Legendre Spectral Projection Methods for Fredholm–Hammerstein Integral Equations

Abstract

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Hammerstein integral equation. The convergence of the approximate solutions to the actual solution is discussed and the rates of convergence are obtained. We are able to obtain similar superconvergence rates for the iterated Legendre Galerkin solution for Hammerstein integral equations with smooth kernel as in the case of piecewise polynomial based Galerkin method.

1 Introduction

In this section, we consider the following Hammerstein integral equation

$$\begin{aligned} x(t)-\int _{-1}^{1}k(t,s)\psi (s,x(s))ds=f(t),\quad -1\le t \le 1, \end{aligned}$$

(1.1)

where k, f and $\psi$ are known functions and x is the unknown solution to be found in a Banach space $\mathbb {X}$. Hammerstein integral equations (1.1) arises as a reformulation of boundary value problems with certain nonlinear boundary conditions.

Several numerical methods are available in literature to solve nonlinear integral equations. Various spectral methods for solving different type of integral equations are present in literature (see [3, 4, 15–19, 21, 22]). The Galerkin, collocation, Petrov–Galerkin, degenerate kernel and Nystr$\ddot{o}$m methods are commonly used projection methods for finding numerical solutions of the equation of type (1.1) (see [2, 6–12, 14]). In [12, 13] Kumar and Sloan discussed a new type of collocation method and established superconvergence results for the solution of Hammerstein integral equations. Some recent results on the numerical solutions of the Hammerstein equations can be found in [11].

In the case of piecewise polynomial based projection methods, we consider $-1=t_0 < t_1 < \cdots < t_n=1$, a partition of $[-1,1]$ and let $h=\max \{t_{i+1}-t_i: 0\le t_i\le n-1\}$ denote the norm of the partition. We assume that $h \rightarrow 0$, as $n \rightarrow \infty$. In this case the approximating subspaces $\mathbb {X}_n=S^{\nu }_{r,n}$, the space of all piecewise polynomials of order r (i.e., of degree $\le r-1$) with break points at $t_1, t_2, \ldots , t_{n-1}$ and with $\nu$ continuous derivatives, $-1\le \nu \le r-2$. Let ${{\mathcal {P}}}_n$ be either orthogonal or interpolatory bounded projections from $\mathbb {X}$ onto ${\mathbb {X}}_n$. Then in Galerkin or in collocation method, the Hammerstein integral equation (1.1) is approximated by

$$\begin{aligned} x_n-{\mathcal {P}}_n{\mathcal {K}}\psi (x_n)={\mathcal {P}}_nf, \end{aligned}$$

(1.2)

where ${\mathcal {K}}\psi (x_n)(t)=\int _{-1}^{1} \!k(t,s)\psi (s,x_n(s))\, \mathrm {d} s$. The iterated solution is defined by ${{\tilde{x}}_n}= f+{{\mathcal {K}}}\psi (x_n).$ Under some suitable conditions on the kernel k and the right hand side function f of the Eq. (1.1), it is known that the orders of convergence for Galerkin and collocation solutions are $\mathcal {O}(h^{r})$ and for the iterated Galerkin and iterated collocation solutions are $\mathcal {O}(h^{2r})$ (see [9, 10]). However, to get better accuracy in piecewise polynomial based projection methods, the number of partition points should be increased. Hence in such cases, one has to solve a large system of nonlinear equations, which is computationally very much expensive.

In this paper, we have applied Galerkin and collocation method to solve Eq. (1.1) using global polynomial basis functions. Use of global polynomials will imply smaller nonlinear systems, something which is highly desirable in practical computations. Hence we choose to use global polynomials rather than piecewise polynomial basis functions in this paper. In particular, we use Legendre polynomials, which can be generated recursively with ease and possess nice property of orthogonality. Further, these Legendre polynomials are less expensive computationally compared to piecewise polynomial basis functions. However, if ${{\mathcal {P}}}_n$ denotes either orthogonal or interpolatory projection from $\mathbb {X}$ into a subspace of global polynomials of degree $\le n$, then $\Vert {{\mathcal {P}}}_n\Vert _{\infty }$ is unbounded. It is the purpose of this work to obtain similar convergence results for the approximate solutions in both $L^2$-norm and infinity norm using Legendre polynomial bases as in the case of piecewise polynomial bases.

We organize this paper as follows. In Sect. 2, we discuss the Legendre spectral Galerkin and Legendre spectral collocation methods to obtain convergence results. In Sect. 3, numerical results are given to illustrate the theoretical results. Throughout this paper, we assume that c is a generic constant.

2 Legendre Spectral Galerkin and Collocation Methods: Hammerstein Integral Equations with Smooth Kernel

In this section, we describe the Galerkin and collocation methods for solving Hammerstein integral equations using Legendre polynomial basis functions.

Let $\mathbb {X}= {\mathcal {C}}[-1,1]$ and consider the following Hammerstein integral equation

$$\begin{aligned} x(t) - \int _{-1}^{1} k(t,s) \psi (s,x(s))\,ds = f(t),\quad -1 \le t\le 1, \end{aligned}$$

(2.1)

where k, f and $\psi$ are known functions and x is the unknown function to be determined. For a fixed $t\in [-1,1]$, we denote $k_t(s)=k(t,s).$

Throughout the paper, the following assumptions are made on f, k(., .) and $\psi (.,x(.))$:

• (i) $f\in {\mathcal {C}}[-1,1]$.

• (ii) $\underset{t\rightarrow t^{'}}{\lim }\Vert k(t,.)-k(t',.)\Vert _\infty =0,~~t, t'\in [-1,1]$.

• (iii) $M = \Vert k\Vert _\infty = \underset{{t,s\in [-1,1]}}{\sup }|k(t,s)|<{\infty }$.

• (iv) The nonlinear function $\psi (s,x)$ is bounded and continuous over $[-1,1]\times \mathbb {R}$. $\psi (s,x)$ is Lipschitz continuous in x, i.e., for any $x_{1}$, $x_{2}\in \mathbb {R}$, $\exists$ $c_{1}>0$ such that

  $$\begin{aligned} |\psi (s,x_{1})-\psi (s,x_{2})|\le c_{1}|x_{1}-x_{2}|,~ \forall s\in [-1,1]. \end{aligned}$$

• (v) The partial derivative $\psi ^{(0,1)}(s,x(s))$ of $\psi$ w.r.t the second variable exists and is Lipschitz continuous in x, i.e., for any $x_{1}$, $x_{2}\in \mathbb {R}$, $\exists$ $c_{2}>0$ such that

  $$\begin{aligned} \left| \psi ^{(0,1)}(s,x_{1})-\psi ^{(0,1)}(s,x_{2})|\le c_{2}\right| x_{1}-x_{2}|,~ \forall s\in [-1,1]. \end{aligned}$$

  From this, we have $\psi ^{(0,1)}(.,.) \in {\mathcal {C}}([-1,1]\times \mathbb {R})$.

• (vi) We assume that M and $c_1$ satisfy the condition that $2Mc_1 <1$.

Note that under the above assumptions on f, k and $\psi$, for a sufficiently small number $h>0$, we have

$$\begin{aligned} |x(t+h)-x(t)|= & {} \left| f(t+h)+\int _{-1}^{1} k(t+h,s)\psi (s,x(s))\,ds-f(t)\right. \\&\left. -\int _{-1}^{1} k(t,s)\psi (s,x(s))\,ds\right| \\\le & {} |f(t+h)-f(t)|+\left| \int _{-1}^{1}[k(t+h,s)-k(t,s)]\psi (s,x(s))\,ds\right| \\\le & {} |f(t+h)-f(t)|+\sup _{s\in [-1,1]}|k(t+h,s)-k(t,s)|\int _{-1}^{1}|\psi (s,x(s))|\,ds\\\rightarrow & {} 0 ~as~ h\rightarrow 0. \end{aligned}$$

This implies $x\in {\mathcal {C}}[-1,1]$.

Let ${\mathcal {C}}^r[-1,1]$ denote the space of r-times continuously differentiable functions. For the rest of the paper we assume that the kernel $k(.,.) \in {\mathcal {C}}^r([-1,1]\times [-1,1])$, the nonlinear function $\psi (.,.)\in {\mathcal {C}}^r([-1,1]\times \mathbb {R})$ and $f\in {\mathcal {C}}^r[-1,1]$. Denote

$$\begin{aligned} (D^{i,j}k)(t,s) = \frac{\partial ^{i+j}}{\partial t^i\partial s^j}k(t,s), \quad t,s \in [-1,1], \end{aligned}$$

and

$$\begin{aligned} \Vert k\Vert _{r,\infty }=max\Big \{\big \Vert D^{(i,j)}k\big \Vert _\infty : 0\le i\le r, 0\le j\le r\Big \}. \end{aligned}$$

Now for $j=1, 2, \ldots r$, we have from estimate (2.1) that

$$\begin{aligned} x^{(j)}(t)=f^{(j)}(t)+\int _{-1}^{1}\left\{ \frac{{\partial }^j}{\partial t^j}k(t,s)\right\} \psi (s,x(s))ds. \end{aligned}$$

Hence by our assumptions on f, k and $\psi$, it follows that $x \in {\mathcal {C}}^r[-1,1]$. We write

$$\begin{aligned} \Vert x\Vert _{r,\infty }= \max \Big \{\big \Vert x^{(j)}\big \Vert _\infty : 0\le j\le r\Big \}, \end{aligned}$$

where $x^{(j)}$ denotes the j-th derivative of x.

