Abstract
In this paper, we introduce a new extension of the double controlled metric-type spaces, called double controlled metric-like spaces, by assuming that the “self-distance” may not be zero On the other hand, if the value of the metric is zero, then it has to be a “self-distance” (i.e., we replace [varsigma(g,h)=0 Leftrightarrow g=h] by [varsigma(g,h)=0 Rightarrow g=h]). Using this new type of metric spaces, we generalize many results in the literature. We prove fixed point results along with examples illustrating our theorems. Also, we present double controlled metric-like spaces endowed with a graph along with an open question.
Highlights
In 1922, Banach [1] proved the existence and uniqueness of a fixed point for selfcontractive mapping in metric spaces
That was the starting point for researchers in the field of analysis to generalize his result, whether by changing the contractions or by generalizing the type of metric spaces covering a wider class of metrics, for example, extension of metric spaces to partial metric spaces or b-metric spaces; see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
One interesting extension of metric spaces is b-metric spaces introduced by Bakhtin [18]
Summary
In 1922, Banach [1] proved the existence and uniqueness of a fixed point for selfcontractive mapping in metric spaces. Suppose that a function ς : F × F → R+ satisfies the following conditions for all g, h, w ∈ F : (1) ς(g, h) = 0 ⇐⇒ g = h; Mlaiki Journal of Inequalities and Applications The pair (F , ς) is called a double controlled metric-like space (DCMLS).
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