d-Neighborhood system and generalized F-contraction in dislocated metric space (original) (raw)

Background Metrics appear everywhere in Mathematics: Geometry, Probability, statistics, coding theory, graph theory, pattern recognition, networks, computer graphics, molecular biology, theory of information and computer semantics are some of the fields in which metrics and/or their cousins play a significant role. The notion of metric spaces introduced by Frechet (1906), is one of the helpful topic in Analysis. Banach (1922) proved a fixed point theorem for contraction mapping in a complete metric space. The Banach contraction theorem is one of the primary result of functional analysis. After Banach contraction theorem, huge number of fixed point theorems have been established by various authors and they made different generalizations of this theorem. Matthews (1985) generalized Banach contraction mapping theorem in dislocated metric space. Hitzler (2001) introduce the notion of dislocated metric (d-metric) space and presented variants of Banach contraction principle for various modified forms of a metric space including dislocated metric space and applied them to semantic analysis of logic programs. Hitzler (2001) has applied fixed point theorems for self maps on dislocated metric spaces, quasi dislocated metric spaces, generalized ultra metric spaces in his thesis "Generalized Metrics and Topology in Logic Programming Semantics". In this context, Hitzler raised some related questions on the topological aspects of dislocated metrics. Recently, Sarma and Kumari (2012) initiated the concept of d-balls and established topological properties on d-metric space. In the context of d-metric space,