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\begin{center}
{\Large \textbf{\ Approximation by Kantorovich type $(p,q)$-Bernstein-Schurer Operators }}

\bigskip

\textbf{M. Mursaleen}, \textbf{Faisal Khan}

Department of\ Mathematics, Aligarh Muslim University, Aligarh--202002, India%
\\[0pt]

mursaleenm@gmail.com; faisalamu2011@gmail.com \\[0pt]

\bigskip

\bigskip

\textbf{Abstract}
\end{center}

\parindent=8mm {\footnotesize {In this paper, we introduce a Shurer type genaralization of
$(p,q)$-Bernstein-Kantorovich operators based on $(p,q)$-integers and we call it as
$(p,q)$-Bernstein-Schurer Kantorovich operators. We study approximation properties for
these operators based on Korovkin's type approximation theorem and also study some direct
theorems. Furthermore, we give comparisons and some illustrative graphics for the convergence
of operators to some function.}}

\bigskip

{\footnotesize \emph{Keywords and phrases}: $(p,q)$-Bernstein-Schurer Kantorovich
operators; $(p,q)$-Bernstein Kantorovich operators; $q$-Bernstein-Schurer Kantorovich operators; modulus of continuity; Positive
linear operator; Korovkin's type approximation theorem.}

{\footnotesize \emph{AMS Subject Classifications (2010)}: {41A10, 41A25,
41A36, 40A30}}

\section{Introduction and preliminaries}

\hspace{8mm}The applications of $q$-calculus emerged as a new area in the field of approximation
theory from last two decades. The development of $q$-calculus has led to the discovery of
various modifications of Bernstein polynomials involving $q$-integers. The aim of these
generalizations is to provide appropriate and powerful tools to application
areas such as numerical analysis, computer-aided geometric design and
solutions of differential equations.

\parindent8mmIn 1987, Lupa\c{s} \cite{lp} introduced the first $q$-analogue of Bernstein
operators \cite{brn} and investigated its approximating and shape-preserving properties.
Another $q$-generalization of the classical Bernstein polynomials is due to Phillips \cite{pl}.
Several generalizations of well-known positive linear operators based on $q$-integers were
introduced and their approximation properties have been studied by several authors. For instance,
the approximation properties of $q$-Bleimann, Butzer and Hahn operators \cite{ar1}; $q$-analogue
of Sz\'{a}sz-Kantorovich operators \cite{m3}; Approximation by Kantorovich type $q$-Bernstein
operators \cite{dal} and $q$-analogue of generalized Berstein-Shurer operators {\cite{ma1} were studied.

Mursaleen et al studied approximation properties of the $q$-analogue of generalized Berstein
Shurer operators  {\cite{ma1}.\\

 Recently, Mursaleen et al applied $(p,q)$-calculus in approximation theory and introduced $(p,q)$-analogue of Bernstein Operators \cite{mka1} and $(p,q)$-analogue of Bernstein-Stancu Operators \cite{mka2}, $(p,q)$-analogue of
Bernstein-Kantorovich Operators \cite{mka3} respectively. Recently these works got attention in computer aided geometric design (CAGD)\\

   As in computer aided geometric design , the basis of Bernstein polynomials plays a significant role in order to preserve the shape of the curves or surfaces. The classical Bezier curve \cite{bezier} and $q$-bezier \cite{ph1,hp} constructed with Bernstein basis functions are the most important curve in CAGD. Thus motivated by the work of Mursaleen et al in \cite{mka1}, very recently, Khalid et al in \cite{kl} introduced  Bezier curves and surfaces in Computer aided geometric design defined by $(p,q)$-integers.\\














Ozarslan and Vedi \cite{ov} introduced  Kantorovich type genelaization of the $q$-Bernstein-Schurer
operators and studied some approximation properties of these operators. These operators are given as follows:
\begin{equation}\label{e1.1}
K^q_{n,\ell}(f;x)=\sum\limits_{k=0}^{n+\ell}b^q_{n,\ell,k}(x)~\int_{0}^{1}f\left(\frac{[k]_{q}}{[n+1]_{q}}+
\frac{1+(q-1)[k]_{q}}{[n+1]_q}t\right)d_{q}t ,~~x\in \lbrack 0,1],
\end{equation}
\begin{equation*}
  b^q_{n,\ell,k}(x):= \left[
\begin{array}{c}
n+\ell \\
k
\end{array}
\right] _{q}x^{k}\prod\limits_{s=0}^{n+\ell-k-1}(1-q^{s}x).
\end{equation*}
Where $\ell\in\mathbb{N_{\circ}}=\mathbb{N}\cup\{0\},~~0<q<1$ is fixed and
$K_{n,\ell}^q:C[0,1+\ell]\rightarrow C[0,1+\ell]$ are defined for all $n\in\mathbb{N}$
and for any function $f\in C[0,1+\ell].$\\

\parindent=8mm Details on the $q$-calculus can be found in \cite{kac} and for the applications of
$q$-calculus in approximation theory, one can refer \cite{gp1}.\\


Mursaleen et al introduced $(p,q)$-Bernstein-kantorovich operators as follows in \cite{mka3}

\begin{equation*}
K_{n}^{(p,q)}(f;x)=[n+1]_{p,q}\sum\limits_{k=0}^{n}\frac{(p-q)}{p^k(p-1)-q^k(k-1)} b_{n,k}^{(p,q)}(x)\int_{\frac{[k]_{p,q}}{[n+1]_{p,q}}}^{\frac{[k+1]_{p,q}}{[n+1]_{p,q}}}f(t)d_{p,q}t ,~~x\in \lbrack 0,1]
\end{equation*}
where
\begin{equation*}
   b_{n,k}^{(p,q)}(x)= \left[
\begin{array}{c}
n \\
k%
\end{array}%
\right] _{p,q}x^k(1-x)_{p,q}^{n-k}= \left[
\begin{array}{c}
n \\
k%
\end{array}%
\right] _{p,q}x^{k}\prod\limits_{s=0}^{n-k-1}(p^s-q^{s}x).
\end{equation*}



