Appendix A
Using Leibniz rule, we have
$$\begin{aligned}&\Vert [(\mathcal {K}_1\psi _1)'(x_0)x]^{1}\Vert _\infty \nonumber \\&\quad =\sup _{t\in [0,1]}|[(\mathcal {K}_1\psi _1)'(x_0)x]^{1}(t)|\nonumber \\&\quad =\sup _{t\in [0,1]}|\frac{\partial }{\partial t}\int _{0}^{t}k_1(t,s)\psi _1^{(0,1)}(s,x_0(s))x(s)ds|\nonumber \\&\quad \le \sup _{t\in [0,1]}\left[ |k_1(t,t)||\psi ^{(0,1)}(t,x_0(t))||x(t)|+\left| \int _{0}^{t}\frac{\partial }{\partial t}k_1(t,s)\psi _1^{(0,1)}(s,x_0(s))x(s)ds\right| \right] \nonumber \\&\quad \le \sup _{t\in [0,1]}[|k_1(t,t)||\psi _1^{(0,1)}(t,x_0(t))||x(t)|]\nonumber \\&\qquad +\sup _{t,s\in [0,1]}|\frac{\partial }{\partial t}k_1(t,s)|\sup _{s\in [0,1]}|\psi _1^{(0,1)}(s,x_0(s))|\int _{0}^{t}|x(s)|ds\nonumber \\&\quad \le M_1d_1\Vert x\Vert _\infty +\Vert k_1\Vert _{1,\infty }d_1\Vert x\Vert _\infty <\infty . \end{aligned}$$
(A.1)
For \(j=0, 1, 2,..., r\), we have
$$\begin{aligned} \Vert [(\mathcal {K}_2\psi _2)'(x_0)x]^{(j)}\Vert _\infty= & {} \sup _{t\in [0,1]}|[(\mathcal {K}_2\psi _2)'(x_0)x]^{(j)}(t)|\nonumber \\= & {} \sup _{t\in [0,1]}|\frac{\partial ^j}{\partial t^j}\int _{0}^{1}k_2(t,s)\psi _2^{(0,1)}(s,x_0(s))x(s)ds|\nonumber \\\le & {} \sup _{t, s\in [0,1]}|\frac{\partial ^j}{\partial t^j}k_2(t,s)|\sup _{s\in [0,1]}[|\psi _2^{(0,1)}(s,x_0(s))||x(s)|]\nonumber \\\le & {} \Vert k_2\Vert _{j,\infty }d_2\Vert x\Vert _\infty <\infty . \end{aligned}$$
(A.2)
Appendix B
Lemma 5
For any \(x, y\in L^2[0,1]\) or \(\mathcal {C}[0,1]\), the following hold
$$\begin{aligned}&\Vert [(\mathcal {K}_{1,n}^M\psi _1)'(x)-(\mathcal {K}_{1,n}^M\psi _1)'(y)]z\Vert _\infty \le [p_1M_1c_1+ch(M_1+2\Vert k_1\Vert _{1,\infty })c_1p_1^2]\Vert x\nonumber \\&\quad -y\Vert _{\infty }\Vert z\Vert _{\infty }, \\&\Vert [(\mathcal {K}_{2,n}^M\psi _2)'(x)-(\mathcal {K}_{2,n}^M\psi _2)'(y)]z\Vert _\infty \le [p_1M_2c_2+ch^rc_2\Vert k_2\Vert _{r,\infty }p_1^2]\Vert x\nonumber \\&\quad -y\Vert _{\infty }\Vert z\Vert _{\infty }, \end{aligned}$$
where c is a constant independent of n.
Proof
For any x, y, z, using (2.5), (3.2), we have for \(i=1, 2\),
$$\begin{aligned}&\Vert [(\mathcal {K}_{i,n}^M\psi _i)'(x)-(\mathcal {K}_{i,n}^M\psi _i)'(y)]z\Vert _\infty \nonumber \\&\quad = \Vert [\mathcal {P}_n(\mathcal {K}_i\psi _i)'(x)+(\mathcal {I}-\mathcal {P}_n)(\mathcal {K}_i\psi _i)'(\mathcal {P}_nx)\mathcal {P}_n\nonumber \\&\qquad -\mathcal {P}_n(\mathcal {K}_i\psi _i)'(y)-(\mathcal {I}-\mathcal {P}_n)(\mathcal {K}_i\psi _i)'(\mathcal {P}_ny)\mathcal {P}_n]z\Vert _{\infty }\nonumber \\&\quad \le \Vert \mathcal {P}_n[(\mathcal {K}_i\psi _i)'(x)-(\mathcal {K}_i\psi _i)'(y)]z\Vert _{\infty }+\Vert (\mathcal {I}-\mathcal {P}_n)[(\mathcal {K}_i\psi _i)'(\mathcal {P}_nx)\nonumber \\&\qquad -(\mathcal {K}_i\psi _i)'(\mathcal {P}_ny)]\mathcal {P}_nz\Vert _{\infty }\nonumber \\&\quad \le p_1M_ic_i\Vert x-y\Vert _{\infty }\Vert z\Vert _{\infty }+\Vert (\mathcal {I}-\mathcal {P}_n)[(\mathcal {K}_i\psi _i)'(\mathcal {P}_nx)-(\mathcal {K}_i\psi _i)'(\mathcal {P}_ny)]\mathcal {P}_nz\Vert _{\infty }. \qquad \qquad \end{aligned}$$
