In this section, we describe the Galerkin and collocation methods for solving Hammerstein integral equation of mixed type using Legendre polynomial basis functions.
Let \(\mathbb {X}= L^{2}[-1,\,1]\) or \(\mathcal {C}[-1,\,1]\) and consider the following Hammerstein integral equation of mixed type
$$\begin{aligned} x(t)-\sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\psi _{i}(s,\,x(s))ds=f(t),\quad -1\le t \le 1, \end{aligned}$$
(2.1)
where \(k_{i},\,f\) and \(\psi _{i}\) are known functions and x is the unknown function to be determined. Let
$$\begin{aligned} \left( \mathcal {K}_{i}\psi _{i}\right) (x)(t)=\int \nolimits _{-1}^{1} k_{i}(t,\,s)\psi _{i}(s,\,x(s))ds,\quad x\in \mathbb {X}. \end{aligned}$$
Then the Eq. (2.1) can be written as
$$\begin{aligned} x-\sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}(x)=f. \end{aligned}$$
(2.2)
Next, we define the operator \(\mathcal {T}\) on \(\mathbb {X}\) by
$$\begin{aligned} \mathcal {T}x{:=}f+\underset{i=1}{\overset{m}{\sum }}\mathcal {K}_{i}\psi _{i}(x),\quad x\in \mathbb {X}, \end{aligned}$$
then the Eq. (2.2) can be written as
$$\begin{aligned} x=\mathcal {T}x. \end{aligned}$$
(2.3)
Throughout the paper, the following assumptions are made on \(f,\,k_{i}(.,\,.)\) and \(\psi _{i}(.,\,x(.)):\)
-
(i)
\(f\in \mathcal {C}[-1,\,1],\)
-
(ii)
\(\lim \nolimits _{t\rightarrow t^{'}}\Vert k_{i}(t,\,.)-k_{i}(t^{'},\,.)\Vert _\infty =0,\quad t,\,t^{'}\in [-1,\,1],\quad 1\le i \le m,\)
-
(iii)
\(m_{i} =\sup \nolimits _{t,\,s\in [-1,\,1]}|k_{i}(t,\,s)|<{\infty }\) and \(M=\sum \nolimits _{i=1}^{m}m_{i},\)
-
(iv)
the nonlinear functions \(\psi _{i}(s,\,x)\) are continuous in \(s\in [-1,\,1]\) and Lipschitz continuous in x, i.e., for any \(x_{1},\,x_{2} \in \mathbb {R},\) there exists constants \(c_{i} > 0, i=1,\,2,\ldots ,m\) such that
$$\begin{aligned} \left| \psi _{i}\left( s,\,x_{1}\right) -\psi _{i}\left( s,\,x_{2}\right) \right| \le c_{i}\left| x_{1}-x_{2}\right| , \end{aligned}$$
and \(l_{1}= \sup \nolimits _{i=1,2,\ldots ,m}c_{i},\)
-
(v)
the partial derivatives \(\psi ^{(0,1)}_{i}(s,\,x)\) of \(\psi _{i}(s,\,x)\) with respect to the second variable exist and Lipschitz continuous in x, i.e., for any \(x_{1},\,x_{2} \in \mathbb {R},\) there exists constants \(q_{i} > 0,\, i=1,\,2,\ldots ,m\) such that
$$\begin{aligned} \left| \psi ^{(0,1)}_{i}\left( s,\,x_{1}\right) -\psi ^{(0,1)}_{i}\left( s,\,x_{2}\right) \right| \le q_{i}\left| x_{1}-x_{2}\right| , \end{aligned}$$
and \(l_{2}= \underset{i=1,2,\ldots ,m}{\sup }q_{i},\)
-
(vi)
\(\psi _{i}(s,\,y)=\psi _{i}(s,\,y_{0})+\psi _{i}^{(0,1)}(s,\,y_{0}+\theta (y-y_{0}))(y-y_{0}),\,\forall y,\,y_{0} \in \mathbb {R},\) and \(0 < \theta < 1.\)
The following theorem gives the condition for the existence of unique solution of the Eq. (2.3) in \(\mathbb {X}.\)
Theorem 2.1
Let \(\mathbb {X}= L^{2}[-1,\,1]\) or \(\mathcal {C}[-1,\,1]\) and \(f\in \mathbb {X},\) and let \(k_{i}(.,\,.) \in \mathcal {C}([-1,\,1]\times [-1,\,1])\) with \(m_{i}= \sup \nolimits _{t,s\in [-1,\,1]}|k_{i}(t,\,s)|<{\infty }.\) Let \(\psi _{i}(s,\,x(s))\in \mathcal {C}([-1,\,1]\times \mathbb {R})\) satisfy the Lipschitz condition in the second variable, i.e.,
$$\begin{aligned} \left| \psi _{i}\left( s,\,y_{1}\right) -\psi _{i}\left( s,\,y_{2}\right) \right| \le c_{i}\left| y_{1}-y_{2}\right| ,\quad y_{1},\,y_{2}\in \mathbb {R},\quad i=1,\,2,\ldots ,m, \end{aligned}$$
with \(2Ml_{1} < 1,\) where \(M=\sum \nolimits _{i=1}^{m} m_{i}\) and \(l_{1}=max\{c_{1},\,c_{2},\ldots ,c_{m}\}.\) Then the operator equation \(x = \mathcal {T}x\) has a unique solution \(x_{0} \in \mathbb {X},\) i.e., we have \(x_{0}=\mathcal {T}x_{0}.\)
Proof of the above theorem can be easily done using similar technique given in Theorem 2.4 of [9].
We set the following notations. Let
$$\begin{aligned} \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x(t)=\sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) x(s)ds. \end{aligned}$$
For the rest of the paper we assume that the kernel \(k_{i}(.,\,.) \in \mathcal {C}^{r}([-1,\,1]\times [-1,\,1]).\) Let \(\Vert k\Vert _{j,\infty }=\sum \limits _{i=1}^{m}\Vert k_{i}\Vert _{j,\infty },\) where
$$\begin{aligned} \left\| k_{i}\right\| _{j,\infty }=\sum \limits _{p=0}^{j}\sum \limits _{q=0}^{j}\left| \frac{\partial ^{p+q}}{\partial t^{p}\partial s^{q}}k_{i}(t,\,s)\right| ,\quad t,\,s \in [-1,\,1]. \end{aligned}$$
For \(j=0,\,1,\ldots ,r,\) we have
$$\begin{aligned}&\left| \sum \limits _{i=1}^{m}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x\right] ^{(j)}(t)\right| =\left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}\frac{\partial ^{j}}{\partial t^{j}}k_{i}(t,\,s)\psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) x(s)ds\right| \nonumber \\&\quad \le \sum \limits _{i=1}^{m}\underset{s,t\in [-1,1]}{\sup }\left| \frac{\partial ^{j}}{\partial t^{j}}k_{i}(t,\,s)\right| \underset{i=1,2,\ldots ,m}{\sup }\underset{s\in [-1,1]}{\sup }\left| \psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) \right| \int \nolimits _{-1}^{1}|x(s)|ds\nonumber \\&\quad \le \sqrt{2}\underset{i=1,2,\ldots ,m}{\sup }d_{i}\sum \limits _{i=1}^{m}\left\| k_{i}\right\| _{j,\infty }\Vert x\Vert _{L^{2}}\nonumber \\&\quad \le \sqrt{2}d\Vert k\Vert _{j,\infty }\Vert x\Vert _{L^{2}}\le 2d\Vert k\Vert _{j,\infty }\Vert x\Vert _{\infty }, \end{aligned}$$
(2.4)
where \(d_{i}=\sup \nolimits _{s\in [-1,1]}|\psi _{i}^{(0,1)}(s,\,x_{0}(s))|\) and \(d=\sup \nolimits _{i=1,2,\ldots ,m}d_{i}.\)
Hence, for \(j=0,\,1,\,2,\ldots ,r,\) we have
$$\begin{aligned} \left\| \sum \limits _{i=1}^{m}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x\right] ^{(j)}\right\| _\infty \le \sqrt{2}d\Vert k\Vert _{j,\infty }\Vert x\Vert _{L^{2}}\le 2d\Vert k\Vert _{j,\infty }\Vert x\Vert _\infty , \end{aligned}$$
(2.5)
and
$$\begin{aligned} \left\| \sum \limits _{i=1}^{m}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x\right] ^{(j)}\right\| _{L^{2}}\le \sqrt{2}\left\| \sum \limits _{i=1}^{m}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x\right] ^{(j)}\right\| _\infty \le 2d\Vert k\Vert _{j,\infty }\Vert x\Vert _{L^{2}}. \end{aligned}$$
(2.6)
Next we prove the following lemma, which we need in our convergence analysis.
