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\title[Some Results on  PCNTPS over Banach algebras]
{ Some Results on the  Projective Cone Normed \\Tensor Product Spaces Over Banach Algebras}
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\author[Das, Goswami and Mishra]{Dipankar Das, Nilakshi Goswami  and Vishnu Narayan Mishra$^1$}
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\address{Dipankar Das\\Department of Mathematics, \\
Gauhati University,\\
  Guwahati-781014,\\
   Assam,  India\\
\email: dipankardasguw@yahoo.com}
%----------Author 2
\address{Nilakshi Goswami\\Department of Mathematics,\\
 Gauhati University,\\
  Guwahati-781014,\\
   Assam,  India\\
\email: nila$_-$g2003@yahoo.co.in}
%----------Author 3
\address {Vishnu Narayan Mishra\\
Department of Mathematics, \\
Indira Gandhi National Tribal University,\\
 Lalpur, Amarkantak, Anuppur 484 887, India\\[.1cm]
L. 1627 Awadh Puri Colony Beniganj,\\
 Phase -III, Opposite - Industrial Training Institute (I.T.I.), \\
 Ayodhya Main Road Faizabad 224 001, \\
 Uttar Pradesh, India\\
\email:  vishnunarayanmishra@gmail.com\\ vishnu$_-$narayanmishra@yahoo.co.in}

\maketitle

\begin{abstract}
For two real Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$, let $K_p$ be the projective cone in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$. Using this we define a cone norm  on the algebraic tensor product of two vector spaces over the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ and discuss some properties. We derive some fixed point theorems in this projective cone normed tensor product space over Banach algebra with a suitable example.  For two self mappings $S$ and $T$ on a cone Banach space over Banach algebra, the   stability of the iteration scheme $x_{2n+1}=Sx_{2n}$, $x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...$  converging to the common fixed point  of $S$ and $T$ is also discussed here.
\end{abstract}
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\keywords cone normed space, stability of fixed points, projective tensor product.


\tableofcontents



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\2000mathclass{37C25, 47L07,26A18, 47A80}





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\footnotetext[1]{Corresponding author.}

\section{ Introduction}
$\;\;\;$In 2007, Huang  and  Zhang~\cite{4} introduced cone metric spaces and gave application of  fixed point theory  in such spaces.  Since then, a number of researchers (~\cite{1}, \cite{30},  \cite{19},~\cite{4},  \cite{28}, \cite{liu},\cite{18}) developed the fixed point theory in cone metric spaces. In 2010, the concept of  cone normed spaces was initiated by Turkoglu et al.~\cite{7}.  In ~\cite{5}, Karapinar derived some fixed point theorems in cone Banach spaces.\\
\par However, recently some authors viz.  Amini-Harandi et al. \cite{ami}, Asadi et al.\cite{asadi}, Du\cite{du}, Ercan \cite{ercan}, Feng and Mao\cite{feng},  Khamsi \cite{kham} etc., have shown the equivalence of fixed point results between cone metric spaces and metric spaces, and also between cone $b$-metric spaces and $b$-metric spaces. So study of fixed point theorems in cone metric spaces is no more interesting in this sense. But in \cite{liu},    Liu and Xu introduced cone metric space over Banach algebra and initiated a new study by defining generalized Lipschitz mapping, where the contractive coefficient is a vector instead of usual real constant. They provided an example to explain the non equivalence of fixed point results between the vectorial versions and scalar versions. In present days, some researchers viz.,  Huang and Radenovi\'{c} (\cite{huang1}, \cite{huang2}),  Huang et al. \cite{huang3}, Xu and  Radenovi\'{c} \cite{xu} etc., observed the interest and need for research in the field of studying fixed point theorems. They developed many important results in the framework of cone metric spaces and cone $b$-metric spaces over  Banach algebra. \\[.2cm]
Let $\mathbb{A}_1$ be a real Banach algebra, $\Vert . \Vert$ be its norm and $e_1$ be its unit element. A nonempty
closed subset $P_1$ of $\mathbb{A}_1$ is called a cone if
\begin{enumerate}
\item[(i)] $P_1$ is closed, non empty and $\{0,e_1\}\subset P_1 $.
\item[(ii)] $\alpha P_1 + \beta P_1 \subset P_1$ for all non negative real numbers $\alpha, \beta$.
\item[(iii)] $P_1^2=P_1P_1\subset P_1$.
\item[(iv)] $ P_1\cap (-P_1) = \{0\}$.\\
\end{enumerate}
For a given a cone $P_1\subset \mathbb{A}_1$, a partial ordering $``\preceq"$ on $\mathbb{A}_1$ with respect to $P_1$ is defined by $x \preceq y$ if and only if $y-x \in P_1$ . $x \prec y$  will indicate  $x \preceq y$ and $x \neq y$, while $x \ll y$ will stand for $y-x \in int P_1$(interior of $P_1$). If $int P_1\neq \phi$, then $P_1$ is called a solid cone. Here   $``\prec"$ and $``\ll"$ are also partial orderings with respect to $P_1$.\\[.2cm]
\begin{Defn}\cite{liu} Let $X$ be a non empty set. Suppose the mapping $d:X\times X\to \mathbb{A}_1$ satisfies:
\begin{enumerate}
\item[(i)] $0\preceq d(x,y)$ for all $x,y\in X$,
\item[(ii)] $d(x,y)= 0$ if and only if $x=y$,
\item[(iii)] $d(x,y)= d(y,x)$ for all $x,y\in X$ and
\item[(iv)] $d(x,y)\preceq d(x,z)+d(z,y)$ for all $x,y,z\in X$.
\end{enumerate}
Then the pair $(X,d)$ is called a cone metric space over Banach algebra.\end{Defn}
\begin{Defn}\cite{7} Let $X$   be a vector space over $\mathbb{R}$ and $\Vert . \Vert_{P_1}:X\to \mathbb{A}_1$  be a mapping satisfying:
\begin{enumerate}
\item[(i)] $0\preceq \Vert x \Vert_{P_1}\;\forall x\in X$
\item[(ii)] $\Vert x \Vert_{P_1}=0\Leftrightarrow x=0\;\forall x\in X$
\item[(iii)] $\Vert kx \Vert_{P_1}=\vert k\vert\Vert x \Vert_{P_1}\;\forall\; k\in \mathbb{R},\;\forall x\in X$
\item[(iv)] $\Vert x+y \Vert_{P_1}\preceq \Vert x \Vert_{P_1}+\Vert y \Vert_{P_1}\;\forall\; x,y\in X$
\end{enumerate}
Then the pair $(X,\Vert . \Vert_{P_1})$ is called a cone normed space over the Banach algebra $\mathbb{A}_1$ and $\Vert . \Vert_{P_1}$ is called a cone norm.\\
Every cone normed spaces is a cone metric space over Banach algebra with $d(x,y)=\Vert x-y\Vert_{P_1}$.
\end{Defn}
\begin{Defn}~\cite{liu} A cone $P_1$ is called a normal cone if there is a number $K>0$ such that $\forall x,y\in \mathbb{A}_1$ $$0\preceq x \preceq y\Rightarrow \Vert x\Vert\leqslant K\Vert y\Vert.$$
\end{Defn}
\begin{Defn} \cite{4}
The cone $P_1$ is called regular if every increasing sequence in $\mathbb{A}_1$  which is bounded from above is convergent. That is, if $\{x_n\}$ is a sequence such that
$$x_1\preceq x_2\preceq ... \preceq  x_n \preceq ... \preceq y$$
for some $y \in \mathbb{A}_1$, then there is $x \in \mathbb{A}_1$ such that $\Vert x_n-x\Vert \to 0\; as\;n\to\infty$.
\end{Defn}
\begin{Exmp}~\cite{liu} Let $\mathbb{A}_1 =l^1=\{x=\{x_n\}_{n\geqslant 1}:\sum_{n=1}^{\infty}\vert x_n\vert<\infty \}$ with convolution as multiplication:
$$xy=\{x_n\}_{n\geqslant 1}\{y_n\}_{n\geqslant 1}=\{\sum_{i+j=n}x_iy_j\}_{n\geqslant 1}$$
Thus $\mathbb{A}_1$ is a Banach algebra with unit $e_1=\{1,0,0,...\}$.
Let $P_1 = \{x =\{x_n\}_{n\geqslant 1}\in \mathbb{A}_1 : x_n\geqslant 0\; \forall\;n \}$, which is a normal cone in $\mathbb{A}_1$.
Let $X = l^1$ with the metric $d : X \times X \to \mathbb{A}_1$ defined by
$$d(x, y) = d (\{x_n\}_{n\geqslant 1},\{y_n\}_{n\geqslant 1}) = \{\vert x_n -y_n \vert\}_{n\geqslant 1}$$
Then $(X, d)$ is a cone metric space over the Banach algebra $\mathbb{A}_1$.
\end{Exmp}
\begin{Defn}\cite{liu}  Let $(X, d)$ be a cone metric space over Banach algebra $\mathbb{A}_1$, $x \in X$, $\{x_n\}$ a sequence in $X$. Then
\begin{enumerate}
\item[(i)] $\{x_n\}$ converges to $x$ whenever for every $c\in \mathbb{A}_1$ with $0\ll c$ there is a natural number $N$ such that $d(x_n , x)\ll c$ for all $n \geqslant N$. We denote this by $\lim_{n\to\infty} x_n = x$ or $ x_n \to x (n\to\infty)$.
\item[(ii)]  $\{x_n\}$ is a Cauchy sequence whenever for each $0\ll c$ there is a natural number $N$ such that $d(x_n , x_m)\ll c$ for all $n,m \geqslant N$.
\end{enumerate}
\end{Defn}
\begin{Defn}\cite{xu} Let $P_1$ be a solid cone in a Banach algebra $\mathbb{A}_1$. A sequence $\{u_n\}\subset P_1$ is said to be a
$c$-sequence if for each $0\ll  c$ there exists a natural number ${N}$ such that $u_n\ll c$ for all $ n > {N}$.