(B.1)
For \(i=1\), using (2.7), for \(r=1\), we have
$$\begin{aligned}&\Vert (\mathcal {I}-\mathcal {P}_n)[(\mathcal {K}_1\psi _1)'(\mathcal {P}_nx)-(\mathcal {K}_1\psi _1)'(\mathcal {P}_ny)]\mathcal {P}_nz\Vert _{\infty }\nonumber \\&\quad \le ch\Vert [((\mathcal {K}_1\psi _1)'(\mathcal {P}_nx)-(\mathcal {K}_1\psi _1)'(\mathcal {P}_ny))\mathcal {P}_nz]^{(1)}\Vert _{\infty }. \end{aligned}$$
(B.2)
Now using Leibniz rule, Lipschitz continuity of \(\psi _1^{(0,1)} (.,.)\) and estimate (2.5), we get
$$\begin{aligned}&\Vert [((\mathcal {K}_1\psi _1)'(\mathcal {P}_nx)-(\mathcal {K}_1\psi _1)'(\mathcal {P}_ny))\mathcal {P}_nz]^{(1)}\Vert _{\infty }\nonumber \\&\quad =\sup _{t\in [0,1]}|[((\mathcal {K}_1\psi _1)'(\mathcal {P}_nx)-(\mathcal {K}_1\psi _1)'(\mathcal {P}_ny))\mathcal {P}_nz]^{(1)}(t)|\nonumber \\&\quad \le \sup _{t\in [0,1]}\left| \frac{\partial }{\partial t}\int _{0}^{t} k_1(t,s)[{\psi _1}^{(0,1)}(s, \mathcal {P}_nx(s))-{\psi _1}^{(0,1)}(s,\mathcal {P}_ny(s))]\mathcal {P}_nz(s)ds\right| \nonumber \\&\quad \le \sup _{t\in [0,1]}\left[ |k_1(t,t)||{\psi _1}^{(0,1)}(t, \mathcal {P}_nx(t))-{\psi _1}^{(0,1)}(t,\mathcal {P}_ny(t))||\mathcal {P}_nz(t)|\right] \nonumber \\&\quad \quad + \sup _{t\in [0,1]}\left[ \left| \int _{0}^{t}\{\frac{\partial }{\partial t}k_1(t,s)\}[{\psi _1}^{(0,1)}(s, \mathcal {P}_n(s))-{\psi _1}^{(0,1)}(s,\mathcal {P}_ny(s))]\mathcal {P}_nz(s)ds\right| \right] \nonumber \\&\quad \le M_1c_1\sup _{t\in [0,1]}|\mathcal {P}_n(x-y)(t)||\mathcal {P}_nz(t)|+c_1\Vert k_1\Vert _{1,\infty }\sup _{s\in [0,1]}|\mathcal {P}_n(x-y)(s)||\mathcal {P}_nz(s)|\int _{0}^{t} ds\nonumber \\&\quad \le M_1c_1\Vert \mathcal {P}_n(x-y)\Vert _{\infty }\Vert \mathcal {P}_nz\Vert _{\infty }+c_1\Vert k_1\Vert _{1,\infty }\Vert \mathcal {P}_n(x-y)\Vert _{\infty }\Vert \mathcal {P}_nz\Vert _{\infty }\nonumber \\&\quad \le (M_1+\Vert k_1\Vert _{1,\infty })c_1p_1^2\Vert x-y\Vert _{\infty }\Vert z\Vert _{\infty }. \end{aligned}$$
(B.3)
Hence from (B.2) and (B.3), it follows that
$$\begin{aligned} \Vert (\mathcal {I}&-\mathcal {P}_n)[(\mathcal {K}_1\psi _1)'(\mathcal {P}_nx)-(\mathcal {K}_1\psi _1)'(\mathcal {P}_ny)]\mathcal {P}_nz\Vert _{\infty } \nonumber \\&\le ch(M_1+\Vert k_1\Vert _{1,\infty })c_1p_1^2\Vert x-y\Vert _{\infty }\Vert z\Vert _{\infty }. \nonumber \\ \end{aligned}$$
(B.4)
Combining estimates (B.1) (for \(i=1\)) and (B.4), the result follows
$$\begin{aligned} \Vert [(\mathcal {K}_{1,n}^M\psi _1)'(x)&-(\mathcal {K}_{1,n}^M\psi _1)'(y)]z\Vert _\infty \nonumber \\&\le [p_1M_1c_1+ch(M_1+\Vert k_1\Vert _{1,\infty })c_1p_1^2]\Vert x-y\Vert _{\infty }\Vert z\Vert _{\infty }. \end{aligned}$$
Now for \(i=2\), using (2.7), we have