Motivated by the work of  Mursaleen et al
for $(p,q)$-Bernstein-Kantorovich  operators based on
$(p,q)$-integers in  \cite{mka3}, now we present a Shurer type generalisation of $(p,q)$-Bernstein-Kantorovich  operators.\\


Let us recall certain notations of $(p,q)$-calculus:

The $(p,q)$-integer was introduced in order to generalize or unify several
forms of $q$-oscillator algebras well known in the earlier physics
literature related to the representation theory of single parameter quantum
algebras \cite{chak}. The $(p,q)$-integer $[n]_{p,q}$ is defined by

\begin{equation*}
[n]_{p,q}:=\frac{p^n-q^{n}}{p-q},~~~n=0,1,2,\cdots,~~0<q<p\leq1.
\end{equation*}

The $(p,q)$-Binomial expansion is
\begin{equation*}
(ax+by)_{p,q}^n :=\sum\limits_{k=0}^{n}\left[
\begin{array}{c}
n \\
k%
\end{array}%
\right] _{p,q}q^{\frac{k(k-1)}{2}}p^{\frac{(n-k)(n-k-1)}{2}}a^{n-k}b^kx^{n-k}y^k
\end{equation*}

\begin{equation*}
(x+y)_{p,q}^n:=(x+y)(px+qy)(p^2x+q^2y)\cdots(p^{n-1}x+q^{n-1}y).
\end{equation*}

Also, the $(p,q)$-binomial coefficients are defined by

\begin{equation*}
\left[
\begin{array}{c}
n\\
k%
\end{array}%
\right] _{p,q}:=\frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!}
\end{equation*}%
and the definite integrals of the function $f$ are defined by
\begin{equation*}
    \int_{0}^{a}f(x)d_{p,q}x=(q-p)a\sum\limits_{k=0}^{\infty}\frac{p^k}{q^{k+1}}
    f\left(\frac{p^k}{q^{k+1}}a\right),~~~\text{when}~~~\left|\frac pq\right|<1,
\end{equation*}
and
\begin{equation*}
    \int_{0}^{a}f(x)d_{p,q}x=(p-q)a\sum\limits_{k=0}^{\infty}\frac{q^k}{p^{k+1}}
    f\left(\frac{q^k}{p^{k+1}}a\right),~~~\text{when}~~~\left|\frac pq\right|>1.
\end{equation*}

Details on $(p,q)$-calculus can be found in \cite{mah,jacob,jag,vivek,sad}.
For $p=1$, all the notions of $(p,q)$-calculus are reduced to $q$-calculus.


\parindent=8mm Now, we introduce a new generalization of $q$-Bernstein-Kantorovich
operators by using the notion of $(p,q)$-calculus and call it as $(p,q)$-Bernstein-Schurer-Kantorovich  operators.
We study the approximation properties based on Korovkin's type approximation theorem and also establish
some direct theorems. Further, we show comparisons and some illustrative graphics for the convergence of
operators to a function.\\



\section{Construction of Operators}

Now, we introduce $(p,q)$-analogue of Kantorovich type Bernstein-Schurer operators as
\begin{equation}\label{e2.1}
K_{n,\ell}^{(p,q)}(f;x)=\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)~\int_{0}^{1}f\left(\frac{[k]_{p,q}}{[n+1]_{p,q}}+
\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t\right)d_{p,q}t ,~~x\in \lbrack 0,1],
\end{equation}
where
\begin{equation*}
   b_{n,\ell,k}^{(p,q)}(x)= \left[
\begin{array}{c}
n+\ell \\
k%
\end{array}
\right] _{p,q}x^k(1-x)_{p,q}^{n+\ell-k}= \left[
\begin{array}{c}
n+\ell \\
k%
\end{array}
\right] _{p,q}x^{k}\prod\limits_{s=0}^{n+\ell-k-1}(p^s-q^{s}x).
\end{equation*}