\end{Defn}
\begin{lem}
\cite{rad} If $\mathbb{A}_1$ is a real Banach algebra with a solid cone $P_1$ and if $0\preceq u \ll c$ for each $0\ll c$, then $u=0$.\end{lem}
\begin{lem}\cite{rad}  If $\mathbb{A}_1$ is a real Banach algebra with a solid cone $P_1$ and if $a,b,c\in \mathbb{A}_1$ and  $a\preceq b \ll c$, then $a\preceq c$.\end{lem}
\begin{lem}\cite{rad}  If $\mathbb{A}_1$ is a real Banach algebra with a solid cone $P_1$ and if $\Vert x_n\Vert \to 0$ $(n\to\infty)$, then for any $0\ll c$, there exists $N \in \mathbb{N}$ such that, for any $n > N$, we have $x_n\ll c$.\end{lem}
\begin{lem}\cite{xu}  Let $(X, d)$ be a complete cone metric space over a Banach algebra $\mathbb{A}_1$ and
let $P_1$ be the underlying solid cone in  $\mathbb{A}_1$. Let $\{x_n\}$ be a sequence in $X$ and $0\ll c$. If $\{x_n\}$ converges to $x \in X$, then we have:\\[.2cm]
(i) $\{d(x_n , x)\}$ is a $c$-sequence.\\
(ii) For any $p \in \mathbb{N}$, $\{d(x_n , x_{n+p} )\}$ is a $c$-sequence.\end{lem}
\begin{lem} \cite{xu}  If $k\in P_1$ with spectral radius $r(k) < 1$, then $\Vert k^n\Vert\to 0\; as\;n\to\infty$.\end{lem}
\begin{lem}\cite{huang1} Let $\mathbb{A}_1$ be a Banach algebra with a unit $e_1$ and $P_1$ be a solid cone in $\mathbb{A}_1$. Let $h \in \mathbb{A}_1$ and $u_n = h^n$. If $r(h) < 1$, then $\{u_n\}$ is a $c$-sequence.\end{lem}
\begin{lem}\cite{huang1} Let $\mathbb{A}_1$ be a Banach algebra with a unit $e_1$ and $u \in \mathbb{A}_1$. If $r(u) < \vert C\vert$ and $C$ is a complex
constant, then
$$r(Ce_1 - u)^{-1}\leqslant \dfrac{1}{\vert C\vert-r(u)} .$$
\end{lem}
\begin{lem} \cite{xu} Let $P_1$ be a solid cone in a Banach algebra $\mathbb{A}_1$. Suppose that $k \in P_1$ and $\{u_n\}$ is a
$c$-sequence in $P$ . Then $\{ku_n\}$ is a $c$-sequence.\end{lem}
\begin{lem}\cite{rudin} Let $\mathbb{A}_1$ be a Banach algebra with a unit $e_1$, $k\in\mathbb{A}_1$, then $\lim_{n\to\infty}\Vert k^n\Vert^{\frac{1}{n}}$ exists and the spectral
radius $r(k)$ satisfies
$r(k) = \lim_{n\to\infty}\Vert k^n\Vert^{\frac{1}{n}}=\inf\Vert k^n\Vert^{\frac{1}{n}}$. If $r(k) < 1$, then $e_1 -k$ is invertible in $\mathbb{A}$, moreover, $$(e_1 - k)^{-1}=\sum_{i=0}^{\infty}k^i.$$
\end{lem}
\begin{lem}\cite{rudin} Let $\mathbb{A}_1$ be a Banach algebra with a unit $e_1$, $a,b\in\mathbb{A}_1$. If $a$ commutes with $b$, then
$r(a + b) \leqslant r(a) + r(b)$, $r(ab) \leqslant r(a)r(b)$.\end{lem}
\begin{lem}\cite{huang2} Let $\mathbb{A}_1$ be a Banach algebra with a unit $e_1$ and $P_1$ be a solid cone in $\mathbb{A}_1$. Let $u, \alpha,\beta \in P_1$ such that $\alpha\preceq \beta$  and $u\preceq \alpha u$. If $r(\beta) < 1$, then $u =0$.
\end{lem}
In 2014, Liu and Xu derived the following fixed point theorem with generalized Lipschitz condition:\\ [.1cm]
\begin{thm}~\cite{xu} Let $( X , d )$ be a  cone Banach space over a Banach algebra $\mathbb{A}_1$, and $P_1$ be the underlying  solid cone with  $k\in P_1$ and $r(k)<1$. Suppose the mapping $T : X \to X$ satisfies generalized Lipschitz condition:
$$d(T x, T y)\preceq kd(x, y),\;\forall\;x, y \in  X.$$
Then $T$ has a unique fixed point in $X$ and for any $x \in X$, iterative sequence $\{T^n x\}$ converges to the fixed point.
\end{thm}
 Let $(X,\Vert .\Vert_{P_1})$ and $(Y,\Vert .\Vert_{P_2})$ be two cone Banach spaces over  Banach algebra, where $P_1$ and $P_2$ are solid normal cones (with normal constant 1).  In this paper, we derive some fixed point theorems for a self mapping $T$ in the projective cone normed tensor product space over  Banach algebra.  We also discuss the stability of an iteration scheme converging to a common fixed point of two self mappings on  a cone Banach space over Banach algebra.
\section{Main Results: (a) Projective Cone Normed Tensor Product Space(PCNTPS) over Banach algebra}
\noindent First, we define a cone norm over Banach algebra for the algebraic tensor product of two vector spaces.
\begin{lem}\cite{3} Let $X, Y$ be normed spaces over $\mathbb{F}$ with dual spaces $X^*$ and $Y^*$ respectively. Given $x\in X, y\in Y$, Let $x\otimes y$ be the element of $BL(X^*,Y^*;\mathbb{F})$ (which is the set of all bounded bilinear forms from $X^*\times Y^*$ to $\mathbb{F}$), defined by
$$x\otimes y(f,g)=f(x)g(y),\;(f\in X^*,g\in Y^*)$$
The algebraic tensor product of $X$ and $Y$, $X\otimes Y$ is defined to be the linear span of
$\{x\otimes y:x\in X, y\in Y\}$ in $BL(X^*,Y^*;\mathbb{F}).$
\end{lem}
\begin{lem}~\cite{3} Given normed spaces $X$ and $Y$, the projective tensor norm $\gamma$ on $X\otimes Y$ is defined by
$$ \Vert u\Vert_\gamma = \inf \{\sum_i \Vert x_i\Vert\Vert y_i\Vert : u=\sum_i x_i\otimes y_i\}$$
where the infimum is taken over all (finite) representations of $u$.\\[.2cm]
The completion of $(X\otimes Y, \Vert .\Vert_{\gamma})$ is called projective tensor product of $X$ and $Y$ and it is denoted by $X\otimes_\gamma Y$.\end{lem}
\begin{lem}~\cite{6} Let $X$ and $Y$ be Banach spaces. Then $\gamma$ is a cross norm on $X\otimes Y$ and $\Vert x\otimes y\Vert_\gamma=\Vert x\Vert\Vert y\Vert$ for every $x\in X,y\in Y$.\end{lem}
\begin{lem}~\cite{3} $X\otimes_\gamma Y$ can be represented as a linear subspace of $BL(X^*,Y^*;\mathbb{F})$ consisting of all elements of the form $u=\sum_i x_i\otimes y_i$ where $\sum_i \Vert x_i\Vert\Vert y_i\Vert<\infty$. Moreover, $ \Vert u\Vert_\gamma = \inf \{\sum_i \Vert x_i\Vert\Vert y_i\Vert\}$ over all such representations of $u$.
\end{lem}
\begin{lem}~\cite{3} Let $X$ and $Y$ be normed algebras over $\mathbb{F}$. There exists a unique product on $X \otimes Y$ with respect to which $X \otimes Y$ is an algebra and $$(a \otimes b)(c \otimes d)=ac \otimes bd\qquad(a,c\in X,b,d\in Y).$$
\end{lem}
\begin{lem}~\cite{3} Let $X$ and $Y$ be normed algebras over $\mathbb{F}$. Then projective tensor norm on $X\otimes Y$ is an algebra norm. \\Clearly, we can conclude that if $X$ and $Y$ are Banach algebras over $\mathbb{F}$ then $X\otimes_\gamma Y$ becomes a Banach algebra.
\end{lem}
Let $\mathbb{A}_1$ and $\mathbb{A}_2$ be two real Banach algebras with the unit elements $e_1$ and $e_2$ respectively. $P_1$ and $P_2$ be two solid cones in $\mathbb{A}_1$ and $\mathbb{A}_2$ respectively. Let $K_p$ be the projective cone (\cite{peressini},\cite{peressini1}) in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ defined by
$$K_p=\{\sum_i x_i\otimes y_i\; : \; x_i\in P_1,\;y_i\in P_2\}$$
[Here,\begin{align*}
K_pK_p=\sum_i x_i\otimes y_i)(\sum_j c_j\otimes d_j)=\sum_i \sum_j  x_ic_j\otimes y_id_j
\end{align*}
Since $P_1P_1\subset P_1$, $P_2P_2\subset P_2$, $x_i,c_j\in P_1$ and $y_i,d_j\in P_2$ so by definition $x_ic_j\in P_1$ and $y_id_j\in P_2$. Thus, $K_pK_p\subset K_p$.]\\
\par For  vector spaces $X$ and $Y$ over $\mathbb{R}$, with the cone norms $\Vert . \Vert_{P_1}:X\to \mathbb{A}_1$ and $\Vert . \Vert_{P_2}:Y\to \mathbb{A}_2$, we define:\\[.2cm]
$\Vert . \Vert_{K_p}:X\otimes Y\to \mathbb{A}_1\otimes_\gamma \mathbb{A}_2\; by \;\Vert u \Vert_{K_p}=\sum_i\Vert x_i \Vert_{P_1}\otimes \Vert y_i \Vert_{P_2},\;u=\sum_i x_i\otimes y_i\in X\otimes  Y$
\begin{enumerate}
\item[(i)] $0\preceq \Vert x_i \Vert_{P_1}\Rightarrow \Vert x_i \Vert_{P_1}\in P_1$, $\;0\preceq \Vert y_i \Vert_{P_2}\Rightarrow \Vert y_i \Vert_{P_2}\in P_2,\;\forall i$\\[.1cm]
$\Rightarrow\sum_i\Vert x_i \Vert_{P_1}\otimes \Vert y_i \Vert_{P_2}\in K_p$ (by definition of projective cone). \\[.1cm]
So, $0\preceq\sum_i\Vert x_i \Vert_{P_1}\otimes \Vert y_i \Vert_{P_2}$ i.e., $0\preceq\Vert u \Vert_{K_p}$.
\item[(ii)]
\begin{align}
 \Vert u \Vert_{K_p}=0\Rightarrow &\sum_i\Vert x_i \Vert_{P_1}\otimes \Vert y_i \Vert_{P_2}=0\nonumber\\
\Rightarrow &\Vert x_1 \Vert_{P_1}\otimes \Vert y_1 \Vert_{P_2}+\Vert x_2 \Vert_{P_1}\otimes \Vert y_2 \Vert_{P_2}+...=0
\end{align}
Each of the terms in $(1)$ is an element of $K_p$. We call these as $a_1,a_2,a_3,...$ etc. So, each $0\preceq a_i \;\forall i$.\\
(If $a,b\in K_p$ such that $a+b=0$, then $a=-b$, i.e., $b,-b\in K_p\Rightarrow b=0$. So, $a=0$. Similarly for any $n$ number of terms this holds.)\\[.2cm]
Therefore, $a_i=0\;\forall i$ i.e., $\Vert x_i \Vert_{P_1}\otimes \Vert y_i \Vert_{P_2}=0\;\forall i$
\begin{align*}
\Rightarrow &\Vert x_i \Vert_{P_1}=0,\; \Vert y_i \Vert_{P_2}=0\;\forall i\\
\Rightarrow &x_i =0,\;y_i =0\;\forall i\\
\Rightarrow &\sum_i x_i\otimes y_i=0\Rightarrow u=0
\end{align*}
Conversely, let $u=\sum_i x_i\otimes y_i=0\Rightarrow (\sum_i x_i\otimes y_i)(f,g)=0\;\forall f\in X^*,\;g\in Y^*$\\
In particular, we take $f:X\to \mathbb{R^+}$, $g:Y\to \mathbb{R^+}\cup\{0\}$ such that $ker g=\{0\}$.
\begin{align*}
(\sum_i x_i\otimes y_i)(f,g)=0&\Rightarrow\sum_i f(x_i)g(y_i)=0\\
&\Rightarrow  g(y_i)=0\;\forall i\Rightarrow y_i=0\;\forall i\\
&\Rightarrow \Vert y_i\Vert_{P_2}=0\;\forall i\;(\text{by cone norm property})\\
&\Rightarrow \Vert x_i\Vert_{P_1}\otimes\Vert y_i\Vert_{P_2}=0\;\forall i\\
&\Rightarrow \sum_i\Vert x_i\Vert_{P_1}\otimes\Vert y_i\Vert_{P_2}=0\\
&\Rightarrow\Vert u\Vert_{K_p}=0
\end{align*}
\item[(iii)] $\Vert ku \Vert_{K_p}=\vert k\vert\Vert u \Vert_{K_p}\;\forall\; u\in X\otimes Y,\;k\in \mathbb{R}$\\
\item[(iv)] $\Vert u+v \Vert_{K_p}\preceq \Vert u \Vert_{K_p}+\Vert v \Vert_{K_p}\;\forall\; u,v\in X\otimes Y$\\
(follows by definition)
\end{enumerate}
Thus, $\Vert .\Vert_{K_p}$ is a cone norm on $X\otimes Y$. We call $(X\otimes Y,\Vert .\Vert_{K_p})$ as projective cone normed tensor product space(PCNTPS) over  Banach algebra.\\[.1cm]
\begin{lem} If $P_1$ and $P_2$ are normal cones, then $K_p$ is also normal.
