Let C and Q be nonempty closed convex sets in N and M , respectively, and A an M × N real matrix. The split feasibility problem (SFP) is to find x ∈ C with Ax ∈ Q, if such x exist. Byrne (2002 Inverse Problems 18 441-53) proposed a CQ algorithm with the following iterative scheme:

Patients with Parkinson's disease (PD) were often observed with gastrointestinal symptoms, which preceded the onset of motor symptoms. Neuropathology of PD has also been found in the enteric nervous system (ENS). Many studies have reported significant PD-related alterations of gut microbiota. This meta-analysis was performed to evaluate the differences of gut microbiota between patients with PD and healthy controls (HCs) across different geographical regions. We conducted a systematic online search for case-control studies detecting gut microbiota in patients with PD and HCs. Mean difference (MD) and 95% confidence interval (CI) were calculated to access alterations in the abundance of certain microbiota families in PD. Fifteen case-control studies were included in this meta-analysis study. Our results showed significant lower abundance levels of Prevotellaceae (MD = −0.37, 95% CI = −0.62 to −0.11), Faecalibacterium (MD = −0.41, 95% CI: −0.57 to −0.24), and Lachnospiraceae (MD = −0.34, 95% CI = −0.59 to −0.09) in patients with PD compared to HCs. Significant higher abundance level of Bifidobacteriaceae (MD = 0.38, 95%; CI = 0.12 to 0.63), Ruminococcaceae (MD = 0.58, 95% CI = 0.07 to 1.10), Verrucomicrobiaceae (MD = 0.45, 95% CI = 0.21 to 0.69), and Christensenellaceae (MD = 0.20, 95% CI = 0.07 to 0.34) was also found in patients with PD. Thus, shared alterations of certain gut microbiota were detected in patients with PD across different geographical regions. These PD-related gut microbiota dysbiosis might lead to the impairment of short-chain fatty acids (SCFAs) producing process, lipid metabolism, immunoregulatory function, and intestinal permeability, which contribute to the pathogenesis of PD.

Absolute value equations, Smoothing Newton method, Global convergence, Convergence rate,

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