Lemma 2.1
For any \(x,\,y\in L^{2}[-1,\,1]\) or \(\mathcal {C}[-1,\,1],\) the following hold
$$\begin{aligned}&\left\| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) \left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) (x)\right\| _{\infty } \le \sqrt{2}Ml_{1}\left\| x_{0}-x\right\| _{L^{2}},\\&\left\| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right] y\right\| _{\infty }\le \sqrt{2}Ml_{2}\left\| x_{0}-x\right\| _{L^{2}}\Vert y\Vert _{\infty }. \end{aligned}$$
Proof
Using Lipschitz’s continuity of \(\psi _{i}^{(0,1)}(.,\,x_{0}(.))\) and Cauchy–Schwarz inequality, we have
$$\begin{aligned}&\left| \left[ \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) \left( x_{0}\right) -\underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) (x)\right] (t)\right| \\&\quad \le \left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\left[ \psi _{i}\left( s,\,x_{0}(s)\right) -\psi _{i}(s,\,x(s))\right] ds\right| \nonumber \\&\quad \le \sum \limits _{i=1}^{m}\sup \limits _{s,t\in [-1,1]}\left| k_{i}(t,\,s)\right| \int \nolimits _{-1}^{1}\left| \psi _{i}\left( s,\,x_{0}(s)\right) -\psi _{i}(s,\,x(s))\right| ds\nonumber \\&\quad \le \sum \limits _{i=1}^{m}m_{i}\int \nolimits _{-1}^{1}c_{i}\left| \left( x_{0}-x\right) (s)\right| ds\\&\quad \le Ml_{1}\int \nolimits _{-1}^{1}\left| \left( x_{0}-x\right) (s)\right| ds\nonumber \\&\quad \le \sqrt{2}Ml_{1}\left\| x_{0}-x\right\| _{L^{2}}. \end{aligned}$$
This implies that
$$\begin{aligned} \left\| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) \left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) (x)\right\| _{\infty } \le \sqrt{2}Ml_{1}\left\| x_{0}-x\right\| _{L^{2}}. \end{aligned}$$
(2.7)
On the similar lines, using Lipschitz’s continuity of \(\psi _{i}^{(0,1)}(.,\,x_{0}(.))\) and Cauchy–Schwarz inequality, we obtain
$$\begin{aligned} \left\| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right] y\right\| _{\infty }&\le Ml_{2}\left\| x_{0}-x\right\| _{L^{2}}\Vert y\Vert _{L^{2}}\end{aligned}$$
(2.8)
$$\begin{aligned}&\le \sqrt{2}Ml_{2}\left\| x_{0}-x\right\| _{L^{2}}\Vert y\Vert _{\infty }.\qquad \end{aligned}$$
(2.9)
Hence the proof follows.\(\square \)
Next we will apply Legendre Galerkin and Legendre collocation methods to the Eq. (2.1). To do this, we let \(\mathbb {X}_{n}\) be the sequence of polynomial subspaces of \(\mathbb {X}\) of degree \(\le \)
n and we choose Legendre polynomials \(\{\phi _{0},\,\phi _{1},\,\phi _{2},\ldots ,\phi _{n}\}\) as an orthonormal basis for the subspace \(\mathbb {X}_{n},\) where,
$$\begin{aligned} \phi _{0}(x)=1,\quad \phi _{1}(x)=x,\quad x\in [-1,\,1], \end{aligned}$$
and for \(i = 1,\,2,\ldots ,n-1\)
$$\begin{aligned} (i+1)\phi _{i+1}(x)= (2i+1)x\phi _{i}(x)-i\phi _{i-1}(x),\quad x\in [-1,\,1]. \end{aligned}$$
(2.10)
Orthogonal projection operator let \(\mathbb {X}=L^{2}[-1,\,1]\) or \(\mathcal {C}[-1,\,1]\) and let the operator \({\mathcal {P}}^{G}_{n}{\text {:}}\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) be the orthogonal projection defined by
$$\begin{aligned} {\mathcal {P}}^{G}_{n} x=\sum \limits ^{n}_{j=0} \langle x, \,\phi _{j} \rangle \phi _{j},\quad x \in \mathbb {X}, \end{aligned}$$
(2.11)
where \(\langle x,\, \phi _{j} \rangle = \int \nolimits ^{1}_{-1}x(t)\phi _{j}(t)dt.\)
We quote the following lemmas which follows from (Canuto et al. [4], pp. 283–287).
Lemma 2.2
Let \(\mathcal {P}_{n}^{G}{\text {:}}\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) denote the orthogonal projection defined by (2.11). Then the projection \(\mathcal {P}_{n}^{G}\) satisfies the following properties.
-
(i)
\(\{\mathcal {P}_{n}^{G}{\text {:}}\,n\in \mathbb {N}\}\) is uniformly bounded in \({L^{2}}\)-norm.
-
(ii)
There exists a constant \(c>0\) such that for any \(n\in \mathbb {N}\) and \(u\in \mathbb {X},\)
$$\begin{aligned} \left\| \mathcal {P}_{n}^{G}u-u\right\| _{L^{2}}\le c\inf _{\phi \in \mathbb {X}_{n}}\Vert u-\phi \Vert _{L^{2}}. \end{aligned}$$
Lemma 2.3
Let \(\mathcal {P}_{n}^{G}\) be the orthogonal projection defined by (2.11). Then for any \(u\in \mathcal {C}^{r}[-1,\,1],\) there hold
$$\begin{aligned}&\left\| u-\mathcal {P}_{n}^{G}u\right\| _{L^{2}}\le cn^{-r}\left\| u^{(r)}\right\| _{L^{2}},\end{aligned}$$
(2.12)
$$\begin{aligned}&\left\| u-\mathcal {P}_{n}^{G}u\right\| _{\infty }\le cn^{\frac{3}{4}-r}\left\| u^{(r)}\right\| _{L^{2}},\end{aligned}$$
(2.13)
$$\begin{aligned}&\left\| u-\mathcal {P}_{n}^{G}u\right\| _\infty \le cn^{\frac{1}{2}-r}V\left( u^{(r)}\right) , \end{aligned}$$
(2.14)
where c is a constant independent of n and \(V(u^{(r)})\) denotes the total variation of \(u^{(r)}.\)
Interpolatory 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
$$\begin{aligned} \mathcal {P}_{n}^{C}u \in \mathbb {X}_{n},\quad \mathcal {P}_{n}^{C} u\left( \tau _{i}\right) = u\left( \tau _{i}\right) ,\quad i = 0,\,1, \ldots , n,\quad u\in \mathbb {X}. \end{aligned}$$
(2.15)
According to the analysis of (Canuto et al. [4], p. 289), \(\mathcal {P}_{n}^{C}\) satisfies the following lemmas.
Lemma 2.4
Let \(\mathcal {P}_{n}^{C}:\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) be the interpolatory projection defined by (2.15). Then there hold
-
(i)
\(\{\mathcal {P}_{n}^{C}:\,n\in \mathbb {N}\}\) is uniformly bounded in \(L^{2}\)-norm.
-
(ii)
There exists a constant \(c >0\) such that for any \(n\in \mathbb {N}\) and \(u\in \mathbb {X},\)
$$\begin{aligned} \left\| \mathcal {P}_{n}^{C} u-u\right\| _{L^{2}}\le c \underset{\phi \in \mathbb {X}_{n}}{\inf }\Vert u-\phi \Vert _{L^{2}}\rightarrow 0,\quad n \rightarrow \infty . \end{aligned}$$
Lemma 2.5
Let \(\mathcal {P}_{n}^{C}:\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) be the interpolatory projection defined by (2.15). Then for any \(u\in \mathcal {C}^{r}[-1,\,1],\) there exists a constant c independent of n such that
$$\begin{aligned}&\left\| u-\mathcal {P}_{n}^{C} u\right\| _{L^{2}}\le cn^{-r}\left\| u^{(r)}\right\| _{L^{2}}. \end{aligned}$$
(2.16)
Noting that
$$\begin{aligned} \left\| \mathcal {P}_{n}^{C}\right\| _{\infty }=1+\frac{2^{3/2}}{\sqrt{\pi }}n^{1/2}+B_{0}+\mathcal {O}\left( n^{-1/2}\right) , \end{aligned}$$
where \(B_{0}\) is a bounded constant (see Tang et al. [12]), we have
$$\begin{aligned} \left\| (I-\mathcal {P}_{n}^{C})u\right\| _\infty&\le \left( 1+\left\| \mathcal {P}_{n}^{C}\right\| _{\infty }\right) \underset{\chi \in \mathbb {X}_{n}}{\inf }\Vert u-\chi \Vert _{\infty }\nonumber \\&\le cn^{\frac{1}{2}}n^{-r}\left\| u^{(r)}\right\| _{\infty }\le cn^{\frac{1}{2}-r}\left\| u^{(r)}\right\| _{\infty }. \end{aligned}$$
(2.17)
Throughout this paper, we assume that the projection operator \({\mathcal P}_{n}:\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) is either orthogonal projection \(\mathcal {P}_{n}^{G}\) defined by (2.11) or interpolatory projection operator \(\mathcal {P}_{n}^{C}\) defined by (2.15). By Lemmas 2.2 and 2.4, we have that \(\Vert {\mathcal {P}}_{n}\Vert _{L^{2}}\) is uniformly bounded. We denote, \(\Vert {\mathcal {P}}_{n}\Vert _{L^{2}}\le p,\) for all \(n\in N\) and \(\Vert {\mathcal {P}}_{n}x\Vert _{L^{2}} \le p_{1}\Vert x\Vert _{\infty },\) where p and \(p_{1}\) are constants independent of n. Further, we have from Lemmas 2.3 and 2.5 and estimate (2.17) that
$$\begin{aligned}&\displaystyle \left\| u-{\mathcal {P}}_{n}u\right\| _{L^{2}} \le c n^{-r} \left\| u^{(r)}\right\| _{L^{2}},\end{aligned}$$
(2.18)
$$\begin{aligned}&\displaystyle \left\| u-{\mathcal {P}}_{n}u\right\| _{\infty } \le cn^{\beta -r}\left\| u^{(r)}\right\| _{\infty },\quad 0<\beta < 1,\quad \text{ and }\quad r=0,\,1,\,2,\ldots ,\qquad \end{aligned}$$
(2.19)
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 \(\Vert \mathcal {P}_{n}u -u\Vert _{\infty } \nrightarrow 0,\) as \(n \rightarrow \infty \) for any \(u \in \mathcal {C}[-1,\,1].\)
The projection method for Eq. (2.2) is seeking an approximate solution \(x_{n} \in \mathbb {X}_{n}\) such that
$$\begin{aligned} x_{n}-\sum \limits _{i=1}^{m}\mathcal {P}_{n}\mathcal {K}_{i}\psi _{i}\left( x_{n}\right) =\mathcal {P}_{n}f. \end{aligned}$$
(2.20)
If \(\mathcal {P}_{n}\) is chosen to be \(\mathcal {P}_{n}^{G},\) the above scheme (2.20) leads to 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
$$\begin{aligned} \mathcal {T}_{n}u:=\sum \limits _{i=1}^{m}\mathcal {P}_{n}\mathcal {K}_{i}\psi _{i}(u)+\mathcal {P}_{n}f,\quad u\in \mathbb {X}. \end{aligned}$$
(2.21)
Then the Eq. (2.20) can be written as
$$\begin{aligned} x_{n}=\mathcal {T}_{n}x_{n}. \end{aligned}$$
(2.22)
In order to obtain more accurate approximate solution, we further consider the iterated projection method for (2.2). To this end, we define the iterated solution as
$$\begin{aligned} {\tilde{x}_{n}}=f+\sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}\left( x_{n}\right) . \end{aligned}$$
(2.23)
Applying \({\mathcal {P}}_{n}\) on both sides of the Eq. (2.23), we obtain
$$\begin{aligned} {\mathcal {P}}_{n}{\tilde{x}_{n}}={\mathcal {P}}_{n}f+\sum \limits _{i=1}^{m}\mathcal {P}_{n}\mathcal {K}_{i}\psi _{i}\left( x_{n}\right) . \end{aligned}$$
(2.24)
From Eqs. (2.20) and (2.24), it follows that \({\mathcal {P}}_{n}{\tilde{x}_{n}}=x_{n}.\) Using this, we see that the iterated solution \({\tilde{x}_{n}}\) satisfies the following equation
$$\begin{aligned} {\tilde{x}_{n}}-\sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}\left( \mathcal {P}_{n}\tilde{x}_{n}\right) =f. \end{aligned}$$
(2.25)
Letting \(\widetilde{\mathcal {T}}_{n}(u):= f+\sum \nolimits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}(\mathcal {P}_{n}u),\,u\in \mathbb {X},\) the Eq. (2.25) can be written as \({\tilde{x}_{n}}=\widetilde{\mathcal {T}}_{n}{\tilde{x}_{n}}.\)
Next we prove one lemma which we will use further in our analysis.
