\in {\mathbf{L}}_{-\gamma ,-\delta }^2,$

$${∥{\mathcal{G}}_{n}f∥}_{{\tilde{\mathbf{R}}}_{{\gamma }_0,{\delta }_0}\to \mathbb{C}}=\sum _{k=1}^n{\displaystyle \frac{\left|{\lambda }_{nk}^F\left(f\right)\right|}{v^{{\gamma }_0,{\delta }_0}\left(x_{nk}^{\alpha ,\beta }\right)}}\le {\mathcal{C}}_1={\mathcal{C}}_1\left(f\right)<\infty \forall n\in \mathbb{N}.$$

(82)

Considering ${\mathcal{G}}_n$ as a linear map from ${\mathbf{L}}_{-\gamma ,-\delta }^2$ into the dual space of ${\tilde{\mathbf{R}}}_{{\gamma }_0,{\delta }_0}$ and again applying the principle of uniform boundedness to (82), we get

$${\mathcal{C}}_2:=sup\left\{{∥{\mathcal{G}}_n∥}_{{\mathbf{L}}_{-\gamma ,-\delta }^2\to {\left({\tilde{\mathbf{R}}}_{{\gamma }_0,{\delta }_0}\right)}^{\ast }}:n\in \mathbb{N}\right\}<\infty .$$

Together with (82), this implies

$$\sum _{k=1}^n{\displaystyle \frac{\left|{\lambda }_{nk}^F\left(f\right)\right|}{v^{{\gamma }_0,{\delta }_0}\left(x_{nk}^{\alpha ,\beta }\right)}}\le {\mathcal{C}}_2{∥f∥}_{-\gamma ,-\delta }\forall f\in {\mathbf{L}}_{-\gamma ,-\delta }^2.$$

(83)

We conclude

$$\begin{array}{cc}& \sum _{k=1}^n\left|{\Lambda }_{nk}^{\alpha ,\beta }\left(x\right)-{\Lambda }_{nk}^{\alpha ,\beta }\left(x_0\right)\right|\hfill \\ & =\sum _{k=1}^n\left|\left[{\lambda }_{nk}^F(H(x,·)-H(x_0,·))\right]S(x,x_{nk}^{\alpha ,\beta })+{\lambda }_{nk}^F\left(H(x_0,·)\right)\left[S(x,x_{nk}^{\alpha ,\beta })-S(x_0,x_{nk}^{\alpha ,\beta })\right]\right|\hfill \\ & \le \sum _{k=1}^n{\displaystyle \frac{\left|{\lambda }_{nk}^F(H(x,·)-H(x_0,·))\right|}{v^{{\gamma }_0,{\delta }_0}\left(x_{nk}^{\alpha ,\beta }\right)}}{∥S(x,·)∥}_{{\gamma }_0,{\delta }_0,\infty }\hfill \\ & +\sum _{k=1}^n{\displaystyle \frac{\left|{\lambda }_{nk}^F(H(x_0,·)\right|}{v^{{\gamma }_0,{\delta }_0}\left(x_{nk}^{\alpha ,\beta }\right)}}{∥S(x,·)-S(x_0,·)∥}_{{\gamma }_0,{\delta }_0,\infty }\hfill \\ & \le {\mathcal{C}}_2[{∥H(x,·)-H(x_0,·)∥}_{-\gamma ,-\delta }{∥S(x,·)∥}_{{\gamma }_0,{\delta }_0,\infty }\hfill \\ & +{∥H(x_0,·)∥}_{-\gamma ,-\delta }{∥S(x,·)-S(x_0,·)∥}_{{\gamma }_0,{\delta }_0,\infty }].\hfill \end{array}$$

From the estimate of the terms $I_1$ and $I_3$ above (cf. (71) and (73)), we infer that, for $0\le x_1<x_2\le 1,$

$$\begin{array}{cc}\hfill & {∥H(x_2,·)-H(x_1,·)∥}_{-\gamma ,-\delta }^2\hfill \\ \hfill & ={\int }_{x_1}^{x_2}{(x_2-y)}^{2\kappa }{v}^{2\alpha -\gamma ,2\beta -\delta }\left(y\right)dy\hfill \\ \hfill & +{\int }_0^{x_1}{\left|{(x_2-y)}^{\kappa }-{(x_1-y)}^{\kappa }\right|}^2{v}^{2\alpha -\gamma ,2\beta -\delta }\left(y\right)dy\hfill \\ \hfill & \le \mathcal{C}{(x_2-x_1)}^{\eta },\hfill \end{array}$$

