Double-Composed Metric Spaces

The double-controlled metric-type space ( X , D ) is a metric space in which the triangle inequality has the form D ( η , μ ) ≤ ζ 1 ( η , θ ) D ( η , θ ) + ζ 2 ( θ , μ ) D ( θ , μ ) for all η , θ , μ ∈ X . The maps ζ 1 , ζ 2 : X × X → [ 1 , ∞ ) are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form D ( η , μ ) ≤ α D ( η , θ ) + β D ( θ , μ ) for all η , θ , μ ∈ X . In our new space, the control functions α , β : [ 0 , ∞ ) → [ 0 , ∞ ) are composed of the metric D in the triangle inequality, where the control functions ζ 1 , ζ 2 : X × X → [ 1 , ∞ ) in a double-controlled metric-type space are multiplied with the metric D . We establish some fixed-point theorems along with the examples and applications.