Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^2-2 \langle x, Jy \rangle + \Vert y \Vert^2$. Here, $J$ is the normalized duality mapping $J:X \to X^{'}$ given by $$ J(x)=\{ Jx \in X^{'} : \langle x, Jx \rangle = \Vert x \Vert^2=\Vert Jx \Vert^2 \}. $$ It should be noted that $J$ is single-valued when $X$ is smooth. I have come to an inequality that $\phi(x_{n+1}, x_n) \leq \phi(x_n, x_{n-1})$. Then, using the above criteria, we will further reduce to the inequality $\phi(x_{n+1}, x_n) \leq \phi(x_1, x_0)$. Then we will get $$h(\Vert x_{n+1} - x_n \Vert) \leq \phi(x_{n+1}, x_n) \leq W(x_1, x_0);$$ where $h:[0, \infty) \to [0, \infty)$ is a continuous, strictly increasing, convex function with $h(0)=0$, does $(x_n)$ behaves like a Cauchy sequence here. If not, what extra condition is required to show that $\{x_n\}$ is Cauchy? Here $X^{'}$ is the dual of $X$. Kindly help me out. Thank you.

Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^2-2 \langle x, Jy \rangle + \Vert y \Vert^2$. Here, $J$ is the normalized duality mapping $J:X \to X^{'}$ given by $$ J(x)=\{ Jx \in X^{'} : \langle x, Jx \rangle = \Vert x \Vert^2=\Vert Jx \Vert^2 \}. $$ It should be noted that $J$ is single-valued when $X$ is smooth. I have come to an inequality that $W(x_{n+1}, x_n) \leq W(x_n, x_{n-1})$. Then, using the above criteria, we will further reduce to the inequality $W(x_{n+1}, x_n) \leq W(x_1, x_0)$. Then we will get $$h(\Vert x_{n+1} - x_n \Vert) \leq W(x_{n+1}, x_n) \leq W(x_1, x_0);$$ where $h:[0, \infty) \to [0, \infty)$ is a continuous, strictly increasing, convex function with $h(0)=0$, does $(x_n)$ behaves like a Cauchy sequence here. If not, what extra condition is required to show that $\{x_n\}$ is Cauchy? Here $X^{'}$ is the dual of $X$. Kindly help me out. Thank you.

Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^2-2 \langle x, Jy \rangle + \Vert y \Vert^2$. Here, $J$ is the normalized duality mapping $J:X \to X^{'}$ given by $$ J(x)=\{ Jx \in X^{'} : \langle x, Jx \rangle = \Vert x \Vert^2=\Vert Jx \Vert^2 \}. $$ It should be noted that $J$ is single-valued when $X$ is smooth. If we consider, $h(\Vert x_{n+1} - x_n \Vert) \leq W(x_1, x_0)$; where $h:[0, \infty) \to [0, \infty)$ is a continuous, strictly increasing, convex function with $h(0)=0$, does $(x_n)$ behaves like a Cauchy sequence here. If not, what extra condition is required to show that $\{x_n\}$ is Cauchy? Here $X^{'}$ is the dual of $X$. Kindly help me out. Thank you.

Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^2-2 \langle x, Jy \rangle + \Vert y \Vert^2$. Here, $J$ is the normalized duality mapping $J:X \to X^{'}$ given by $$ J(x)=\{ Jx \in X^{'} : \langle x, Jx \rangle = \Vert x \Vert^2=\Vert Jx \Vert^2 \}. $$ It should be noted that $J$ is single-valued when $X$ is smooth. I have come to an inequality that $\phi(x_{n+1}, x_n) \leq \phi(x_n, x_{n-1})$. Then, using the above criteria, we will further reduce to the inequality $\phi(x_{n+1}, x_n) \leq \phi(x_1, x_0)$. Then we will get $$h(\Vert x_{n+1} - x_n \Vert) \leq \phi(x_{n+1}, x_n) \leq W(x_1, x_0);$$ where $h:[0, \infty) \to [0, \infty)$ is a continuous, strictly increasing, convex function with $h(0)=0$, does $(x_n)$ behaves like a Cauchy sequence here. If not, what extra condition is required to show that $\{x_n\}$ is Cauchy? Here $X^{'}$ is the dual of $X$. Kindly help me out. Thank you.