Controlled Metric Type Spaces and the Related Contraction Principle

Abstract

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.

A Mizoguchi–Takahashi Type Fixed Point Theorem in Complete Extended b-Metric Spaces

“…The idea of generalizing metric spaces into b-metric spaces was initiated from the works of Bakhtin [11], Bourbaki [12], and Czerwik [13,14]. In [15], the notion of b-metric space was generalized further by introducing the concept of extended b-metric spaces (see also [16][17][18]) as follows: Definition 1 ( [15]). Let U be a non empty set and φ :…”

Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations

“…Shukla [10] extended the notion of a partial metric to a partial b-metric. Afterwards, numerous research articles have been dealt with fixed point theorems for various classes of single-valued and multi-valued operators in b-metric and partial b-metric spaces (see, for example, [3,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]). In this article, we shall investigate fixed points ofĆirić type rational graphic (Υ, Λ)-contraction pair mappings on partial b-metric spaces endowed with a directed graph G.…”

Fixed Point Results for Multi-Valued Contractions in b−Metric Spaces and an Application

“…The Banach contraction principle is considered to be one of the most useful tools in fixed point theory. It has been extended or generalized in different directions and many (common) fixed point theorems have been provided (see References [1][2][3][4][5][6][7][8][9][10][11][12][13]). The theory of fixed points for multi-valued mappings has been developed after the famous paper of Nadler [14].…”