Let

$$\begin{aligned} {\mathcal {K}}y(t)=\int _{-1}^{1} \!k(t,s)y(s)\, \mathrm {d} s,~~t\in [-1,1], ~y\in \mathbb {X}. \end{aligned}$$

Note that, using Holder’s inequality we have for any $y\in \mathbb {X}$,

$$\begin{aligned} \big \Vert {\mathcal {K}}y\big \Vert _\infty =\sup _{t\in [-1,1]}|{\mathcal {K}}y(t)|=\sup _{t\in [-1,1]}\left| \int _{-1}^{1} k(t,s)y(s)ds\right|\le & {} \sup _{t,s\in [-1,1]}|k(t,s)|\int _{-1}^{1}|y(s)|ds\nonumber \\\le & {} \sqrt{2}M\Vert y\Vert _{L^2}, \end{aligned}$$

(2.2)

and

$$\begin{aligned} \big \Vert {\mathcal {K}}y\big \Vert _{L^2}\le \sqrt{2}\big \Vert {\mathcal {K}}y\big \Vert _\infty \le 2M\Vert y\Vert _{L^2}. \end{aligned}$$

(2.3)

This implies

$$\begin{aligned} \Vert {\mathcal {K}}\Vert _{L^2} \le 2M. \end{aligned}$$

(2.4)

We will use Kumar and Sloan [12] technique for finding the approximate solution of the Eq. (2.1). The projection method will now be applied to an equivalent equation for the function z defined by

$$\begin{aligned} z(t):= \psi (t,x(t)),~~t\in [-1,1]. \end{aligned}$$

(2.5)

Note that, since $\psi (.,.)\in {\mathcal {C}}^r([-1,1]\times \mathbb {R})$ and $x\in {\mathcal {C}}^{r}[-1,1]$, using chain rule for higher derivatives it is easy to obtain that $z\in {\mathcal {C}}^r[-1,1]$.

The desired exact solution x of (2.1) is obtained by the equation

$$\begin{aligned} x(t) = f(t) + \int _{-1}^{1} \!k(t,s)z(s)\, \mathrm {d} s,~~t\in [-1,1]. \end{aligned}$$

(2.6)

For our convenience, we consider a nonlinear operator $\Psi :\mathbb {X}\rightarrow \mathbb {X}$ defined by

$$\begin{aligned} \Psi (x)(t):=\psi (t,x(t)). \end{aligned}$$

(2.7)

Then the Eq. (2.1) will take the form

$$\begin{aligned} x = \mathcal {K}z+f, \end{aligned}$$

(2.8)

and Eq. (2.5) becomes

$$\begin{aligned} z=\Psi ({\mathcal {K}}z+f). \end{aligned}$$

(2.9)

Let ${\mathcal {T}}(u):=\Psi ({\mathcal {K}}u+f)$, $u\in \mathbb {X}$, then the Eq. (2.9) can be written as

$$\begin{aligned} z={\mathcal {T}}z. \end{aligned}$$

(2.10)

Theorem 2.1

Let $\mathbb {X}= {\mathcal {C}}[-1,1]$, $f\in \mathbb {X}$ and $k(.,.) \in {\mathcal {C}}([-1,1]\times [-1,1])$ with $M = \underset{t,s\in [-1,1]}{\sup }|k(t,s)| < \infty$. Let $\psi (s,y(s)) \in {\mathcal {C}}([-1,1]\times \mathbb {R})$ satisfies the Lipschitz condition in the second variable, i.e.,

$$\begin{aligned} \big |\psi (s,y_1)-\psi (s,y_2)\big |\le c_1 \big |y_1-y_2\big |, ~~~y_1, y_2\in \mathbb {X}, \end{aligned}$$

with $2Mc_1 < 1$. Then the operator equation $z = {\mathcal {T}}z$ has a unique solution $z_0 \in \mathbb {X}$, i.e., we have $z_0={\mathcal {T}}z_0$.

Proof

Let $z_1, z_2 \in {\mathcal {C}}[-1,1]$. Using Lipschitz’s continuity of $\psi (.,x(.))$ and the estimate (2.2), we have

$$\begin{aligned} \big \Vert {\mathcal {T}}z_1 - {\mathcal {T}}z_2\big \Vert _\infty= & {} \big \Vert \Psi ({\mathcal {K}}z_{1}+f) - \Psi ({\mathcal {K}}z_{2}+f)\big \Vert _\infty \nonumber \\\le & {} c_1\big \Vert {\mathcal {K}}(z_1-z_2)\big \Vert _\infty \nonumber \\\le & {} c_1 M \sqrt{2}\big \Vert z_1-z_2\big \Vert _{L^2}\nonumber \\\le & {} 2Mc_1\big \Vert z_1-z_2\big \Vert _\infty . \end{aligned}$$

(2.11)

By assumption $2 Mc_1 < 1$, hence ${\mathcal {T}}$ is a contraction mapping on $\mathbb {X}$. Since $\mathbb {X}={\mathcal {C}}[-1,1]$ with $\Vert .\Vert _\infty$ norm is a Banach space, ${\mathcal {T}}$ has a unique fixed point in $\mathbb {X}$, by Banach contraction theorem. We denote this unique solution as $z_0$. Hence the proof follows. $\square$

Next we will apply Legendre Galerkin and Legendre collocation methods to the Eq. (2.9). To do this, we let $\mathbb {X}_n =$ span{$\phi _{0}$, $\phi _{1}$, $\phi _{2}$, $\ldots$, $\phi _{n}$} be the sequence of Legendre polynomial subspaces of $\mathbb {X}$ of degree $\le n$, where {$\phi _{0}$, $\phi _{1}$, $\phi _{2}$, $\ldots$, $\phi _{n}$} forms an orthonormal basis for $\mathbb {X}_n$. Here $\phi _i$’s are given by

$$\begin{aligned} \phi _i(s)=\sqrt{\frac{2i+1}{2}}L_i(s),\quad i=0,1,\ldots ,n, \end{aligned}$$

(2.12)

where $L_i$’s are the Legendre polynomials of degree $\le i$. These Legendre polynomials can be generated by the following three-term recurrence relation

$$\begin{aligned} L_{0}(s)=1, L_{1}(s)=s,\quad s\in [-1,1], \end{aligned}$$

(2.13)

and for $i = 1, 2, \ldots , n-1$

$$\begin{aligned} (i+1)L_{i+1}(s)= (2i+1)sL_{i}(s)-iL_{i-1}(s),\quad s\in [-1,1]. \end{aligned}$$

(2.14)

Orthogonal projection operator: Let $\mathbb {X}={\mathcal {C}}[-1,1]$ and let the operator ${{\mathcal {P}}}^G_n : \mathbb {X}\rightarrow \mathbb {X}_n$ be the orthogonal projection defined by

$$\begin{aligned} {{\mathcal {P}}}^G_n x=\sum ^{n}_{j=0} \langle x, \; \phi _j \rangle \phi _j,\ \ x \in \mathbb {X}, \end{aligned}$$

(2.15)

where $\langle x, \; \phi _j \rangle = \int ^1_{-1}x(t)\phi _j(t)dt.$

We quote the following proposition and lemma which follows from (Canuto et al. [5], pp 283-287).

Proposition 2.1

Let ${\mathcal {P}}_n^{G}:\mathbb {X}\rightarrow \mathbb {X}_n$ denote the orthogonal projection defined by (2.15). Then the projection ${\mathcal {P}}_n^{G}$ satisfies the following properties.

• (i) $\Vert {\mathcal {P}}_n^Gu\Vert _{L^2}\le p_1\Vert u\Vert _{\infty }$, where $p_1$ is a constant independent of n.

• (ii) There exists a constant $c>0$ such that for any $n\in \mathbb {N}$ and $u\in \mathbb {X}$,

  $$\begin{aligned} \big \Vert {\mathcal {P}}_n^{G}u-u\big \Vert _{L^2}\le c\inf _{\phi \in \mathbb {X}_n}\Vert u-\phi \Vert _{L^2}\rightarrow 0,~ as~ n\rightarrow \infty . \end{aligned}$$

  (2.16)

Lemma 2.1

Let ${\mathcal {P}}_n^{G}$ be the orthogonal projection defined by (2.15). Then for any $u\in {\mathcal {C}}^{r}[-1, 1]$, there hold

$$\begin{aligned}&\big \Vert u-{\mathcal {P}}_n^{G}u\big \Vert _{L^2}\;\le \;cn^{-r} \big \Vert u^{(r)}\big \Vert _{L^2}, \end{aligned}$$

(2.17)

$$\begin{aligned}&\big \Vert u-{\mathcal {P}}_n^{G}u\big \Vert _\infty \;\le \; cn^{\frac{3}{4}-r}\big \Vert u^{(r)}\big \Vert _{L^2}, \end{aligned}$$

(2.18)

where c is a constant independent of n.