Lemma 2.6
Let \(\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0})\) and \(\widetilde{\mathcal {T}}_{n}^{'}(x_{0})\) be the Frechet derivatives of \(\sum \nolimits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}(x)\) and \(\widetilde{\mathcal {T}}_{n}(x),\) respectively at \(x_{0}.\) Then \(\Vert (\mathcal {I}-\mathcal {P}_{n})\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0})\Vert _{L^{2}}\rightarrow 0\) and \(\Vert (\mathcal {I}-\mathcal {P}_{n})\widetilde{\mathcal {T}}_{n}^{'}(x_{0})\Vert _{L^{2}}\rightarrow 0,\) as \(n\rightarrow \infty .\)
Proof
Using the estimates (2.6) and (2.18), we have
$$\begin{aligned} \left\| \left( \mathcal {I}-\mathcal {P}_{n}\right) \left[ \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right] (x)\right\| _{L^{2}}&\le cn^{-r}\left\| \sum \limits _{i=1}^{m}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x\right] ^{(r)}\right\| _{L^{2}}\nonumber \\&\le cn^{-r}2d\Vert k\Vert _{r,\infty }\Vert x\Vert _{L^{2}}\nonumber \\&\rightarrow 0,\quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$
(2.26)
Again using the estimate (2.18), we have
$$\begin{aligned} \left\| \left( \mathcal {I}-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) (x)\right\| _{L^{2}}\le cn^{-r}\left\| \left[ \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x\right] ^{(r)}\right\| _{L^{2}}. \end{aligned}$$
(2.27)
The Frechet derivative of \(\widetilde{\mathcal {T}}_{n}(x)\) at \(x_{0}\) is given by
$$\begin{aligned} \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(t)&= \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \mathcal {P}_{n}x(t)\\&= \underset{i=1}{\overset{m}{\sum }}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\psi _{i}^{(0,1)}\left( s,\,\mathcal {P}_{n}x_{0}(s)\right) \mathcal {P}_{n}x(s)ds. \end{aligned}$$
Consider
$$\begin{aligned}&\left| \underset{i=1}{\overset{m}{\sum }}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \mathcal {P}_{n}x\right] ^{(r)}(t)\right| \nonumber \\&\quad =\left| \sum _{i=1}^{m}\int \nolimits _{-1}^{1}\frac{\partial ^{r}}{\partial t^{r}}k_{i}(t,\,s) \psi _{i}^{(0,1)}\left( s,\,\mathcal {P}_{n}x_{0}(s)\right) \mathcal {P}_{n}x(s)ds\right| \nonumber \\&\quad \le \sum _{i=1}^{m}\sup _{t,s\in [-1,1]}\left| \frac{\partial ^{r}}{\partial t^{r}}k_{i}(t,\,s)\right| \left[ \int \nolimits _{-1}^{1}\left| \left[ \psi _{i}^{(0,1)}\left( s,\,\mathcal {P}_{n}x_{0}(s)\right) \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. -\,\psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) \right] \mathcal {P}_{n}x(s)\right| ds+\int \nolimits _{-1}^{1}\left| \psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) \mathcal {P}_{n}x(s)\right| ds\right] \nonumber \\&\quad \le \sum \limits _{i=1}^{m}\left\| k_{i}\right\| _{r,\infty }\left[ \int \nolimits _{-1}^{1}q_{i}\left| \left( \mathcal {P}_{n}x_{0}(s)-x_{0}(s)\right) \right| \left| \mathcal {P}_{n}x(s)\right| ds\nonumber \right. \\&\qquad \left. +\sup _{s\in [-1,1]}\left| \psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) \right| \int \nolimits _{-1}^{1}\left| \mathcal {P}_{n}x(s)\right| ds\right] \nonumber \\&\quad \le l_{2}\Vert k\Vert _{r,\infty }\left( \left\| \mathcal {P}_{n}x_{0}-x_{0}\right\| _{L^{2}}\left\| \mathcal {P}_{n}x\right\| _{L^{2}}+\sqrt{2}d\left\| \mathcal {P}_{n}x\right\| _{L^{2}}\right) \nonumber \\&\quad \le \left( l_{2}p\left\| \mathcal {P}_{n}x_{0}-x_{0}\right\| _{L^{2}}+\sqrt{2}d p\right) \Vert x\Vert _{L^{2}}\Vert k\Vert _{r,\infty }. \end{aligned}$$
(2.28)
Hence, we have
$$\begin{aligned} \left\| \left[ \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) x\right] ^{(r)}\right\| _{L^{2}}&\le \sqrt{2}\left\| \left[ \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) x\right] ^{(r)}\right\| _{\infty }\\&= \sqrt{2}\left\| \sum _{i=1}^{m}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \mathcal {P}_{n}x\right] ^{(r)}\right\| _{\infty }\\&\le \left( \sqrt{2}l_{2}p\left\| \mathcal {P}_{n}x_{0}-x_{0}\right\| _{L^{2}}+2d p\right) \Vert k\Vert _{r,\infty }\Vert x\Vert _{L^{2}}. \end{aligned}$$
Combining this with the estimate (2.27), we have
$$\begin{aligned} \left\| \left( \mathcal {I}-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'} \left( x_{0}\right) \right\| _{L^{2}}&\le cn^{-r}\left( \sqrt{2}l_{2}p\left\| \mathcal {P}_{n}x_{0}-x_{0}\right\| _{L^{2}}+2d p\right) \Vert k\Vert _{r,\infty }\nonumber \\&\rightarrow 0,\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$
(2.29)
This completes the proof. \(\square \)
We quote the following theorem from Vainikko [13] which gives us the conditions under which the solvability of one equation leads to the solvability of other equation.
Theorem 2.2
Let \(\widehat{T}\) and \(\widetilde{T}\) be continuous operators over an open set \(\varOmega \) in a Banach space \(\mathbb {X}.\) Let the equation \(x=\widetilde{T}x\) has an isolated solution \(\tilde{x}_{0} \in \varOmega \) and let the following conditions be satisfied.
-
(a)
The operator \(\widehat{T}\) is Frechet differentiable in some neighbourhood of the point \(\tilde{x}_{0},\) while the linear operator \(\mathcal {I}-\widehat{T}^{'}(\tilde{x}_{0})\) 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 \(\Vert x-\tilde{x}_{0}\Vert \le \delta \) is contained within \(\varOmega ).\)
$$\begin{aligned}&\sup _{\Vert x-\tilde{x}_0\Vert \le \delta }\left\| {\left( \mathcal {I}-\widehat{T}^{'}\left( \tilde{x}_{0}\right) \right) }^{-1}\left( \widehat{T}^{'}(x)-\widehat{T}^{'}\left( \tilde{x}_{0}\right) \right) \right\| \le q,\end{aligned}$$
(2.30)
$$\begin{aligned}&\alpha =\left\| {\left( \mathcal {I}-\widehat{T}^{'}\left( \tilde{x}_{0}\right) \right) }^{-1}\left( \widehat{T}\left( \tilde{x}_{0}\right) -\widetilde{T}\left( \tilde{x}_{0}\right) \right) \right\| \le \delta (1-q). \end{aligned}$$
(2.31)
Then the equation \(x=\widehat{T}x\) has a unique solution \(\hat{x}_{0}\) in the sphere \(\Vert x-\tilde{x}_{0}\Vert \le \delta .\) Moreover, the inequality
$$\begin{aligned} \frac{\alpha }{1+q}\le \left\| \hat{x}_{0}-\tilde{x}_{0}\right\| \le \frac{\alpha }{1-q} \end{aligned}$$
(2.32)
is valid.
Next we discuss the existence of approximate and iterated approximate solutions and their error bounds. To do this, first we recall the following definition of \(\nu \)-convergence and a theorem from Ahues et al. [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 \({{\mathcal T}_{n}},\,{\mathcal {T}} \in {\mathbb {B}}{\mathbb {L}}(\mathbb {X}).\) We say \({{\mathcal T}_{n}}\) is \(\nu \)-convergent to \({\mathcal {T}}\) if
$$\begin{aligned} \left\| {\mathcal T}_{n}\right\| \le C,\quad \left\| \left( {\mathcal T}_{n}-{\mathcal T}\right) {\mathcal T} \right\| \rightarrow 0,\quad \left\| \left( {\mathcal T}_{n}-{\mathcal T} \right) {\mathcal T}_{n}\right\| \rightarrow 0,\quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$
Lemma 2.7
Let \(\mathbb {X}\) be a Banach space and \({\mathcal T},\,{\mathcal T}_{n}\) be bounded linear operators on \(\mathbb {X}.\) If \({{\mathcal T}_{n}}\) is norm convergent to \({\mathcal {T}}\) or \({{\mathcal {T}}_{n}}\) is \(\nu \)-convergent to \({\mathcal T}\) and \((\mathcal {I}-\mathcal {T})^{-1}\) exists, then \((\mathcal {I}-\mathcal {T}_{n})^{-1}\) exists and uniformly bounded on \(\mathbb {X},\) for sufficiently large n.
