Uniform continuity

Extension (predicate logic)

Pure mathematics

Discrete mathematics

Computer science

Fubini–Study metric

Contraction (grammar)

Metric space

Mathematics

Injective metric space

Internal medicine

Medicine

Programming language

Triangle inequality

Contraction principle

Metric map

In this article, we introduce a new extension of b-metric spaces, called controlled metric type spaces, by employing a control function α ( x , y ) of the right-hand side of the b-triangle inequality. Namely, the triangle inequality in the new defined extension will have the form, d ( x , y ) ≤ α ( x , z ) d ( x , z ) + α ( z , y ) d ( z , y ) , f o r a l l x , y , z ∈ X . Examples of controlled metric type spaces that are not extended b-metric spaces in the sense of Kamran et al. are given to show that our extension is different. A Banach contraction principle on controlled metric type spaces and an example are given to illustrate the usefulness of the structure of the new extension.

Controlled Metric Type Spaces and the Related Contraction Principle