Interpoaltory projection operator: Let $\{\tau _0, \tau _1, \ldots , \tau _n\}$ be the zeros of the Legendre polynomial of degree $n+1$ and define interpolatory projection ${\mathcal {P}}_n^C :\mathbb {X}\rightarrow {\mathbb {X}}_n$ by

$$\begin{aligned} {\mathcal {P}}_n^Cu \in \mathbb {X}_n,~~ {\mathcal {P}}_n^C u(\tau _i) = u(\tau _i), ~i = 0, 1, \ldots , n,~ u\in \mathbb {X}. \end{aligned}$$

(2.19)

According to the analysis of (Canuto et al. [5]), ${\mathcal {P}}_n^C$ satisfies the following lemmas.

Lemma 2.2

Let ${\mathcal {P}}_n^C :\mathbb {X}\rightarrow \mathbb {X}_n$ be the interpolatory projection defined by (2.19). Then there hold

• (i) $\big \Vert {\mathcal {P}}_n^Cu\big \Vert _{L^2}\le p_2\big \Vert u\big \Vert _{\infty }$, where $p_2$ is a constant independent of n.

• (ii) There exists a constant $c >0$ such that for any $n\in \mathbb {N}$ and $u\in \mathbb {X}$,

  $$\begin{aligned} \big \Vert {\mathcal {P}}_n^C u-u\big \Vert _{L^2}\le c \underset{\phi \in \mathbb {X}_n}{\inf }\big \Vert u-\phi \big \Vert _{L^2}\rightarrow 0,~as~n \rightarrow \infty . \end{aligned}$$

  (2.20)

Lemma 2.3

Let ${\mathcal {P}}_n^C :\mathbb {X}\rightarrow \mathbb {X}_n$ be the interpolatory projection defined by (2.19). Then for any $u\in {\mathcal {C}}^r[-1,1]$, there exists a constant c independent of n such that

$$\begin{aligned}&\big \Vert u-{\mathcal {P}}_n^C u\big \Vert _{L^2}\;\le \; cn^{-r} \big \Vert u^{(r)}\big \Vert _{L^2}, \end{aligned}$$

(2.21)

$$\begin{aligned}&\big \Vert u-{\mathcal {P}}_n^{C}u\big \Vert _\infty \;\le \; cn^{\frac{1}{2}-r}\big \Vert u^{(r)}\big \Vert _{L^2}. \end{aligned}$$

(2.22)

Throughout this paper, we assume that the projection operator ${\mathcal P}_n:\mathbb {X}\rightarrow \mathbb {X}_n$ is either orthogonal projection ${\mathcal {P}}_n^G$ or interpolatory projection operator ${\mathcal {P}}_n^C$ defined as above. From the above discussed properties of ${\mathcal {P}}_n^G$ and ${\mathcal {P}}_n^C$, we have

$$\begin{aligned} \big \Vert {{\mathcal {P}}}_nu\big \Vert _{L^2}\le & {} p\big \Vert u\big \Vert _\infty ,~ u\in \mathbb {X}, \end{aligned}$$

(2.23)

where p is a constant independent of n. Also estimates (2.16) and (2.20) imply that

$$\begin{aligned} \big \Vert {\mathcal {P}}_{n}u -u\big \Vert _{L^2}\rightarrow 0, ~\text {as}~ n \rightarrow \infty ,~ \forall u \in {\mathcal {C}}[-1,1]. \end{aligned}$$

(2.24)

Further we have from Lemmas 2.1 and 2.3 that

$$\begin{aligned} \big \Vert u-{\mathcal {P}}_nu\big \Vert _{L^2}\le & {} cn^{-r} \big \Vert u^{(r)}\big \Vert _{L^2}, \end{aligned}$$

(2.25)

$$\begin{aligned} \big \Vert u-{{\mathcal {P}}}_nu\big \Vert _\infty\le & {} cn^{\beta -r}\big \Vert u^{(r)}\big \Vert _{L^2}, ~0<\beta < 1, \text{ and } r=0, 1, 2, \cdots \end{aligned}$$

(2.26)

where c is a constant independent of n, $\beta =\frac{3}{4}$ for orthogonal projection operators and $\beta =\frac{1}{2}$ for interpolatory projections. Note that $\big \Vert {\mathcal {P}}_{n}u -u\big \Vert _\infty \nrightarrow 0$, as $n \rightarrow \infty$ for any $u \in {\mathcal {C}}[-1,1]$.

The projection method for Eq. (2.9) is seeking an approximate solution $z_n \in \mathbb {X}_n$ such that

$$\begin{aligned} z_n={\mathcal {P}}_n\Psi ({\mathcal {K}}z_n+f). \end{aligned}$$

(2.27)

If ${\mathcal {P}}_n$ is chosen to be ${\mathcal {P}}_n^G$, the above scheme (2.27) leads to the Legendre Galerkin method, whereas if ${\mathcal {P}}_n$ is replaced by ${\mathcal {P}}_n^C$ we get the Legendre collocation method.

Let ${\mathcal {T}}_n$ be the operator defined by

$$\begin{aligned} {\mathcal {T}}_n(u):={\mathcal {P}}_n\Psi ({\mathcal {K}}u+f),~u\in \mathbb {X}. \end{aligned}$$

(2.28)

Then the Eq. (2.27) can be written as

$$\begin{aligned} z_n={\mathcal {T}}_nz_n. \end{aligned}$$

(2.29)

Corresponding approximate solution $x_n$ of x is given by

$$\begin{aligned} x_n = {\mathcal {K}}z_n +f. \end{aligned}$$

(2.30)

In order to obtain more accurate approximation solution, we further consider the iterated projection method for (2.9). To this end, we define the iterated solution as

$$\begin{aligned} {{\tilde{z}}_n}=\Psi ({\mathcal {K}}z_n+f). \end{aligned}$$

(2.31)

Applying ${{\mathcal {P}}}_n$ on both sides of the Eq. (2.31), we obtain

$$\begin{aligned} {{\mathcal {P}}}_n{{\tilde{z}}_n}={{\mathcal {P}}}_n\Psi ({\mathcal {K}}z_n+f). \end{aligned}$$

(2.32)

From Eqs. (2.27) and (2.32), it follows that ${{\mathcal {P}}}_n{{\tilde{z}}_n}=z_n.$ Using this, we see that the iterated solution ${{\tilde{z}}_n}$ satisfies the following equation

$$\begin{aligned} {{\tilde{z}}_n} = \Psi ({\mathcal {K}}{\mathcal {P}}_n{\tilde{z}}_n+f). \end{aligned}$$

(2.33)

Letting $\widetilde{{\mathcal {T}}}_n(u):= \Psi ({\mathcal {K}}{\mathcal {P}}_nu+f)$, $u\in \mathbb {X}$, the Eq. (2.33) can be written as ${{\tilde{z}}_n}=\widetilde{{\mathcal {T}}}_n{{\tilde{z}}_n}.$ Corresponding approximate solution ${\tilde{x}}_n$ of x is given by

$$\begin{aligned} {\tilde{x}}_n = {\mathcal {K}}{\tilde{z}}_n +f. \end{aligned}$$

(2.34)

We quote the following theorem from [20] which gives us the condition under which the solvability of one equation leads to the solvability of other equation.

Theorem 2.2

(Vainikko [20]) Let $\widehat{\mathcal {F}}$ and $\widetilde{\mathcal {F}}$ be continuous operators over an open set $\Omega$ in a Banach space $\mathbb {X}$. Let the equation $x=\widetilde{\mathcal {F}}x$ has an isolated solution ${\tilde{x}}_0 \in \Omega$ and let the following conditions be satisfied.

• (a) The operator $\widehat{\mathcal {F}}$ is Frechet differentiable in some neighborhood of the point ${\tilde{x}}_0$, while the linear operator $\mathcal {I}-\widehat{\mathcal {F}}'({\tilde{x}}_0)$ is continuously invertible.

• (b) Suppose that for some $\delta > 0$ and $0<q<1$, the following inequalities are valid (the number $\delta$ is assumed to be so small that the sphere $\big \Vert x-{\tilde{x}}_0\big \Vert \le \delta$ is contained within $\Omega$).

  $$\begin{aligned}&\sup _{\big \Vert x-{\tilde{x}}_0\big \Vert \le \delta }\big \Vert {(\mathcal {I}-\widehat{\mathcal {F}}' ({\tilde{x}}_0))}^{-1}(\widehat{\mathcal {F}}'(x)- \widehat{\mathcal {F}}'({\tilde{x}}_0))\big \Vert \le q,\end{aligned}$$

  (2.35)
  $$\begin{aligned}&\alpha =\big \Vert {(\mathcal {I}-\widehat{\mathcal {F}}' ({\tilde{x}}_0))}^{-1}(\widehat{\mathcal {F}} ({\tilde{x}}_0)-\widetilde{\mathcal {F}}({\tilde{x}}_0))\big \Vert \le \delta (1-q). \end{aligned}$$

  (2.36)

Then the equation $x=\widehat{\mathcal {F}}x$ has a unique solution $\hat{x}_0$ in the sphere $\big \Vert x-{\tilde{x}}_0\big \Vert \le \delta$. Moreover, the inequality

$$\begin{aligned} \frac{\alpha }{1+q}\le \big \Vert \hat{x}_0-{\tilde{x}}_0\big \Vert \le \frac{\alpha }{1-q} \end{aligned}$$

(2.37)

is valid.