\end{lem}
\begin{pf}
 Let $u,v\in \mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ be such that $0\preceq u\preceq v\Rightarrow v-u\in K_p$. Let
 $$v=\sum_{i=1}^n x_i\otimes y_i,\;u=\sum_{j=1}^m p_j\otimes q_j.$$
 For $n<m:$
 \begin{align*}
 v-u\in K_p\Rightarrow& \sum_{i=1}^n x_i\otimes y_i-\sum_{j=1}^m p_j\otimes q_j\in K_p\\
 \Rightarrow&\sum_{i=1}^n(x_i-p_i)\otimes y_i+\sum_{i=1}^n p_i\otimes (y_i-q_i)+\sum_{j=n+1}^m (-p_j)\otimes q_j\in K_p
 \end{align*}
So, by the form of elements of $K_p$, we get,\\[.1cm]
$x_i-p_i,p_i\in P_1$; $y_i-q_i,y_i\in P_2$ $\forall i=1,2,...,n$ and
\\[.2cm]
 $-p_j\in P_1$; $q_j\in P_2$ $\forall j=n+1,n+2,...,m$.\\
Again, $$v-u\in K_p\Rightarrow\sum_{i=1}^n x_i\otimes (y_i-q_i)+\sum_{i=1}^n (x_i-p_i)\otimes q_i+\sum_{j=n+1}^m p_j\otimes (-q_j)\in K_p$$
So, $x_i-p_i,x_i\in P_1$; $y_i-q_i,q_i\in P_2$ $\forall i=1,2,...,n$ and\\[.1cm]
 $p_j\in P_1$; $-q_j\in P_2$ $\forall j=n+1,n+2,...,m$.\\[.2cm]
Hence, $p_j=0,q_j=0$ $\forall j=n+1,n+2,...,m$.\\[.1cm]
Now, $p_i\preceq x_i $ and $q_i\preceq y_i$. Since $P_1$ and $P_2$ are normal cones, so, there exist constants $K_1,K_2\geqslant 1$ such that
$$ \Vert p_i\Vert\leqslant K_1\Vert x_i\Vert,\;\Vert q_i\Vert\leqslant K_2\Vert y_i\Vert\;\forall i.$$
Since, $\Vert u\Vert= \Vert \sum_{j=1}^m p_j\otimes q_j\Vert\leqslant \sum_{j=1}^m \Vert p_j\Vert\Vert q_j\Vert\leqslant K_1K_2\sum_{j=1}^m \Vert x_j\Vert\Vert y_j\Vert$, so,\\[.1cm]
 for the projective tensor norm in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$, we have
$\Vert u\Vert\leqslant K_1K_2\Vert v\Vert$. \\[.1cm]
For $n\geqslant m$, the condition is obviously satisfied. \\[.1cm]
Therefore, $K_p$ is a normal cone with the normal constant $K_1K_2(\geqslant 1)$.
\end{pf}
\begin{lem} If $P_1$ and $P_2$ are regular cones, then $ K_p$ is also regular.
\end{lem}
\begin{pf}
Let $\{u_n\}_{n\geqslant 1}$ be a sequence in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ such that $u_1\preceq u_2,...,\preceq y$ for some $y\in \mathbb{A}_1\otimes_\gamma \mathbb{A}_2$.\\[.2cm]
To show that $\{u_n\}_{n\geqslant 1}$  is convergent in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$:\\[.2cm]
Let $u_1=\sum_{i} p_{1_i}\otimes q_{1_i}$, $u_2=\sum_{i} p_{2_i}\otimes q_{2_i}$, ..., $u_n=\sum_{i} p_{n_i}\otimes q_{n_i}$, ... and $y=\sum_{i} a_i\otimes b_i\in \mathbb{A}_1\otimes_\gamma \mathbb{A}_2$.\\[.2cm]
Since $u_1\preceq u_2\preceq ... \preceq y$, as in above Lemma 2.7, we can show that \\
$p_{1_i}\preceq p_{2_i}\preceq ... \preceq a_i\;\forall i$ and $q_{1_i}\preceq q_{2_i}\preceq ... \preceq b_i\;\forall i$. \\[.2cm]
For each $i$, $\{p_{n_i}\}_{n\geqslant 1}$ is a sequence in $\mathbb{A}_1$ (increasing) which is bounded from above, and so also $\{q_{n_i}\}_{n\geqslant 1}$ in $\mathbb{A}_2$. Since $P_1$ and $P_2$ are regular, there exist $r_i\in \mathbb{A}_1$ and $s_i\in \mathbb{A}_2$ such that $\lim_{n\to\infty}\Vert p_{n_i}-r_i\Vert=0$ and $\lim_{n\to\infty}\Vert q_{n_i}-s_i\Vert=0$ for each $i$.\\[.2cm]
Now, $\sum_i r_i\otimes s_i=u(say)\in \mathbb{A}_1\otimes_\gamma \mathbb{A}_2$.
\begin{align*}
\Vert u_n-u\Vert&=\Vert \sum_i p_{n_i}\otimes q_{n_i}-\sum_i r_i\otimes s_i\Vert\\
&\leqslant \sum_i\Vert p_{n_i}-r_i\Vert\Vert q_{n_i}\Vert+\sum_i\Vert q_{n_i}-s_i\Vert\Vert r_i\Vert\\
&\to 0\;as\;n\to\infty
\end{align*}
Hence, $K_p$ is regular.
\end{pf}
\begin{lem}
 For normal cones $P_1$ and $P_2$, if $(X,\Vert .\Vert_{P_1})$ and $(Y,\Vert .\Vert_{P_2})$ are two cone Banach spaces over  Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$ respectively, then $(X\otimes Y,\Vert .\Vert_{K_p})$ is also a cone Banach space over  the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$.
\end{lem}
 \begin{pf}
 Let $\{u_n\}$ (where $u_n=\sum_ix_{n_i}\otimes y_{n_i}$) be a Cauchy sequence in $X\otimes Y$. Since $P_1$ and $P_2$ are normal cones, so, by Lemma 2.7, $K_p$  is also normal.\\[.1cm]
  Since $\{u_n\}$ is Cauchy, so, $\Vert\Vert u_n-u_m\Vert_{K_p}\Vert\to 0\;as\;m,n\to \infty$. Now,
  \begin{align*}
  \Vert\Vert u_n-u_m\Vert_{K_p}\Vert&=\Vert\Vert \sum_ix_{n_i}\otimes y_{n_i}-\sum_ix_{m_i}\otimes y_{m_i}\Vert_{K_p}\Vert\\
  &=\Vert\Vert \sum_i(x_{n_i}-x_{m_i})\otimes y_{n_i}+\sum_ix_{m_i}\otimes (y_{n_i}-y_{m_i})\Vert_{K_p}\Vert\\
    &\leqslant \sum_i\Vert\Vert x_{n_i}-x_{m_i}\Vert_{P_1}\Vert \Vert\Vert y_{n_i}\Vert_{P_2}\Vert+\sum_i \Vert\Vert x_{m_i} \Vert_{P_1}\Vert \Vert\Vert y_{n_i}-y_{m_i}\Vert_{P_2}\Vert\\
    &\to 0 \;as\; n,m\to\infty
  \end{align*}
 $\Rightarrow (\Vert\Vert x_{n_i}-x_{m_i}\Vert_{P_1}\Vert\to 0 \;as\; m,n\to\infty$ or $\Vert\Vert y_{n_i}\Vert_{P_2}\Vert\to 0\;as\; m,n\to\infty$) and \\
 ($\Vert\Vert x_{m_i}\Vert_{P_1}\Vert\to 0\;as\; m,n\to\infty$ or $\Vert\Vert y_{n_i}-y_{m_i}\Vert_{P_2}\Vert\to 0 \;as\; m,n\to\infty$) for each $i$.\\[.1cm]
 $\Rightarrow (\{x_{n_i}\}_n$ is a Cauchy sequence in $(X,\Vert .\Vert_{P_1})$ or $\{y_{n_i}\}_n\to 0$ in $(Y,\Vert .\Vert_{P_2})$) and \\
 ($\{y_{n_i}\}_n$ is a Cauchy sequence in $(Y,\Vert .\Vert_{P_2})$  or  $\{x_{n_i}\}_n\to 0$ in $(X,\Vert .\Vert_{P_1})$) for each $i$.\\[.1cm]
 $\Rightarrow \{x_{n_i}\}_n$and $\{y_{n_i}\}_n$ are convergent sequences in $(X,\Vert .\Vert_{P_1})$ and $(Y,\Vert .\Vert_{P_2})$ respectively for each $i$, since these are cone Banach spaces.\\[.2cm]
 Let  $x_{n_i}\to a_i\in X$ and $y_{n_i}\to b_i\in Y$ for each $i$. We take $u=\sum_ia_i\otimes b_i\in X\otimes Y$. It can be easily shown that $\Vert \Vert u_n-u\Vert_{K_p}\Vert\to 0\;as\;n\to\infty$, i.e., $\{u_n\}$ is a convergent sequence in $X\otimes Y$.\\[.1cm]
 Hence, $(X\otimes Y,\Vert .\Vert_{K_p})$ is a cone Banach space over the  Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$.