$$\begin{aligned}&\Vert (\mathcal {I}-\mathcal {P}_n)[(\mathcal {K}_2\psi _2)'(\mathcal {P}_nx)-(\mathcal {K}_2\psi _2)'(\mathcal {P}_ny)]\mathcal {P}_nz\Vert _{\infty } \nonumber \\&\quad \le ch^r\Vert [((\mathcal {K}_2\psi _2)'(\mathcal {P}_nx)-(\mathcal {K}_2\psi _2)'(\mathcal {P}_ny))\mathcal {P}_nz]^{(r)}\Vert _{\infty }. \end{aligned}$$
(B.5)
Using Lipschitz continuity of \(\psi _2^{(0,1)} (.,.)\) and estimate (2.5), we get for \(j = 0,1,2,...,r\)
$$\begin{aligned}&\Vert [((\mathcal {K}_2\psi _2)'(\mathcal {P}_nx)-(\mathcal {K}_2\psi _2)'(\mathcal {P}_ny))\mathcal {P}_nz]^{(j)}\Vert _{\infty }\nonumber \\= & {} \sup _{t\in [0,1]}| [((\mathcal {K}_2\psi _2)'(\mathcal {P}_nx)-(\mathcal {K}_2\psi _2)'(\mathcal {P}_ny))\mathcal {P}_nz]^{(j)}(t)|\nonumber \\= & {} \sup _{t\in [0,1]}\left| \int _{0}^{1}\frac{\partial ^j}{\partial t^j}k_2(t,s)[\psi _2^{(0,1)}(s,\mathcal {P}_nx(s))-\psi _2^{(0,1)}(s,\mathcal {P}_ny(s))]\mathcal {P}_nz(s)ds\right| \nonumber \\\le & {} c_2\Vert k_2\Vert _{r,\infty }\Vert \mathcal {P}_n(x-y)\Vert _{\infty }\Vert \mathcal {P}_nz\Vert _{\infty }\nonumber \\\le & {} c_2\Vert k_2\Vert _{r,\infty }p_1^2\Vert x-y\Vert _{\infty }\Vert z\Vert _{\infty }. \end{aligned}$$
(B.6)
Hence (B.5) and (B.6) gives,
$$\begin{aligned}&\Vert (\mathcal {I}-\mathcal {P}_n)[(\mathcal {K}_2\psi _2)'(\mathcal {P}_nx)-(\mathcal {K}_2\psi _2)'(\mathcal {P}_ny)]\mathcal {P}_nz\Vert _{\infty } \nonumber \\&\le ch^rc_2\Vert k_2\Vert _{r,\infty }p_1^2\Vert x-y\Vert _{\infty }\Vert z\Vert _{\infty }. \end{aligned}$$
(B.7)
Combining estimates (B.1) (for \(i=2\)) and (B.7), we get
$$\begin{aligned}&\Vert [(\mathcal {K}_{2,n}^M\psi _2)'(x)-(\mathcal {K}_{2,n}^M\psi _2)'(y)]z\Vert _\infty \\&\le [p_1M_2c_2+ch^rc_2\Vert k_2\Vert _{r,\infty }p_1^2]\Vert x-y\Vert _{\infty }\Vert z\Vert _{\infty }. \end{aligned}$$
Hence the result follows. \(\square \)
Appendix C
We have
$$\begin{aligned}&\Vert [(\mathcal {K}_1\psi _1)^{'}(\mathcal {P}_nx_0)\mathcal {P}_nx]^{(1)}\Vert _{\infty }\nonumber \\&\quad = \sup _{t\in [0,1]}|(\mathcal {K}_1\psi _1)^{'}(\mathcal {P}_nx_0)\mathcal {P}_nx]^{(1)}(t)|\nonumber \\&\quad =\sup _{t\in [0,1]}\left| \frac{\partial }{\partial t} \int _{0}^{t}k_1(t,s)\psi _1^{(0,1)}(s, \mathcal {P}_nx_0(s))\mathcal {P}_nx(s)ds\right| \nonumber \\&\quad \le \sup _{t\in [0,1]}\left| \frac{\partial }{\partial t}\int _{0}^{t} k_1(t,s)[{\psi _1}^{(0,1)}(s, \mathcal {P}_nx_0(s))-{\psi _1}^{(0,1)}(s,x_0(s))]\mathcal {P}_nx(s)ds\right| \nonumber \\&\qquad + \sup _{t\in [0,1]}\left| \frac{\partial }{\partial t }\int _{0}^{t} k_1(t,s){\psi _1}^{(0,1)}(s,x_0(s))\mathcal {P}_nx(s)ds\right| . \end{aligned}$$
(C.1)
Using Leibniz rule and estimates (2.5), (2.7), we get
