To model complex systems with discrete-time features and memory effects in the uncertain environment, a definition of an uncertain fractional forward difference equation with Riemann-Liouville-like forward difference is introduced. Moreover, analytic solutions to a type of special linear uncertain fractional difference equations are presented by the Picard iteration method. Then, an existence and uniqueness theorem of the solutions is proved by applying Banach contraction mapping theorem. Finally, two examples are provided to illustrate the validity of the existence and uniqueness theorem.

We evaluated how metabolic disorders affected antihypertension therapy. 2,912 rural Chinese patients with hypertension who provided blood samples, demographic and clinical data at baseline and after 1 year of antihypertension therapy were evaluated. At baseline, 1,515 patients (52.0%) were already receiving drug therapy and 11.4% of them had controlled blood pressure (BP). After 1 year, all 2,912 patients were receiving antihypertension therapy that was administered by community physicians, and 59.42% of them had controlled BP. Central obesity and abnormal triglyceride, high-density lipoprotein cholesterol, and glucose were associated with 15–70% higher risks of uncontrolled hypertension. Metabolic syndrome using the JIS criteria was associated with poor BP control (odds ratio: 1.71 and 1.54 for the baseline and follow-up datasets, respectively). The risk of uncontrolled hypertension increased with the number of metabolic disorders (p for trend <0.01). The presence of ≥3 metabolic disorder factors was associated with higher risks of poor BP control. The associations of metabolic factors and uncontrolled hypertension were stronger for the standard and modified ATP III criteria, compared to the IDF and JIS criteria. Metabolic factors were associated with less effective antihypertension therapy, and all definitions of metabolic syndrome helped identify patients with elevated risks of uncontrolled hypertension.

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Uncertain fractional differential equation driven by Liu process plays a significant role in depicting the memory effects of uncertain dynamical systems. This paper mainly investigates the stability problems for the Caputo type of uncertain fractional differential equations with the order [Formula: see text]. The concept of stability in measure of solutions to uncertain fractional differential equation is proposed based on uncertainty theory. Several sufficient conditions for ensuring the stability of the solutions are derived, respectively, in which the systems are divided into two cases with order [Formula: see text] and [Formula: see text]. Some illustrative examples are performed to display the effectiveness of the proposed results.

In this paper, the finite-time stability in mean for the uncertain fractional order linear time-invariant discrete systems is investigated. First, the uncertain fractional order difference equations with the nabla operators are introduced. Then, some conditions of finite-time stability in mean for the systems driven by the nabla uncertain fractional order difference equations with the fractional order [Formula: see text] are obtained by the property of Riemann–Liouville-type nabla difference and the generalized Gronwall inequality. Furthermore, based on these conditions, the state feedback controllers are designed. Finally, some examples are presented to illustrate the effectiveness of the results.