Theorem 2.3
Let \(x_{0}\in \mathcal {C}^{r}[-1,\,1]\) be an isolated solution of the Eq. (2.2). Assume that 1 is not an eigenvalue of the linear operator \(\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0}),\) where \(\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0})\) denotes the Frechet derivative of \(\sum \nolimits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}(x)\) at \(x_{0}.\) Let \(\mathcal {P}_{n}:\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) be either orthogonal or interpolatory projection operator defined by (2.11) or (2.15), respectively. Then the Eq. (2.20) has a unique solution \(x_{n}\in B(x_{0},\,\delta )=\{x:\,\Vert x-x_{0}\Vert _{L^{2}} < \delta \}\) for some \(\delta >0\) and for sufficiently large n. Moreover, there exists a constant \(0< q <1,\) independent of n such that
$$\begin{aligned} \frac{\alpha _{n}}{1+q}\le {\left\| x_{n}-x_{0}\right\| }_{L^{2}} \le \frac{\alpha _{n}}{1-q}, \end{aligned}$$
where \(\alpha _{n}={\Vert (\mathcal {I}-{{\mathcal {T}}_{n}}^{'}(x_{0}))^{-1}({\mathcal {T}}_{n}(x_{0})-{\mathcal {T}}(x_{0}))\Vert }_{L^{2}}.\)
Proof
Using estimate (2.26), we have
$$\begin{aligned} {\left\| \left( {{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) -{{\mathcal {T}}}^{'}\left( x_{0}\right) \right) (x)\right\| }_{L^{2}}&= \left\| \left( \sum \limits _{i=1}^{m}\mathcal {P}_{n}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right) (x)\right\| _{L^{2}}\\&= \left\| \left( \mathcal {P}_{n}-\mathcal {I}\right) \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right] (x)\right\| _{L^{2}}\\&\le 2dcn^{-r}\Vert k\Vert _{r,\infty }\Vert x\Vert _{L^{2}}. \end{aligned}$$
Hence
$$\begin{aligned} {\left\| {{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) -{{\mathcal {T}}}^{'}\left( x_{0}\right) \right\| }_{L^{2}}\rightarrow 0,\quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$
This shows that \(\mathcal {T}_{n}^{'}(x_{0})\) is norm convergent to \(\mathcal {T}^{'}(x_{0}).\) Hence by Lemma 2.7, we have \({(\mathcal {I}-{\mathcal {T}}_{n}^{'}(x_{0}))}^{-1}\) exists and uniformly bounded on \(\mathbb {X},\) for some sufficiently large n, i.e., there exists some \(A_{1} > 0\) such that \(\Vert {(\mathcal {I}-{\mathcal {T}}_{n}^{'}(x_{0}))}^{-1}\Vert _{L^{2}} \le A_{1} <\infty .\)
Now from the estimate (2.8), we have for any \(x\in B(x_{0},\,\delta ),\)
$$\begin{aligned} {\left\| \left( {{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) -{{\mathcal {T}}_{n}}^{'}(x)\right) y\right\| }_{L^{2}}&= \left\| \sum \limits _{i=1}^{m}\mathcal {P}_{n}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) (y)-\sum \limits _{i=1}^{m}\mathcal {P}_{n}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)(y)\right\| _{L^{2}}\nonumber \\&\le \!\left\| {\mathcal {P}}_{n}\right\| _{L^{2}} \left\| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right] (y)\right\| _{L^{2}}\nonumber \\&\le \sqrt{2}\!\left\| {\mathcal {P}}_{n}\right\| _{L^{2}}\! \left\| \left[ \sum \limits _{i=1}^{m}\!\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\!\left( x_{0}\right) \!-\! \sum \limits _{i=1}^{m}\!\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\!\!\right] (y)\right\| _{\infty }\nonumber \\&\le \sqrt{2}pMl_{2}\delta \Vert y\Vert _{L^{2}}. \end{aligned}$$
(2.33)
This implies
$$\begin{aligned} \left\| {{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) -{{\mathcal {T}}_{n}}^{'}(x)\right\| _{L^{2}}\le \sqrt{2}pMl_{2}\delta . \end{aligned}$$
Hence, we have
$$\begin{aligned} \sup _{\Vert x-{x_{0}}\Vert _{L^{2}}\le \delta } {\left\| {\left( \mathcal {I}-{{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) }^{-1}\left( {{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) -{{\mathcal {T}}_{n}}^{'}(x)\right) \right\| }_{L^{2}}\le A_{1}\sqrt{2}pMl_{2}\delta \le q,\,(\mathrm{say}) \end{aligned}$$
where \( 0 < q <1,\) which proves the Eq. (2.30) of Theorem 2.2.
Taking use of (2.18), we have
$$\begin{aligned} \alpha _{n}&= \left\| \left( \mathcal {I}-{{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\left( {\mathcal {T}}_{n}\left( x_{0}\right) -{\mathcal {T}}\left( x_{0}\right) \right) \right\| _{L^{2}}\nonumber \\&\le A_{1}\left\| {\mathcal {T}}_{n}\left( x_{0}\right) -{\mathcal {T}}\left( x_{0}\right) \right\| _{L^{2}}\nonumber \\&= A_{1}\left\| \sum \limits _{i=1}^{m}{\mathcal {P}}_{n}\left( \mathcal {K}_{i}\psi _{i}x_{0}+f\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}x_{0}+f\right) \right\| _{L^{2}}\nonumber \\&= A_{1}\left\| \left( {\mathcal {P}}_{n}-\mathcal {I}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}x_{0}+f\right) \right\| _{L^{2}}\nonumber \\&= A_{1}{\left\| \left( {\mathcal {P}}_{n}-\mathcal {I}\right) x_{0}\right\| }_{L^{2}}\nonumber \\&\le A_{1}cn^{-r}\left\| x_{0}^{(r)}\right\| _{L^{2}}\rightarrow 0,\quad \mathrm{as}\quad n \rightarrow \infty . \end{aligned}$$
(2.34)
By choosing n large enough such that \(\alpha _{n} \le \delta (1-q),\) the Eq. (2.31) of Theorem 2.2 is satisfied. Hence by applying Theorem 2.2, we obtain
$$\begin{aligned} \frac{\alpha _{n}}{1+q}\le {\left\| x_{n}-x_{0}\right\| }_{L^{2}} \le \frac{\alpha _n}{1-q}. \end{aligned}$$
This completes the proof.\(\square \)
Theorem 2.4
Let \(x_{0}\in \mathcal {C}^{r}[-1,\,1],\,r \ge 1,\) be an isolated solution of the Eq. (2.2). Assume that 1 is not an eigenvalue of the linear operator \(\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0}),\) where \(\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0})\) denotes the Frechet derivative of \(\sum \nolimits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}(x)\) at \(x_{0}.\) Let \(\mathcal {P}_{n}:\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) be either orthogonal or interpolatory projection operator defined by (2.11) or (2.15), respectively. Then the Eq. (2.20) has a unique solution \(x_{n}\in B(x_{0},\,\delta )=\{x:\,\Vert x-x_{0}\Vert _{\infty } < \delta \}\) for some \(\delta >0\) and for sufficiently large n. Moreover, there exists a constant \(0< q <1,\) independent of n such that
$$\begin{aligned} \frac{\alpha _{n}}{1+q}\le {\left\| x_{n}-x_{0}\right\| }_{\infty } \le \frac{\alpha _{n}}{1-q}, \end{aligned}$$
where \(\alpha _{n}={\Vert (\mathcal {I}-{{\mathcal {T}}_{n}}^{'}(x_{0}))^{-1}({\mathcal {T}}_{n}(x_{0})-{\mathcal {T}}(x_{0}))\Vert }_{\infty }.\)
Proof
Using estimates (2.5) and (2.19), we have
$$\begin{aligned} {\left\| \left( {{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) -{{\mathcal {T}}}^{'}\left( x_{0}\right) \right) x\right\| }_{\infty }&= \left\| \left[ \mathcal {P}_{n}\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right] x\right\| _{\infty }\nonumber \\&= \left\| \left( \mathcal {P}_{n}-\mathcal {I}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x\right\| _{\infty }\nonumber \\&\le cn^{\beta -r}\Vert [(\mathcal {K}_i\psi _i)'(x_0)x]^{(r)}\Vert _{\infty }\nonumber \\&\le 2cn^{\beta -r}d\Vert k\Vert _{r,\infty }\Vert x\Vert _\infty . \end{aligned}$$
(2.35)
Since \(0 < \beta < 1,\) for \(\beta < r = 1,\,2,\,3,\ldots ,\) it follows that
$$\begin{aligned} {\left\| \mathcal {T}_{n}^{'}\left( x_{0}\right) -\mathcal {T}^{'}\left( x_{0}\right) \right\| }_{\infty } =\mathcal {O}\left( n^{\beta -r}\right) \rightarrow 0,\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$
Hence by applying Lemma 2.7, we see that \({(\mathcal {I}-{\mathcal {T}}_{n}^{'}(x_{0}))}^{-1}\) exists and uniformly bounded on \(\mathbb {X},\) for some sufficiently large n, i.e., there exists some \(A_{2} > 0\) such that \(\Vert {(\mathcal {I}-{\mathcal {T}}_{n}^{'}(x_{0}))}^{-1}\Vert _{\infty } \le A_{2} <\infty .\)
Now, we have for any \(x\in B(x_{0},\,\delta ),\)
$$\begin{aligned} {\left\| \left( {{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) -{{\mathcal {T}}_{n}}^{'}(x)\right) y\right\| }_{\infty }&\le \left\| \left[ \sum \limits _{i=1}^{m}\mathcal {P}_{n}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\mathcal {P}_{n}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right] (y)\right\| _{\infty } \nonumber \\&\le \left\| \left( \mathcal {P}_{n}-I\right) \sum \limits _{i=1}^{m}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right] (y)\right\| _{\infty }\nonumber \\&\quad +\left\| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right] (y)\right\| _{\infty }\nonumber \\&\le cn^{\beta -r}\left\| \left[ \left( \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right) (y)\right] ^{(r)}\right\| _{\infty }\nonumber \\&\quad +\left\| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right] (y)\right\| _{\infty }. \end{aligned}$$
(2.36)
Now using the Lipschitz’s continuity of \(\psi _{i}^{(0,1)}(.,\,x_{0}(.)),\) we have
$$\begin{aligned}&\left\| \left[ \left( \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right) (y)\right] ^{(r)}\right\| _{\infty }\nonumber \\&\quad =\sup _{t\in [-1,1]}\left| \left[ \left( \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}(x)\right) (y)\right] ^{(r)}(t)\right| \nonumber \\&\quad =\sup _{t\in [-1,1]}\left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}\frac{\partial ^{r}}{\partial t^{r}}k_{i}(t,\,s)\left[ \psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) -\psi _{i}^{(0,1)}(s,\,x(s))\right] y(s)ds\right| \nonumber \\&\quad \le \sum \limits _{i=1}^{m}\sup _{t,s\in [-1,1]}\left| \frac{\partial ^{r}}{\partial t^{r}}k_{i}(t,\,s)\right| \int \nolimits _{-1}^{1}q_{i}\left| \left( x_{0}-x\right) (s)\right| |y(s)|ds\nonumber \\&\quad \le 2l_{2}\Vert k\Vert _{r,\infty }\left\| x_{0}-x\right\| _{\infty }\Vert y\Vert _{\infty }\le 2l_{2}\Vert k\Vert _{r,\infty }\delta \Vert y\Vert _{\infty }. \end{aligned}$$
(2.37)
Hence combining the estimates (2.8), (2.36) and (2.37), we obtain
$$\begin{aligned}&\sup _{\Vert x-{x_{0}}\Vert _{\infty }\le \delta } {\left\| {\left( \mathcal {I}-{{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) }^{-1}\left( {{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) -{{\mathcal {T}}_{n}}^{'}(x)\right) \right\| }_{\infty }\\&\quad \le A_{2}\left( cn^{\beta -r}2l_{2}\Vert k\Vert _{r,\infty }\delta +2l_{2}M\delta \right) \\&\quad \le 2A_{2}l_{2}\left( M+cn^{\beta -r}\Vert k\Vert _{r,\infty }\right) \delta \nonumber \\&\quad \le q,\,(\mathrm{say}) \end{aligned}$$
where \( 0 < q <1,\) which proves the Eq. (2.30) of Theorem 2.2.