For $p=1$, equation (\ref{e2.1}) turns out to be the classical $q$-Bernstein-Kantorovich operators (\ref{e1.1}).\\
\\
%\newpage
Now, we have the following basic lemmas:
 \begin{lemma}\label{lem2.1} For $x\in \lbrack 0,1],~0<q<p\leq 1$
\begin{enumerate}
\item [(i)] $K_{n,\ell}^{(p,q)}(1;x)=~1$;
\item[(ii)]$K_{n,\ell}^{(p,q)}(t;x)=~\frac{(px+1-x)_{p,q}^{n+\ell}}{[2]_{p,q}[n+1]_{p,q}}
+\frac{(p+2q-1)[n+\ell]_{p,q}}{[2]_{p,q}[n+1]_{p,q}}x$;
 \item[(iii)] $K_{n,\ell}^{(p,q)}(t^2;x)=~\frac{(p^2x+1-x)_{p,q}^{n+\ell}}{[3]_{p,q}[n+1]_{p,q}^2}
+\left(1+\frac{2q}{[2]_{p,q}}+\frac{q^2-1}{[3]_{p,q}}\right)\frac{[n+\ell]_{p,q}}{[n+1]_{p,q}^2}(px+1-x)_{p,q}^{n+\ell-1}x\\
~~~~~~~~~~~~~~~~~~~~+\left(1+\frac{2(q-1)}{[2]_{p,q}}+\frac{(q-1)^2}{[3]_{p,q}}\right)\frac{[n+\ell]_{p,q}[n+\ell-1]_{p,q}}{[n+1]_{p,q}^2}x^2;$\\
 \item[(iv)]$K_{n,\ell}^{(p,q)}\big{(}(t-x);x\big{)}=\frac{(p^2x+1-x)_{p,q}^{n+\ell}}{[2]_{p,q}[n+1]_{p,q}}+\left(\frac{(p+2q-1)}{[2]_{p,q}[n+1]_{p,q}}-1\right)x;$
\item[(v)]$K_{n,\ell}^{(p,q)}\big{(}(t-x)^2;x\big{)}=\frac{(p^2x+1-x)_{p,q}^{n+\ell}}{[3]_{p,q}[n+1]_{p,q}^2}
~+~\left\{\left(1+\frac{2q}{[2]_{p,q}}+\frac{q^2-1}{[3]_{p,q}}\right)\frac{[n+\ell]_{p,q}(px+1-x)_{p,q}^{n+\ell-1}}{[n+1]_{p,q}^2}
-\frac{2(px+1-x)_{p,q}^{n+\ell}}{[2]_{p,q}[n+1]_{p,q}}\right\}x\\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\left\{q\left(1+\frac{2(q-1)}{[2]_{p,q}}+\frac{(q-1)^2}{[3]_{p,q}}\right)\frac{[n+\ell]_{p,q}[n+\ell-1]_{p,q}}{[n+1]_{p,q}^2}
-\frac{2(p+2q-1)[n+\ell]_{p,q}}{[2]_{p,q}[n+1]_{p,q}}+1\right\}x^2$.
\end{enumerate}
\end{lemma}

%\parindent=0mm\textbf{Proof}. (i)
%\begin{equation*}
%K_{n,\ell}^{(p,q)}(1;x)=\sum\limits_{k=0}^{n+\ell}b_{n,\ell,k}^{(p,q)}(x)\int_{0}^{1}d_{p,q}t=1.
%\end{equation*}
%(ii)
%\begin{eqnarray*}
%K_{n,\ell}^{(p,q)}(t;x)&=&\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)~\int_{0}^{1}\left(\frac{[k]_{p,q}}{[n+1]_{p,q}}+
%\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t\right)d_{p,q}t \\
%&=&\frac{1}{[2]_{p,q}[n+1]_{p,q}}\sum\limits_{k=0}^{n+\ell}b_{n,\ell,k}^{(p,q)}(x)\big{(}([2]_{p,q}-1)[k]_{p,q}+[k+1]_{p,q}\big{)}\\
%\end{eqnarray*}%
%Using $[k+1]_{p,q}={p^{k}+q[k]_{p,q}}$, we have
%\begin{eqnarray*}
%K_{n,\ell}^{(p,q)}(t;x)&=&\frac{1}{[2]_{p,q}[n+1]_{p,q}}\bigg{\{}\sum\limits_{k=0}^{n+\ell}\left[
%\begin{array}{c}
%n+\ell \\
%k%
%\end{array}%
%\right] _{p,q}(px)^k(1-x)_{p,q}^{n+\ell-k}\\
%&&+(q+[2]_{p,q}-1)\sum\limits_{k=0}^{n+\ell}\left[
%\begin{array}{c}
%n+\ell \\
%k%
%\end{array}%
%\right] _{p,q}x^k(1-x)_{p,q}^{n+\ell-k}[k]_{p,q}\bigg{\}}\\
%&=&\frac{1}{[2]_{p,q}[n+1]_{p,q}}\big{\{}(px+1-x)_{p,q}^{n+\ell}+(2q+p-1)[n+\ell]_{p,q}\sum\limits_{k=0}^{n-1}\left[
%\begin{array}{c}
%n+\ell \\
%k%
%\end{array}%
%\right] _{p,q}x^{k+1}(1-x)_{p,q}^{n+\ell-k-1}\big{\}}\\
%&=&\frac{(px+1-x)_{p,q}^{n+\ell}}{[2]_{p,q}[n+1]_{p,q}}+\frac{(2q+p-1)[n+\ell]_{p,q}x}{[2]_{p,q}[n+1]_{p,q}}.
%\end{eqnarray*}
%
%(iii)
%\begin{eqnarray*}
%K_{n,\ell}^{(p,q)}(t^2;x)&=&K_{n,\ell}^{(p,q)}(f;x)=\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)~\int_{0}^{1}\left(\frac{[k]_{p,q}}{[n+1]_{p,q}}+
%\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t\right)^2d_{p,q}t \\
%&=&\sum\limits_{k=0}^{n+\ell}b_{n,\ell,k}^{(p,q)}(x)\bigg{(}\frac{[k]_{p,q}^2}{[n+1]_{p,q}^2}
%+\frac{2[k]_{p,q}([k+1]_{p,q}-[k]_{p,q})}{[2]_{p,q}[n+1]_{p,q}^2}+\frac{2[k]_{p,q}([k+1]_{p,q}-[k]_{p,q})}{[2]_{p,q}[n+1]_{p,q}^2}\\
%&&+\frac{([k+1]_{p,q}-[k]_{p,q})^2}{[3]_{p,q}[n+1]_{p,q}^2}\bigg{)}\\
%&=&\sum\limits_{k=0}^{n+\ell}b_{n,\ell,k}^{(p,q)}(x)\bigg[\left(1+\frac{2(q-1)}{[2]_{p,q}}+\frac{(q-1)^2}{[3]_{p,q}}\right)\frac{[k]^2_{p,q}}{[n+1]^2_{p,q}}\\
%&&+2\left(\frac{1}{[2]_{p,q}}+\frac{(q-1)}{[3]_{p,q}}\right)\frac{[k]_{p,q}p^k}{[n+1]^2_{p,q}}+\frac{p^{2k}}{[3]_{p,q}[n+1]^2_{p,q}}\bigg]\\
%&=&\frac{(p^2x+1-x)^{n+\ell}_{p,q}}{[3]_{p,q}[n+1]_{p,q}}+\left(1+\frac{2q}{[2]_{p,q}}+\frac{(q^2-1)}{[3]_{p,q}}\right)
