This paper is concerned with the solution for impulsive differential equations with Hadamard fractional derivatives. The general solution of this impulsive fractional system is found by considering the limit case in which impulses approach zero. Next, an example is provided to expound the theoretical result.

Motivated by some preliminary works about general solution of impulsive system with fractional derivative, the generalized impulsive differential equations with Caputo-Hadamard fractional derivative ofq∈C  (R(q)∈(1,2)) are further studied by analyzing the limit case (as impulses approach zero) in this paper. The formulas of general solution are found for the impulsive systems.

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Previous studies have suggested that the herbal medicine simiaosan has beneficial effects on gouty arthritis (GA), for which conventional Western medicines are insufficient (particularly in cases of multiple episodes). The objective of the present study was to investigate the mechanism by which simiaosan alleviated the symptoms of GA. Sprague-Dawley rat models of acute GA were successfully established, as verified by pathological analyses. Additionally, an NLR family pyrin domain containing 3 (NLRP3) overexpression vector was constructed and a high transfection efficiency was confirmed by reverse transcription PCR. The following five treatment groups were established: i) Normal control; ii) model + saline; iii) model + simiaosan; iv) model + NALP3-overexpressing adenovirus + simiaosan; and v) model + empty vector adenovirus + simiaosan. The samples from mice in each group were subjected to hematoxylin and eosin (H&E) staining for assessing the histopathological changes, enzyme-linked immunosorbent assays for determining IL-1β and TGF-β1 levels and western blotting for evaluating NALP3 expression. H&E staining indicated that simiaosan could reduce the infiltration of inflammatory cells, while NALP3 overexpression aggravated the inflammatory response in tissues. Expression levels of IL-1β, TGF-β1 and NALP3 were significantly higher in the model and the model + NALP3-overexpressing adenovirus + simiaosan groups compared with the normal control group. Levels of IL-1β, TGF-β1 and NALP3 were significantly lower in the model + simiaosan and model + empty vector adenovirus + simiaosan groups compared with the model group. These results indicated that the effects of simiaosan were mediated through NALP3 inhibition. Therefore, the herbal medicine simiaosan was revealed to possess an ability to alleviate the symptoms of GA by regulating the NALP3/IL-1β signaling pathway.

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