Since \(\beta < r=1,\,2,\ldots ,\) using the estimate (2.19), we have
$$\begin{aligned} \alpha _n&= \left\| \left( \mathcal {I}-{{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\left( {\mathcal {T}}_{n}\left( x_{0}\right) -{\mathcal {T}}\left( x_{0}\right) \right) \right\| _{\infty }\nonumber \\&\le A_{2}\left\| {\mathcal {T}}_{n}\left( x_{0}\right) -{\mathcal {T}}\left( x_{0}\right) \right\| _{\infty }\nonumber \\&= A_{2}\left\| {\mathcal {P}}_{n}\left( \sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}x_{0}+f\right) -\left( \sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}x_{0}+f\right) \right\| _{\infty }\nonumber \\&= A_{2}\left\| \left( {\mathcal {P}}_{n}-\mathcal {I}\right) \left( \sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}x_{0}+f\right) \right\| _{\infty }\nonumber \\&= A_{2}{\left\| \left( {\mathcal {P}}_{n}-\mathcal {I}\right) x_{0}\right\| }_{\infty }\le cn^{\beta -r}\left\| x_{0}^{(r)}\right\| _{\infty } \rightarrow 0,\quad \mathrm{as}\quad n \rightarrow \infty . \end{aligned}$$
(2.38)
By choosing n large enough such that \(\alpha _{n} \le \delta (1-q),\) the Eq. (2.31) of Theorem 2.2 is satisfied. Hence by applying Theorem 2.2, we obtain
$$\begin{aligned} \frac{\alpha _{n}}{1+q}\le {\left\| x_{n}-x_{0}\right\| }_{\infty } \le \frac{\alpha _{n}}{1-q}. \end{aligned}$$
This completes the proof.\(\square \)
Next we discuss the existence and convergence of the iterated approximate solution \({\tilde{x}_{n}}\) to \(x_{0}.\)
Theorem 2.5
\(\widetilde{\mathcal {T}}_{n}^{'}(x_{0})\) is \(\nu \)-convergent to \(\mathcal {T}^{'}(x_{0})\) in both \(L^{2}\)-norm and infinity norm.
Proof
Consider
$$\begin{aligned} \left| {\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) x(t)\right|&= \left| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \mathcal {P}_{n}x(t)\right| \nonumber \\&\le \left| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right] \mathcal {P}_{n}x(t)\right| \nonumber \\&\quad +\left| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \mathcal {P}_{n}x(t)\right| . \end{aligned}$$
(2.39)
Now using estimates (2.8), (2.18) and the fact that \(\Vert \mathcal {P}_{n}x\Vert _{L^{2}}\le p_{1}\Vert x\Vert _{\infty },\) we have
$$\begin{aligned} \left\| \left[ \sum \limits _{i=1}^{m} \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \!-\!\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right] \mathcal {P}_{n}x\right\| _{\infty } \!&\le Ml_{2}\left\| \mathcal {P}_{n}x_{0}\!-\!x_{0}\right\| _{L^{2}}\left\| \mathcal {P}_{n}x\right\| _{L^{2}}\nonumber \\&\le cMl_{2}p_{1}n^{-r} \left\| x_{0}^{(r)}\right\| _{L^{2}}\Vert x\Vert _{\infty }\nonumber \\&\rightarrow 0,\quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$
(2.40)
Again using the assumption (iii), we see that
$$\begin{aligned} \left\| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \mathcal {P}_{n}x\right\| _{\infty }&= \underset{t\in [-1,1]}{\max }\left| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \mathcal {P}_{n}x(t)\right| \nonumber \\&= \underset{t\in [-1,1]}{\max } \left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) \mathcal {P}_{n}x(s)ds\right| \nonumber \\&\le M\sup _{1\le i\le m}d_{i} \int \nolimits _{-1}^{1}\left| \mathcal {P}_{n}x(s)\right| ds\nonumber \\&\le \sqrt{2}Md\left\| \mathcal {P}_{n}x\right\| _{L^{2}}\le \sqrt{2}Mdp_{1}\Vert x\Vert _{\infty }. \end{aligned}$$
(2.41)
Now combining the estimates (2.39)–(2.41) we obtain
$$\begin{aligned} \left\| \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right\| _{\infty } \le \left( cMl_{2}p_{1}n^{-r}\left\| x_{0}^{(r)}\right\| _{L^{2}}+\sqrt{2}Mdp_{1}\right) <\infty . \end{aligned}$$
This shows that \(\Vert \widetilde{\mathcal {T}}_{n}^{'}(x_{0})\Vert _\infty \) is uniformly bounded.
Consider
$$\begin{aligned}&\left| \left[ \mathcal {T}^{'}\left( x_{0}\right) -\widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right] \mathcal {T}^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&=\left| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'} \left( \mathcal {P}_{n}x_{0}\right) \mathcal {P}_{n}\right] \sum \limits _{i=1}^{m} \left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&=\left| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \left[ \sum \limits _{i=1}^{m}\mathcal {P}_{n}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) +\left( I-\mathcal {P}_{n}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right] x(t)\right. \nonumber \\&\quad \left. -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \mathcal {P}_{n}\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&\le \left| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \right] \mathcal {P}_{n}\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&\quad +\left| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \left( I-\mathcal {P}_{n}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x(t)\right| . \end{aligned}$$
(2.42)
Now for the second term in the above estimate (2.42), we have
$$\begin{aligned}&\left| \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \left( I-\mathcal {P}_{n}\right) \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&=\left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) \left( I-\mathcal {P}_{n}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x(s)ds\right| \nonumber \\&\le \left| <g_{t}(.),\,\left( I-\mathcal {P}_{n}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x(.)>\right| \nonumber \\&\le \sqrt{2}\left\| g_{t}\right\| _{L^{2}}\left\| \left( I-\mathcal {P}_{n}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right\| _{L^{2}}\Vert x\Vert _{\infty }, \end{aligned}$$
(2.43)
where \(g_{t}(s)=\sum \nolimits _{i=1}^{m}k_{i}(t,\,s)\psi _{i}^{(0,1)}(s,\,x_{0}(s)).\)
Hence we have
$$\begin{aligned}&\left\| \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \left( I-\mathcal {P}_{n}\right) \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right\| _{\infty }\nonumber \\&\quad \le \sqrt{2}\left\| g_{t}\right\| _{L^{2}} \left\| \left( I-\mathcal {P}_{n}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right\| _{L^{2}}. \end{aligned}$$
(2.44)
Now for the first term in the estimate (2.42), using (2.8) we have
$$\begin{aligned}&\left\| \left[ \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \right] \mathcal {P}_{n}\underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x\right\| _{\infty }\nonumber \\&\quad \le Ml_{2}\left\| \mathcal {P}_{n}x_{0}-x_{0}\right\| _{L^{2}}\left\| \mathcal {P}_{n}\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) x\right\| _{L^{2}} \nonumber \\&\quad \le Ml_{2}p_{1}\left\| \left( I-\mathcal P_{n}\right) x_{0}\right\| _{L^{2}}\left\| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right\| _{\infty }\Vert x\Vert _{\infty }. \end{aligned}$$
(2.45)
Hence combining estimates (2.18), (2.26), (2.42), (2.44) and (2.45), we have
$$\begin{aligned} \left\| \left[ \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) -\mathcal {T}^{'}\left( x_{0}\right) \right] \mathcal {T}^{'}\left( x_{0}\right) \right\| _{\infty }&\le Ml_{2}p_{1}\left\| \mathcal {P}_{n}x_{0}-x_{0}\right\| _{L^{2}} \left\| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right\| _{\infty } \\&\quad +\,\sqrt{2}\left\| g_{t}\right\| _{L^{2}} \left\| \left( I-\mathcal {P}_{n}\right) \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \right\| _{L^{2}}\\&\quad \rightarrow 0,\quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$
Next consider
$$\begin{aligned}&\left| \left[ \mathcal {T}^{'}\left( x_{0}\right) -\widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right] \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(t) \right| \nonumber \\&\quad =\left| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'} \left( \mathcal {P}_{n}x_{0}\right) \mathcal {P}_{n}\right] \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&\quad =\left| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \left[ \mathcal {P}_{n}\widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) +\left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right] x(t)\right. \nonumber \\&\qquad -\left. \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \mathcal {P}_{n}\widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&\quad \le \left| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \right] \mathcal {P}_{n}\widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&\qquad +\left| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(t)\right| . \end{aligned}$$
(2.46)
For the second term in the above estimate (2.46), using Cauchy–Schwartz inequality, we get
$$\begin{aligned}&\left| \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(t)\right| \nonumber \\&\quad = \left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\psi _{i}^{(0,1)}\left( s,\,x_{0}(s)\right) \left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(s)ds\right| \nonumber \\&\quad =\left| <g_{t}(.),\,\left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x(.)>\right| \nonumber \\&\quad \le \sqrt{2}\left\| g_{t}\right\| _{L^{2}}\left\| \left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right\| _{L^{2}}\Vert x\Vert _{\infty }. \end{aligned}$$
(2.47)
This implies
$$\begin{aligned} \left\| \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) \left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x\right\| _{\infty } \!\le \sqrt{2}\left\| g_{t}\right\| _{L^{2}}\left\| \left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right\| _{L^{2}}\Vert x\Vert _{\infty }.\nonumber \\ \end{aligned}$$
(2.48)
Using the estimate (2.8), we have
$$\begin{aligned}&\left\| \left[ \underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( x_{0}\right) -\underset{i=1}{\overset{m}{\sum }}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \right] \mathcal {P}_{n} \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x\right\| _{\infty }\nonumber \\&\quad \le Ml_{2}\left\| \mathcal {P}_{n}x_{0}-x_{0}\right\| _{L^{2}}\left\| \mathcal {P}_{n}\widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) x\right\| _{L^{2}}\nonumber \\&\quad \le Ml_{2}p_{1}\left\| \left( I-\mathcal P_{n}\right) x_{0}\right\| _{L^{2}}\left\| \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right\| _{\infty }\Vert x\Vert _{\infty }. \end{aligned}$$
(2.49)
Hence combining estimates (2.18), (2.29), (2.46), (2.48) and (2.49) we have
$$\begin{aligned} \left\| \left[ \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) -\mathcal {T}^{'}\left( x_{0}\right) \right] \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right\| _{\infty }&\le Ml_{2}p_{1}\left\| \mathcal {P}_{n}x_{0}-x_{0}\right\| _{L^{2}}\left\| \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right\| _{\infty } \\&\quad +\,\sqrt{2}\left\| g_{t}\right\| _{L^{2}}\left\| \left( I-\mathcal {P}_{n}\right) \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right\| _{L^{2}}\\&\quad \rightarrow 0,\quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$
This shows that \(\widetilde{\mathcal {T}}_{n}^{'}(x_{0})\) is \(\nu \)-convergent to \(\mathcal {T}^{'}(x_{0})\) in infinity norm.
