Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are finite dimensional compact and convex Hilbert spaces of same dimension. Furthermore $T_1(\cdot)$ is nonexpansive, i.e. has Lipschitz constant equal to $1$, $T_2(\cdot, y)$ is nonexpansive for all fixed $y \in Y$ and $T_2(x,\cdot)$ is uniformly Lipschitz continuous, i.e. there exists one Lipschitz constant for all $x \in X$ .

We know that for any $\lambda \in (0,1)$ and any $y_0 \in Y$ the iteration $y_{n+1} = (1-\lambda) y_n + \lambda T_1(y_n)$ converges to a fixed point $\bar{y} = T_(\bar{y})$.

We also know that for any $\rho \in (0,1)$ and any $x_0 \in X$ and a fixed $y \in Y$ the iteration $x_{n+1} = (1-\rho) x_n + \rho T_2(x_n, y)$ converges to a fixed point $\bar{x} = T_(\bar{x}, y)$.

My question is: Does the iteration

$x_{n+1} = (1-\rho) x_n + \rho T_2(x_n, y_n)$

converge to a fixed point $\bar{x} = T_2(\bar{x}, \bar{y})$?

Does anybody have a hint in proving or disproving it?

Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are compact and convex subsets of the same Euclidean space. Furthermore $T_1(\cdot)$ is nonexpansive, i.e. has Lipschitz constant equal to $1$, $T_2(\cdot, y)$ is nonexpansive for all fixed $y \in Y$ and $T_2(x,\cdot)$ is uniformly Lipschitz continuous, i.e. there exists one Lipschitz constant for all $x \in X$ .

We know that for any $\lambda \in (0,1)$ and any $y_0 \in Y$ the iteration $y_{n+1} = (1-\lambda) y_n + \lambda T_1(y_n)$ converges to a fixed point $\bar{y} = T_(\bar{y})$.

We also know that for any $\rho \in (0,1)$ and any $x_0 \in X$ and a fixed $y \in Y$ the iteration $x_{n+1} = (1-\rho) x_n + \rho T_2(x_n, y)$ converges to a fixed point $\bar{x} = T_(\bar{x}, y)$.

My question is: Does the iteration

$x_{n+1} = (1-\rho) x_n + \rho T_2(x_n, y_n)$

converge to a fixed point $\bar{x} = T_2(\bar{x}, \bar{y})$?

Does anybody have a hint in proving or disproving it?

Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are finite dimensional comapct and covex hilbert spaces of same dimension. Furthermore $T_1(\cdot)$ is nonexpansive, i.e. has Lipschitz constant equal to $1$, $T_2(\cdot, y)$ is nonexpansive for all fixed $y \in Y$ and $T_2(x,\cdot)$ is uniformly Lipschitz continuous, i.e. ther exists one Lipschitzconstant for all $x \in X$ .

We know that for any $\lambda \in (0,1)$ and any $y_0 \in Y$ the itearation $y_{n+1} = (1-\lambda) y_n + \lambda T_1(y_n)$ converges to a fixed point $\bar{y} = T_(\bar{y})$.

We also know that for any $\rho \in (0,1)$ and any $x_0 \in X$ and a fixed $y \in Y$ the itearation $x_{n+1} = (1-\rho) x_n + \rho T_2(x_n, y)$ converges to a fixed point $\bar{x} = T_(\bar{x}, y)$.

My question is: Does the iteration

$x_{n+1} = (1-\rho) x_n + \rho T_2(x_n, y_n)$

converge to a fixed point $\bar{x} = T_2(\bar{x}, \bar{y})$?

Does anybody have a hint in proving or disproving it?

Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are finite dimensional compact and convex Hilbert spaces of same dimension. Furthermore $T_1(\cdot)$ is nonexpansive, i.e. has Lipschitz constant equal to $1$, $T_2(\cdot, y)$ is nonexpansive for all fixed $y \in Y$ and $T_2(x,\cdot)$ is uniformly Lipschitz continuous, i.e. there exists one Lipschitz constant for all $x \in X$ .

We know that for any $\lambda \in (0,1)$ and any $y_0 \in Y$ the iteration $y_{n+1} = (1-\lambda) y_n + \lambda T_1(y_n)$ converges to a fixed point $\bar{y} = T_(\bar{y})$.

We also know that for any $\rho \in (0,1)$ and any $x_0 \in X$ and a fixed $y \in Y$ the iteration $x_{n+1} = (1-\rho) x_n + \rho T_2(x_n, y)$ converges to a fixed point $\bar{x} = T_(\bar{x}, y)$.

My question is: Does the iteration

$x_{n+1} = (1-\rho) x_n + \rho T_2(x_n, y_n)$

converge to a fixed point $\bar{x} = T_2(\bar{x}, \bar{y})$?

Does anybody have a hint in proving or disproving it?