A new contraction condition and its application to weakly singular Volterra integral equations of the second kind

In this paper, we prove some results concerning the existence and uniqueness of solutions for a large class of nonlinear Volterra integral equations of the second kind, especially singular Volterra integral equations, in the Banach space X:=C([0,1])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X:=C([0,1])$$\end{document} consisting of real functions defined and continuous on the interval [0, 1]. The main idea used in the proof is that by using a new contraction condition we can construct a Cauchy sequence in the complete metric space X such that it is convergent to a unique element of this space. Finally, we present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results.