Third-order family of methods in Banach spaces

Recently, Parida and Gupta [P.K. Parida, D.K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. Appl. Math., 206 (2007) 873-877] used Rall's recurrence relation approach (from 1961) to approximate roots of nonlinear equation, by developing several methods, the latest of which is free of second derivative and it is of third order. In this paper, we use an idea of Kou and Li [J.-S. Kou, Y._T. Li, Modified Chebyshev's method free from second derivative for non-linear equations, Appl. MAth. Comput. 187. (2007), 1027-1032] and modify the approach of Parida and Cupta, obtaining yet another third order method to approximate a solution of a nonlinear equation in a Banach space. We give several applications to our method

Endothelial dysfunction – a major mediator of diabetic vascular disease

The vascular endothelium is a multifunctional organ and is critically involved in modulating vascular tone and structure. Endothelial cells produce a wide range of factors that also regulate cellular adhesion, thromboresistance, smooth muscle cell proliferation, and vessel wall inflammation. Thus, endothelial function is important for the homeostasis of the body and its dysfunction is associated with several pathophysiological conditions, including atherosclerosis, hypertension and diabetes. Patients with diabetes invariably show an impairment of endothelium-dependent vasodilation. Therefore, understanding and treating endothelial dysfunction is a major focus in the prevention of vascular complications associated with all forms of diabetes mellitus. The mechanisms of endothelial dysfunction in diabetes may point to new management strategies for the prevention of cardiovascular disease in diabetes. This review will focus on the mechanisms and therapeutics that specifically target endothelial dysfunction in the context of a diabetic setting. Mechanisms including altered glucose metabolism, impaired insulin signaling, low-grade inflammatory state, and increased reactive oxygen species generation will be discussed. The importance of developing new pharmacological approaches that upregulate endothelium-derived nitric oxide synthesis and target key vascular ROS-producing enzymes will be highlighted and new strategies that might prove clinically relevant in preventing the development and/or retarding the progression of diabetes associated vascular complications