%\frac{[n+\ell]_{p,q}(px+1-x)^{n+\ell-1}_{p,q}}{[n+1]^2_{p,q}}x\\
%&&+q\left(1+\frac{2(q-1)}{[2]_{p,q}}+\frac{(q-1)^2}{[3]_{p,q}}\right)\frac{[n+\ell]_{p,q}[n+\ell-1]_{p,q}}{[n+1]^2}x^2.
%\end{eqnarray*}
%
%(iv) Using the linearity of the operators $K_{n,\ell}^{(p,q)},$ we have
%\begin{eqnarray*}
%K_{n,\ell}^{(p,q)}\big{(}(t-x)^2;x\big{)}&=& K_{n,\ell}^{(p,q)}(t^2;x)-2xK_{n,\ell}^{(p,q)}(t;x)+x^2K_{n,\ell}^{(p,q)}(1;x) \\
%   &=&\frac{(p^2x+1-x)_{p,q}^{n+\ell}}{[3]_{p,q}[n+1]_{p,q}^2}+\left(1+\frac{2q}{[2]_{p,q}}+\frac{q^2-1}{[3]_{p,q}}\right)
%   \frac{[n+\ell]_{p,q}}{[n+1]_{p,q}^2}(px+1-x)_{p,q}^{n+\ell-1}x\\
%   &&+\bigg(1+\frac{2(q-1)}{[2]_{p,q}}+\frac{(q-1)^2}{[3]_{p,q}}\bigg)\frac{[n+\ell]_{p,q}[n+\ell-1]_{p,q}}{[n+1]_{p,q}^2}x^2;\\
%   &&-2x\bigg\{\frac{(px+1-x)_{p,q}^{n+\ell}}{[2]_{p,q}[n+1]_{p,q}}
%+\frac{(p+2q-1)[n+\ell]_{p,q}}{[2]_{p,q}[n+1]_{p,q}}x\bigg\}+x^2\\
%&=&\frac{(p^2x+1-x)_{p,q}^{n+\ell}}{[3]_{p,q}[n+1]_{p,q}^2}+\bigg\{\left(1+\frac{2q}{[2]_{p,q}}+\frac{q^2-1}{[3]_{p,q}}\right)
%\frac{[n+\ell]_{p,q}(px+1-x)_{p,q}^{n+\ell-1}}{[n+1]_{p,q}^2}\\
%&&-\frac{2(px+1-x)_{p,q}^{n+\ell}}{[2]_{p,q}[n+1]_{p,q}}\bigg\}x+\bigg\{q\left(1+\frac{2(q-1)}{[2]_{p,q}}+\frac{(q-1)^2}{[3]_{p,q}}\right)\\
%&&\frac{[n+\ell]_{p,q}[n+\ell-1]_{p,q}}{[n+1]_{p,q}^2}-\frac{2(p+2q-1)[n+\ell]_{p,q}}{[2]_{p,q}[n+1]_{p,q}}+1\bigg\}x^2
%\end{eqnarray*}

\section{Approximation results}

\hspace{8mm}In this section, we find out the rate of convergence of the operators (\ref{e2.1})
using the modulus of continuity and Lipschitz classes. Furthermore, we clculate the rate convergence
in terms of the first modulus of continuity of the function.

\parindent=8mmLet $C[0,1+\ell]$ be the linear space of all real valued continuous functions $f$
on $[0,1+\ell]$ and let $T$ be a linear operator defined on $C[0,1+\ell].$
We say that $T$ is $positive$ if for every non-negative $f\in C[0,1+\ell],$ we
have $T(f,x)\geq 0$ for all $x\in $ $[0,1+\ell].$

\parindent=8mmThe classical Korovkin approximation theorem \cite{alt, pp,sr} states as follows:

\parindent=8mmLet $(T_{n})$ be a sequence of positive linear operators from
$C[0,1+\ell]$ into $C[a,b].$ Then $\lim_{n}\Vert T_{n}(f,x)-f(x)\Vert_{C[0,1+\ell]}=0$,
for all $f\in C[0,1+\ell]$ if and only if $\lim_{n}\Vert T_{n}(f_{i},x)-f_{i}(x)\Vert _{C[0,1+\ell]}=0$,
for $i=0,1,2$, where $f_{0}(x)=1,~f_{1}(x)=x$ and $f_{2}(x)=x^{2}.$

\begin{theorem}\label{theorem3.1} {\em Let $0<q_{n}<p_{n}\leq 1$ such that
$\lim\limits_{n\rightarrow \infty }p_{n}=1$ and $\lim\limits_{n\rightarrow
\infty }q_{n}=1$. Then for each $f\in [0,1+\ell],~K_{n,\ell}^{(p_{n},q_{n})}(f;x)$
converges uniformly to $f$ on $[0,1+\ell]$.}
\end{theorem}

\parindent=0mm\textbf{Proof}. By the Korovkin Theorem it is sufficient to
show that
\begin{equation*}
\lim\limits_{n\rightarrow \infty }\Vert
K_{n,\ell}^{(p_{n},q_{n})}(t^{m};x)-x^{m}\Vert _{C[0,1+\ell]}=0,~~~m=0,1,2.
\end{equation*}%
By Lemma \ref{lem2.1} (i), it is clear that
\begin{equation*}
\lim\limits_{n\rightarrow \infty }\Vert K_{n,\ell}^{(p_{n},q_{n})}(1;x)-1\Vert
_{C[0,1+\ell]}=0.
\end{equation*}%
Now, by Lemma \ref{lem2.1}  (ii)
\begin{equation*}
 |K_{n,\ell}^{(p_{n},q_{n})}(t;x)-x|\leq \frac{(p_nx+1-x)_{p_n,q_n}^{n+\ell}}{[2]_{p,q}[n+1]_{p_n,q_n}}
 +\left(\frac{(p_n+2q_n-1)}{[2]_{p_n,q_n}[n+1]_{p_n,q_n}}-1\right)x
\end{equation*}
Taking maximum of both sides of the above inequality, we get
\begin{equation*}
    \|K_{n,\ell}^{(p_{n},q_{n})}(t;x)-x\|_{[0,1+\ell]}\leq\frac{p_n^n}{[2]_{p_n,q_n}[n+1]_{p_n,q_n}}
    +\frac{(p_n+2q_n-1)[n+\ell]_{p_n,q_n}}{[2]_{p_n,q_n}[n+1]_{p_n,q_n}}-1
\end{equation*}
which yields
\begin{equation*}
\lim\limits_{n\rightarrow \infty }\Vert
K_{n,\ell}^{(p_{n},q_{n})}(t;x)-x\Vert _{C[0,1+\ell]}=0.
\end{equation*}%
Similarly we can show that
\begin{equation*}
\lim\limits_{n\rightarrow \infty }\Vert
