A New Contraction-Type Mapping on a Vectorial Dislocated Metric Space over Topological Modules

Abstract

One recent and prolific direction in the development of fixed point theory is to consider an operator T:X→X defined on a metric space (X,d) which is an F—contraction, i.e., T verifies a condition of type τ+F(d(T(x),T(y))≤F(d(x,y)), for all x,y∈X, T(x)≠T(y), where τ>0 and F:(0,∞)→R satisfies some suitable conditions which ensure the existence and uniqueness for the fixed point of operator T. Moreover, the notion of F-contraction over a metric space (X,d) was generalized by considering the notion of (G,H)—contraction, i.e., a condition of type G(d(Tx,Ty))≤H(d(x,y)), for all x,y∈X, Tx≠Ty for some appropriate G,H:(0,∞)→R functions. Recently, the abovementioned F-contraction theory was extended to the setup of cone metric space over the topological left modules. The principal objective of this paper is to introduce the concept of vectorial dislocated metric space over a topological left module and the notion of A-Cauchy sequence, as a generalization of the classical Cauchy sequence concept. Furthermore, based on the introduced concept, a fixed point result is provided for an operator T:X→X, which satisfies the condition (G,H)—contraction, where G,H are defined on the interior of a solid cone.

Keywords

• vectorial dislocated metric spaces
• topological left modules
• fixed point theorems
• generalized contraction