Contractive Fixed Points in a Rectangular Metric Space and Applications

Branciari (2000) introduced the notion of a rectangular metric space ({\it rms}) as a generalization of a metric space and proved the well-known Banach's contraction mapping theorem in an {\it rms}, which was further generalized by Sarma et al. (2009) through a Ciric contraction. A fixed point $p$ of a self-map $f$ is a contractive fixed point (Edelstein, 1962), provided all the Picard's iterates $x,fx,f^2x, \ldots $ converge to $p$. In the first part of the paper, contractive fixed points of Banach and Ciric contractions are established in a rectangular metric space. Usually, it is shown that an appropriate Picard's iterative sequence with an arbitrary seed converges to a point, which is a unique fixed point. Rather than relying on the standard iterative procedure, in the next part of the paper, unique fixed points are obtained for Banach, Hardy-Roger and Ciric's contractions in a rectangular metric space through the rectangle inequality and the greatest lower bound property of real numbers. In the last part of the paper, two elegant problems of Volterra integral equations are presented with the necessary MATLAB interpretation.