K_{n,\ell}^{(p_{n},q_{n})}(t^2;x)-x^2\Vert _{C[0,1+\ell]}=0.
\end{equation*}%
Thus the proof is completed.\newline
\newline

Now we will compute the rate of convergence in terms of modulus of
continuity.\newline

\parindent=8mm Let $f\in [0,1+\ell]$. The modulus of continuity of $f$ denoted
by $\omega_{f}(\delta )$ gives the maximum oscillation of $f$ in any interval of
length not exceeding $\delta >0$ and it is given by the relation
\begin{equation*}
\omega_{f}(\delta )=\sup\limits_{|x-y|\leq \delta }|f(x)-f(y)|,~~x,y\in [0,1+\ell].
\end{equation*}%
It is known that $\lim\limits_{\delta \rightarrow 0^+}\omega_{f}(\delta )=0$ for $%
f\in C[0,1+\ell]$ and for any $\delta >0$ one has
\begin{equation}\label{e3.1}
|f(y)-f(x)|\leq \omega_{f}(\delta)\biggl{(}\frac{|y-x|}{\delta}+1%
\biggl{)}.
\end{equation}

\begin{theorem}\label{theorem3.2} {\em  If $f\in C[0,1]$, then
\begin{equation*}
\bigl{|}K_{n,\ell}^{(p,q)}(f;x)-f(x)\bigl{|}\leq 2\omega\left(f,\sqrt{\delta _{n,\ell}^{(p,q)}(x)}\right)
\end{equation*}%
takes place, where $\delta_{n,\ell}^{(p,q)}=K_{n,\ell}^{(p,q)}\big{(}(t-x)^2;x\big{)}$}
\end{theorem}

\parindent=0mm \textbf{Proof}. Since $K_{n,\ell}^{(p,q)}(1;x)=1$, we have

\begin{align*}
\big|K_{n,\ell}^{(p,q)}(f;x)-f(x)\big|&\leq K_{n,\ell}^{(p,q)}\big(|f(t)-f(x)|;x\big)\\
&=\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)~\int_{0}^{1}f\left|\left(\frac{[k]_{p,q}}{[n+1]_{p,q}}+
\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t\right)-f(x)\right|d_{p,q}t.
\end{align*}
In view of (\ref{e3.1}), we get
\begin{align*}
  \big|K_{n,\ell}^{(p,q)}(f;x)-f(x)\big|&\leq\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)~\int_{0}^{1}
\left(\frac{\left|\frac{[k]_{p,q}}{[n+1]_{p,q}}+\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t-x\right|}{\delta}+1\right)\omega(f,\delta)~d_{p,q}t\\
&=\omega(f,\delta)+\frac{\omega(f,\delta)}{\delta}\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)
\int_{0}^{1}\left|\frac{[k]_{p,q}}{[n+1]_{p,q}}+\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t-x\right|d_{p,q}t
\end{align*}
Now using Cauchy-Schwartz inequality, we get
\begin{align*}
\big|K_{n,\ell}^{(p,q)}(f;x)-f(x)\big|&\leq\omega(f,\delta)+\frac{\omega(f,\delta)}{\delta}\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)\\
&\hspace{3cm}\times\bigg\{\int_{0}^{1}\bigg(\frac{[k]_{p,q}}{[n+1]_{p,q}}+\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t-x\bigg)^2d_{p,q}t\bigg\}^{\frac{1}{2}}.
\end{align*}

Again applying the cauchy-Schwartz inequality, we have

\begin{equation*}
\big|K_{n,\ell}^{(p,q)}(f;x)-f(x)\big|\leq\omega(f,\delta)+\frac{\omega(f,\delta)}{\delta}\bigg\{\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)\hspace{5.5cm}
\end{equation*}
\begin{equation*}
\times\int_{0}^{1}\bigg(\frac{[k]_{p,q}}{[n+1]_{p,q}}+\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t-x\bigg)^2d_{p,q}t\bigg\}^{\frac{1}{2}}
\bigg\{\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)\bigg\}^{\frac{1}{2}}
\end{equation*}

\begin{align*}
\hspace{1.5cm}&=\omega(f,\delta)+\frac{\omega(f,\delta)}{\delta}\bigg\{\sum\limits_{k=0}^{n+\ell}b_{n,\ell,k}^{(p,q)}(x)
\int_{0}^{1}\bigg(\frac{[k]_{p,q}}{[n+1]_{p,q}}+\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t-x\bigg)^2d_{p,q}t\bigg\}^{\frac{1}{2}}
\end{align*}
\begin{equation*}
=\omega(f,\delta)+\frac{\omega(f,\delta)}{\delta}\bigg\{K_{n,\ell}^{(p,q)}\big{(}(t-x)^2;x\big{)}\bigg\}^{\frac{1}{2}}.\hspace{4cm}
\end{equation*}
Now on taking $\delta:=\delta_{n,\ell}^{(p,q)}(x)=K_{n,\ell}^{(p,q)}\big{(}(t-x)^2;x\big{)},$ we obtain
\begin{equation*}
  \big|K_{n,\ell}^{(p,q)}(f;x)-f(x)\big|\leq2\omega\left(f,\sqrt{K_{n,\ell}^{(p,q)}\big((t-x)^2;x\big)}\right)
\end{equation*}
 This completes the proof of the theorem.\\


\hspace{8mm}Now we give the rate of convergence of the operators $K_{n,\ell}^{(p,q)}$ in terms
of the elements of the usual Lipschitz class $\text{Lip}_M(\alpha)$.\\

