In this paper, we provide explicit formulas for the exact inverses of the symmetric tridiagonal near-Toeplitz matrices characterized by weak diagonal dominance in the Toeplitz part. Furthermore, these findings extend to scenarios where the corners of the near Toeplitz matrices lack diagonal dominance ($|\widetilde{b}| < 1$). Additionally, we compute the row sums and traces of the inverse matrices, thereby deriving upper bounds for their infinite norms. To demonstrate the practical applicability of our theoretical results, we present numerical examples addressing numerical solution of the Fisher problem using the fixed point method. Our findings reveal that the convergence rates of fixed-point iterations closely align with the expected rates, and there is minimal disparity between the upper bounds and the infinite norm of the inverse matrix. Specifically, this observation holds true for $|b| = 2$ with $|\widetilde{b}| \geq 1$. In other cases, there exists potential to enhance the obtained upper bounds.

翻译：本文针对Toeplitz部分具有弱对角占优特性的对称三对角近Toeplitz矩阵，给出了其精确逆矩阵的显式表达式。此外，这些结论可进一步推广至近Toeplitz矩阵角部元素不满足对角占优条件（$|\widetilde{b}| < 1$）的情形。我们还计算了逆矩阵的行和与迹，由此推导出其无穷范数的上界。为验证理论结果的实际应用价值，我们通过数值算例展示了采用不动点法求解Fisher问题的数值解。研究发现，不动点迭代的收敛速率与预期速率高度吻合，且逆矩阵无穷范数与所得上界之间的差异极小。特别地，当$|b| = 2$且$|\widetilde{b}| \geq 1$时，这一现象尤为显著。在其他情况下，所得上界仍存在进一步优化的空间。