On similar lines it can be shown that \(\widetilde{\mathcal {T}}_n'(x_0)\) is \(\nu \)-convergent to \(\mathcal {T}'(x_0)\) in \(L^2\)- norm. This completes the proof . \(\square \)
The following theorem is obtained by direct application of Lemma 2.7 and Theorem 2.5.
Theorem 2.6
Let \(x_{0}\in \mathcal {C}^{r}[-1,\,1]\) be an isolated solution of the Eq. (2.2). Assume that 1 is not an eigenvalue of \(\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0}).\) Then for sufficiently large n, the operator \(\mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}(x_{0})\) is invertible on \(\mathcal {C}[-1,\,1]\) and there exist constants \(L,\, L_{1} > 0\) independent of n such that \({\Vert (\mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}(x_{0}))^{-1}\Vert }_{\infty } \le L\) and \({\Vert (\mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}(x_{0}))^{-1}\Vert }_{L^{2}} \le L_{1}.\)
Theorem 2.7
Let \(x_{0}\in \mathcal {C}^{r}[-1,\,1]\) be an isolated solution of the Eq. (2.2). Let \(\mathcal {P}_{n}:\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) be either orthogonal or interpolatory projection operator defined by (2.11) or (2.15), respectively. Assume that 1 is not an eigenvalue of \(\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0}),\) then for sufficiently large n, the iterated solution \({\tilde{x}}_{n}\) defined by (2.25) is the unique solution in the sphere \(B(x_{0},\,\delta )=\{x:\,\Vert x-x_{0}\Vert _{\infty } < \delta \}.\) Moreover, there exists a constant \(0<q<1,\) independent of n such that
$$\begin{aligned} \frac{\beta _{n}}{1+q}\le {\left\| {\tilde{x}_{n}}-x_{0}\right\| }_{\infty } \le \frac{\beta _{n}}{1-q}, \end{aligned}$$
where
$$\begin{aligned} \beta _{n}={\left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\left( \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right) \right\| }_{\infty }. \end{aligned}$$
Proof
From Theorem 2.6, there exists a constant \(L > 0\) such that
$$\begin{aligned} {\left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\right\| }_{\infty }\le L, \end{aligned}$$
for sufficiently large value of n.
Using the estimate (2.8) of Lemma 2.1 and the fact that \(\Vert \mathcal {P}_{n}x\Vert _{L^{2}}\le p_{1}\Vert x\Vert _{\infty },\) for any \(x \in B(x_{0},\,\delta ),\) we have
$$\begin{aligned} \left\| \left[ \widetilde{\mathcal {T}}_{n}^{'}(x) - \widetilde{\mathcal {T}}_{n}^{'}\left( x_{0}\right) \right] y\right\| _{\infty }&= \left\| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x\right) -\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\left( \mathcal {P}_{n}x_{0}\right) \right] \mathcal {P}_{n}y\right\| _{\infty }\nonumber \\&\le Ml_{2}\left\| \mathcal {P}_{n}\left( x-x_{0}\right) \right\| _{L^{2}}\left\| \mathcal {P}_{n}y\right\| _{L^{2}}\nonumber \\&\le Ml_{2}p_{1}^{2}\left\| x-x_{0}\right\| _{\infty }\Vert y\Vert _{\infty }\le Ml_{2}p_{1}^{2}\delta \Vert y\Vert _{\infty }. \end{aligned}$$
(2.50)
This implies
$$\begin{aligned} \sup _{\Vert x-{x_{0}}\Vert _{\infty } \le \delta } {\left\| {\left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) }^{-1}\left( {\widetilde{\mathcal {T}}_{n}}^{'}(x)-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) \right\| }_{\infty }\le LMl_{2}p_{1}^{2}\delta \le q,\,(\mathrm{say}) \end{aligned}$$
where \(0<q<1,\) which proves the Eq. (2.30) of Theorem 2.2.
Now using the estimate (2.7) and (2.18), we have
$$\begin{aligned} \left\| \left[ \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right] \right\| _{\infty }&\le \left\| \left[ \sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}\left( \mathcal {P}_{n}x_{0}\right) -\sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}\left( x_{0}\right) \right] \right\| _{\infty }\nonumber \\&\le l_{1}M\sqrt{2}\left\| \left( \mathcal {I}-\mathcal {P}_{n}\right) x_{0}\right\| _{L^{2}}\nonumber \\&\le l_{1}M\sqrt{2}cn^{-r}\left\| x_{0}^{(r)}\right\| _{L^{2}} \rightarrow 0, \quad \mathrm{as}\quad n\rightarrow \infty .\qquad \end{aligned}$$
(2.51)
Hence
$$\begin{aligned} \beta _{n}={\left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\left( \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right) \right\| }_{\infty }&\le Ll_{1}M\sqrt{2}cn^{-r}\left\| x_{0}^{(r)}\right\| _{L^{2}}\\&\rightarrow 0, \quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$
Choose n large enough such that \(\beta _{n}\le \delta (1-q).\) Then the Eq. (2.31) of Theorem 2.2 is satisfied. Thus by applying Theorem 2.2, we obtain
$$\begin{aligned} \frac{\beta _{n}}{1+q}\le {\left\| {\tilde{x}_{n}}-x_{0}\right\| }_{\infty } \le \frac{\beta _{n}}{1-q}, \end{aligned}$$
where
$$\begin{aligned} \beta _{n}={\left\| {\left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) }^{-1}\left( {\widetilde{\mathcal {T}}_{n}}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right) \right\| }_{\infty }. \end{aligned}$$
This completes the proof. \(\square \)
Theorem 2.8
Let \(x_{0}\in \mathcal {C}^r[-1,\,1]\) be an isolated solution of the Eq. (2.2). Let \(\mathcal {P}_{n}:\,\mathbb {X}\rightarrow \mathbb {X}_{n}\) be either orthogonal or interpolatory projection operator defined by (2.11) and (2.15), respectively. Assume that 1 is not an eigenvalue of \(\sum \nolimits _{i=1}^{m}(\mathcal {K}_{i}\psi _{i})^{'}(x_{0}),\) then for sufficiently large n, the iterated solution \({\tilde{x}}_{n}\) defined by (2.25) is the unique solution in the sphere \(B(x_{0},\,\delta )=\{x:\,\Vert x-x_{0}\Vert _{L^{2}} < \delta \}.\) Moreover, there exists a constant \(0<q<1,\) independent of n such that
$$\begin{aligned} \frac{\beta _{n}}{1+q}\le {\left\| {\tilde{x}_{n}}-x_{0}\right\| }_{L^{2}} \le \frac{\beta _{n}}{1-q}, \end{aligned}$$
where
$$\begin{aligned} \beta _{n}={\left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\left( \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right) \right\| }_{L^{2}}. \end{aligned}$$
Proof
From Theorem 2.6, we have \((\mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}(x_{0}))^{-1} \) exists and it is uniformly bounded in \(L^{2}\)-norm on \(\mathcal {C}[-1,\,1],\) i.e., there exists a constant \(L_{1}>0\) such that \({\Vert (\mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}(x_{0}))^{-1}\Vert }_{L^{2}}\le L_{1}.\)
Now using the estimate (2.8) and the fact that \(\Vert \mathcal {P}_{n}\Vert _{L^{2}}\le p,\) we have for any \(x\in B(x_{0},\,\delta ),\)
$$\begin{aligned} \left\| \left[ {\widetilde{\mathcal {T}}_{n}}^{'}(x)-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right] (y)\right\| _{L^{2}}&\le \sqrt{2}\left\| \left[ {\widetilde{\mathcal {T}}_{n}}^{'}(x) - {\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right] (y)\right\| _{\infty }\\&\le \sqrt{2}\left\| \left[ \sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\!\left( \mathcal {P}_{n}x\right) \!-\!\sum \limits _{i=1}^{m}\left( \mathcal {K}_{i}\psi _{i}\right) ^{'}\!\left( \mathcal {P}_{n}x_{0}\right) \right] \!\mathcal {P}_{n}y\right\| _{\infty }\nonumber \\&\le \sqrt{2}Ml_{2}\left\| \mathcal {P}_{n}\left( x-x_{0}\right) \right\| _{L^{2}}\left\| \mathcal {P}_{n}y\right\| _{L^{2}}\nonumber \\&\le \sqrt{2}Ml_{2}p^{2}\left\| x-x_{0}\right\| _{L^{2}}\Vert y\Vert _{L^{2}}\nonumber \\&\le \sqrt{2}Ml_{2}p^{2}\delta \Vert y\Vert _{L^{2}}. \end{aligned}$$
Thus we obtain
$$\begin{aligned} \sup _{\Vert x-{x_{0}}\Vert _{L^{2}}\le \delta } {\left\| {\left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) }^{-1}\left( {\widetilde{\mathcal {T}}_{n}}^{'}(x)-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) \right\| }_{L^{2}}\le L_{1}\sqrt{2}Ml_{2}p^{2}\delta \le q,\,(\mathrm{say}) \end{aligned}$$
where \(0<q<1,\) which proves the Eq. (2.30) of Theorem 2.2.
