Abstract
AbstractThe aim of this paper to present fixed point results for single-valued operators in b-metric spaces. The case of scalar metric and the case of vector-valued metric approaches are considered. As an application, a system of integral equations is studied.
Highlights
Introduction and preliminariesIt is well known that Banach’s contraction principle for single-valued contractions was extended to several types of generalized metric spaces.An interesting extension to the case of spaces endowed with vector-valued metrics was done by Perov [ ]
Many other contributions on this topic are known ; see, for example, [ – ]. Another extension of the Banach contraction principle was given for the case of socalled b-metric spaces, starting with some results given by Czerwik; see [ ]
The concept of coupled fixed point and the study of coupled fixed point problems appeared for the first time in some papers of Opoitsev, while the topic expanded with the work of Guo and Lakshmikantham, where the monotone iterations technique is exploited
Summary
Introduction and preliminariesIt is well known that Banach’s contraction principle for single-valued contractions was extended to several types of generalized metric spaces.An interesting extension to the case of spaces endowed with vector-valued metrics was done by Perov [ ]. If (X, d) is a metric space and T : X × X → X is an operator, by definition, a coupled fixed point for T is a pair (x∗, y∗) ∈ X × X satisfying x∗ = T(x∗, y∗), y∗ = T(y∗, x∗). The theory of coupled fixed points in the setting of an ordered metric space and under some contractive type conditions on the operator T was re-considered by Gnana Bhaskar and Lakshmikantham in [ ] (see Lakshmikantham and Ćirić in [ ]).
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