\hspace{8mm}Let $f\in C[0,1+\ell]$, $M>0$ and $0<\alpha\leq1$.We recall that $f$ belongs to the class
$\text{Lip}_M(\alpha)$ if the inequality
\begin{equation*}
    |f(t)-f(x)|\leq M|t-x|^\alpha~~~(t,x\in[0,1+\ell])
\end{equation*}
is satisfied.

\begin{theorem}\label{theorem3.3} {\em Let $0<q<p\leq1$. Ten for each $f\in \text{Lip}_M(\alpha)$ we have
\begin{equation*}
    |K_{n,\ell}^{(p,q)}(f;x)-f(x)|\leq M\left(\delta_{n,\ell}^{(p,q)}(x)\right)^{\frac{\alpha}{2}}
\end{equation*}
where
\begin{equation*}
\delta_{n,\ell}^{(p,q)}(x)={K_{n,\ell}^{(p,q)}\big{(}(t-x)^2;x\big{)}}.
\end{equation*}}
\end{theorem}

\parindent=0mm\textbf{Proof}. By the monotonicity of the operators $K_{n,\ell}^{(p,q)}$, we can write
\begin{align*}
\big|K_{n,\ell}^{(p,q)}(f;x)-f(x)\big|&\leq K_{n,\ell}^{(p,q)}\big(|f(t)-f(x)|;x\big)\\
&=\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)~\int_{0}^{1}f\left|\left(\frac{[k]_{p,q}}{[n+1]_{p,q}}+
\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t\right)-f(x)\right|d_{p,q}t\\
 &\leq M\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)~\int_{0}^{1}\left|\left(\frac{[k]_{p,q}}{[n+1]_{p,q}}+
\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t\right)-x\right|^{\alpha}d_{p,q}t\\.
\end{align*}
Now applying the H\"{o}lder's inequality for the sum with $p=\frac2{\alpha}$ and $q=\frac2{2-\alpha}$ and taking into consideration Lemma 2.1(i) and Lemma 2.2(ii), we have
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
  |K_{n,\ell}^{(p,q)}(f;x)-f(x)| &\leq& M\sum\limits_{k=0}^{n+\ell}\Bigg{\{} b_{n,\ell,k}^{(p,q)}(x)\int_{0}^{1}\left(\left(\frac{[k]_{p,q}}{[n+1]_{p,q}}+
\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t\right)-x\right)^{2}d_{p,q}t \Bigg{\}}^{\frac\alpha2}\\
  &&\times\Bigg{\{} b_{n,\ell,k}^{(p,q)}(x)\int_{0}^{1} 1 d_{p,q}t \Bigg{\}}^{\frac{2-\alpha}2}\\
 &\leq& M\Bigg{\{}\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)\int_{0}^{1}\left(\left(\frac{[k]_{p,q}}{[n+1]_{p,q}}+
\frac{[k+1]_{p,q}-[k]_{p,q}}{[n+1]_q}t\right)-x\right)^{2}d_{p,q}t \Bigg{\}}^{\frac\alpha2}\\
  &&\times\Bigg{\{}\sum\limits_{k=0}^{n+\ell} b_{n,\ell,k}^{(p,q)}(x)\int_{0}^{1} 1 d_{p,q}t \Bigg{\}}^{\frac{2-\alpha}2}\\
  &=&M\bigl{\{}K_{n,\ell}^{(p,q)}\bigl{(}(t-x)^2;x\bigl{)}\bigl{\}}^\frac{\alpha}{2}.
\end{eqnarray*}
Choosing $\delta_{n,\ell}^{(p,q)}(x)={K_{n,\ell}^{(p,q)}\big{(}(t-x)^2;x\big{)}}$, we arrive at the desired result.
\\

\parindent=8mm Next we prove the local approximation property for the operators $K_{n,\ell}^{(p,q)}$.
The Peetre's $K$-functional is defined by \newline
\begin{equation*}
K_{2}(f,\delta )=\inf [\{\Vert f-g\Vert +\delta \Vert g^{\prime \prime}\Vert \}:g\in W^{2}],
\end{equation*}%
where%
\begin{equation*}
W^{2}=\{g\in C[0,1+\ell]:g^{\prime },g^{\prime \prime }\in C[0,1+\ell]\}.
\end{equation*}

By \cite{dl}, there exists a positive constant $C>0$ such that $%
K_{2}(f,\delta )\leq C\omega_{2}(f,\delta ^{\frac{1}{2}}),~~\delta >0;$ where the
second order modulus of continuity is given by
\begin{equation*}
\omega_{2}(f,\delta ^{\frac{1}{2}})=\sup\limits_{0<h\leq \delta ^{\frac{1}{2}%
}}\sup\limits_{x\in \lbrack 0,1+\ell]}\mid f(x+2h)-2f(x+h)+f(x)\mid .
\end{equation*}%
\newline

Also for $f\in [0,1+\ell]$ the usual modulus of continuity is given by
\begin{equation*}
\omega(f,\delta )=\sup\limits_{0<h\leq \delta ^{\frac{1}{2}}}\sup\limits_{x\in
\lbrack 0,1+\ell]}\mid f(x+h)-f(x)\mid .
\end{equation*}

\begin{theorem}\label{theorem3.4} {\em Let $f\in C[0,1+\ell]$ and $0<q<p\leq1$.
Then for all $n\in \mathbb{N}$, there exists an absolute constant $C>0$
such that
\begin{equation*}
\mid K_{n,\ell}^{(p,q)}(f;x)-f(x)\mid \leq C\omega_{2}\big(f,\sqrt{a_{n,\ell}^{(p,q)}(x)}\big)
+\omega\big(f,c_{n,\ell}^{(p,q)}\big),
\end{equation*}%
where%
\begin{equation*}