$$\begin{aligned}&\sup _{t\in [0,1]}\left| \frac{\partial }{\partial t}\int _{0}^{t} k_1(t,s)[{\psi _1}^{(0,1)}(s, \mathcal {P}_nx_0(s))-{\psi _1}^{(0,1)}(s,x_0(s))]\mathcal {P}_nx(s)ds\right| \nonumber \\&\quad \le \sup _{t\in [0,1]}\left[ |k_1(t,t)||{\psi _1}^{(0,1)}(t, \mathcal {P}_nx_0(t))-{\psi _1}^{(0,1)}(t,x_0(t))||\mathcal {P}_nx(t)|\right] \nonumber \\&\quad \quad +\sup _{t\in [0,1]}\left[ \left| \int _{0}^{t}\{\frac{\partial }{\partial t}k_1(t,s)\}[{\psi _1}^{(0,1)}(s, \mathcal {P}_nx_0(s))-{\psi _1}^{(0,1)}(s,x_0(s))]\mathcal {P}_nx(s)ds\right| \right] \nonumber \\&\quad \le M_1c_1\sup _{t\in [0,1]}|(\mathcal {P}_nx_0-x_0)(t)||\mathcal {P}_nx(t)|\nonumber \\&\qquad +c_1\Vert k_1\Vert _{1,\infty }\sup _{s\in [0,1]}|(\mathcal {P}_n-\mathcal {I})x_0(s)||\mathcal {P}_nx(s)|\int _{0}^{t} ds \nonumber \\&\quad \le M_1c_1\Vert (\mathcal {P}_n-\mathcal {I})x_0\Vert _{\infty }\Vert \mathcal {P}_nx\Vert _{\infty }+c_1\Vert k_1\Vert _{1,\infty }\Vert (\mathcal {P}_n-\mathcal {I})x_0\Vert _{\infty }\Vert \mathcal {P}_nx\Vert _{\infty }\nonumber \\&\quad \le (M_1+\Vert k_1\Vert _{1,\infty })cc_1p_1h^r\Vert x_0^{(r)}\Vert _{\infty }\Vert x\Vert _{\infty }. \end{aligned}$$
(C.2)
Again by using Leibniz rule and estimate (2.5), we obtain
$$\begin{aligned}&\sup _{t\in [0,1]}\left| \frac{\partial }{\partial t }\int _{0}^{t} k_1(t,s){\psi _1}^{(0,1)}(s,x_0(s))\mathcal {P}_nx(s)ds\right| \nonumber \\&\quad \le \sup _{t\in [0,1]}\left| \frac{\partial }{\partial t }\int _{0}^{t} k_1(t,s){\psi _1}^{(0,1)}(s,x_0(s))(\mathcal {P}_n-\mathcal {I})x(s)ds\right| \nonumber \\&\quad \quad + \sup _{t\in [0,1]}\left| \frac{\partial }{\partial t }\int _{0}^{t} k_1(t,s){\psi _1}^{(0,1)}(s,x_0(s))x(s)ds\right| \nonumber \\&\quad \le \sup _{t\in [0,1]}\left[ |k_1(t,t)||{\psi _1}^{(0,1)}(t,x_0(t))||(\mathcal {P}_n-\mathcal {I})x(t)| \right. \nonumber \\&\quad \quad \left. +\left| \int _{0}^{t}{\frac{\partial }{\partial t}k_1(t,s)}{\psi _1}^{(0,1)}(s,x_0(s))(\mathcal {P}_n-\mathcal {I})x(s)ds\right| \right] \nonumber \\&\quad \quad + \sup _{t\in [0,1]}\left[ |k_1(t,t)||{\psi _1}^{(0,1)}(t,x_0(t))||x(t)|+\left| \int _{0}^{t}{\frac{\partial }{\partial t}k_1(t,s)}{\psi _1}^{(0,1)}(s,x_0(s))x(s)ds\right| \right] \nonumber \\&\quad \le M_1d_1\Vert (\mathcal {P}_n-\mathcal {I})x\Vert _{\infty }\nonumber \\&\quad \quad +\sup _{t,s\in [0,1]}\left| \frac{\partial }{\partial t}k_1(t,s)\right| \sup _{s\in [0,1]}\left| {\psi _1}^{(0,1)}(s,x_0(s))\right| \left| (\mathcal {P}_n-\mathcal {I})x(s)\right| \int _{0}^{t} ds+M_1d_1\Vert x\Vert _{\infty }\nonumber \\&\quad \quad + \sup _{t,s\in [0,1]}\left| \frac{\partial }{\partial t}k_1(t,s)\right| \sup _{s\in [0,1]}\left| {\psi _1}^{(0,1)}(s,x_0(s))\right| \left| x(s)\right| \int _{0}^{t} ds\nonumber \\&\quad \le [M_1d_1+\Vert k_1\Vert _{1,\infty }d_1](2+p_1)\Vert x\Vert _{\infty }. \end{aligned}$$
(C.3)
Combining estimates (C.1), (C.2) and (C.3), we get
$$\begin{aligned}&\Vert [(\mathcal {K}_1\psi _1)^{'}(\mathcal {P}_nx_0)\mathcal {P}_nx]^{(1)}\Vert _{\infty }\nonumber \\&\quad \le [(M_1+\Vert k_1\Vert _{1,\infty })cc_1p_1h^r\Vert x_0^{(r)}\Vert _{\infty }+{(M_1d_1+\Vert k_1\Vert _{1,\infty }d_1)(2+p_1)}]\Vert x\Vert _{\infty }. \end{aligned}$$
(C.4)
Also using Lipschitz continuity of \(\psi _2^{(0,1)}(.,.)\), estimates (2.5) and (2.7), we have