Now using the estimate (2.51), we have
$$\begin{aligned} \beta _{n}&= {\left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\left( \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right) \right\| }_{L^{2}}\\&\le {\left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\right\| }_{L^{2}}{\left\| \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal T\left( x_{0}\right) \right\| }_{L^{2}}\\&\le L_{1}\sqrt{2}{\left\| \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right\| }_{\infty }\rightarrow 0, \quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$
Choose n large enough such that \(\beta _{n}\le \delta (1-q).\) Hence the Eq. (2.31) of Theorem 2.2 is satisfied. Then applying Theorem 2.2, we get
$$\begin{aligned} \frac{\beta _{n}}{1+q}\le {\left\| {\tilde{x}_{n}}-x_{0}\right\| }_{L^{2}} \le \frac{\beta _{n}}{1-q}, \end{aligned}$$
where
$$\begin{aligned} \beta _{n}={\left\| {\left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) }^{-1}\left( {\widetilde{\mathcal {T}}_{n}}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right) \right\| }_{L^{2}}. \end{aligned}$$
This completes the proof. \(\square \)
In the following theorem we give the error bounds for the iterated solutions in both infinity norm and \(L^{2}\)-norm.
Theorem 2.9
Let \(x_{0}\in \mathcal {C}[-1,\,1]\) be an isolated solution of the Eq. (2.2). Let \({\tilde{x}_{n}}\) defined by the iterated scheme (2.25). Then the following hold
$$\begin{aligned} {\left\| {\tilde{x}_{n}}-x_{0}\right\| }_{\infty }\le c \left\{ \left\| x_{0}-\mathcal {P}_{n}x_{0}\right\| _{L^{2}}^{2}+\left| < g_{t},\,\left( I-\mathcal {P}_{n}\right) x_{0}>\right| \right\} , \end{aligned}$$
(2.52)
and
$$\begin{aligned} {\left\| {\tilde{x}_{n}}-x_{0}\right\| }_{L^{2}}\le c\left\{ \left\| x_{0}-\mathcal {P}_{n}x_{0}\right\| _{L^{2}}^2+\left| < g_{t},\,\left( I-\mathcal {P}_{n}\right) x_{0}>\right| \right\} , \end{aligned}$$
(2.53)
where \(g_{t}(s)= \sum \nolimits _{i=1}^{m}k_{i}(t,\,s)\psi _{i}^{(0,\,1)}(s,\,x_{0}(s))\) and c is a constant independent of n.
Proof
From Theorem 2.7, we have
$$\begin{aligned} \frac{\beta _{n}}{1+q}\le {\left\| {\tilde{x}_{n}}-x_{0}\right\| }_{\infty } \le \frac{\beta _{n}}{1-q}, \end{aligned}$$
where,
$$\begin{aligned} \beta _{n}={\left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\left( \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right) \right\| }_{\infty }. \end{aligned}$$
Hence using Theorem 2.6, we get
$$\begin{aligned} \left\| \tilde{x}_{n}-x_{0}\right\| _{\infty } \le \beta _{n}&= {\left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\left( \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right) \right\| }_{\infty }\nonumber \\&\le \left\| \left( \mathcal {I}-{\widetilde{\mathcal {T}}_{n}}^{'}\left( x_{0}\right) \right) ^{-1}\right\| _{\infty } \left\| \widetilde{\mathcal {T}}_{n}\left( x_{0}\right) -\mathcal {T}\left( x_{0}\right) \right\| _{\infty }\nonumber \\&\le L \left\| \sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}\left( \mathcal {P}_{n}x_{0}\right) -\sum \limits _{i=1}^{m}\mathcal {K}_{i}\psi _{i}\left( x_{0}\right) \right\| _{\infty }. \end{aligned}$$
(2.54)
We denote \(g(t,\,s,\,x_{0},\,x,\,\theta ) = \sum \nolimits _{i=1}^{m}k_{i}(t,\,s)\psi _{i}^{(0,1)}(s,\,x_{0}(s)+\theta (x(s)-x_{0}(s)))\) and \(g_{t}(s) = \sum \nolimits _{i=1}^{m}k_{i}(t,\,s)\psi _{i}^{(0,1)}(s,\,x_{0}(s)).\)
We have
$$\begin{aligned}&\left| \underset{i=1}{\overset{m}{\sum }}\left[ \left( \mathcal {K}_{i}\psi _{i}\right) \left( \mathcal {P}_{n}x_{0}\right) -\left( \mathcal {K}_{i}\psi _{i}\right) \left( x_{0}\right) \right] (t)\right| \nonumber \\&=\left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\left[ \psi _{i}\left( s,\,\mathcal {P}_{n}x_{0}(s)\right) -\psi _{i}\left( s,\,x_{0}(s)\right) \right] ds\right| \nonumber \\&= \left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\left[ \psi _{i}\left( s,\,x_{0}(s)+\theta \left( \mathcal {P}_{n}x_{0}(s)-x_{0}(s)\right) \right) \right] \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)ds\right| \nonumber \\&=\left| \int \nolimits _{-1}^{1}g\left( t,\,s,\,x_{0},\,\mathcal {P}_{n}x_{0},\,\theta \right) \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)ds\right| \nonumber \\&=\left| \int \nolimits _{-1}^{1}\left[ g\left( t,\,s,\,x_{0},\,\mathcal {P}_{n}x_{0},\,\theta \right) -g_{t}(s)+g_{t}(s)\right] \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)ds\right| \nonumber \\&\le \left| \int \nolimits _{-1}^{1}\left[ g\left( t,\,s,\,x_{0},\,\mathcal {P}_{n}x_{0},\,\theta \right) -g_{t}(s)\right] \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)ds\right| \nonumber \\&\quad +\left| \int \nolimits _{-1}^{1}g_{t}(s)\left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)ds\right| . \end{aligned}$$
(2.55)
For the first term of the above estimate (2.55), using Lipschitz’s continuity of \(\psi _{i}^{(0,1)}(.,\,x(.)),\) we have
$$\begin{aligned}&\left| \int \nolimits _{-1}^{1}\left[ g\left( t,\,s,\,x_{0},\,\mathcal {P}_{n}x_{0},\,\theta \right) -g_{t}(s)\right] \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)ds\right| \nonumber \\&=\left| \sum \limits _{i=1}^{m}\int \nolimits _{-1}^{1}k_{i}(t,\,s)\left[ \psi _{i}^{(0,1)}\left( s,\,x_{0}(s)+\theta \left( \mathcal {P}_{n}x_{0}(s)-x_{0}(s)\right) \right) \right. \right. \nonumber \\&\quad \left. \left. -\psi _{i}\left( s,\,x_{0}(s)\right) \right] \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)ds\right| \nonumber \\&\le \sum \limits _{i=1}^{m}\sup _{s,t\in [-1,1]}\left| k_{i}(t,\,s)\right| \int \nolimits _{-1}^{1}q_{i}\left| \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)\right| \left| \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)\right| ds\nonumber \\&=Ml_{2}\int \nolimits _{-1}^{1}\left| \left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)\right| ^{2}ds \le Ml_{2}\left\| x_{0}-\mathcal {P}_{n}x_{0}\right\| _{L^{2}}^{2}. \end{aligned}$$
(2.56)
Now for the second term of (2.55), we have
$$\begin{aligned} \left| \int \nolimits _{-1}^{1}g_{t}(s)\left( x_{0}-\mathcal {P}_{n}x_{0}\right) (s)ds\right| =\left| <g_{t}(.),\,\left( I-\mathcal {P}_{n}\right) \left( x_{0}\right) >\right| . \end{aligned}$$
(2.57)
Hence combining estimates (2.55)–(2.57), we have
$$\begin{aligned}&\left\| \underset{i=1}{\overset{m}{\sum }}\mathcal {K}_{i}\psi _{i}\left( \mathcal {P}_{n}x_{0}\right) -\underset{i=1}{\overset{m}{\sum }}\mathcal {K}_{i}\psi _{i}\left( x_{0}\right) \right\| _{\infty }\nonumber \\&\quad \le Ml_{2}\left\| x_{0}-\mathcal {P}_{n}x_{0}\right\| _{L^{2}}^{2} + \left| < g_{t}(.),\,\left( I-\mathcal {P}_{n}\right) \left( x_{0}\right) >\right| . \end{aligned}$$
(2.58)
Therefore from estimates (2.54) and (2.58), we have
$$\begin{aligned} \left\| \tilde{x}_{n}-x_{0}\right\| _{\infty }&\le L \left\| \mathcal {K}\left( \mathcal {P}_{n}x_{0}\right) -\mathcal {K}\left( x_{0}\right) \right\| _{\infty }\nonumber \\&\le c\left\{ \left\| x_{0}-\mathcal {P}_{n}x_{0}\right\| _{L^{2}}^{2} + \left| < g_{t}(.),\,\left( I-\mathcal {P}_{n}\right) \left( x_{0}\right) >\right| \right\} . \end{aligned}$$
(2.59)
where c is a constant independent of n, this proves the estimate (2.52).