a_{n,\ell}^{(p,q)}(x)= K_{n,\ell}^{(p,q)}\left( (t-x)^{2};x\right) +\left(\alpha_{n,\ell}^{(p,q)}(x)-x\right)^2,~{c_{n,\ell}^{(p,q)}(x)=\left(\alpha_{n,\ell}^{(p,q)}(x)-x\right)}
\end{equation*}
and
\begin{equation*}
 \alpha_{n,\ell}^{(p,q)}(x)=\frac{(px+1-x)_{p,q}^{n+\ell}}{[2]_{p,q}[n+1]_{p,q}}
+\frac{(p+2q-1)[n+\ell]_{p,q}}{[2]_{p,q}[n+1]_{p,q}}x.
\end{equation*}}
\end{theorem}

\parindent=0mm \textbf{Proof}. Using the operators (\ref{e2.1}), we define the following operators
\begin{equation}\label{e3.2}
  \tilde{K}_{n,\ell}^{(p,q)}(f;x):=K_{n,\ell}^{(p,q)}(f;x)-f\bigg(\frac{(px+1-x)_{p,q}^{n+\ell}}{[2]_{p,q}[n+1]_{p,q}}
+\frac{(p+2q-1)[n+\ell]_{p,q}}{[2]_{p,q}[n+1]_{p,q}}x\bigg)+f(x).
\end{equation}

 Then, by the Lemma \ref{lem2.1} (ii) For $x\in \lbrack 0,1],~0<q<p\leq 1$
\begin{enumerate}
\item [] $\tilde{K}_{n,\ell}^{(p,q)}(1;x)=~1$;
\item[]$\tilde{K}_{n,\ell}^{(p,q)}(t-x;x)=0$.
\end{enumerate}
Then, for a given $g\in W^{2}.$ From Taylor's expansion, we get
\begin{equation*}
g(t)=g(x)+g^{\prime }(x)(t-x)+\int_{x}^{t}(t-u)~g^{\prime \prime
}(u)~du,~~~t\in \lbrack 0,A],~A>0,~~
\end{equation*}
and for the operators (\ref{e3.2}), we get

\begin{eqnarray*}
\tilde{K}_{n,\ell}^{(p,q)}(g;x)&=&g(x)+\tilde{K}_{n,\ell}^{(p,q)}\left( \int_{x}^{t}(t-u)~g^{\prime \prime
}(u)~du;x\right),\\
 \mid \tilde{K}_{n,\ell}^{(p,q)}(g;x)-g(x)\mid& =& \biggl{|}\tilde{K}_{n,\ell}^{(p,q)}\left(
\int_{x}^{t}(t-u)~g^{\prime \prime }(u)~du;x\right) \biggl{|}\\
{}&=&\left|{K}_{n,\ell}^{(p,q)}\left(\int_{x}^{t}(t-u)~g^{\prime \prime }(u)~du;x\right)-\int_{x}^{\alpha_{n,\ell}^{(p,q)}(x)}\left(\alpha_{n,\ell}^{(p,q)}(x)-u\right)g^{\prime\prime}(u)~du\right|\\
{}&\leq&\left|{K}_{n,\ell}^{(p,q)}\left(\int_{x}^{t}(t-u)~g^{\prime \prime }(u)~du;x\right)\right|+\left|\int_{x}^{\alpha_{n,\ell}^{(p,q)}(x)}\left(\alpha_{n,\ell}^{(p,q)}(x)-u\right)g^{\prime\prime}(u)~du\right|\\
&\leq& \Vert g^{\prime \prime }\Vert K_{n,\ell}^{(p,q)}\left( (t-x)^{2};x\right) +\|g^{\prime\prime}\|\left(\alpha_{n,\ell}^{(p,q)}(x)-x\right)^2\\
&=&\Vert g^{\prime \prime }\Vert\left( K_{n,\ell}^{(p,q)}\left( (t-x)^{2};x\right) +\left(\alpha_{n,\ell}^{(p,q)}(x)-x\right)^2\right)
= \Vert g^{\prime \prime}\Vert a_{n,\ell}^{(p,q)}(x).
\end{eqnarray*}

On the other hand, by the definition of $\tilde{K}_{n,\ell}^{(p,q)}(f;x),$ we have
\begin{equation*}
\mid \tilde{K}_{n,\ell}^{(p,q)}(f;x)\mid \leq 4\Vert f\Vert .
\end{equation*}
Now
\begin{eqnarray*}
\mid K_{n,\ell}^{(p,q)}(f;x)-f(x)\mid &\leq& \mid \tilde{K}_{n,\ell}^{(p,q)}\bigl{(}(f-g);x\bigl{)}
-(f-g)(x)\mid +\mid \tilde{K}_{n,\ell}^{(p,q)}(g;x)-g(x)\mid\\
  &&+\left|f\big(\alpha_{n,\ell}^{(p,q)}(x)\big)-f(x)\right|\\
&\leq& 4\Vert f-g\Vert +a_{n,\ell}^{(p,q)}(x)\Vert g^{\prime \prime }\Vert+\left|f\big(\alpha_{n,\ell}^{(p,q)}(x)\big)-f(x)\right|\\
&\leq&4(\Vert f-g\Vert +a_{n,\ell}^{(p,q)}(x)\Vert g^{\prime \prime} \Vert)+\omega\big(f,c_{n,\ell}^{(p,q)}\big).
\end{eqnarray*}

In view of the property of $K$-functional, it can be easily seen that

\begin{eqnarray*}
\mid K_{n,\ell}^{(p,q)}(f;x)-f(x)\mid &\leq& 4 K\big(f,\sqrt{a_{n,\ell}^{(p,q)}(x)}\big)
+\omega\big(f,c_{n,\ell}^{(p,q)}\big)\\
&\leq & C\omega_{2}\big(f,\sqrt{a_{n,\ell}^{(p,q)}(x)}\big)
+\omega\big(f,c_{n,\ell}^{(p,q)}\big).
\end{eqnarray*}

\parindent=8mm This completes the proof of the theorem.

%\begin{theorem}\label{theorem3.5} {\em Let $f\in C[0,1+\ell]$ and $0<q<p\leq1$. If $f(x)$ has continuous derivative $f^{\prime}(x)$ and $\omega{\big(f^{\prime},\delta\big)}$ is the modulus of continuity of $f^{\prime}(x)$ on $[0,1+\ell],$
%Then for all $n\in \mathbb{N}$, there exists an absolute constant $M>0$
%such that
%\begin{equation*}
%\mid K_{n,\ell}^{(p,q)}(f;x)-f(x)\mid \leq M c_{n,\ell}^{(p,q)}\omega_{2}\big(f,\sqrt{a_{n,\ell}^{(p,q)}(x)}\big)
%+\omega\big(f,\big),
%\end{equation*}}
%\end{theorem}

























\newpage

\section{Graphical analysis}

With the help of Matlab, we show comparisons and some illustrative graphics