 \end{pf}
\begin{exmp}
We take $X$ as any normed space and $\mathbb{A}_1=(l^1,\Vert .\Vert)$, over $\mathbb{R}$.\\ Let $P_1=\{\{x_n\}_{n\geqslant 1}\in \mathbb{A}_1,x_n\geqslant 0\;\forall n\}$. Then $\Vert .\Vert_{P_1} :X\to \mathbb{A}_1$  defined by:\\
$\Vert x\Vert_{P_1}=\{\dfrac{\Vert x\Vert}{2^n}\}_{n\geqslant 1}$ is a cone norm on $\mathbb{R}$. Then clearly, $(X,\Vert .\Vert_{P_1})$ is a cone Banach space over $\mathbb{A}_1$.\\[.1cm]
 Next we take, $Y=\mathbb{R}$, $\mathbb{A}_2=(\mathbb{R},\Vert .\Vert)$, $P_2=\{y:y\geqslant 0\}$. Then $\Vert .\Vert_{P_2} :Y\to \mathbb{A}_2$  defined by:\\ $\Vert y\Vert_{P_2}=\vert y\vert$ is a cone norm on $\mathbb{R}$.
 Clearly, $(\mathbb{R},\Vert .\Vert_{P_2})$ is also a cone Banach space over  $\mathbb{A}_2$.\\[.1cm]
 Now, $\Vert .\Vert_{K_p} :X\otimes Y\to \mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ i.e., $\Vert .\Vert_{K_p} :X\otimes \mathbb{R}\to l^1\otimes_\gamma \mathbb{R}$ is defined by:\\
 $\Vert u\Vert_{K_p}=\Vert \sum_ix_i\otimes y_i\Vert_{K_p}= \sum_i\Vert x_i\Vert_{P_1}\otimes \Vert y_i\Vert_{P_2}=\sum_i\{\dfrac{\Vert x_i\Vert}{2^n}\}_{n\geqslant 1}\otimes \vert y_i\vert=\sum_i\{\dfrac{\Vert x_i\Vert\vert y_i\vert}{2^n}\}_{n\geqslant 1}$. (Since $l^1\otimes_\gamma \mathbb{R}=l^1(\mathbb{R})$ \cite{6})\\
 Thus $(X\otimes Y,\Vert .\Vert_{K_p})$ is a cone Banach space over   the Banach algebra $l^1\otimes_\gamma \mathbb{R}$.
 \end{exmp}
\begin{lem}
 For a cone normed space $(X,\Vert .\Vert_p)$, where $P$ is a normal cone with normal constant $K=1$, if
 $\Vert \Vert u\Vert_P\Vert<\Vert \Vert v\Vert_P\Vert$ then $\Vert u\Vert_P\prec \Vert v\Vert_P$, $u,v\in X$.
 \end{lem}
 \begin{pf}
 Given $\Vert \Vert u\Vert_P\Vert<\Vert \Vert v\Vert_P\Vert$. \\
 If possible, let $\Vert v\Vert_P\preceq \Vert u\Vert_P$. Since $P$ is normal, so, $$\Vert \Vert v\Vert_P\Vert\leqslant K\Vert \Vert u\Vert_P\Vert=\Vert \Vert u\Vert_P\Vert,\;\text{a contradiction.}$$
  Hence, $\Vert u\Vert_P\preceq \Vert v\Vert_P$ but $\Vert u\Vert_P\neq \Vert v\Vert_P$. So,  $\Vert u\Vert_P\prec \Vert v\Vert_P$.
 \end{pf}
 Now, we want to establish some fixed point theorems in projective cone normed tensor product space over  Banach algebra.\\
(In all the following results, we take $d(x,0)=\Vert x\Vert_{P}$, for the cone $P$.)
 \section{Main Results: (b) Some Fixed Point Theorems in PCNTPS}
\begin{thm}
 Let $(X,\Vert .\Vert_{P_1})$ and $(Y,\Vert .\Vert_{P_2})$ be two cone Banach spaces over  Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$ respectively, where $P_1$ and $P_2$ are solid normal cones (with normal constant 1) and
 $(X\otimes Y,\Vert .\Vert_{K_p})$ is the projective cone Banach space over Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$.
 Let $T_1:X \otimes Y \to X$ and $T_2:X \otimes Y \to Y$ be two mappings satisfying:
\begin{enumerate}
\item[(i)] $\Vert \Vert T_1u - T_1v\Vert_{P_1} \Vert  \leqslant \dfrac{1}{M_2}\Vert k\Vert u-v\Vert_{K_p} \Vert$,
\item[(ii)] $\Vert \Vert T_2u - T_2v\Vert_{P_2} \Vert  \leqslant \dfrac{1}{M_1}\Vert k\Vert u-v\Vert_{K_p} \Vert$,
\item[(iii)]$\Vert \Vert T_1u\Vert_{P_1} \Vert< M_1\;and\; \Vert \Vert T_2u\Vert_{P_2} \Vert<M_2,\; \forall u,v\in X \otimes Y.$
\end{enumerate}
 Then the mapping $ T: X \otimes Y \to X \otimes Y$ defined by $ Tu=T_1u\otimes T_2u$, $u\in X \otimes Y$ has a unique fixed point in $X \otimes Y$ if $k\in K_p$ with $r(k)<\frac{1}{2}$.
 \end{thm}
\begin{pf}
Let $u,v\in X \otimes Y$. We have,
\begin{align*}
\Vert Tu-Tv\Vert_{K_p} &= \Vert T_1u \otimes T_2u - T_1v \otimes T_2v \Vert_{K_p} \\
&=\Vert (T_1u - T_1v) \otimes T_2u+  T_1v\otimes (T_2u - T_2v)\Vert_{K_p}\\
&\preceq \Vert T_1u - T_1v\Vert_{P_1} \otimes \Vert T_2u\Vert_{P_2} + \Vert T_1v\Vert_{P_1} \otimes \Vert T_2u - T_2v\Vert_{P_2}
\end{align*}
Since $K_p$ is normal with normal constant $1$, so,
\begin{align}
\Vert\Vert  Tu-Tv\Vert_{K_p}\Vert  &\leqslant \Vert \Vert T_1u - T_1v\Vert_{P_1}\Vert \Vert \Vert T_2u\Vert_{P_2}\Vert  +\Vert\Vert T_1v\Vert_{P_1}\Vert \Vert \Vert T_2u - T_2v\Vert_{P_2}\Vert\nonumber\\
&\qquad\qquad(\text{taking projective norm in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$}).\nonumber\\
&< \Vert k \Vert u-v\Vert_{K_p} \Vert +\Vert k\Vert u-v\Vert_{K_p} \Vert\nonumber\\
&=2 \Vert k\Vert u-v\Vert_{K_p} \Vert\nonumber\\
\Rightarrow\Vert\Vert  Tu-Tv\Vert_{K_p}\Vert  &< \Vert 2k\Vert u-v\Vert_{K_p}\Vert
\end{align}
So by Lemma 2.10, equation (2) implies
$$\Vert Tu-Tv\Vert_{K_p} \prec 2k\Vert u-v\Vert_{K_p}$$
Now, by  Theorem 1.1 the mapping $T$  has a unique fixed point in $X\otimes Y$.
\end{pf}
\begin{exmp} We take $X=l^1$ and $\mathbb{A}_1=(l^1,\Vert .\Vert)$ over $\mathbb{R}$.\\ Let $P_1=\{\{x_n\}_{n\geqslant 1}\in l^1,x_n\geqslant 0\;\forall n\}$. Then $\Vert .\Vert_{P_1} :l^1\to \mathbb{A}_1$ defined by:\\
  $\Vert a_i\Vert_{P_1}=\{\vert  a_{i_k}\vert\}_{k}$ ($a_i=\{a_{i_k}\}_k$) is a cone norm.  Clearly, $(X,\Vert .\Vert_{P_1})$ is also a cone Banach space over  the unital Banach algebra $\mathbb{A}_1$.\\[.2cm]
 Next we take, $Y=\mathbb{R}$, $\mathbb{A}_2=(\mathbb{R},\Vert .\Vert)$, $P_2=\{y:y\geqslant 0\}$. Then $\Vert .\Vert_{P_2} :Y\to \mathbb{A}_2$ defined by:\\ $\Vert y\Vert_{P_2}=\vert y\vert$ is a cone norm on $\mathbb{R}$.
 Clearly, $(\mathbb{R},\Vert .\Vert_{P_2})$ is also a cone Banach space over  the unital Banach algebra $\mathbb{A}_2$.\\[.2cm]
 Now, $\Vert .\Vert_{K_p} :X\otimes Y\to \mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ i.e., $\Vert .\Vert_{K_p} :l^1\otimes \mathbb{R}\to l^1\otimes_\gamma \mathbb{R}$ is defined by:
 $$\Vert u\Vert_{K_p}=\Vert \sum_ia_{i}\otimes y_i\Vert_{K_p}= \sum_i\Vert \{a_{i_k}\}_k\Vert_{P_1}\otimes \Vert y_i\Vert_{P_2}=\sum_i\{\vert a_{i_k}\vert\}_{k}\otimes \vert y_i\vert=\sum_i\{\vert a_{i_k}\vert\vert y_i\vert\}_{k}$$
( Since, $l^1\otimes_\gamma \mathbb{R}=l^1(\mathbb{R})$ \cite{6})\\
 Thus $(l^1\otimes \mathbb{R},\Vert .\Vert_{K_p})$ is a cone Banach space over  $l^1(\mathbb{R})$ with unit $e=\{1,0,0,...\}$.\\[.2cm]
  Let $D_{l^1}$, $D_{ \mathbb{R}}$ and $D_{l^1\otimes \mathbb{R}}$ (containing $D_{l^1}\otimes D_\mathbb{R}$) be  subsets of $l^1$, $ \mathbb{R}$ and $l^1\otimes \mathbb{R}$ bounded (strictly) by constants $c$, $c$ and $c^2$ respectively.