$$\begin{aligned} \Vert [(\mathcal {K}_2\psi _2)^{'}(\mathcal {P}_nx_0)\mathcal {P}_nx]^{(r)}\Vert _{\infty }= & {} \sup _{t\in [0,1]}|(\mathcal {K}_2\psi _2)^{'}(\mathcal {P}_nx_0)\mathcal {P}_nx]^{(r)}(t)|\nonumber \\= & {} \sup _{t\in [0,1]}\left| \frac{\partial ^{r}}{\partial t^{r}}\int _{0}^{1} k_2(t,s)\psi _2^{(0,1)}(s, \mathcal {P}_nx_0(s))\mathcal {P}_nx(s)ds\right| \nonumber \\\le & {} \sup _{t\in [0,1]}\left| \frac{\partial ^{r}}{\partial t^{r}}\int _{0}^{1} k_2(t,s)[{\psi _2}^{(0,1)}(s, \mathcal {P}_nx_0(s)) \right. \nonumber \\&\left. -{\psi _2}^{(0,1)}(s,x_0(s))]\mathcal {P}_nx(s)ds\right| \nonumber \\&+ \sup _{t\in [0,1]}\left| \frac{\partial ^{r}}{\partial t^{r}}\int _{0}^{1} k_2(t,s){\psi _2}^{(0,1)}(s,x_0(s))\mathcal {P}_nx(s)ds\right| \nonumber \\\le & {} \sup _{t,s\in [0,1]}\left| \frac{\partial ^{r}}{\partial t^{r}} k_2(t,s)\right| \left| \int _{0}^{1}\left[ {\psi _2}^{(0,1)}(s, \mathcal {P}_nx_0(s)) \right. \right. \nonumber \\&\left. \left. -{\psi _2}^{(0,1)}(s,x_0(s))\right] \mathcal {P}_nx(s)ds\right| \nonumber \\&+ \sup _{t,s\in [0,1]}\left| \frac{\partial ^{r}}{\partial t^{r}} k_2(t,s)\right| \left| \int _{0}^{1}{\psi _2}^{(0,1)}(s,x_0(s))\mathcal {P}_nx(s)ds\right| \nonumber \\\le & {} \sup _{t,s\in [0,1]}\left| \frac{\partial ^{r}}{\partial t^{r}} k_2(t,s)\right| \int _{0}^{1}c_2\left| (\mathcal {P}_nx_0(s)-x_0(s))\right| \left| \mathcal {P}_nx(s)\right| ds\nonumber \\&+ \sup _{t,s\in [0,1]}\left| \frac{\partial ^{r}}{\partial t^{r}} k_2(t,s)\right| \sup _{s\in [0,1]}\left| {\psi _2}^{(0,1)}(s,x_0(s))\right| \left| \int _{0}^{1}\mathcal {P}_nx(s)ds\right| \nonumber \\\le & {} d_r\left[ c_2\Vert (\mathcal {P}_n-\mathcal {I})x_0\Vert _{\infty }+ d_2\right] \Vert \mathcal {P}_nx\Vert _{\infty }\nonumber \\\le & {} d_r\left[ c_2ch^{r}\Vert (x_0)^{r}\Vert _{\infty }+ d_2\right] p_1\Vert x\Vert _{\infty }. \end{aligned}$$
(C.5)
Appendix D
Using Lipschitz continuity of \(\psi _1^{(0,1)}(.,x(.))\) and Leibniz rule, and estimate (2.5), for any \(x\in B(x_0,\delta )\), we obtain
$$\begin{aligned}&\Vert [(\mathcal {K}_1\psi _1)'(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)'(\mathcal {P}_nx)\mathcal {P}_ny]^{1}\Vert _\infty \nonumber \\&\quad =\sup _{t\in [0,1]}|[((\mathcal {K}_1\psi _1)'(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)'(\mathcal {P}_nx))\mathcal {P}_ny]^{1}(t)|\nonumber \\&\quad =\sup _{t\in [0,1]}\left| \frac{\partial }{\partial t}\int _{0}^{t}k_1(t,s)[\psi _1^{(0,1)}(s,\mathcal {P}_nx_0(s))-\psi _1^{(0,1)}(s,\mathcal {P}_nx(s))]\mathcal {P}_ny(s)ds\right| \nonumber \\&\quad \le c_1 \sup _{t\in [0,1]}\left[ |k_1(t,t)||\mathcal {P}_n(x_0-x)(t))||\mathcal {P}_ny(t)| \right. \nonumber \\&\quad \quad \left. +\left| \int _{0}^{t}\frac{\partial }{\partial t}k_1(t,s)\mathcal {P}_n(x_0-x)(s)\mathcal {P}_ny(s)ds\right| \right] \nonumber \\&\quad \le c_1\left[ \sup _{t\in [0,1]}[|k_1(t,t)||\mathcal {P}_n(x_0-x)(t)||\mathcal {P}_ny(t)|] \right. \nonumber \\&\quad \quad \left. +\sup _{t,s\in [0,1]}\left| \frac{\partial }{\partial t}k_1(t,s)\right| \sup _{s\in [0,1]}|\mathcal {P}_n(x_0-x)(s)|\int _{0}^{t}|\mathcal {P}_ny(s)|ds\right] \nonumber \\&\quad \le c_1M_1\Vert \mathcal {P}_n(x_0-x)\Vert _\infty \Vert \mathcal {P}_ny\Vert _\infty +c_1\Vert k_1\Vert _{1,\infty }\Vert \mathcal {P}_n(x_0-x)\Vert _\infty \Vert \mathcal {P}_ny\Vert _\infty \nonumber \\&\quad \le c_1(M_1+\Vert k_1\Vert _{1,\infty })p_1^2\Vert x_0-x\Vert _{\infty }\Vert y\Vert _{\infty }\nonumber \\&\quad \le c_1(M_1+\Vert k_1\Vert _{1,\infty })p_1^2\delta \Vert y\Vert _{\infty }. \end{aligned}$$
(D.1)
Again we have,
$$\begin{aligned}&\Vert [((\mathcal {K}_2\psi _2)'(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)'(\mathcal {P}_nx))\mathcal {P}_ny]^{(r)}\Vert _\infty \nonumber \\&\quad = \sup _{t\in [0,1]}|[(\mathcal {K}_2\psi _2)'(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)'(\mathcal {P}_nx))\mathcal {P}_ny]^{(r)}(t)|\nonumber \\&\quad =\sup _{t\in [0,1]}\left| \frac{\partial ^r}{\partial t^r}\int _{0}^{1}k_2(t,s)[\psi _2^{(0,1)}(s,\mathcal {P}_nx_0(s))-\psi _2^{(0,1)}(s,\mathcal {P}_nx(s))]\mathcal {P}_ny(s)ds\right| \nonumber \\&\quad \le c_2\sup _{t, s\in [0,1]}\left| \frac{\partial ^r}{\partial t^r}k_2(t,s)\right| \sup _{s\in [0,1]}|\mathcal {P}_n(x_0-x)(s)|\int _{0}^{1}|\mathcal {P}_ny(s)|ds\nonumber \\&\quad \le c_2\Vert k_2\Vert _{r,\infty }\Vert \mathcal {P}_n(x_0-x)\Vert _\infty \Vert \mathcal {P}_ny\Vert _\infty \le c_2\Vert k_2\Vert _{r,\infty }p_1^2\delta \Vert y\Vert _\infty . \end{aligned}$$
(D.2)
Appendix E
Using Lipschitz continuity of \(\psi _1^{(0,1)}(.,x(.))\), Leibniz rule and estimate (2.7), we obtain
$$\begin{aligned}&\Vert [(\mathcal {K}_1\psi _1)(x_0)-(\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)y]^{1}\Vert _\infty \nonumber \\&\quad =\sup _{t\in [0,1]}|[((\mathcal {K}_1\psi _1)(x_0)-(\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0))y]^{1}(t)|\nonumber \\&\quad =\sup _{t\in [0,1]}\left| \frac{\partial }{\partial t}\int _{0}^{t}k_1(t,s)[\psi _1(s,x_0(s))-\psi _1(s,\mathcal {P}_nx_0(s))]y(s)ds\right| \nonumber \\&\quad \le l_1 \sup _{t\in [0,1]}\left[ |k_1(t,t)||(x_0-\mathcal {P}_nx_0)(t))||y(t)|+\left| \int _{0}^{t}\frac{\partial }{\partial t}k_1(t,s)(x_0-\mathcal {P}_nx_0)(s)y(s)ds\right| \right] \nonumber \\&\quad \le l_1[\sup _{t\in [0,1]}[|k_1(t,t)||(x_0-\mathcal {P}_nx_0)(t)||y(t)|]\nonumber \\&\quad \quad +\sup _{t,s\in [0,1]}|\frac{\partial }{\partial t}k_1(t,s)|\sup _{s\in [0,1]}|(x_0-\mathcal {P}_nx_0)(s)|\int _{0}^{t}|y(s)|ds]\nonumber \\&\quad \le l_1M_1\Vert x_0-\mathcal {P}_nx_0\Vert _\infty \Vert y\Vert _\infty +l_1\Vert k_1\Vert _{1,\infty }\Vert x_0-\mathcal {P}_nx_0\Vert _\infty \Vert y\Vert _\infty \nonumber \\&\quad \le l_1(M_1+\Vert k_1\Vert _{1,\infty })ch^r\Vert x_0^{(r)}\Vert _\infty \Vert y\Vert _\infty . \end{aligned}$$
(E.1)
Lipschitz continuity of \(\psi _2^{(0,1)}(.,x(.))\) and estimate (2.7) imply