Next we discuss the existence of approximate and iterated approximate solutions and their error bounds. To do this, we first recall the following definition of $\nu$-convergence and a lemma from [1].

Definition 2.1

Let $\mathbb {X}$ be Banach space and ${\mathbb {B}}{\mathbb {L}}(\mathbb {X})$ be space of bounded linear operators from $\mathbb {X}$ into $\mathbb {X}$. Let ${\mathcal{K}_n},\;{{\mathcal {K}}} \in {\mathbb {B}}{\mathbb {L}}(\mathbb {X})$. We say ${\mathcal{K}_n}$ is $\nu$-convergent to ${{\mathcal {K}}}$ if

$$\begin{aligned} \big \Vert \mathcal{K}_n\big \Vert \le c<\infty , \big \Vert (\mathcal{K}_n-\mathcal{K} )\mathcal{K} \big \Vert \rightarrow 0, \big \Vert (\mathcal{K}_n-\mathcal{K} )\mathcal{K}_n\big \Vert \rightarrow 0, ~as ~ n\rightarrow \infty . \end{aligned}$$

Lemma 2.4

(Ahues et al. [1]) Let $\mathbb {X}$ be a Banach space and ${\mathcal K}$, $\mathcal{K}_n$ be bounded linear operators on $\mathbb {X}.$ If $\big \Vert \mathcal{K}_n-\mathcal{K}\big \Vert \rightarrow 0$, as $n\rightarrow \infty$ or ${{{\mathcal {K}}}_n}$ is $\nu$-convergent to ${\mathcal K}$ and $(\mathcal {I}-{\mathcal {K}})^{-1}$ exists, then $(\mathcal {I}-{\mathcal {K}}_n)^{-1}$ exists and uniformly bounded on $\mathbb {X}$, for sufficiently large n.

Lemma 2.5

Let $z_0\in {\mathcal {C}}^{r}[-1,1]$, then the following hold

$$\begin{aligned} \big \Vert {\mathcal {K}}(\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _\infty = \sup _{t\in [-1,1]}|< k_t(.), (\mathcal {I}-{\mathcal {P}}_n)z_0 >| \le M\sqrt{2}\big \Vert (\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _{L^2}. \end{aligned}$$

In particular we have $\big \Vert {\mathcal {K}}(\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _\infty \rightarrow 0, ~ as ~ n\rightarrow \infty$.

Proof

Using Cauchy-Schwarz inequality and the estimate (2.25), we have

$$\begin{aligned} \big \Vert {\mathcal {K}}(\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _\infty= & {} \sup _{t\in [-1,1]}|{\mathcal {K}}(\mathcal {I}-{\mathcal {P}}_n)z_0(t)|\nonumber \\= & {} \sup _{t\in [-1,1]}\left| \int _{-1}^{1}k(t,s)(\mathcal {I}-{\mathcal {P}}_n)z_0(s)ds\right| \end{aligned}$$

(2.38)

$$\begin{aligned}= & {} \sup _{t\in [-1,1]}|< k_t(.), (\mathcal {I}-{\mathcal {P}}_n)z_0 >|\nonumber \\\le & {} \sup _{t\in [-1,1]}\big \Vert k_t(.)\big \Vert _{L^2}\big \Vert (\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _{L^2}\nonumber \\\le & {} \sqrt{2}M\big \Vert (\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _{L^2} \end{aligned}$$

(2.39)

$$\begin{aligned}\le & {} c\sqrt{2}Mn^{-r}\big \Vert z_0^{(r)}\big \Vert _{L^2} \rightarrow 0,~ as~ n \rightarrow \infty . \end{aligned}$$

(2.40)

Hence the proof follows.$\square$

Lemma 2.6

Let ${\mathcal {T}}'(z_0)$ and $\widetilde{{{\mathcal {T}}}}_n'(z_0)$ be the Frechet derivatives of ${\mathcal {T}}(z)$ and $\widetilde{{{\mathcal {T}}}}_n(z)$, respectively at $z_0$. Then

$$\begin{aligned}&\big \Vert (\mathcal {I}-{{\mathcal {P}}}_n)\widetilde{{{\mathcal {T}}}}_n' (z_0)\big \Vert _{L^2} \rightarrow 0,~ as~ n \rightarrow \infty ,\\&\big \Vert (\mathcal {I}-{{\mathcal {P}}}_n){\mathcal {T}}'(z_0)\big \Vert _{L^2} \rightarrow 0, ~ as ~ n\rightarrow \infty . \end{aligned}$$

Proof

We have $\widetilde{{\mathcal {T}}}_n'(z_0) = \Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f){\mathcal {K}}{\mathcal {P}}_n$.

Now using the Lipschitz’s continuity of $\psi ^{(0,1)}(.,x(.))$, Lemma 2.5 and boundedness of $\big \Vert \Psi '({\mathcal {K}}z_0+f)\big \Vert _\infty$, we have

$$\begin{aligned} \big \Vert \Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)\big \Vert _\infty\le & {} \big \Vert \Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)-\Psi ' ({\mathcal {K}}z_0+f)\big \Vert _\infty +\big \Vert \Psi '({\mathcal {K}}z_0+f)\big \Vert _\infty \nonumber \\\le & {} c_2\big \Vert {\mathcal {K}}({\mathcal {P}}_n-\mathcal {I})z_0 \big \Vert _\infty +\big \Vert \Psi '({\mathcal {K}}z_0+f)\big \Vert _\infty \le B < \infty , \end{aligned}$$

(2.41)

where B is a constant independent of n.

This implies

$$\begin{aligned} \big \Vert \Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)\big \Vert _{L^2} \le \sqrt{2}\big \Vert \Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f) \big \Vert _\infty \le \sqrt{2}B < \infty . \end{aligned}$$

(2.42)

Next, Let ${\bar{B}}:=\{x\in \mathbb {X}: \big \Vert x\big \Vert _{L^2}\le 1\}$ be the closed unit ball in $\mathbb {X}$. We have $\widetilde{{\mathcal {T}}}_n'(z_0)=\Psi ' ({\mathcal {K}}{\mathcal {P}}_nz_0+f){\mathcal {K}}{\mathcal {P}}_n$. Since $\{{\mathcal {K}}{\mathcal {P}}_n\}$ is a sequence of compact operators and $\Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)$ is uniformly bounded, $\widetilde{{\mathcal {T}}}_n'(z_0)$ are compact operators. Thus $S=\{\widetilde{{\mathcal {T}}}_n'(z_0)x:x\in {\bar{B}}, n\in N\}$ is relatively compact set. Using estimate (2.24), we can conclude

$$\begin{aligned} \big \Vert (\mathcal {I}-{\mathcal {P}}_n)\widetilde{{\mathcal {T}}}_n' (z_0)\big \Vert _{L^2}= & {} \sup \big \{\big \Vert (\mathcal {I}-{\mathcal {P}}_n) \widetilde{{\mathcal {T}}}_n'(z_0)x\big \Vert _{L^2}:x\in {\bar{B}}\big \}\nonumber \\= & {} \sup \{\big \Vert (\mathcal {I}-{\mathcal {P}}_n)y\big \Vert _{L^2}:y\in S\}\rightarrow 0, ~ n\rightarrow \infty . \end{aligned}$$

(2.43)

Similarly, since $\Psi '({\mathcal {K}}z_0+f)$ is bounded and ${\mathcal {K}}$ is compact, ${\mathcal {T}}'(z_0)=\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}$ is also compact and we have

$$\begin{aligned} \big \Vert (\mathcal {I}-{\mathcal {P}}_n){{\mathcal {T}}}'(z_0)\big \Vert _{L^2}\rightarrow 0, ~as ~n\rightarrow \infty . \end{aligned}$$

This completes the proof. $\square$

Theorem 2.3

Let $z_0\in {\mathcal {C}}^{r}[-1,1]$ be an isolated solution of the Eq. (2.9). Assume that 1 is not an eigenvalue of the linear operator $\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}$, where $\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}$ denotes the Frechet derivative of $\Psi ({\mathcal {K}}z+f)$ at $z_0$. Let ${\mathcal {P}}_n :\mathbb {X}\rightarrow \mathbb {X}_n$ be either orthogonal or interpolatory projection operator defined by (2.15) and (2.19) respectively. Then the Eq. (2.27) has a unique solution $z_n\in B(z_0,\delta )=\{z \,:\big \Vert z-z_0\big \Vert _{L^2} < \delta \}$ for some $\delta >0$ and for sufficiently large n. Moreover, there exists a constant $0< q <1$, independent of n such that

$$\begin{aligned} \frac{\alpha _n}{1+q}\le {\big \Vert z_n-z_0\big \Vert }_{L^2} \le \frac{\alpha _n}{1-q}, \end{aligned}$$

where $\alpha _n={\big \Vert (\mathcal {I}-{{{\mathcal {T}}}_n}' (z_0))^{-1}({{\mathcal {T}}}_n(z_0)-{{\mathcal {T}}}(z_0))\big \Vert }_{L^2}.$ Further, we obtain

$$\begin{aligned} {\big \Vert z_n-z_0\big \Vert }_{L^2}\le & {} c{\big \Vert ({{\mathcal {P}}}_n-\mathcal {I})z_0\big \Vert }_{L^2}=\mathcal {O}(n^{-r}), \end{aligned}$$

where c is a constant independent of n.