  \\We define $T_1:D_{l^1\otimes \mathbb{R}}\to D_{l^1}$ by
$T_1(\sum_i a_i \otimes y_i)=\dfrac{1}{2c^2}\sum_i\{a_{i_k}y_i\}_{k}$, where $a_i=\{a_{i_k}\}_k$ and
\\$T_2:D_{l^1\otimes \mathbb{R}}\to D_{\mathbb{R}}$ by
$T_2(\sum_i a_i \otimes y_i)=\dfrac{1}{3}\sum_i\Vert \{\vert a_{i_k}\vert\}_k\Vert\vert y_i\vert$. \\
Then
\begin{align*}
\Vert\Vert T_1(\sum_i a_i \otimes y_i)\Vert_{P_1}\Vert&=\Vert \Vert \dfrac{1}{2c^2}\sum_i\{a_{i_k}y_i\}_{k}\Vert_{P_1}\Vert
\leqslant \dfrac{1}{2c^2} \Vert \sum_i\{\vert a_{i_k}y_i\vert \}_{k}\Vert\\
&\leqslant \dfrac{1}{2c^2} \sum_i(\sum_k \vert a_{i_k}\vert )\vert y_i\vert\;(\text{using norm in $l^1$})\\
&= \dfrac{1}{2c^2} \sum_i \Vert a_i\Vert\vert y_i\vert
\end{align*}
Taking projective tensor norm in $l^1(\mathbb{R})$,
\begin{align*}
\Vert\Vert T_1(\sum_i a_i \otimes y_i)\Vert_{P_1}\Vert&\leqslant\dfrac{1}{2c^2} \Vert \sum_ia_i\otimes y_i\Vert\\
&< \dfrac{1}{2c^2}.c^2=\dfrac{1}{2}(=M_1)
\end{align*}
and
\begin{align*}
\Vert\Vert T_2(\sum_i a_i \otimes y_i)\Vert_{P_2}\Vert&=\Vert\Vert \dfrac{1}{3}\sum_i\Vert \{\vert a_{i_k}\vert\}_k\Vert\vert y_i\vert\Vert_{P_2}\Vert \\
&\leqslant\Vert \dfrac{1}{3}\sum_i\Vert \{\vert a_{i_k}\vert\}_k\Vert\vert y_i\vert\Vert\leqslant \dfrac{1}{3}\sum_i(\sum_k\vert a_{i_k}\vert)\vert y_i\vert\\
&= \dfrac{1}{3}\sum_i\Vert a_i\Vert\vert y_i\vert
\end{align*}
So, for projective tensor norm in $l^1(\mathbb{R})$,
\begin{align*}
\Vert\Vert T_2(\sum_i a_i \otimes y_i)\Vert_{P_2}\Vert\leqslant \dfrac{1}{3}\Vert \sum_i a_i\otimes y_i\Vert< \dfrac{c^2}{3}(=M_2)
\end{align*}
For $u=\sum_i a_i\otimes y_i,\;v=\sum_i b_i\otimes x_i \in D_{l^1\otimes \mathbb{R}}$, we have,
\begin{align*}
\Vert T_1(u)-T_1(v) \Vert_{P_1}&=\Vert \dfrac{1}{2c^2}\sum_i\{a_{i_k}y_i\}_{k}-\dfrac{1}{2c^2}\sum_i\{b_{i_k}x_i\}_{k} \Vert_{P_1}\\
&=\dfrac{1}{2c^2}\Vert\sum_i\{a_{i_k}y_i-b_{i_k}x_i\}_{k}\Vert_{P_1}\\
&=\dfrac{1}{2c^2}\sum_i\{\vert a_{i_k}y_i-b_{i_k} x_i\vert\}_k
\end{align*}
\begin{align}
\Rightarrow \Vert \Vert T_1(u)-T_1(v) \Vert_{P_1}\Vert&=\Vert \dfrac{1}{2c^2}\sum_i\{\vert a_{i_k}y_i-b_{i_k} x_i\vert\}_k\Vert\nonumber\\
&\leqslant \dfrac{1}{2c^2}\sum_i\sum_k\vert a_{i_k}y_i-b_{i_k} x_i\vert\nonumber\\
&\leqslant  \dfrac{1}{2c^2}\sum_{i,k}(\vert a_{i_k}y_i\vert+\vert b_{i_k} x_i\vert)\nonumber\\
&=\dfrac{1}{2c^2}\sum_{i,k}(\vert a_{i_k}\vert \vert y_i\vert+\vert b_{i_k}\vert \vert x_i\vert)
\end{align}
\begin{align}
\Vert\Vert u-v\Vert_{K_p}\Vert&=\Vert\Vert \sum_i a_i\otimes y_i-\sum b_i\otimes x_i\Vert_{K_p}\Vert\nonumber\\
&=\Vert\Vert \sum_i a_i\otimes y_i+\sum (-b_i)\otimes x_i\Vert_{K_p}\Vert\nonumber\\
&=\Vert\sum_i\{\vert a_{i_k}\vert\vert y_i\vert\}_k+\sum_i\{\vert b_{i_k}\vert\vert x_i\vert\}_k\Vert\;\text{(by definition of $\Vert .\Vert_{K_p}$)}\nonumber\\
&=\sum_i\sum_k(\vert a_{i_k}\vert\vert y_i\vert+\vert b_{i_k}\vert\vert x_i\vert)
\end{align}
From (3) and (4),
\begin{align*}
\Rightarrow \Vert\Vert T_1(u)-T_1(v) \Vert_{P_1}\Vert  &\leqslant\dfrac{1}{2c^2}\Vert \Vert u-v\Vert_{K_p}\Vert\\
 &\leqslant \dfrac{1}{c^2}\Vert e\Vert u-v\Vert_{K_p} \Vert =\dfrac{1/3}{c^2/3}\Vert e\Vert u-v\Vert_{K_p} \Vert\\
 &=\dfrac{1}{M_2}\Vert k \Vert u-v\Vert_{K_p} \Vert ,\;k=\dfrac{1}{3}e\in K_p
 \end{align*}
 Similarly,
 \begin{align*}
\Vert T_2(u)-T_2(v) \Vert_{P_2}&=\Vert \dfrac{1}{3}\sum_i\Vert \{\vert a_{i_k}\vert\}_k\Vert \vert y_i\vert-\dfrac{1}{3}\sum_i\Vert \{\vert b_{i_k}\vert\}_k\Vert \vert x_i\vert \Vert_{P_2}\\
&=\dfrac{1}{3}\vert\sum_i(\Vert \{\vert a_{i_k}\vert\}_k\Vert \vert y_i\vert-\Vert \{\vert b_{i_k}\vert\}_k\Vert \vert x_i\vert)\vert\\
&\preceq\dfrac{1}{3}\sum_i\Vert \{\vert a_{i_k}\vert\}_k\Vert \vert y_i\vert+\sum_i\Vert \{\vert b_{i_k}\vert\}_k\Vert \vert x_i\vert\\
\Rightarrow  \Vert \Vert T_2(u)-T_2(v) \Vert_{P_2}\Vert&\leqslant\dfrac{1}{3}\sum_i\sum_k\vert a_{i_k}\vert \vert y_i\vert+\sum_i\sum_k\vert b_{i_k}\vert \vert x_i\vert\\
&\qquad (\text{$P_2$ being a normal cone with normal constant 1})\\
&=\dfrac{1}{3}\Vert \Vert u-v\Vert_{K_p}\Vert\;(\text{from (4)})\\
& \leqslant \dfrac{1/3}{1/2}\Vert e\Vert u-v \Vert_{K_p}\Vert=\dfrac{1}{M_1}\Vert k\Vert u-v \Vert_{K_p}\Vert ,\;k=\dfrac{1}{3}e\in K_p.\\
  \text{Also}\;\; & r(k)=\lim_{n\to\infty}\Vert \left( \frac{1}{3}e\right)^n \Vert^{\frac{1}{n}}= \dfrac{1}{3}<\dfrac{1}{2}.
\end{align*}
So by Theorem 3.1, the mapping $T:D_{l^1\otimes  \mathbb{R}} \to D_{l^1\otimes  \mathbb{R}}$ defined by $$T(\sum_i a_i \otimes x_i)=\dfrac{1}{6c}\sum_i\{M{a_{i_n}x_i}\}_n,\;where\;M=\sum_i\Vert a_i \Vert . \vert x_i\vert $$
has a unique fixed point in $D_{l^1\otimes  \mathbb{R}}$.
\end{exmp}
\begin{thm}
 Let $(X,\Vert .\Vert_{P_1})$ and $(Y,\Vert .\Vert_{P_2})$ be two cone Banach spaces over Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$, where $P_1$ and $P_2$ are normal cones (with normal constant 1) and  $(X\otimes Y,\Vert .\Vert_{K_p})$ is the projective cone Banach space over $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$.
 Let $T_1:X \otimes Y \to X$ and $T_2:X \otimes Y \to Y$ be  two mappings satisfying:
\begin{align*}
(i)\;\Vert \Vert T_1u - T_1v\Vert_{P_1} \Vert & \leqslant \dfrac{1}{M_2}[\Vert k (\Vert u-Tu\Vert_{K_p}+\Vert v-Tv\Vert_{K_p}) \Vert ] \\
(ii)\;\Vert \Vert T_2u - T_2v\Vert_{P_2} \Vert & \leqslant \dfrac{1}{M_1}[\Vert k(\Vert u-Tu\Vert_{K_p} + \Vert v-Tv\Vert_{K_p}) \Vert ]\\
(iii)\;\Vert \Vert T_1u\Vert_{P_1} \Vert< M_1&\;and\; \Vert \Vert T_2u\Vert_{P_2} \Vert< M_2\;\forall u,v\in X \otimes Y.
\end{align*}
 Then the mapping $ T: X \otimes Y \to X \otimes Y$ defined by $ Tu=T_1u\otimes T_2u$, $u\in X \otimes Y$ has a unique fixed point if $k\in K_p$ with $r(k)<\frac{1}{4}$.
 \end{thm}
\begin{thm}
Let $P_1$ and $P_2$ be two solid normal cones (with normal constant 1)  in Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$ respectively. Let $T_1:X \otimes Y \to X$ and $T_2:X \otimes Y \to Y$ be two mappings satisfying:
\begin{align*}
&(i)\;\Vert T_1u - T_1v\Vert_{P_1}\otimes e_2  \preceq \dfrac{k}{M_2}\Vert u-v\Vert_{K_p},\\
&(ii)\;e_1\otimes \Vert T_2u - T_2v\Vert_{P_2}  \preceq \dfrac{k}{M_1}\Vert u-v\Vert_{K_p}\;\forall u,v\in X \otimes Y,\\
&(iii)\;e_1\in P_1,\;e_2\in P_2,\\
&(iv)\;\Vert \Vert T_1u\Vert_{P_1} \Vert< M_1\;and\; \Vert \Vert T_2u\Vert_{P_2} \Vert< M_2\;\forall u,v\in X \otimes Y.
\end{align*}
 where $k\in K_p$ and $r(k)<\frac{1}{2}$. Then $T:X\otimes Y\to X\otimes Y$ defined by $Tu=T_1u\otimes T_2u$, $u\in X \otimes Y$ has a unique fixed point in $X \otimes Y$.
 \end{thm}
\begin{pf}
Let $u,v\in X \otimes Y$. We have,
\begin{align*}
 \Vert  Tu-Tv\Vert_{K_p}  &\preceq  \Vert T_1u - T_1v\Vert_{P_1}\otimes \Vert T_2u\Vert_{P_2}+\Vert T_1v\Vert_{P_1}\otimes \Vert T_2u - T_2v\Vert_{P_2}\\
 &=(\Vert T_1u - T_1v\Vert_{P_1}\otimes e_2)(e_1\otimes \Vert T_2u\Vert_{P_2})\\
 &\quad+(\Vert T_1v\Vert_{P_1}\otimes e_2)(e_1\otimes \Vert T_2u - T_2v\Vert_{P_2})\\
&\preceq \dfrac{k}{M_2}\Vert u-v\Vert_{K_p}(e_1\otimes \Vert T_2u\Vert_{P_2})+(\Vert T_1v\Vert_{P_1}\otimes e_2)\dfrac{k}{M_1}\Vert u-v\Vert_{K_p}
\end{align*}
Since $K_p$ is normal, so, taking projective norm in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$,
\begin{align*}
 \Vert\Vert  Tu-Tv\Vert_{K_p}\Vert & \leqslant \dfrac{1}{M_2}\Vert k\Vert u-v\Vert_{K_p}\Vert\Vert e_1\otimes \Vert T_2u\Vert_{P_2}\Vert+\Vert\Vert T_1v\Vert_{P_1}\otimes e_2\Vert\dfrac{1}{M_1}\Vert k\Vert u-v\Vert_{K_p}\Vert\\
 &= \dfrac{1}{M_2}\Vert k\Vert u-v\Vert_{K_p}\Vert\Vert   \Vert T_2u\Vert_{P_2}\Vert+\Vert\Vert T_1v\Vert_{P_1}\Vert\dfrac{1}{M_1}\Vert k\Vert u-v\Vert_{K_p}\Vert\\
 &< \Vert 2k\Vert (u-v)\Vert_{K_p}\Vert
 \end{align*}
Now, proceeding as in Theorem 3.1, we can show that $T$ has a unique fixed point in $X\otimes Y$.