$$\begin{aligned}&\Vert [((\mathcal {K}_2\psi _2)(x_0)-(\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0))y]^{(r)}\Vert _\infty \nonumber \\&\quad =\sup _{t\in [0,1]}|[(\mathcal {K}_2\psi _2)(x_0)-(\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0))y]^{(r)}(t)|\nonumber \\&\quad =\sup _{t\in [0,1]}\left| \frac{\partial ^r}{\partial t^r}\int _{0}^{1}k_2(t,s)[\psi _2(s,x_0(s))-\psi _2(s,\mathcal {P}_nx_0(s))]y(s)ds\right| \nonumber \\&\quad \le l_2\sup _{t, s\in [0,1]}\left| \frac{\partial ^r}{\partial t^r}k_2(t,s)\right| \sup _{s\in [0,1]}|(x_0-\mathcal {P}_nx_0)(s)|\int _{0}^{1}|y(s)|ds\nonumber \\&\quad \le l_2\Vert k_2\Vert _{r,\infty }\Vert x_0-\mathcal {P}_nx_0\Vert _\infty \Vert y\Vert _\infty \le l_2ch^r\Vert k_2\Vert _{r,\infty }\Vert x_0^{(r)}\Vert _{\infty }\Vert y\Vert _\infty . \end{aligned}$$
(E.2)
Appendix F
We have
$$\begin{aligned} x_n^M-x_0= & {} (\mathcal {K}_{1,n}^M\psi _1)(x_n^M)+(\mathcal {K}_{2,n}^M\psi _2)(x_n^M)-(\mathcal {K}_1\psi _1)(x_0)-(\mathcal {K}_2\psi _2)(x_0)\nonumber \\= & {} \sum _{j=1}^{2}[(\mathcal {K}_{j,n}^M\psi _j)(x_n^M)-(\mathcal {K}_j\psi _j)(x_0)]\nonumber \\= & {} \sum _{j=1}^{2}[(\mathcal {K}_{j,n}^M\psi _j)(x_n^M)-(\mathcal {K}_{j,n}^M\psi _j)(x_0)+(\mathcal {K}_{j,n}^M\psi _j)(x_0)-(\mathcal {K}_j\psi _j)(x_0)]\nonumber \\= & {} \sum _{j=1}^{2}[(\mathcal {K}_{j,n}^M\psi _j)(x_n^M)-(\mathcal {K}_{j,n}^M\psi _j)(x_0)-(\mathcal {K}_{j,n}^M\psi _j)'(x_0)(x_n^M-x_0)]\nonumber \\&+ \sum _{j=1}^{2}[(\mathcal {K}_{j,n}^M\psi _j)'(x_0)(x_n^M-x_0)+(\mathcal {K}_{j,n}^M\psi _j)(x_0)-(\mathcal {K}_j\psi _j)(x_0)]. \end{aligned}$$
(F.1)
This implies
$$\begin{aligned}&(\mathcal {I}-(\sum _{j=1}^2(\mathcal {K}_{j,n}^M\psi _j)'(x_0)))(x_n^M-x_0)\nonumber \\&\quad = \sum _{j=1}^{2}[(\mathcal {K}_{j,n}^M\psi _j)(x_n^M)-(\mathcal {K}_{j,n}^M\psi _j)(x_0)-(\mathcal {K}_{j,n}^M\psi _j)'(x_0)(x_n^M-x_0)]\nonumber \\&\qquad +\sum _{j=1}^{2}[(\mathcal {K}_{j,n}^M\psi _j)(x_0)-(\mathcal {K}_j\psi _j)(x_0)]. \end{aligned}$$
(F.2)
Using mean value theorem, from estimate (F.1) and (F.2), we have
$$\begin{aligned} x_n^M-x_0= & {} (\mathcal {I}-(\sum _{j=1}^2(\mathcal {K}_{j,n}^M\psi _j)'(x_0)))^{-1}\left[ \sum _{j=1}^{2}(\mathcal {K}_{j,n}^M\psi _j)(x_n^M)\right. \nonumber \\&\left. -(\mathcal {K}_{j,n}^M\psi _j)(x_0)-(\mathcal {K}_{j,n}^M\psi _j)'(x_0)(x_n^M-x_0)\right] \nonumber \\&+(\mathcal {I}-(\sum _{j=1}^2(\mathcal {K}_{j,n}^M\psi _j)'(x_0)))^{-1}\sum _{j=1}^2\left[ (\mathcal {K}_{j,n}^M\psi _j)(x_0)-(\mathcal {K}_j\psi _j)(x_0)\right] \nonumber \\= & {} (\mathcal {I}-{\mathcal {T}_n^M}'(x_0))^{-1}\sum _{j=1}^{2}[{(\mathcal {K}_{j,n}^M\psi _j)'(x_0+\theta _{j1}(x_n^M-x_0))-(\mathcal {K}_{j,n}^M\psi _j)'(x_0)}(x_n^M-x_0)]\nonumber \\&+ (\mathcal {I}-{\mathcal {T}_n^M}'(x_0))^{-1}\sum _{j=1}^{2}[(\mathcal {K}_{j,n}^M\psi _j)(x_0)-(\mathcal {K}_j\psi _j)(x_0)], \end{aligned}$$
(F.3)
where \(0<\theta _{j1}<1\) for \(j=1,2\).