Proof

Using Lemma 2.6, we have

$$\begin{aligned} {\big \Vert {{{\mathcal {T}}}_n}'(z_0)-{{{\mathcal {T}}}}'(z_0)\big \Vert }_{L^2}= & {} {\big \Vert {\mathcal {P}}_n\Psi '({{\mathcal {K}}}z_0+f){\mathcal {K}} -\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}\big \Vert }_{L^2}\\= & {} \big \Vert ({\mathcal {P}}_n-\mathcal {I})\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}\big \Vert _{L^2}\\= & {} \big \Vert ({\mathcal {P}}_n-\mathcal {I}){\mathcal {T}}'(z_0)\big \Vert _{L^2}\rightarrow 0, ~as ~n \rightarrow \infty . \end{aligned}$$

Since we assume that 1 is not an eigen value of ${\mathcal {T}}'(z_{0})$, $(\mathcal {I}-{{\mathcal {T}}}'(z_0))$ is invertible. Hence by applying Lemma 2.4, we have ${(\mathcal {I}-{{\mathcal {T}}}_n'(z_0))}^{-1}$ exists and uniformly bounded on $\mathbb {X}$, for some sufficiently large n, i.e., there exists some $A_1 > 0$ such that $\big \Vert {(\mathcal {I}-{{\mathcal {T}}}_n'(z_0))}^{-1}\big \Vert _{L^2} \le A_1 <\infty .$

Now from estimates (2.2) and (2.23), we have for any $z\in B(z_0,\delta )$,

$$\begin{aligned} {\big \Vert [{{{\mathcal {T}}}_n}'(z_0)-{{{\mathcal {T}}}_n}'(z)]v\big \Vert }_{L^2}= & {} {\big \Vert [{\mathcal {P}}_n\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}-{{\mathcal {P}}}_n\Psi '({\mathcal {K}}z+f){\mathcal {K}}]v\big \Vert }_{L^2}\nonumber \\\le & {} p\big \Vert [\Psi '({\mathcal {K}}z_0+f)-\Psi '({\mathcal {K}}z+f)]{\mathcal {K}}v\big \Vert _{\infty }\nonumber \\\le & {} p\big \Vert \Psi '({\mathcal {K}}z_0+f)-\Psi '({\mathcal {K}}z+f)\big \Vert _{\infty }\big \Vert {\mathcal {K}}v\big \Vert _{\infty }\nonumber \\\le & {} \sqrt{2}pM\big \Vert \Psi '({\mathcal {K}}z_0+f)-\Psi '({\mathcal {K}}z+f)\big \Vert _{\infty }\big \Vert v\big \Vert _{L^2}.\qquad \end{aligned}$$

(2.44)

Taking use of the Lipschtiz’s continuity of $\psi ^{(0,1)}(.,x(.))$ and estimate (2.2), we have

$$\begin{aligned} \big \Vert \Psi '({\mathcal {K}}z_0+f)-\Psi '({\mathcal {K}}z+f)\big \Vert _{\infty }\le & {} c_2\big \Vert {\mathcal {K}}(z_0-z)\big \Vert _\infty \nonumber \\\le & {} \sqrt{2}c_2M\big \Vert z_0-z\big \Vert _{L^2}\le \sqrt{2}Mc_2\delta . \end{aligned}$$

(2.45)

Using the estimate (2.45) in (2.44), we obtain

$$\begin{aligned} {\big \Vert [{{{\mathcal {T}}}_n}'(z_0)-{{{\mathcal {T}}}_n}'(z)]v\big \Vert }_{L^2}\le 2pM^{2}c_2\delta \big \Vert v\big \Vert _{L^2}. \end{aligned}$$

Thus we have

$$\begin{aligned} \sup _{\big \Vert z-{z_0}\big \Vert _{L^2}\le \delta } {\big \Vert {(\mathcal {I}-{{{\mathcal {T}}}_n}'(z_0))}^{-1} ({{{\mathcal {T}}}_n}'(z_0)-{{{\mathcal {T}}}_n}'(z))\big \Vert }_{L^2}\le 2A_1pM^{2}c_2\delta \le q ~(say). \end{aligned}$$

Here we choose $\delta$ in such a way that, $0 < q <1$. This proves the Eq. (2.35) of Theorem 2.2.

Taking use of (2.25), we have

$$\begin{aligned} \alpha _n= & {} {\big \Vert (\mathcal {I}-{{{\mathcal {T}}}_n}'(z_0))^{-1} ({{\mathcal {T}}}_n(z_0)-{{\mathcal {T}}}(z_0))\big \Vert }_{L^2}\\\le & {} A_1{\big \Vert {{\mathcal {T}}}_n(z_0)-{{\mathcal {T}}}(z_0)\big \Vert }_{L^2}\\= & {} A_1{\big \Vert {{\mathcal {P}}}_n\Psi ({\mathcal {K}}z_0+f)-\Psi ({\mathcal {K}}z_0+f)\big \Vert }_{L^2}\\= & {} A_1\big \Vert ({{\mathcal {P}}}_n-\mathcal {I})\Psi ({\mathcal {K}}z_0+f)\big \Vert _{L^2}\\= & {} A_1{\big \Vert ({{\mathcal {P}}}_n-\mathcal {I})z_0\big \Vert }_{L^2}\rightarrow 0,~as~n \rightarrow \infty . \end{aligned}$$

By choosing n large enough such that $\alpha _n \le \delta (1-q),$ the Eq. (2.36) of Theorem 2.2 is satisfied. Hence by applying Theorem 2.2, we obtain

$$\begin{aligned} \frac{\alpha _n}{1+q}\le {\big \Vert z_n-z_0\big \Vert }_{L^2} \le \frac{\alpha _n}{1-q}, \end{aligned}$$

and

$$\begin{aligned} {\big \Vert z_n-z_0\big \Vert }_{L^2} \le \frac{\alpha _n}{1-q}\le c{\big \Vert ({{\mathcal {P}}}_n-\mathcal {I})z_0\big \Vert }_{L^2}. \end{aligned}$$

Hence from estimate (2.25), we have

$$\begin{aligned} \big \Vert z_n-z_0\big \Vert _{L^2}=\mathcal {O}(n^{-r}). \end{aligned}$$

This completes the proof. $\square$

Next we discuss the existence and convergence of the iterated approximate solution ${{\tilde{z}}_n}$ to $z_0$.

Theorem 2.4

$\widetilde{{\mathcal {T}}}_n'(z_0)$ is $\nu$-convergent to ${\mathcal {T}}'(z_0)$ in both infinity norm and $L^2$-norm.

Proof

Consider

$$\begin{aligned} \Big |{\widetilde{{\mathcal {T}}}_n}'(z_0)z(t)\Big |= & {} \Big |\Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f){\mathcal {K}}{\mathcal {P}}_nz(t)\Big |\nonumber \\\le & {} \Big |\Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)-\Psi '({\mathcal {K}}z_0+f)||{\mathcal {K}}{\mathcal {P}}_nz(t)\Big |\nonumber \\&+\Big |\Psi '({\mathcal {K}}z_0+f)||{\mathcal {K}}{\mathcal {P}}_nz(t)\Big |. \end{aligned}$$

(2.46)

Now using the Lipschtiz’s continuity of $\psi ^{(0,1)}(.,x(.))$ and Lemma 2.5, we have

$$\begin{aligned} \Big \Vert \Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)-\Psi ' ({\mathcal {K}}z_0+f)\Big \Vert _\infty\le & {} c_2\big \Vert {\mathcal {K}} (\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _\infty \nonumber \\\rightarrow & {} 0, ~ as~ n\rightarrow \infty . \end{aligned}$$

(2.47)

Using estimate (2.2) and (2.23), we have

$$\begin{aligned} \big \Vert {\mathcal {K}}{\mathcal {P}}_nz\big \Vert _\infty \le \sqrt{2}M\big \Vert {\mathcal {P}}_nz\big \Vert _{L^2}\le \sqrt{2}Mp\big \Vert z\big \Vert _\infty , \end{aligned}$$

(2.48)

which implies

$$\begin{aligned} \big \Vert {\mathcal {K}}{\mathcal {P}}_n\big \Vert _\infty \le \sqrt{2}Mp. \end{aligned}$$

(2.49)

Now combining the estimates (2.46), (2.47) and (2.49), we obtain

$$\begin{aligned} \big \Vert \widetilde{{\mathcal {T}}}_n'(z_0)\big \Vert _\infty \le \sqrt{2}M p(c_2\big \Vert {\mathcal {K}}(\mathcal {I}-{\mathcal {P}}_n)z_0 \big \Vert _\infty +\big \Vert \Psi '({\mathcal {K}}z_0+f)\big \Vert _\infty ) < \infty . \end{aligned}$$

This shows that $\big \Vert \widetilde{{\mathcal {T}}}_n'(z_0)\big \Vert _\infty$ is uniformly bounded.