\end{pf}
\begin{thm}
In the above theorem, if the conditions $(i)$ and $(ii)$ are replaced by
\begin{align*}
&(i)\;\Vert T_1u - T_1v\Vert_{P_1}\otimes e_2  \preceq \dfrac{k}{M_2}[\Vert u-Tv\Vert_{K_p}+ \Vert v-Tu\Vert_{K_p}],\\
&(ii)\;e_1\otimes \Vert T_2u - T_2v\Vert_{P_2}  \preceq \dfrac{k}{M_1}[\Vert u-Tv\Vert_{K_p}+ \Vert v-Tu\Vert_{K_p}]\;\forall u,v\in X \otimes Y,\\
&(iii)\;e_1\in P_1,\;e_2\in P_2,\\
&(iv)\;\Vert \Vert T_1u\Vert_{P_1} \Vert< M_1\;and\; \Vert \Vert T_2u\Vert_{P_2} \Vert< M_2\;\forall u,v\in X \otimes Y.
\end{align*}
 then $T$ has a unique fixed point in $X \otimes Y$, where $k\in K_p$   and $r(k)<\frac{1}{4}$.
 \end{thm}
\section{Main Results: (c) Stability of   iteration scheme converging to the fixed point  in cone normed spaces over Banach algebra:}
 \par In \cite{2}, Asadi et al. generalized  the results of Qing and Rhoades  \cite{17}(regarding the $T$-stability of Picard's iteration scheme  in metric spaces) to cone metric spaces.\\[.1cm]
 An iteration procedure $x_{n+1}=f(T,x_n)$ is said to be $T$-stable with respect to $T$ on a metric space $X$, if $\{x_n\}$ converges to a fixed point $q$ of $T$ and whenever $\{y_n\}$ is a sequence in $X$ with $\lim_{n\to\infty}d(y_{n+1},f(T,y_n))=0$, we have $\lim_{n\to\infty}y_n=q$.\\[.1cm]
\begin{lem} Let $P$ be a normal cone with normal constant $K$ in a Banach algebra $\mathbb{A}$. Let $\{a_n\}$ and $\{b_n\}$ be two sequences in $\mathbb{A}$ satisfying the following inequality:
$$a_{n+1}\preceq ha_n+b_n,$$
where $h\in P$ with $r(h)<1$ and $b_n \to 0$ as $n\to\infty$. Then $ a_n\to 0\;as\;n\to\infty$.
\end{lem}
\begin{pf}
Let $m$ be a positive integer. By recursion we have
\begin{align*}
a_{n+1}\preceq b_n+hb_{n-1}+...+h^mb_{n-m}+h^{m+1}a_{n-m}
\end{align*}
Since $P$ is normal,
\begin{align*}
\Vert a_{n+1}\Vert\leqslant K\Vert b_n+hb_{n-1}+...+h^mb_{n-m}\Vert +K\Vert h^{m+1}\Vert \Vert a_{n-m}\Vert
\end{align*}
By Lemma 1.5, and hypothesis $ a_n\to 0\;as\;n\to\infty$.
\end{pf}
\begin{thm} Let $( X , \Vert .\Vert_p)$ be a  cone Banach  space over the unital Banach algebra $\mathbb{A}$, and $P$ be a solid cone (not necessarily normal cone) in $\mathbb{A}$. Suppose $S$ and $T$ be self mappings on $X$ satisfying
\begin{align*}
\Vert Sx-Ty\Vert_{p}\preceq \alpha\Vert x-y\Vert_{p}+\beta(\Vert x-Sx\Vert_{p}+ \Vert y-Ty\Vert_{p})+\gamma(\Vert x-Ty\Vert_{p}+ \Vert y-Sx\Vert_{p})
\end{align*}
for all $x,y\in X$ , where $\alpha, \beta, \gamma\in P$ commute with each other and $r(\beta + \gamma)+r(\alpha +  \beta + \gamma)< 1$. Then $S$ and $T$ have a common unique  fixed point $q$ in $X$.
\end{thm}
\begin{pf}
Let $x_0\in X$. We define a sequence $\{x_n\}$ by $x_{2n+1}=Sx_{2n},\;x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...$ . Now,
\begin{align*}
\Vert x_{2n+1}-x_{2n+2}\Vert_p&=\Vert Sx_{2n} - Tx_{2n+1}\Vert_p\\
&\preceq \alpha\Vert x_{2n}-x_{2n+1}\Vert_p+\beta(\Vert x_{2n}-Sx_{2n}\Vert_p+\Vert x_{2n+1}-Tx_{2n+1}\Vert_p)\\
&\qquad+\gamma(\Vert x_{2n+1}-Sx_{2n}\Vert_p+\Vert x_{2n}-Tx_{2n+1}\Vert_p)\\
&=(\alpha +  \beta + \gamma)\Vert x_{2n}-x_{2n+1}\Vert_p+(\beta + \gamma)\Vert x_{2n+1}-x_{2n+2}\Vert_p\\
\Rightarrow \Vert x_{2n+1}-x_{2n+2}\Vert_p&\preceq  (e^*-\beta - \gamma)^{-1}(\alpha +  \beta + \gamma)\Vert x_{2n}-x_{2n+1}\Vert_p\;
\end{align*}
($e^*$ being the unit element of $\mathbb{A}$.) Similarly it can be shown that
\begin{align*}
\Vert x_{2n+3}-x_{2n+2}\Vert_p\preceq  (e^*-\beta - \gamma)^{-1}(\alpha +  \beta + \gamma)\Vert x_{2n+2}-x_{2n+1}\Vert_p
\end{align*}
Therefore for all $n$,
\begin{align*}
\Vert x_{n+1}-x_{n+2}\Vert_p\preceq  (e^*-\beta - \gamma)^{-1}(\alpha +  \beta + \gamma)\Vert x_{n}-x_{n+1}\Vert_p
\end{align*}
Now,
\begin{align*}
r((e^*-\beta - \gamma)^{-1}(\alpha +  \beta + \gamma))&\leqslant r(e^*-\beta - \gamma)^{-1})r(\alpha +  \beta + \gamma)\\
&\leqslant \dfrac{r(\alpha +  \beta + \gamma)}{1-r(\beta + \gamma)}<1
\end{align*}
Hence from Lemma 1.2, Lemma 1.3 , Lemma 1.5 and Lemma 1.8(see  \cite{huang3}) we have, $\{x_n\}$ is a Cauchy sequence and converges to some $z$ as $n\to \infty$. Now
\begin{align*}
\Vert z - Tz\Vert_p&\preceq \Vert z-x_{2n+1}\Vert_p+\Vert x_{2n+1}-Tz\Vert_p\\
&\preceq \Vert z-x_{2n+1}\Vert_p+\Vert Sx_{2n}-Tz\Vert_p\\
&\preceq  \Vert z-x_{2n+1}\Vert_p+\alpha\Vert x_{2n}-z\Vert_p+\beta(\Vert x_{2n}-Sx_{2n}\Vert_p+\Vert z-Tz\Vert_p)\\
&\qquad+\gamma(\Vert z-Sx_{2n}\Vert_p+\Vert x_{2n}-Tz\Vert_p)\\
\Rightarrow\Vert z - Tz\Vert_p&\preceq (\beta + \gamma)^{-1} [\Vert z-x_{2n+1}\Vert_p+\alpha\Vert x_{2n}-z\Vert_p+\beta \Vert x_{2n}-x_{2n+1}\Vert_p\\
&\qquad+\gamma \Vert z-x_{2n+1}\Vert_p+\gamma\Vert x_{2n}-z\Vert_p)]
\end{align*}
By Lemma 1.1, Lemma 1.4 and Lemma 1.8, it is clear that right hand side of the above inequality  is a $c$-sequence, this means $z = T z$.
\begin{align*}
\Vert Sz - z\Vert_p&=\Vert Sz - Tz\Vert_p\\
&\preceq \alpha\Vert z-z\Vert_p+\beta(\Vert z-Sz\Vert_p+\Vert z-Tz\Vert_p)\\
&\qquad+\gamma(\Vert z-Sz\Vert_p+\Vert z-Tz\Vert_p)\\
\Rightarrow\Vert Sz-z\Vert_p&\preceq (\beta + \gamma) \Vert z-Sz\Vert_p
\end{align*}
Since, $\beta+ \gamma \preceq \alpha+\beta+\gamma$ and $r(\alpha+\beta+\gamma)<1$. Hence by Lemma 1.11,
$\Vert z-Sz\Vert_p=0$, so $Sz=z$.\\
To show uniqueness: Let $z$ and $q$ be two distinct common fixed points of $S$ and $T$.
\begin{align*}
\Vert z - q\Vert_p&=\Vert Sz - Tq\Vert_p\\
&\preceq \alpha\Vert z-q\Vert_p+\beta(\Vert z-Sz\Vert_p+\Vert q-Tq\Vert_p)\\
&\qquad+\gamma(\Vert q-Sz\Vert_p+\Vert z-Tq\Vert_p)\\
\Rightarrow\Vert z-q\Vert_p&\preceq (\alpha + 2\gamma) \Vert z-q\Vert_p
\end{align*}
Since, $\alpha + 2\gamma\preceq (\beta+\gamma)+(\alpha+\beta+\gamma)$, by Lemma 1.10 we have\\
$r(\beta+\gamma+\alpha+\beta+\gamma)\leqslant r(\beta+\gamma)+r(\alpha+\beta+\gamma)<1$. Hence by Lemma 1.11,
$\Vert z-q\Vert_p=0$, so $z=q$.
\end{pf}
\begin{thm} Let $( X , \Vert .\Vert_p)$ be a  cone Banach  space over the unital Banach algebra $\mathbb{A}$ (having unit $e^*$), and $P$ be a solid cone in $\mathbb{A}$. Suppose $T$ be a self mapping on $X$ satisfying
\begin{align*}
\Vert Tx-Ty\Vert_{p}\preceq \alpha\Vert x-y\Vert_{p}+\beta(\Vert x-Tx\Vert_{p}+ \Vert y-Ty\Vert_{p})+\gamma(\Vert x-Ty\Vert_{p}+ \Vert y-Tx\Vert_{p})
\end{align*}
for all $x,y\in X$ , where $\alpha, \beta, \gamma\in P$ commute with each other and $r(\beta + \gamma)+r(\alpha +  \beta + \gamma)< 1$. Then  $T$ has a unique  fixed point $q$ in $X$.
\end{thm}
Now, we discuss stability of an iteration scheme (see \cite{1}) converging to the  common fixed point of $S$ and $T$ on the cone Banach space over Banach algebra. For $x_0\in X$, we consider the following iteration scheme:
\begin{align}
x_{2n+1}=Sx_{2n},\;x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...