Similarly for \(L^{2}\)-norm, we can show that
$$\begin{aligned} \left\| \tilde{x}_{n}-x_{0}\right\| _{L^{2}}&\le \sqrt{2}\left\| \tilde{x}_{n}-x_{0}\right\| _{\infty }\nonumber \\&\le c\left\{ \left\| x_{0}-\mathcal {P}_{n}x_{0}\right\| _{L^{2}}^{2} + \left| <g_{t}(.),\,\left( I-\mathcal {P}_{n}\right) \left( x_{0}\right) >\right| \right\} , \end{aligned}$$
(2.60)
c being a constant independent of n. This completes the proof. \(\square \)
Now we discuss the convergence rates for the approximate and iterated approximate solutions. To distinguish between the Legendre Galerkin solutions and Legendre collocation solutions, we set the following notations. In case of Legendre Galerkin method, we denote the approximate solution and the iterated approximate solutions by \(x_{n}=x_{n}^{G}\) and \(\tilde{x}_{n}=\tilde{x}_{n}^{G},\) respectively. For Legendre collocation method, we write the approximate solution and iterated approximate solution as \(x_{n} ={x}_{n}^{C}\) and \(\tilde{x}_{n} =\tilde{x}_{n}^{C},\) respectively.
Theorem 2.10
Let \(x_{0}\in \mathcal {C}^{r}[-1,\,1]\) be a isolated solution of the Eq. (2.1) and \({x}_{n} = x_{n}^{G}\) be the Legendre Galerkin solution or \(x_{n} = x_{n}^{C}\) be the Legendre collocation approximation of \(x_{0}.\) Then the following hold
$$\begin{aligned} \left\| x_{0}- {x}_{n}^{G}\right\| _{L^{2}},\quad \left\| x_{0}- {x}_{n}^{C}\right\| _{L^{2}} =\mathcal {O}\left( n^{-r}\right) , \end{aligned}$$
and
$$\begin{aligned} \left\| x_{0}- {x}_{n}^{G}\right\| _{\infty },\quad \left\| x_{0}- {x}_{n}^{C}\right\| _{\infty }=\mathcal {O}\left( n^{\frac{1}{2}-r}\right) . \end{aligned}$$
Proof
From Theorem 2.3, we have
$$\begin{aligned} \frac{\alpha _{n}}{1+q}\le {\left\| x_{n}-x_{0}\right\| }_{L^{2}} \le \frac{\alpha _{n}}{1-q}, \end{aligned}$$
where \(\alpha _{n}={\Vert (\mathcal {I}-{{\mathcal {T}}_{n}}^{'}(x_{0}))^{-1}({\mathcal {T}}_{n}(x_{0})-{\mathcal {T}}(x_{0}))\Vert }_{L^{2}}.\)
Hence we have from the estimate (2.34)
$$\begin{aligned} \left\| x_{n}-x_{0}\right\| _{L^{2}} \le \alpha _{n}&\le A_{1}\left\| \left( \mathcal {P}_{n}-I\right) x_{0}\right\| _{L^{2}}\\ {}&\le A_{1}cn^{-r}\left\| x_{0}^{(r)}\right\| _{L^{2}}=\mathcal {O}\left( n^{-r}\right) , \end{aligned}$$
where c is a constant independent of n.
Now for the error in infinity norm, using the estimate (2.38), we have
$$\begin{aligned} \left\| x_{n}-x_{0}\right\| _{\infty } \le A_{2}\left\| \left( \mathcal {P}_{n}-I\right) x_{0}\right\| _{\infty }. \end{aligned}$$
Hence for Legendre Galerkin solution \(x_{n} = x_{n}^{G},\) using estimate (2.14) of Lemma 2.3, we have
$$\begin{aligned} {\left\| x_{n}^{G}-x_{0}\right\| }_{\infty } \le A_{2}\left\| \left( \mathcal {P}_{n}^{G}-I\right) x_{0}\right\| _{\infty }\le A_{2}cn^{\frac{1}{2}-r}V\left( x_{0}^{(r)}\right) =\mathcal {O}\left( n^{\frac{1}{2}-r}\right) ,\qquad \end{aligned}$$
(2.61)
and for Legendre collocation solution \(x_{n}^{C},\) using estimate (2.17), we have
$$\begin{aligned} {\left\| x_{n}^{C}-x_{0}\right\| }_{\infty }\le A_{2}\left\| \left( \mathcal {P}_{n}^{C}-I\right) x_{0}\right\| _{\infty }\le A_{2}cn^{\frac{1}{2}-r}\left\| x_{0}^{(r)}\right\| _{\infty }=\mathcal {O}\left( n^{\frac{1}{2}-r}\right) .\qquad \end{aligned}$$
(2.62)
Hence the proof follows.\(\square \)
Next we will discuss the error bounds for the iterated Legendre Galerkin and iterated Legendre collocation solutions separately.
Theorem 2.11
Let \(x_{0}\in \mathcal {C}^r[-1,1]\) be a isolated solution of the equation (2.1) and \(\tilde{x}_{n}^G\) be the iterated Legendre Galerkin approximations of \(x_{0}\). Then there hold
$$\begin{aligned}&\Vert x_0- \tilde{x}_n^G\Vert _{L^2} =\mathcal {O}(n^{-2r}),\\&\Vert x_0- \tilde{x}_n^G\Vert _{\infty } =\mathcal {O}(n^{-2r}). \end{aligned}$$
Proof
From Theorem 2.9, we have
$$\begin{aligned} \left\| \tilde{x}^{G}_{n}-x_{0}\right\| _{\infty } \le c\left\{ \left\| x_{0}-\mathcal {P}_{n}^{G}x_{0}\right\| _{L^{2}}^{2} +\left| <g_{t}(.),\,\left( I-\mathcal {P}_{n}^{G}\right) \left( x_{0}\right) >\right| \right\} , \end{aligned}$$
(2.63)
where c is a constant independent of n.
Using the orthogonality of the projection operators \({\mathcal {P}}_{n}^{G}\) and Cauchy–Schwarz inequality, we obtain
$$\begin{aligned} \left| <g_{t}(.),\,\left( I-\mathcal {P}_{n}^{G}\right) \left( x_{0}\right) >\right|&= \left| <\left( I-\mathcal {P}_{n}^{G}\right) g_{t}(.),\,\left( I-\mathcal {P}_{n}^{G}\right) \left( x_{0}\right) >\right| \nonumber \\&\le {\left\| \left( \mathcal {I}-{\mathcal {P}}_{n}^{G}\right) g_{t}(.)\right\| }_{L^{2}}{\left\| x_{0}-{\mathcal {P}}_{n}^{G}x_{0}\right\| }_{L^{2}}.\qquad \end{aligned}$$
(2.64)
Hence using estimates (2.12), (2.63) and (2.64), we have
$$\begin{aligned} {\left\| {\tilde{x}_{n}^{G}}-x_{0}\right\| }_{\infty }&\le c\left\{ \left\| x_{0}-\mathcal {P}_{n}^{G}x_{0}\right\| _{L^{2}}^{2} + {\left\| \left( \mathcal {I}-{\mathcal {P}}_{n}^{G}\right) g_{t}(.)\right\| }_{L^{2}}{\left\| x_{0}-{\mathcal {P}}_{n}^{G}x_{0}\right\| }_{L^{2}}\right\} \nonumber \\&\le cn^{-2r}\left\| x_{0}^{(r)}\right\| _{L^{2}}^{2}+c n^{-2r} \left\| x_{0}^{(r)}\right\| _{L^{2}} \left\| (g_{t}(.))^{(r)}\right\| _{L^{2}}\nonumber \\&= \mathcal {O}\left( n^{-2r}\right) . \end{aligned}$$
(2.65)
And also
$$\begin{aligned} \left\| {\tilde{x}_{n}^{G}}-x_{0}\right\| _{L^{2}} \le \sqrt{2}{\left\| {\tilde{x}_{n}^{G}}-x_{0}\right\| }_{\infty } = \mathcal {O}\left( n^{-2r}\right) . \end{aligned}$$
(2.66)
Hence the proof follows. \(\square \)
Theorem 2.12
Let \(x_{0}\in \mathcal {C}^{r}[-1,\,1]\) be a isolated solution of the Eq. (2.1) and \(\tilde{x}_{n}^{C}\) be the iterated Legendre collocation approximations of \(x_{0}.\) Then there hold
$$\begin{aligned}&\left\| x_{0}- \tilde{x}_{n}^{C}\right\| _{L^{2}} =\mathcal {O}\left( n^{-r}\right) ,\\&\left\| x_{0}- \tilde{x}_{n}^{C}\right\| _{\infty } =\mathcal {O}\left( n^{-r}\right) . \end{aligned}$$
Proof
Using Theorem 2.9, Lemma 2.5, we have for the interpolatory projection operator \(\mathcal {P}_{n}^{C}\)
$$\begin{aligned} \left\| \tilde{x}_{n}^{C}-x_{0}\right\| _{\infty }&\le c\left\{ \left\| x_{0}-\mathcal {P}_{n}^{C}x_{0}\right\| _{L^{2}}^{2} + \left| <g_{t}(.),\,\left( I-\mathcal {P}_{n}^{C}\right) \left( x_{0}\right) >\right| \right\} \nonumber \\&\le c\left\{ \left\| x_{0}-\mathcal {P}_{n}^{C}x_{0}\right\| _{L^{2}}^{2}+\left\| g_{t}\right\| _{L^{2}}{\left\| x_{0}-{\mathcal {P}}_{n}^{C}x_{0}\right\| }_{L^{2}}\right\} \nonumber \\&\le c\left\{ n^{-2r}\left\| x_{0}^{(r)}\right\| _{L^{2}}^{2}+n^{-r}\left\| g_{t}\right\| _{L^{2}}\left\| x_{0}^{(r)}\right\| _{L^{2}}\right\} \nonumber \\&= \mathcal {O}\left( n^{-r}\right) , \end{aligned}$$
(2.67)
and
$$\begin{aligned} \left\| {\tilde{x}_{n}^{C}}-x_{0}\right\| _{L^{2}} \le \sqrt{2}{\left\| {\tilde{x}_{n}^{C}}-x_{0}\right\| }_{\infty } = \mathcal {O}\left( n^{-r}\right) . \end{aligned}$$
(2.68)
Hence the proof follows. \(\square \)
Remark
From Theorems 2.10–2.12 we observe that the Legendre Galerkin and Legendre collocation solutions of Hammerstein integral equation of mixed type have same order of convergence, \(\mathcal {O}(n^{-r})\) in \(L^{2}\)-norm and \(\mathcal {O}(n^{\frac{1}{2}-r})\) in infinity norm. The iterated Legendre Galerkin solution converges with the order \(\mathcal {O}(n^{-2r})\) in both \(L^{2}\)-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order \(\mathcal {O}(n^{-r})\) in both \(L^{2}\)-norm and in infinity norm. This shows that iterated Legendre Galerkin method improves over the iterated Legendre collocation method.
