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We consider the convergence behavior using the relaxed Peaceman-Rachfordsplitting method to solve the monotone inclusion problem $0 \in (A + B)(u)$,where $A, B: \Re^n \rightrightarrows \Re^n$ are maximal $\beta$-stronglymonotone operators, $n \geq 1$ and $\beta > 0$. Under a technical assumption,convergence of iterates using the method on the problem is proved when either$A$ or $B$ is single-valued, and the fixed relaxation parameter $\theta$ liesin the interval $(2 + \beta, 2 + \beta + \min \{ \beta, 1/\beta \})$. With thisconvergence result, we address an open problem that is not settled in [20] onthe convergence of these iterates for $\theta \in (2 + \beta, 2 + \beta + \min\{ \beta, 1/\beta\})$. Pointwise convergence rate results and $R$-linearconvergence rate results when $\theta$ lies in the interval $[2 + \beta, 2 +\beta + \min \{ \beta, 1/\beta\})$ are also provided in the paper. Our analysisto achieve these results is atypical and hence novel. Numerical experiments areconducted on the weighted Lasso minimization problem to test the validity ofthe assumption .

Solution to a Monotone Inclusion Problem using the Relaxed Peaceman-Rachford Splitting Method: Convergence and its Rates