\end{align}
 For the cone Banach space $X$ with a normal cone $P$ (with normal constant 1), the above iteration scheme is said to be stable with respect to $S$ and $T$, if $\{x_n\}$ converges to the unique common fixed point $q$ of $S$ and $T$, and whenever $\{y_n\}$ is a sequence in $X$ with
\begin{align}
\lim_{n\to\infty}\Vert y_{2n+1}-Sy_{2n}\Vert_p(=\epsilon_{2n})&=0,\;\text{and}\\
 \lim_{n\to\infty}\Vert y_{2n+2}-Ty_{2n+1}\Vert_p(=\epsilon_{2n+1})&=0,
\end{align}
we have $\lim_{n\to\infty}y_n=q$.\\[.1cm]
Here we derive the following condition for stability of the iteration scheme $(5)$.
\begin{thm} Let $(X,\Vert .\Vert_p)$ be a cone Banach space over the  Banach algebra  $\mathbb{A}$ (having unit $e^*$) with the normal cone $P$(normal constant 1). If $S$ and $T$ are self mappings on $X$ satisfying the condition
\begin{align*}
\Vert Su-Tv\Vert_{p}\preceq \alpha\Vert u-v\Vert_{p}+\beta(\Vert u-Su\Vert_{p}+ \Vert v-Tv\Vert_{p})+\gamma(\Vert u-Tv\Vert_{p}+ \Vert v-Su\Vert_{p})
\end{align*}
 then the iteration scheme (5) is stable with respect to $S$ and $T$ if  $\alpha,\beta,\gamma\in P$ commute with each other and $r(\beta+3\gamma)+r(\alpha+\beta+\gamma)<1$.
 \end{thm}
\begin{pf}
We have, for the common unique fixed point $q$ of $S$ and $T$, $\lim_{n\to\infty}x_n=q$. So,
$$\lim_{n\to\infty}\Vert x_{2n}-Sx_{2n}\Vert_p=0 \;\text{and} \;\lim_{n\to\infty}\Vert x_{2n+1}-Tx_{2n+1}\Vert_p=0.$$
Now,
\begin{align}
\Vert y_{2n+1}-q\Vert_p&\preceq \Vert y_{2n+1}-x_{2n+2}\Vert_p+\Vert x_{2n+2}-q\Vert_p\nonumber\\
&\preceq \Vert y_{2n+1}-Sy_{2n}\Vert_p+\Vert Sy_{2n}-x_{2n+2}\Vert_p+\Vert x_{2n+2}-q\Vert_p\nonumber\\
&= \epsilon_{2n}+\Vert Sy_{2n}-Tx_{2n+1}\Vert_p+\Vert x_{2n+2}-q\Vert_p
\end{align}
\begin{align*}
\Vert Sy_{2n}-Tx_{2n+1}\Vert_p&\preceq \alpha\Vert y_{2n}-x_{2n+1}\Vert_p+\beta(\Vert y_{2n}-Sy_{2n}\Vert_p+\Vert x_{2n+1}-Tx_{2n+1}\Vert_p)\\
&\qquad+\gamma(\Vert x_{2n+1}-Sy_{2n}\Vert_p+\Vert y_{2n}-Tx_{2n+1}\Vert_p)
\end{align*}
\begin{align*}
&\preceq \alpha\Vert y_{2n}-x_{2n+1}\Vert_p+\beta(\Vert y_{2n}-Sy_{2n}\Vert_p+\Vert x_{2n+1}-Tx_{2n+1}\Vert_p)\\
&\quad+\gamma(\Vert x_{2n+1}-Tx_{2n+1}\Vert_p+\Vert Tx_{2n+1}-Sy_{2n}\Vert_p+\Vert y_{2n}-Sy_{2n}\Vert_p+\Vert Sy_{2n}-Tx_{2n+1}\Vert_p)\\
&= \alpha\Vert y_{2n}-x_{2n+1}\Vert_p+(\beta+\gamma)\Vert y_{2n}-Sy_{2n}\Vert_p+(\beta+\gamma)\Vert x_{2n+1}-Tx_{2n+1}\Vert_p\\
&\quad+2\gamma \Vert Sy_{2n}-Tx_{2n+1}\Vert_p\\
&\preceq \alpha\Vert y_{2n}-x_{2n+1}\Vert_p+(\beta+\gamma)(\Vert y_{2n}-x_{2n+1}\Vert_p+\Vert x_{2n+1}-Tx_{2n+1}\Vert_p\\
&\quad+\Vert Tx_{2n+1}-Sy_{2n}\Vert_p)+(\beta+\gamma)\Vert x_{2n+1}-Tx_{2n+1}\Vert_p+2\gamma \Vert Sy_{2n}-Tx_{2n+1}\Vert_p\\
&=(\alpha+\beta+\gamma)\Vert y_{2n}-x_{2n+1}\Vert_p+(2\beta+2\gamma)\Vert x_{2n+1}-Tx_{2n+1}\Vert_p\\
&\quad+(\beta+3\gamma) \Vert Sy_{2n}-Tx_{2n+1}\Vert_p\\
&\preceq(\alpha+\beta+\gamma)\Vert y_{2n}-q\Vert_p+(\alpha+\beta+\gamma)\Vert q-x_{2n+1}\Vert_p\\
&\quad+(2\beta+2\gamma)\Vert x_{2n+1}-Tx_{2n+1}\Vert_p+(\beta+3\gamma) \Vert Sy_{2n}-Tx_{2n+1}\Vert_p
\end{align*}
\begin{align*}
\Rightarrow\Vert Sy_{2n}-Tx_{2n+1}\Vert_p&\preceq \dfrac{\alpha+\beta+\gamma}{e^*-\beta-3\gamma}(\Vert y_{2n}-q\Vert_p+\Vert q-x_{2n+1}\Vert_p)\\
&\quad+\dfrac{2\beta+2\gamma}{e^*-\beta-3\gamma}\Vert x_{2n+1}-Tx_{2n+1}\Vert_p
\end{align*}
So, from $(8)$, we have,
\begin{align*}
\Vert y_{2n+1}-q\Vert_p&\preceq \epsilon_{2n}+\Vert x_{2n+2}-q\Vert_p\\
&\quad+\dfrac{\alpha+\beta+\gamma}{e^*-\beta-3\gamma}[\Vert y_{2n}-q\Vert_p+\Vert q-x_{2n+1}\Vert_p]\\
&\quad+\dfrac{2\beta+2\gamma}{e^*-\beta-3\gamma}\Vert x_{2n+1}-Tx_{2n+1}\Vert_p
\end{align*}
We take $a_{n}=\Vert y_{n}-q\Vert_p$ and
\begin{align*}
b_{n}&= \epsilon_{2n}+\Vert x_{2n+2}-q\Vert_p+\dfrac{\alpha+\beta+\gamma}{e^*-\beta-3\gamma}\Vert q-x_{2n+1}\Vert_p\\
&\quad+\dfrac{2\beta+2\gamma}{e^*-\beta-3\gamma}\Vert x_{2n+1}-Tx_{2n+1}\Vert_p
\end{align*}
Now, $b_{n}\to 0$ as $n\to \infty$. Also,
\begin{align*}
r((e^*-\beta-3\gamma)^{-1}(\alpha+\beta+\gamma))&\leqslant r(e^*-\beta-3\gamma)^{-1}r(\alpha+\beta+\gamma)\\
&\leqslant \dfrac{r(\alpha+\beta+\gamma)}{1-r(\beta+3\gamma)}<1
\end{align*}
So, by the Lemma 4.1, $$\lim_{n\to\infty}a_{2n+1}=\lim_{n\to\infty}\Vert y_{2n+1}-q\Vert_p=0.$$
Again,
\begin{align}
\Vert y_{2n+2}-q\Vert_p&\preceq \Vert y_{2n+2}-x_{2n+1}\Vert_p+\Vert x_{2n+1}-q\Vert_p\nonumber\\
&\preceq \Vert y_{2n+2}-Ty_{2n+1}\Vert_p+\Vert Ty_{2n+1}-Sx_{2n}\Vert_p+\Vert x_{2n+1}-q\Vert_p
\end{align}
\begin{align*}
\Vert Sx_{2n}-Ty_{2n+1}\Vert_p&\preceq \alpha\Vert x_{2n}-y_{2n+1}\Vert_p+\beta(\Vert x_{2n}-Sx_{2n}\Vert_p+\Vert y_{2n+1}-Ty_{2n+1}\Vert_p)\\
&\qquad\qquad+\gamma(\Vert y_{2n+1}-Sx_{2n}\Vert_p+\Vert x_{2n}-Ty_{2n+1}\Vert_p)\\
&\preceq \alpha\Vert x_{2n}-y_{2n+1}\Vert_p+\beta(\Vert x_{2n}-Sx_{2n}\Vert_p+\Vert y_{2n+1}-x_{2n}\Vert_p\\
&\quad+\Vert x_{2n}-Sx_{2n}\Vert_p+\Vert Sx_{2n}-Ty_{2n+1}\Vert_p)\\
&\qquad+\gamma(\Vert y_{2n+1}-x_{2n}\Vert_p+\Vert x_{2n}-Sx_{2n}\Vert_p\\
&\quad+\Vert x_{2n}-Sx_{2n}\Vert_p+\Vert Sx_{2n}-Ty_{2n+1}\Vert_p)\\
&= (\alpha+\beta+\gamma)\Vert x_{2n}-y_{2n+1}\Vert_p+(2\beta+2\gamma)\Vert x_{2n}-Sx_{2n}\Vert_p\\
&\quad+(\beta+\gamma) \Vert Sx_{2n}-Ty_{2n+1}\Vert_p\\
&\preceq(\alpha+\beta+\gamma)\Vert x_{2n}-q\Vert_p+(\alpha+\beta+\gamma)\Vert q-y_{2n+1}\Vert_p\\
&\quad+(2\beta+2\gamma)\Vert x_{2n}-Sx_{2n}\Vert_p+(\beta+\gamma) \Vert Sx_{2n}-Ty_{2n+1}\Vert_p
\end{align*}
\begin{align*}
\Rightarrow \Vert Sx_{2n}-Ty_{2n+1}\Vert_p&\preceq \dfrac{\alpha+\beta+\gamma}{e^*-\beta-\gamma}(\Vert x_{2n}-q\Vert_p+\Vert q-y_{2n+1}\Vert_p)\\
&\quad+\dfrac{2\beta+2\gamma}{e^*-\beta-\gamma}\Vert x_{2n}-Sx_{2n}\Vert_p
\end{align*}
From $(9)$, we have,
\begin{align*}
\Vert y_{2n+2}-q\Vert_p&\preceq \epsilon_{2n+1}+\Vert x_{2n+1}-q\Vert_p\\
&\quad\dfrac{\alpha+\beta+\gamma}{e^*-\beta-\gamma}(\Vert y_{2n+1}-q\Vert_p+\Vert q-x_{2n}\Vert_p)\\
&\quad+\dfrac{2\beta+2\gamma}{e^*-\beta-\gamma}\Vert x_{2n}-Sx_{2n}\Vert_p
\end{align*}
Now, as in the first part, we have
 $$\lim_{n\to\infty}a_{2n+2}=\lim_{n\to\infty}\Vert y_{2n+2}-q\Vert_p=0$$
\par Thus, for all $n$ we have,  $\lim_{n\to\infty}\Vert y_{n}-q\Vert_p=0$. Hence the given iteration is stable with respect to $S$ and $T$.\\[.1cm]
We can also show that, if \\
 $\lim_{n\to\infty}\Vert y_{n}-q\Vert_p=0$, then
$$\lim_{n\to\infty}\Vert y_{2n+1}-Sy_{2n}\Vert_p(=\epsilon_{2n}, say)=0,\;and \;\lim_{n\to\infty}\Vert y_{2n+2}-Ty_{2n+1}\Vert_p(=\epsilon_{2n+1}, say)=0$$
\begin{align*}
\Vert y_{2n+1}-Sy_{2n}\Vert_p&\preceq \Vert y_{2n+1}-q\Vert_p+\Vert q-Sy_{2n}\Vert_p
\end{align*}
\begin{align*}
\Vert q-Sy_{2n}\Vert_p=\Vert Sy_{2n}-Tq\Vert_p&\preceq \alpha \Vert y_{2n}-q\Vert_p+\beta(\Vert q-Tq\Vert_p+\Vert y_{2n}-Sy_{2n}\Vert_p)\\
&+\gamma(\Vert q-Sy_{2n}\Vert_p+\Vert y_{2n}-Tq\Vert_p)\\
&\preceq \alpha \Vert y_{2n}-q\Vert_p+\beta\Vert y_{2n}-q\Vert_p+\gamma\Vert y_{2n}-Tq\Vert_p\\
&+\beta\Vert q-Sy_{2n}\Vert_p+\gamma\Vert q-Sy_{2n}\Vert_p\\
\Rightarrow\Vert q-Sy_{2n}\Vert_p&\preceq \dfrac{\alpha+\beta+\gamma}{e^*-\beta-\gamma}\Vert y_{2n}-q\Vert_p
\end{align*}
\begin{align*}
r(e^*-(\beta+\gamma)^{-1}(\alpha+\beta+\gamma))&\leqslant r(e^*-(\beta+\gamma))^{-1}r(\alpha+\beta+\gamma)\\
&\leqslant\dfrac{r(\alpha+\beta+\gamma)}{1-r(\beta+\gamma)}<1
\end{align*}
Now, using normality condition we get, $\Vert q-Sy_{2n}\Vert_p\to 0\;as\;n\to\infty$.