Next Consider

$$\begin{aligned} |(\widetilde{{\mathcal {T}}}_n'(z_0)-{\mathcal {T}}'(z_0)) \widetilde{{\mathcal {T}}}_n'(z_0)z(t)|= & {} \Big |\{\Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f) {\mathcal {K}}{\mathcal {P}}_n-\Psi '({\mathcal {K}} z_0+f){\mathcal {K}}\}\widetilde{{\mathcal {T}}}_n'(z_0)z(t)\Big |\nonumber \\\le & {} \Big |\Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f) ({\mathcal {K}}{\mathcal {P}}_n-{\mathcal {K}}) \widetilde{{\mathcal {T}}}_n'(z_0)z(t)\Big |\nonumber \\&\quad +\,\Big |\{\Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)- \Psi '({\mathcal {K}}z_0+f)\}{\mathcal {K}}\widetilde{{\mathcal {T}}}_n' (z_0)z(t)\Big |.\nonumber \\ \end{aligned}$$

(2.50)

Now for the first term in the above estimate (2.50), using Lemma 2.5 we have

$$\begin{aligned} \big \Vert {\mathcal {K}}(\mathcal {I}-{\mathcal {P}}_n)\widetilde{{\mathcal {T}}}_n' (z_0)z\big \Vert _\infty= & {} \sup _{t\in [-1,1]}\left| \int _{-1}^{1}k(t,s)(\mathcal {I}-{\mathcal {P}}_n) \widetilde{{\mathcal {T}}}_n' (z_0)z(s)ds\right| \nonumber \\\le & {} \sqrt{2}M \big \Vert (\mathcal {I}-{\mathcal {P}}_n)\widetilde{{\mathcal {T}}}_n' (z_0)\big \Vert _{L^2}\big \Vert z\big \Vert _\infty . \end{aligned}$$

(2.51)

For the second term of the estimate (2.50) using estimates (2.2), (2.47), we have

$$\begin{aligned}&\big \Vert \{\Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)-\Psi ' ({\mathcal {K}}z_0+f)\}{\mathcal {K}}\widetilde{{\mathcal {T}}}_n' (z_0)z\big \Vert _\infty \nonumber \\&\quad \le \big \Vert \Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)-\Psi ' ({\mathcal {K}}z_0+f)\big \Vert _\infty \big \Vert {\mathcal {K}}\widetilde{{\mathcal {T}}}_n'(z_0)z\big \Vert _\infty \nonumber \\&\quad \le \, c_2\big \Vert {\mathcal {K}}(\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _{\infty }\big \Vert {\mathcal {K}}\widetilde{{\mathcal {T}}}_n'(z_0)z\big \Vert _{\infty }\nonumber \\&\quad \le \, 2\sqrt{2}c_2M^2\big \Vert (\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _{L^2}\big \Vert \widetilde{{\mathcal {T}}}_n'(z_0)\big \Vert _\infty \big \Vert z\big \Vert _{\infty }. \end{aligned}$$

(2.52)

Now combining estimates (2.41), (2.50), (2.51) and (2.52), we see that

$$\begin{aligned}&\big \Vert (\widetilde{{\mathcal {T}}}_n'(z_0)-{\mathcal {T}}'(z_0)) \widetilde{{\mathcal {T}}}_n'(z_0)z\big \Vert _\infty \\&\quad \quad \quad \le \left\{ \sqrt{2}MB\big \Vert (\mathcal {I}-{\mathcal {P}}_n) \widetilde{{\mathcal {T}}}_n'(z_0)\big \Vert _{L^2}+2 \sqrt{2}c_2M^2\big \Vert (\mathcal {I}-{\mathcal {P}}_n)z_0 \big \Vert _{L^2}\big \Vert \widetilde{{\mathcal {T}}}_n'(z_0)\big \Vert _{\infty }\right\} \big \Vert z\big \Vert _{\infty }. \end{aligned}$$

Hence using Lemma 2.6, estimate (2.25) and the uniform boundedness of $\big \Vert \widetilde{{\mathcal {T}}}_n'(z_0)\big \Vert _\infty$, we obtain

$$\begin{aligned} \big \Vert (\widetilde{{\mathcal {T}}}_n'(z_0)-{\mathcal {T}}'(z_0))\widetilde{{\mathcal {T}}}_n'(z_0)\big \Vert _\infty \rightarrow 0, ~ as~ n\rightarrow ~\infty . \end{aligned}$$

Following the similar steps we can prove that

$$\begin{aligned} \big \Vert (\widetilde{{\mathcal {T}}}_n'(z_0)-{\mathcal {T}}'(z_0)) {\mathcal {T}}'(z_0)\big \Vert _\infty \rightarrow 0, ~ as~ n\rightarrow ~\infty . \end{aligned}$$

This shows that $\widetilde{{\mathcal {T}}}_n'(z_0)$ is $\nu$-convergent to ${\mathcal {T}}'(z_0)$ in infinity norm.

On similar lines, we can show that $\widetilde{{\mathcal {T}}}_n'(z_0)$ is $\nu$-convergent to ${\mathcal {T}}'(z_0)$ in $L^2$- norm. This completes the proof. $\square$

Hence by applying the Lemma 2.4 and Theorem 2.4, we obtain the following theorem.

Theorem 2.5

Let $z_0\in {\mathcal {C}}^r[-1,1]$ be an isolated solution of the Eq. (2.9). Assume that 1 is not an eigenvalue of $\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}$. Then for sufficiently large n, the operator $\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}'(z_0)$ is invertible on ${\mathcal {C}}[-1,1]$ and there exist constants $L, \; L_1 > 0$ independent of n such that ${\big \Vert (\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}'(z_0))^{-1}\big \Vert }_\infty \le L$ and ${\big \Vert (\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}'(z_0))^{-1}\big \Vert }_{L^2} \le L_1.$

Theorem 2.6

Let $z_0\in {\mathcal {C}}^r[-1,1]$ be an isolated solution of the Eq. (2.9). Let ${\mathcal {P}}_n :\mathbb {X}\rightarrow \mathbb {X}_n$ be either orthogonal or interpolatory projection operator defined by (2.15) and (2.19) respectively. Assume that 1 is not an eigenvalue of $\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}$, then for sufficiently large n, the iterated solution ${{\tilde{z}}}_n$ defined by (2.33) is the unique solution in the sphere $B(z_0,\delta )=\{z \,:\big \Vert z-z_0\big \Vert _\infty < \delta \}$. Moreover, there exists a constant $0<q<1$, independent of n such that

$$\begin{aligned} \frac{\beta _n}{1+q}\le {\big \Vert {{\tilde{z}}_n}-z_0\big \Vert }_\infty \le \frac{\beta _n}{1-q}, \end{aligned}$$

where

$$\begin{aligned} \beta _n={\big \Vert (\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}' (z_0))^{-1}(\widetilde{{\mathcal {T}}}_n(z_0)-{\mathcal {T}}(z_0))\big \Vert }_\infty . \end{aligned}$$

Proof

From Theorem 2.5, we can say, there exists a constant $L > 0$ such that ${\big \Vert (\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}'(z_0))^{-1}\big \Vert }_\infty \le L$, for sufficiently large value of n.

Consider for any $z \in B(z_0,\delta ),$

$$\begin{aligned} ~~\big \Vert [\widetilde{{\mathcal {T}}}_n'(z) - \widetilde{{\mathcal {T}}}_n'(z_0)]v\big \Vert _\infty= & {} {\big \Vert [\{\Psi '({\mathcal {K}}{\mathcal {P}}_nz+f)- \Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)\}{\mathcal {K}} {\mathcal {P}}_n]v\big \Vert }_\infty \nonumber \\\le & {} \big \Vert \Psi '({\mathcal {K}}{\mathcal {P}}_nz+f)-\Psi ' ({\mathcal {K}}{\mathcal {P}}_nz_0+f)\big \Vert _\infty \big \Vert {\mathcal {K}} {\mathcal {P}}_nv\big \Vert _\infty .\qquad \end{aligned}$$

(2.53)

Using Cauchy-Schawrz inequality and estimate (2.48), we have

$$\begin{aligned} \big \Vert \Psi '({\mathcal {K}}{\mathcal {P}}_nz+f)-\Psi '({\mathcal {K}}{\mathcal {P}}_nz_0+f)\big \Vert _\infty\le & {} c_2\big \Vert {\mathcal {K}}{\mathcal {P}}_n(z_0-z)\big \Vert _\infty \nonumber \\\le & {} \sqrt{2}M c_2 p{\big \Vert z-z_0\big \Vert }_\infty \nonumber \\\le & {} \sqrt{2}Mc_2p\delta . \end{aligned}$$

(2.54)

Combining estimates (2.48), (2.53), (2.54), we obtain

$$\begin{aligned} \big \Vert [\widetilde{{\mathcal {T}}}_n'(z)- \widetilde{{\mathcal {T}}}_n'(z_0)]v\big \Vert _\infty \le 2M^2c_2p^2\delta \big \Vert v\big \Vert _\infty . \end{aligned}$$

(2.55)

This implies

$$\begin{aligned} \sup _{\big \Vert z-{z_0}\big \Vert _\infty \le \delta } {\big \Vert {(\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}' (z_0))}^{-1}({\widetilde{{\mathcal {T}}}_n}'(z)- {\widetilde{{\mathcal {T}}}_n}'(z_0))\big \Vert }_\infty \le 2LM^2c_2p^2\delta \le q~ (say). \end{aligned}$$

We choose $\delta$ in such a way that $0<q<1$. Hence this proves the Eq. (2.35) of Theorem 2.2.

