| Summary: | In this article, in the sequel of extending <i>b</i>-metric spaces, we modify controlled metric type spaces via two control functions <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>μ</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> on the right-hand side of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mo>-</mo> </mrow> </semantics> </math> </inline-formula>triangle inequality, that is, <disp-formula> <math display="block"> <semantics> <mrow> <mi>d</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> <mo>≤</mo> <mi>α</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> <mi>d</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> <mo>+</mo> <mi>μ</mi> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> <mi>d</mi> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> <mo>,</mo> <mspace width="3.33333pt"></mspace> <mrow> <mi>for</mi> <mspace width="3.33333pt"></mspace> <mi>all</mi> </mrow> <mspace width="3.33333pt"></mspace> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>X</mi> <mo>.</mo> </mrow> </semantics> </math> </disp-formula> Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula>-nonlinear type contractions in the setting of double controlled metric type spaces.
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