Appendix G
For \(i=1\), taking \(g_1(t,s)=k_1(t,s)\psi _1^{(0,1)}(s,x_0(s))\) and using estimates (E.1), (E.2), we have
$$\begin{aligned}&\Vert (\mathcal {K}_1\psi _1)'(x_0)\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))]\Vert _{\infty }\nonumber \\&\quad =\sup _{t\in [0,1]}|(\mathcal {K}_1\psi _1)'(x_0)\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](t)|\nonumber \\&\quad =\sup _{t\in [0,1]}\left| \int _{0}^{t}k_1(t,s)\psi _1^{(0,1)}(s,x_0(s))\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](s)ds\right| \nonumber \\&\quad \le \sup _{t\in [0,1]}\left| \int _{0}^{1}k_1(t,s)\psi _1^{(0,1)}(s,x_0(s))\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](s)ds\right| \nonumber \\&\quad =\sup _{t\in [0,1]}\left| \int _{0}^{1}g_1(t,s)\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](s)ds\right| \nonumber \\&\quad =\sup _{t\in [0,1]}|<g_1(t,.), \sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](.)>|\nonumber \\&\quad =\sup _{t\in [0,1]}|<g_1(t,.), [(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)(x_0))](.)>|\nonumber \\&\quad \quad +\sup _{t\in [0,1]}|<g_1(t,.), [(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)(x_0))](.)>|\nonumber \\&\quad =\sup _{t\in [0,1]}|<(\mathcal {I}-\mathcal {P}_n)g_1(t,.), [(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)(x_0))](.)>|\nonumber \\&\quad \quad +\sup _{t\in [0,1]}|<(\mathcal {I}-\mathcal {P}_n)g_1(t,.), [(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)(x_0))](.)>|\nonumber \\&\quad \le \Vert (\mathcal {I}-\mathcal {P}_n)g_1(t,.)\Vert _{L^2}\Vert (\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)(x_0))\Vert _{L^2}\nonumber \\&\quad \quad +\Vert (\mathcal {I}-\mathcal {P}_n)g_1(t,.)\Vert _{L^2}\Vert (\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)(x_0))\Vert _{L^2}\nonumber \\&\quad \le ch^{r+1}\Vert (g_1(t,.))^{(r)}\Vert _{\infty }[\Vert ((\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)(x_0))^{(1)}\Vert _{\infty }]\nonumber \\&\quad \quad + ch^{2r}\Vert (g_1(t,.))^{(r)}\Vert _{\infty }\Vert ((\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _1)(x_0))^{(r)}\Vert _{\infty }]\nonumber \\&\quad \le c\Vert (g_1(t,.))^{(r)}\Vert _{\infty }[h^{2r+1}l_1(M_1+\Vert k_1\Vert _{1,\infty })+h^{3r}l_2\Vert k_2\Vert _{r,\infty }]\Vert x_0^{(r)}\Vert _{\infty }. \end{aligned}$$
(G.1)
Appendix H
For \(i=2\), taking \(g_2(t,s)=k_2(t,s)\psi _2^{(0,1)}(s,x_0(s))\) and using estimates (E.1), (E.2), we have
$$\begin{aligned}&\Vert (\mathcal {K}_2\psi _2)'(x_0)\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))]\Vert _{\infty } \nonumber \\&\quad =\sup _{t\in [0,1]}|(\mathcal {K}_2\psi _2)'(x_0)\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](t)|\nonumber \\&\quad =\sup _{t\in [0,1]}\left| \int _{0}^{1}k_2(t,s)\psi _2^{(0,1)}(s,x_0(s))\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](s)ds\right| \nonumber \\&\quad =\sup _{t\in [0,1]}\left| \int _{0}^{1}g_2(t,s)\sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](s)ds\right| \nonumber \\&\quad =\sup _{t\in [0,1]}|<g_2(t,.), \sum _{j=1}^{2}[(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_j\psi _j)(\mathcal {P}_nx_0)-(\mathcal {K}_j\psi _j)(x_0))](.)>|\nonumber \\&\quad =\sup _{t\in [0,1]}|<g_2(t,.), [(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)(x_0))](.)>|\nonumber \\&\quad \quad +\sup _{t\in [0,1]}|<g_2(t,.), [(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)(x_0))](.)>|\nonumber \\&\quad =\sup _{t\in [0,1]}|<(\mathcal {I}-\mathcal {P}_n)g_2(t,.), [(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)(x_0))](.)>|\nonumber \\&\quad \quad +\sup _{t\in [0,1]}|<(\mathcal {I}-\mathcal {P}_n)g_2(t,.), [(\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)(x_0))](.)>|\nonumber \\&\quad \le \Vert (\mathcal {I}-\mathcal {P}_n)g_2(t,.)\Vert _{L^2}\Vert (\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)(x_0))\Vert _{L^2}\nonumber \\&\quad \quad +\Vert (\mathcal {I}-\mathcal {P}_n)g_2(t,.)\Vert _{L^2}\Vert (\mathcal {I}-\mathcal {P}_n)((\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)(x_0))\Vert _{L^2}\nonumber \\&\quad \le ch^{r+1}\Vert (g_2(t,.))^{(r)}\Vert _{\infty }[\Vert ((\mathcal {K}_1\psi _1)(\mathcal {P}_nx_0)-(\mathcal {K}_1\psi _1)(x_0))^{(1)}\Vert _{\infty }]\nonumber \\&\quad \quad +ch^{2r}\Vert (g_2(t,.))^{(r)}\Vert _{\infty }\Vert ((\mathcal {K}_2\psi _2)(\mathcal {P}_nx_0)-(\mathcal {K}_2\psi _2)(x_0))^{(r)}\Vert _{\infty }]\nonumber \\&\quad \le c\Vert (g_2(t,.))^{(r)}\Vert _{\infty }[h^{2r+1}l_1(M_1+\Vert k_1\Vert _{1,\infty })+h^{3r}l_2\Vert k_2\Vert _{r,\infty }]\Vert x_0^{(r)}\Vert _{\infty }. \end{aligned}$$
(H.1)
