The paper by Benjamin Halpern focuses on nonexpanding maps from the unit ball of a real Hilbert space into itself. It establishes that such maps always have at least one fixed point. The main contribution is a method for approximating these fixed points using a recursive sequence $\{x_n\}$ defined by $x_{n+1} = k_{n+1} f(x_n)$, where $f$ is the nonexpanding map and $\{k_n\}$ is a sequence of real numbers. The key results include: 1. **Theorem 1**: Proves that if $y_k$ is the unique element of $B$ satisfying $y_k = kf(y_k)$ for $\|k\| < 1$, then $\lim_{k \to 1, \|k\| < 1} y_k = y$, where $y$ is the unique fixed point of $f$ with the smallest norm. 2. **Theorem 2**: Outlines three necessary conditions for a sequence $\{k_i\}$ to be acceptable, ensuring that the sequence $\{z_n\}$ defined by $z_{n+1} = k_{n+1} f(z_n)$ converges to the fixed point of $f$ with the smallest norm. 3. **Theorem 3**: Provides sufficient conditions for $\{k_i\}$ to be acceptable, including specific inequalities and conditions on the sequence $\{n(i)\}$. 4. **Corollary**: Demonstrates that the sequence $\{k_i\}$ defined by $k_i = 1 - i^{-x}$ with $0 < x < 1$ is acceptable. 5. **Theorem 4**: Extends the results to a more general setting where $x_n$ is defined by $x_{n+1} = k_{n+1} f(x_{n-m_{n+1}})$, showing that $x_n$ converges to the fixed point of $f$. The paper also includes references to previous work by F. E. Browder and other mathematicians, providing a comprehensive background for the research.The paper by Benjamin Halpern focuses on nonexpanding maps from the unit ball of a real Hilbert space into itself. It establishes that such maps always have at least one fixed point. The main contribution is a method for approximating these fixed points using a recursive sequence $\{x_n\}$ defined by $x_{n+1} = k_{n+1} f(x_n)$, where $f$ is the nonexpanding map and $\{k_n\}$ is a sequence of real numbers. The key results include: 1. **Theorem 1**: Proves that if $y_k$ is the unique element of $B$ satisfying $y_k = kf(y_k)$ for $\|k\| < 1$, then $\lim_{k \to 1, \|k\| < 1} y_k = y$, where $y$ is the unique fixed point of $f$ with the smallest norm. 2. **Theorem 2**: Outlines three necessary conditions for a sequence $\{k_i\}$ to be acceptable, ensuring that the sequence $\{z_n\}$ defined by $z_{n+1} = k_{n+1} f(z_n)$ converges to the fixed point of $f$ with the smallest norm. 3. **Theorem 3**: Provides sufficient conditions for $\{k_i\}$ to be acceptable, including specific inequalities and conditions on the sequence $\{n(i)\}$. 4. **Corollary**: Demonstrates that the sequence $\{k_i\}$ defined by $k_i = 1 - i^{-x}$ with $0 < x < 1$ is acceptable. 5. **Theorem 4**: Extends the results to a more general setting where $x_n$ is defined by $x_{n+1} = k_{n+1} f(x_{n-m_{n+1}})$, showing that $x_n$ converges to the fixed point of $f$. The paper also includes references to previous work by F. E. Browder and other mathematicians, providing a comprehensive background for the research.