So, $\epsilon_{2n}\to 0\;as\;n\to\infty$. Again,
\begin{align*}
\Vert y_{2n+2}-Ty_{2n+1}\Vert_p&\preceq \Vert y_{2n+2}-q\Vert_p+\Vert q-Ty_{2n+1}\Vert_p
\end{align*}
\begin{align*}
\Vert q-Ty_{2n+1}\Vert_p&=\Vert Sq-Ty_{2n+1}\Vert_p\\
&\preceq \alpha \Vert q-y_{2n+1}\Vert_p+\beta[\Vert q-Sq\Vert_p+\Vert y_{2n+1}-Ty_{2n+1}\Vert_p]\\
&+\gamma[\Vert q-Ty_{2n+1}\Vert_p+\Vert y_{2n+1}-Sq\Vert_p]\\
&\preceq \alpha \Vert q-y_{2n+1}\Vert_p+\beta\Vert y_{2n+1}-q\Vert_p+\gamma \Vert y_{2n+1}-Sq\Vert_p\\
&+\gamma[\Vert q-Ty_{2n+1}\Vert_p+\beta\Vert q-Ty_{2n+1}\Vert_p\\
\Rightarrow\Vert q-Ty_{2n+1}\Vert_p&\preceq \dfrac{\alpha+\beta+\gamma}{e^*-\beta-\gamma}\Vert y_{2n+1}-q\Vert_p\\
&\to 0\;as\;n\to\infty.
\end{align*}
So, $\epsilon_{2n+1}\to 0\;as\;n\to\infty$.
\end{pf}
\begin{thm} For the cone Banach space $(X\otimes  Y,\Vert .\Vert_{K_p})$ over the unital Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ with the normal cone $K_p$ (normal constant 1), let $T$ be the self mapping on $X\otimes Y$  satisfying the condition:
\begin{align}
\Vert Tu-Tv\Vert_{K_p}&\preceq \alpha\Vert u-v\Vert_{K_p}+\beta(\Vert u-Tu\Vert_{K_p}+ \Vert v-Tv\Vert_{K_p})\nonumber\\
&\quad+\gamma(\Vert u-Tv\Vert_{K_p}+ \Vert v-Tu\Vert_{K_p})
\end{align}
for all $u,v\in X\otimes  Y$, where $\alpha,\beta,\gamma\in K_p$ commute with each other and  $r(\beta+3\gamma)+r(\alpha+\beta+\gamma)<1$. Then the  iteration scheme:
\begin{align*}
x_0&\in X\otimes_\gamma Y,\\
x_{n+1}&=Tx_{n},\;n=0,1,2,...
\end{align*}
 converging to the  fixed point of  $T$ is stable with respect to $T$.
 \end{thm}
\begin{exmp} We take $X=l^1$  and $\mathbb{A}_1=(l^1,\Vert .\Vert)$ over $\mathbb{R}$.\\ Let $P_1=\{\{x_n\}_{n\geqslant 1}\in \mathbb{A}_1,x_n\geqslant 0\;\forall n\}$. Then $\Vert .\Vert_{P_1} :l^1\to \mathbb{A}_1$  defined by $\Vert a_i\Vert_{P_1}=\{\vert a_{i_k}\vert\}_k$,  is a cone norm on $X$. Clearly, $(X,\Vert .\Vert_{P_1})$ is also a cone Banach space over $\mathbb{A}_1$.\\[.1cm]
 Next we take $Y=[0,1]$, $\mathbb{A}_2=([0,1],\Vert .\Vert)$, $P_2=\{y\in [0,1]:y\geqslant 0\}$. Then $\Vert .\Vert_{P_2} :Y\to \mathbb{A}_2$  defined by $\Vert y\Vert_{P_2}=y$ is a cone norm on  $Y$.
 Clearly, $(Y,\Vert .\Vert_{P_2})$ is also a cone Banach space over $\mathbb{A}_2$.\\[.1cm]
 Now, $\Vert .\Vert_{K_p} :X\otimes Y\to \mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ i.e., $\Vert .\Vert_{K_p} :l^1\otimes [0,1]\to l^1\otimes_\gamma [0,1]$ is defined by \\
 $\Vert u\Vert_{K_p}=\Vert \sum_ia_i\otimes y_i\Vert_{K_p}= \sum_i\Vert \{a_{i_k}\}\Vert_{P_1}\otimes \Vert y_i\Vert_{P_2}=\sum_i\{\vert a_{i_k}\vert\}_k\otimes  y_i=\sum_i\{\vert a_{i_k}\vert y_i\}_k$\\[.1cm]
 Thus $(l^1\otimes  [0,1],\Vert .\Vert_{K_p})$ is a cone Banach space over the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ i.e., $l^1([0,1])$ with unit $e=\{1,0,0,...\}$.\\
We define $T_1:l^1\otimes  [0,1]\to l^1$ by
$T_1(\sum_i a_i \otimes y_i)=\dfrac{1}{4}\sum_i\{a_{i_k} y_i\}_k$,  and  $T_2:l^1\otimes  [0,1]\to [0,1]$ by
$T_2(\sum_i a_i \otimes y_i)= \dfrac{1}{2}$. Let $T$ be the self mapping on $l^1\otimes  [0,1]$ such that\\
\begin{align*}
T(\sum_i a_i \otimes y_i)&=T_1(\sum_i a_i \otimes y_i)\otimes T_2(\sum_i a_i \otimes y_i)\\
&=\dfrac{1}{4}\sum_i\{a_{i_k} y_i\}_k\otimes \dfrac{1}{2}\\
&=\dfrac{1}{8}\sum_ia_i\otimes y_i\;(\text{taking norm in $l^1([0,1])$})
\end{align*}
\begin{align*}
\Vert Tu-Tv\Vert_{K_p}&\preceq\Vert \frac{u}{8}-\frac{v}{8}\Vert_{K_p}=2\Vert \frac{u}{16}-\frac{v}{16}\Vert_{K_p}\\
&=2[\Vert \frac{u}{16}-\frac{v}{16}+\frac{v}{16}-\frac{v}{128}+\frac{u}{16}-\frac{u}{128}+\frac{u}{128}-\frac{u}{16}+\frac{v}{128}-\frac{v}{16}\Vert_{K_p}]\\
&\preceq \frac{1}{8}\Vert u-v\Vert_{K_p}+\frac{1}{8}[\Vert u-\frac{u}{8}\Vert_{K_p}+\Vert v-\frac{v}{8}\Vert_{K_p}]\\
&\qquad+\frac{1}{8}[\Vert v-\frac{u}{8}\Vert_{K_p}+\Vert u-\frac{v}{8}\Vert_{K_p}]\\
&= \frac{1}{8}e\Vert u-v\Vert_{K_p}+\frac{1}{8}[e\Vert u-Su\Vert_{K_p}+e\Vert v-Tv\Vert_{K_p}]\\
&\qquad+\frac{1}{8}[e\Vert v-Su\Vert_{K_p}+e\Vert u-Tv\Vert_{K_p}]
\end{align*}
Thus $T$ satisfies $(10)$ with $\alpha=\beta=\gamma=\frac{1}{8}e$. Also  $\alpha,\beta,\gamma\in K_p$,  and  they commute with each  other.\\
 $r(\beta+3\gamma)=\lim_{n\to\infty}\Vert \left(\frac{1}{2}e\right)^n\Vert^{\frac{1}{n}}= \dfrac{1}{2}$ and\\
 $r(\alpha+\beta+\gamma)=\lim_{n\to\infty}\Vert \left(\frac{3}{8}e\right)^n\Vert^{\frac{1}{n}}= \dfrac{3}{8}$. Hence $r(\beta+3\gamma)+ r(\alpha+\beta+\gamma)<1$. \\ Let $b_n=\sum_i\{\dfrac{a_i}{n}\}_{n\geqslant 1} ,y_n=\frac{1}{n+1}\;\forall n=1,2,...$\\
$Tb_n=\dfrac{1}{8}\sum_i\{\dfrac{a_i}{n}\}_{n\geqslant 1}.$
Now,  $q = 0$ is the unique fixed point of $T$ and  $\lim_{n\to\infty}y_n=0$.
Also we have,
\begin{align*}
\lim_{n\to\infty}\Vert b_{n+1}-Tb_{n}\Vert_{K_p}\to 0
 \end{align*}
Hence the  iteration scheme  is $T$-stable for the cone Banach space $l^1\otimes [0,1]$ over the Banach algebra $l^1\otimes_\gamma [0,1]$.\end{exmp}
\begin{Rem} There are many other iteration schemes viz., Picard-Mann hybrid iteration, Mann iteration, Ishikawa iteration etc. \cite{khan}, for which we can discuss the stability with respect to some self mappings on cone normed spaces as well as PCNTPS over Banach algebra.
\end{Rem}
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\end{document}