(84)

where $\eta =2min\left\{1,\kappa \right\}$ if $\kappa >0$ and $\eta =2\chi$ (with some sufficiently small $\chi >0$) if $\kappa <0,$ supposing that $1+2\kappa -max\left\{0,\gamma -2\alpha \right\}-max\left\{0,\delta -2\beta \right\}\ge 0$ and $2\alpha -\gamma ,2\beta -\delta >-1.$ (Note that it is possible to weaken the conditions on the parameters if we use a respective weighted ${\mathbf{L}}^p$-space, $1<p<2,$ instead of the space ${\mathbf{L}}_{-\gamma ,-\delta }^2.$) Relation (84) implies

$$\underset{x\to x_0}{lim}{∥H(x,·)-H(x_0,·)∥}_{-\gamma ,-\delta }^2=0$$

(85)

if the conditions on the parameters $\kappa ,\alpha ,\beta ,\gamma ,\delta$ are fulfilled. The most critical question is under which conditions the limit relation

$$\underset{x\to x_0}{lim}{∥S(x,·)-S(x_0,·)∥}_{{\gamma }_0,{\delta }_0,\infty }=0$$

(86)

is true. Let us assume that the function

$$\left\{(x,y)\in {\mathbb{R}}^2:0\le y\le x\le 1\right\}⟶\mathbb{C},(x,y)\mapsto h(x,y)v^{{\gamma }_0-\alpha ,{\delta }_0-\beta }\left(y\right)$$

is continuous. Since, for $0\le x_1<x_2\le 1,$

$$\begin{array}{cc}\hfill {∥S(x_2,·)-S(x_1,·)∥}_{{\gamma }_0,{\delta }_0,\infty }& =sup\left\{\left|h(x_2,y)-h(x_1,y)\right|v^{{\gamma }_0-\alpha ,{\delta }_0-\beta }\left(y\right):0\le y\le x_1\right\}\hfill \\ & +sup\left\{\left|h(x_2,y)\right|v^{{\gamma }_0-\alpha ,{\delta }_0-\beta }\left(y\right):x_1<y\le x_2\right\},\hfill \end{array}$$

we get

$$\underset{x\to x_0}{lim}{∥S(x,·)-S(x_0,·)∥}_{{\gamma }_0,{\delta }_0,\infty }=h(x_0,x_0)v^{{\gamma }_0-\alpha ,{\delta }_0-\beta }\left(x_0\right),$$

i.e., we can obtain (86) only if

$$h(x,x)v^{{\gamma }_0-\alpha ,{\delta }_0-\beta }\left(x\right)=0,\forall x\in [0,1].$$

(87)

As an example, let us consider the situation where the function $h:\overline{D}⟶\mathbb{C}$ is continuous, and we choose $\alpha =\beta =\gamma =\delta =0.$ Then condition (A) of Corollary 5 is satisfied for $\kappa >-1.$ Also, (76) and (85) hold true. Note that (85) was above proven for $\kappa >-{\displaystyle \frac{1}{2}}.$ But, as already mentioned in the brackets after (85), this condition can be weakened to $\kappa >-1$ in the present situation. Thus, to be able to apply the theory of collectively compact operator sequences, due to (87), we have to assume that $h(x,x)=0$ for all $x\in [0,1].$

If ${\left(a_k^{\ast }\right)}_{k=1}^n$ is the solution of (65) (when $\alpha =\beta =0$) and

$$u_n^{N,\ast }\left(x\right)=f\left(x\right)+\sum _{k=1}^n{q}_{nk}^{0,0}\left(x\right)h(x,x_{nk}^{0,0})a_k^{\ast }$$

is the respective Nyström interpolant, then we have the error estimate (see [3], Theorem 10.9)

$$\begin{array}{cc}\hfill {∥u_n^{N,\ast }-u^{\ast }∥}_{\infty }& \le \mathcal{C}{∥{\mathcal{K}}_{\kappa ,n}^{N,0,0}{u}^{\ast }-{\mathcal{K}}_{\kappa }{u}^{\ast }∥}_{\infty }\hfill \\ & \le \underset{0\le x\le 1}{sup}{\int }_0^x{(x-y)}^{\kappa }\left|{\mathcal{L}}_n^{0,0}\left[h(x,·)u^{\ast }\right]\left(y\right)-h(x,y)u^{\ast }\left(y\right)\right|dy.\hfill \end{array}$$

In order to obtain the convergence rates from this estimate, it is necessary to assume smoothness properties of the function

$${[0,1]}^2⟶\mathbb{C},(x,y)\mapsto \left\{\begin{array}{ccc}h(x,y)& :& 0\le y\le x,\\ 0& :& x<y\le 1,\end{array}\right.$$

while for the collocation–quadrature method studied in Section 3, only smoothness properties of the function

$$\overline{D}⟶\mathbb{C},(x,y)\mapsto h(x,y)$$

are needed (see condition (D)).

5. Conclusions

From a theoretical point of view, concerning stability and convergence, the collocation– quadrature method is better than the Nyström method, since the latter requires additional conditions on the kernel function (see (87)). But, at first glance the collocation–quadrature method is more expensive (i.e., has a higher computational complexity) than the Nyström method. The speed (and cost) of the collocation–quadrature method must be examined in future studies.