Search Results (3)

Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as $(\alpha ,p)$-Liptschitz mappings and studied not only the modified form of the Lipschitz condition, but also the behavior of a finite number of their iterates. The purpose of this paper is to discuss the various types of $(\alpha ,p)$-contractions with cyclic representation that extend the results due to Banach, Kannan, and Chatterjea. Moreover, based on such types of contractions and the property of symmetry, we obtain some related fixed-point results in the setting of metric spaces. Some examples are studied to illustrate the validity of our obtained results. As an application of our results, we establish the existence of the solution to a class of Fredholm integral equations. Full article

In this study, we investigated the relaxant action of galetin 3,6-dimethyl ether (FGAL) on rat aorta. The flavonoid relaxed both PMA‑ and phenylephrine (Phe)-induced contractions (pD₂ = 5.36 ± 0.11 and 4.17 ± 0.10, respectively), suggesting the involvement of PKC and Phe pathways or α₁ adrenergic receptor blockade. FGAL inhibited and rightward shifted Phe-induced cumulative contraction‑response curves, indicating a noncompetitive antagonism of α₁ adrenergic receptors. The flavonoid was more potent in relaxing 30 mM KCl- than 80 mM KCl-induced contractions (pD₂ = 5.50 ± 0.22 and 4.37 ± 0.12). The vasorelaxant potency of FGAL on Phe-induced contraction was reduced in the presence of 10 mM TEA⁺. Furthermore, in the presence of apamin, glibenclamide, BaCl₂ or 4-AP, FGAL-induced relaxation was attenuated, indicating the participation of small conductance calcium-activated K⁺ channels (SK_Ca), ATP-sensitive K⁺ channels (K_ATP), inward rectifier K⁺ channels (K_ir) and voltage-dependent K⁺ channels (K_V), respectively. FGAL inhibited and rightward shifted CaCl₂-induced cumulative contraction-response curves in both depolarizing medium (high K⁺) and in the presence of verapamil and phenylephrine, suggesting inhibition of Ca²⁺ influx through voltage-gated calcium channels (Ca_V) and receptor operated channels (ROCs), respectively. Likewise, FGAL inhibited Phe-induced contractions in Ca²⁺-free medium, indicating inhibition of Ca²⁺ release from the sarcoplasmic reticulum (SR). FGAL potentiated the relaxant effect of aminophylline and sildenafil but not milrinone, suggesting the involvement of phosphodiesterase V (PDE V). Thus, the FGAL vasorelaxant mechanism involves noncompetitive antagonism of α₁ adrenergic receptors, the non-selective opening of K⁺ channels, inhibition of Ca²⁺ influx through Ca_V or ROCs and the inhibition of intracellular Ca²⁺ release. Additionally, there is the involvement of cyclic nucleotide pathway, particularly through PDE V inhibition. Full article

The essential oil of Pectis brevipedunculata (EOPB), a Brazilian ornamental aromatic grass, is characterized by its high content of citral (81.9%: neral 32.7% and geranial 49.2%), limonene (4.7%) and α-pinene (3.4%). Vasodilation induced by EOPB and isolated citral was investigated in pre-contracted vascular smooth muscle, using thoracic aorta from Wistar Kyoto (WKY) rats which was prepared for isometric tension recording. EOPB promoted intense relaxation of endothelium-intact and denuded aortic rings with the concentration to induce 50% of the maximal relaxation (IC₅₀) of 0.044% ± 0.006% and 0.093% ± 0.015% (p < 0.05), respectively. The IC₅₀ values for citral in endothelium-intact and denuded rings were 0.024% ± 0.004% and 0.021% ± 0.004%, respectively (p > 0.05). In endothelium-intact aorta, EOPB-induced vasorelaxation was significantly reduced by L-NAME, a nitric oxide synthase inhibitor. The vasodilator activity of citral was increased in the KCl-contracted aorta and citral attenuated the contracture elicited by Ca²⁺ in depolarized aorta. EOPB and citral elicited vasorelaxation on thoracic aorta by affecting the NO/cyclic GMP pathway and the calcium influx through voltage-dependent L-type Ca²⁺ channels, respectively. Full article