\cite{ma1} for the convergence of $(p,q) $-Bernstein-Schurer-Kantorovich operators
$K_{n,\ell}^{(p,q)}$ to the function $f (x) = 1+cos(5x^2),$ for different values of
parameters $p,q,n$, convergence of the operators $K_{n,\ell}^{(p,q)}$ to the function
is shown in figure $1,2,3,~\text{and}~4.$

\begin{figure*}[htb!]
\begin{center}
\includegraphics[height=6cm, width=10cm]{bsk1.eps}
\end{center}
\caption{ Approximation by $(p,q)$-Bernstein-Schurer-Kantorovich operators to a function}
\end{figure*}


\begin{figure*}[htb!]
\begin{center}
\includegraphics[height=6cm, width=10cm]{bsk2.eps}
\end{center}
\caption{Approximation by $(p,q)$-Bernstein-Schurer-Kantorovich operators to a function}
\end{figure*}



\begin{figure*}[htb!]
\begin{center}
\includegraphics[height=6cm, width=10cm]{bsk5.eps}
\end{center}
\caption{Approximation by $(p,q) $-Bernstein-Schurer-Kantorovich operators to a function}
\end{figure*}


\begin{figure*}[htb!]
\begin{center}
\includegraphics[height=6cm, width=10cm]{bsk3.eps}
\end{center}
\caption{Approximation by $(p,q) $-Bernstein-Schurer-Kantorovich operators to a function}
\end{figure*}

\newpage

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\end{document}


1506.02492.extracted_bib
{ov M.A. Ozarslan, Tuba Vedi, $q$-Bernstein-Schurer-Kantorovich Operators, J. Inequal. Appl. , 256 (2013) 213-444.
1506.02492.extracted_bib
{gp1 A. Aral, V. Gupta and R.P. Agarwal, Applications of $q$-Calculus in Operator Theory, Springer Science$\tiny{+$Business Media New York, 2013.
1506.02492.extracted_bib
{alt F. Altomare, M. Campiti, Korovkin type approximation theory and its applications, de Gruyter Stud. Math. 17, Berlin, 1994.
1506.02492.extracted_bib
{ar1 A. Aral, O. Do\u{gru, Bleimann Butzer and Hahn operators based on $q$-integers, J. Inequal. Appl., (2007) 1-12. Art. ID 79410. %
1506.02492.extracted_bib
{ar2 A. Aral, V. Gupta, On $q$-analogue of Stancu-beta operators, %Appl. Math. Letters, 25 (2012) 67-71.
1506.02492.extracted_bib
{brn S.N. Bernstein, D\'{emostration du th\'{eor\`{eme de Weierstrass fond\'{ee sur le calcul de probabilit\'{es, Comm. Soc. Math. Kharkow (2), 13 (1912-1913) 1-2.
1506.02492.extracted_bib
{bezier P.E. Bezier, Numerical Control-Mathematics and applications, John Wiley and Sons, London, 1972.
1506.02492.extracted_bib
{chak R. Chakrabarti and R. Jagannathan, A $(p,q)$-oscillator realization of two parameter quantum algebras, J. Phys. A: Math. Gen., 24 (1991) 711-718.
1506.02492.extracted_bib
{dal O. Dalmanoglu, Approximation by Kantorovich type q-Bernstein operators, in Proceedings of the 12th WSEAS International Conference on Applied Mathematics, Cairo, Egypt (2007), pp. 113-117.
1506.02492.extracted_bib
{dl R.A. Devore, G.G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. %%
1506.02492.extracted_bib
{ers S. Ersan, O. Do\u{gru, Statistical approximation properties %%of $q$-Bleimann, Butzer and Hahn operators, Math. Comput. Modell., 49 (2009) %%1595-1606. % %
1506.02492.extracted_bib
{gp V. Gupta, C. Radu, Statistical approximation properties of $q$% %-Baskokov--Kantorovich operators, Cent. Eur. J. Math., 7 (4) (2009) 809-818.
1506.02492.extracted_bib
{mah M.N. Hounkonnou, J. D\'{esir\'{e, B. Kyemba, $\mathcal{R% (p,q)$-calculus: differentiation and integration, SUT Journal of Mathematics, Vol. 49, No. 2 (2013), 145-167.
1506.02492.extracted_bib
{kac V. Kac, P. Cheung, Quantum Calculus, Springer-Verlag New York, 2002.
1506.02492.extracted_bib
{kan L. V. Kantorovich, Sur certains d\'{eveloppements suivant les polyn\^{omes de la forme de S. Bernstein, I, II, C. R. Acad. URSS, (1930), 563-568, 595-600.
1506.02492.extracted_bib
{pp P. P. Korovkin, Linear operators and approximation theory, Hindustan Publishing Corporation, Delhi, 1960.
1506.02492.extracted_bib
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