Now using the Lipschtiz’s continuity of $\psi (.,x(.))$ and Lemma 2.5, we have

$$\begin{aligned} \big \Vert \widetilde{{\mathcal {T}}}_n(z_0)-{\mathcal {T}}(z_0)\big \Vert _\infty\le & {} \big \Vert \Psi ({\mathcal {K}}{\mathcal {P}}_nz_0+f)-\Psi ({\mathcal {K}}z_0+f)\big \Vert _\infty \nonumber \\\le & {} c_1\big \Vert {\mathcal {K}}(\mathcal {I}-{\mathcal {P}}_n)z_0\big \Vert _\infty \rightarrow 0, ~ as~ n\rightarrow \infty . \end{aligned}$$

(2.56)

Hence

$$\begin{aligned} \beta _n= & {} {\big \Vert (\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}' (z_0))^{-1}(\widetilde{{\mathcal {T}}}_n(z_0)- {\mathcal {T}}(z_0))\big \Vert }_\infty \nonumber \\ ~\le & {} Lc_1\big \Vert {\mathcal {K}}(\mathcal {I}- {\mathcal {P}}_n)z_0\big \Vert _\infty \rightarrow 0, ~ as~ n\rightarrow \infty . \end{aligned}$$

Choose n large enough such that $\beta _n\le \delta (1-q).$ Then the Eq. (2.36) of Theorem 2.2 is satisfied. Thus by applying Theorem 2.2, we obtain

$$\begin{aligned} \frac{\beta _n}{1+q}\le {\big \Vert {{\tilde{z}}_n}-z_0\big \Vert }_\infty \le \frac{\beta _n}{1-q} \end{aligned}$$

where

$$\begin{aligned} \beta _n={\big \Vert {(\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}' (z_0))}^{-1}({\widetilde{{\mathcal {T}}}_n}(z_0) -{\mathcal {T}}(z_0))\big \Vert }_\infty . \end{aligned}$$

This completes the proof. $\square$

Theorem 2.7

Let $z_0\in {\mathcal {C}}^r[-1,1]$ be an isolated solution of the Eq. (2.9). Let ${\mathcal {P}}_n :\mathbb {X}\rightarrow \mathbb {X}_n$ be either orthogonal or interpolatory projection operator defined by (2.15) and (2.19) respectively. Assume that 1 is not an eigenvalue of $\Psi '({\mathcal {K}}z_0+f){\mathcal {K}}$, then for sufficiently large n, the iterated solution ${{\tilde{z}}}_n$ defined by (2.33) is the unique solution in the sphere $B(z_0,\delta )=\{z \,:\big \Vert z-z_0\big \Vert _{L^2} < \delta \}$. Moreover, there exists a constant $0<q<1$, independent of n such that

$$\begin{aligned} \frac{\beta _n}{1+q}\le {\big \Vert {{\tilde{z}}_n}-z_0\big \Vert }_{L^{2}} \le \frac{\beta _n}{1-q}, \end{aligned}$$

where

$$\begin{aligned} \beta _n={\big \Vert (\mathcal {I}-{\widetilde{{\mathcal {T}}}_n}'(z_0))^{-1}(\widetilde{{\mathcal {T}}}_n(z_0)-{\mathcal {T}}(z_0))\big \Vert }_{L^2}. \end{aligned}$$

Proof

Using the similar steps as in the proof of Theorem 2.6, the proof of the above theorem can be easily done.$\square$

Theorem 2.8

Let $z_0\in {\mathcal {C}}[-1,1]$ be an isolated solution of the Eq. (2.9). Let ${{\tilde{z}}_n}$ defined by the iterated scheme (2.33). Then the following hold

$$\begin{aligned} {\big \Vert {{\tilde{z}}_n}-z_0\big \Vert }_\infty \le c \sup _{t\in [-1,1]}|<k_t,(\mathcal {I}-{{\mathcal {P}}}_n)z_0>|, \end{aligned}$$

(2.57)

and

$$\begin{aligned} {\big \Vert {{\tilde{z}}_n}-z_0\big \Vert }_{L^2}\le c\sup _{t\in [-1,1]}|<k_t,(\mathcal {I}-{{\mathcal {P}}}_n)z_0>|, \end{aligned}$$

(2.58)

where c is a constant independent of n.

Proof

It follows from Theorem 2.6 that

$$\begin{aligned} \frac{\beta _n}{1+q}\le {\big \Vert {{\tilde{z}}_n}-z_0\big \Vert }_\infty \le \frac{\beta _n}{1-q}, \end{aligned}$$

where

$$\begin{aligned} \beta _n={\big \Vert (\mathcal {I}-\widetilde{{\mathcal {T}}}_n' (z_0))^{-1}(\widetilde{{\mathcal {T}}}_n(z_0)-{\mathcal {T}}(z_0))\big \Vert }_\infty . \end{aligned}$$

Hence from Theorem 2.5, estimates (2.39), (2.56), we have

$$\begin{aligned} \big \Vert {\tilde{z}}_n-z_0\big \Vert _\infty \le \frac{\beta _n}{1-q}\le & {} c{\big \Vert (\mathcal {I}-\widetilde{{\mathcal {T}}}_n'(z_0))^{-1}(\widetilde{{\mathcal {T}}}_n(z_0)-{\mathcal {T}}(z_0))\big \Vert }_\infty \\\le & {} c L \big \Vert \Psi ({\mathcal {K}}{\mathcal {P}}_nz_0+f)-\Psi ({\mathcal {K}}z_0+f)\big \Vert _\infty \\\le & {} cLc_1\big \Vert {\mathcal {K}}({\mathcal {P}}_n-\mathcal {I})z_0\big \Vert _\infty \\\le & {} c\sup _{t\in [-1,1]}|<k_t(.),(\mathcal {I}-{\mathcal {P}}_n)z_0>|. \end{aligned}$$

This proves the estimate (2.57).

Similarly for $L^2$- norm we can show that

$$\begin{aligned} {\big \Vert {{\tilde{z}}_n}-z_0\big \Vert }_{L^2}\le \sqrt{2}{\big \Vert {{\tilde{z}}_n}-z_0\big \Vert }_\infty \le c\sup _{t\in [-1,1]} |<k_t,(\mathcal {I}-{{\mathcal {P}}}_n)z_0>|, \end{aligned}$$

where c is a constant independent of n.

This completes the proof. $\square$

Theorem 2.9

Let $x_{0}\in {\mathcal {C}}^r[-1,1]$ be an isolated solution of the Eq. (2.1) and ${x}_{n}$ be the Legendre Galerkin or Legendre collocation approximations of $x_{0}$. Then there hold

$$\begin{aligned}&\big \Vert x_0- {x}_n\big \Vert _{L^2} =\mathcal {O}(n^{-r}),\\&\big \Vert x_0- {x}_n\big \Vert _{\infty } =\mathcal {O}(n^{-r}). \end{aligned}$$

Proof

Using estimates (2.2), (2.8), (2.30) and Theorem 2.3, we have

$$\begin{aligned} \big \Vert x_0- {x}_n\big \Vert _{\infty }=\big \Vert {\mathcal {K}}(z_0- {z}_n)\big \Vert _{\infty }\le & {} \sqrt{2}M\big \Vert z_0- {z}_n\big \Vert _{L^2} = \mathcal {O}(n^{-r}), \end{aligned}$$

and

$$\begin{aligned} \big \Vert x_0- {x}_n\big \Vert _{L^2}\le \sqrt{2}\big \Vert x_0- {x}_n\big \Vert _{\infty }=\mathcal {O}(n^{-r}). \end{aligned}$$

Hence the proof follows. $\square$

Now we discuss the convergence rates for the iterated approximate solutions. To distinguish between the iterated Legendre Galerkin method and iterated Legendre collocation method, we set the following notations. In case of iterated Legendre Galerkin method we denote ${\tilde{z}}_n={\tilde{z}}_n^G$ and ${\tilde{x}}_n={\tilde{x}}_n^G$, and for iterated Legendre collocation method we write ${\tilde{z}}_n = {\tilde{z}}_n^C$ and ${\tilde{x}}_n = {\tilde{x}}_n^C$.

Theorem 2.10

Let $x_{0}\in {\mathcal {C}}^r[-1,1]$ be an isolated solution of the Eq. (2.1) and $\tilde{x}_{n}^G$ be the iterated Legendre Galerkin approximation of $x_{0}$. Then the following superconvergence rates hold

$$\begin{aligned}&\big \Vert x_0- {\tilde{x}}_n^G\big \Vert _{L^2} =\mathcal {O}(n^{-2r}),\\&\big \Vert x_0- {\tilde{x}}_n^G\big \Vert _{\infty } =\mathcal {O}(n^{-2r}). \end{aligned}$$

Proof

From Theorem 2.8, we have

$$\begin{aligned} \big \Vert {\tilde{z}}^G_n-z_0\big \Vert _\infty \le c \sup _{t\in [-1,1]}\big |<k_t(.),(\mathcal {I}-{{\mathcal {P}}}_n^G)z_0(.)>\big |. \end{aligned}$$

(2.59)

Using the orthogonality of the projection operators ${{\mathcal {P}}}_n^G$, Cauchy-Schwarz inequality and estimate (2.17) of Lemma 2.1, we obtain

$$\begin{aligned} \big |<k_t(.),(\mathcal {I}-{{\mathcal {P}}}_n^G)z_0(.)>\big |= & {} |<(\mathcal {I}-{{\mathcal {P}}}_n^G)k_t(.),(\mathcal {I}-{{\mathcal {P}}}_n^G)z_0(.)>|\nonumber \\\le & {} {\big \Vert (\mathcal {I}-{{\mathcal {P}}}_n^G)k_t(.)\big \Vert }_{L^2}{\big \Vert z_0-{{\mathcal {P}}}_n^Gz_0\big \Vert }_{L^2}\nonumber \\\le & {} c n^{-2r} \big \Vert z_0^{(r)}\big \Vert _{L^2} \big \Vert (k_t(.))^{(r)}\big \Vert _{L^2}\nonumber \\\le & {} c n^{-2r} \big \Vert z_0^{(r)}\big \Vert _{L^2} \big \Vert k\big \Vert _{r,\infty }. \end{aligned}$$

(2.60)

Hence from (2.59) and (2.60), we have

$$\begin{aligned} {\big \Vert {{\tilde{z}}_n^G}-z_0\big \Vert }_\infty&\le c n^{-2r} \big \Vert z_0^{(r)}\big \Vert _{L^2} \big \Vert k\big \Vert _{r,\infty } =O(n^{-2r}), \end{aligned}$$

(2.61)

and

$$\begin{aligned} \big \Vert {{\tilde{z}}_n^G}-z_0\big \Vert _{L^2} \le \sqrt{2}{\big \Vert {{\tilde{z}}_n^G}-z_0\big \Vert }_\infty = \mathcal {O}(n^{-2r}). \end{aligned}$$

(2.62)

Using estimates (2.2), (2.8), (2.34) and (2.62), we have

$$\begin{aligned} \big \Vert x_0- {\tilde{x}}_n^G\big \Vert _{\infty }=\big \Vert {\mathcal {K}}(z_0-{\tilde{z}}_n^G) \big \Vert _{\infty }\le \sqrt{2}M\big \Vert {{\tilde{z}}_n^G}-z_0\big \Vert _{L^2} = \mathcal {O}(n^{-2r}), \end{aligned}$$

and

$$\begin{aligned} \big \Vert x_0- {\tilde{x}}_n^G\big \Vert _{L^2}\le \sqrt{2}\big \Vert x_0- {\tilde{x}}_n^G\big \Vert _{\infty }=\mathcal {O}(n^{-2r}). \end{aligned}$$

Hence the proof follows. $\square$

Theorem 2.11

Let $x_{0}\in {\mathcal {C}}^r[-1,1]$ be an isolated solution of the Eq. (2.1) and $\tilde{x}_{n}^C$ be the iterated Legendre collocation approximation of $x_{0}$. Then the following hold

$$\begin{aligned}&\big \Vert x_0- {\tilde{x}}_n^C\big \Vert _{L^2} =\mathcal {O}(n^{-r}),\\&\big \Vert x_0- {\tilde{x}}_n^C\big \Vert _{\infty } =\mathcal {O}(n^{-r}). \end{aligned}$$

Proof

Using Theorem 2.8, estimate (2.21) of Lemma 2.3 and Cauchy-Schwarz inequality, we have for the interpolatory projection operator ${\mathcal {P}}_n^C$

$$\begin{aligned} \big \Vert {\tilde{z}}_n^C-z_0\big \Vert _\infty\le & {} c\sup _{t\in [-1,1]}|<k_t(.),(\mathcal {I}-{{\mathcal {P}}}_n^C)z_0(.)>|\nonumber \\\le & {} c\sup _{t\in [-1,1]}{\big \Vert k_t(.)\big \Vert }_{L^2}{\big \Vert z_0-{{\mathcal {P}}}_n^Cz_0\big \Vert }_{L^2}\nonumber \\\le & {} c\sqrt{2}M{\big \Vert z_0-{{\mathcal {P}}}_n^Cz_0\big \Vert }_{L^2}\nonumber \\\le & {} \sqrt{2}Mc n^{-r} \big \Vert z_0^{(r)}\big \Vert _{L^2} =O(n^{-r}), \end{aligned}$$

(2.63)

and

$$\begin{aligned} \big \Vert {{\tilde{z}}_n^C}-z_0\big \Vert _{L^2} \le \sqrt{2}{\big \Vert {{\tilde{z}}_n^C}-z_0\big \Vert }_\infty = \mathcal {O}(n^{-r}). \end{aligned}$$

(2.64)

Using estimates (2.2), (2.8), (2.34) and (2.64), we have

$$\begin{aligned} \big \Vert x_0- {\tilde{x}}_n^C\big \Vert _{\infty }=\big \Vert {\mathcal {K}}(z_0-{\tilde{z}}_n^C)\big \Vert _{\infty }\le \sqrt{2}M\big \Vert {{\tilde{z}}_n^C}-z_0\big \Vert _{L^2} = \mathcal {O}(n^{-r}), \end{aligned}$$

and

$$\begin{aligned} \big \Vert x_0- {\tilde{x}}_n^C\big \Vert _{L^2}\le \sqrt{2}\big \Vert x_0- {\tilde{x}}_n^C\big \Vert _{\infty }=\mathcal {O}(n^{-r}). \end{aligned}$$

Hence the proof follows. $\square$

Remark

From Theorems 2.9, 2.10, and 2.11 we observe that the Legendre Galerkin and Legendre collocation solutions have same order of convergence, $O(n^{-r})$ both in $L^2$-norm and infinity norm. The iterated Legendre Galerkin solution converges with the order $O(n^{-2r})$ in both $L^2$-norm and infinity norm whereas the iterated Legendre collocation solution converges with the order $O(n^{-r})$ in both $L^2$-norm and in infinity norm.

3 Numerical Example

In this section we present the numerical results. For that we take the Legendre polynomials as the basis functions of $\mathbb {X}_n$ from the three-term recurrence relation

$$\begin{aligned} \phi _0(s)=1, \phi _1(s)=s,\quad s\in [-1,1], \end{aligned}$$

and

$$\begin{aligned} (i+1)\phi _{i+1}(s)= (2i+1)s\phi _{i}(s)-i\phi _{i-1}(s),\quad s\in [-1,1], \quad i = 1, 2,\ldots , n-1. \end{aligned}$$

(3.1)

We present the errors of the approximation solutions and the iterated approximation solutions under the Legendre Galerkin and Legendre collocation methods in both ${L^2}$-norm and infinity norm in Tables 1–4. We use n to represent the highest degree of the Legendre polynomials employed in the computation. The numerical algorithm was run on a PC with Intel Pentium 1.83 GHz CPU, 512MB RAM, and the programs were compiled by using Matlab.

Example 3.1

We consider the following Hammerstein integral equation

$$\begin{aligned} x(t)-\int _{-1}^{1}k(t,s)\psi (s,x(s))ds=f(t),~ -1\le t \le 1, \end{aligned}$$

(3.2)

with the kernel function $k(t,s)=\big (\frac{3\sqrt{2}\pi }{16}\big )\cos \big (\frac{\pi |s-t|}{4}\big )$, $\psi (s,x(s))=[x(s)]^2$ and the function $f(t)=\big (\frac{-1}{4}\big )\cos \big (\frac{\pi t}{4}\big )$ where the exact solution is given by $x(t)=\cos \big (\frac{\pi t}{4}\big )$.

Table 1 Legendre Galerkin method

Table 2 Legendre collocation method

Table 3 Legendre Galerkin method

Table 4 Legendre collocation method

Example 3.2

We consider the following Hammerstein integral equation

$$\begin{aligned} x(t)-\int _{-1}^{1}k(t,s)\psi (s,x(s))ds=f(t), -1\le t \le 1 \end{aligned}$$

(3.3)

with the kernel function $k(t,s)= e^{-2s}\sin (t)$, $\psi (s,x(s))=[x(s)]^2$ and the function $f(t)=t^2$ where the exact solution is given by $x(t)=t^2+1.95778sin(t)$.