Double Controlled Metric Type Spaces and Some Fixed Point Results
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> <m...
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doaj-3ad81981c52246909db7ff2fb65761292020-11-24T20:44:36ZengMDPI AGMathematics2227-73902018-12-0161232010.3390/math6120320math6120320Double Controlled Metric Type Spaces and Some Fixed Point ResultsThabet Abdeljawad0Nabil Mlaiki1Hassen Aydi2Nizar Souayah3Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi ArabiaDepartment of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi ArabiaDepartment of Mathematics, College of Education in Jubail, Imam Abdulrahman Bin Faisal University, P.O. Box 12020, Jubail 31961, Saudi ArabiaDepartment of Natural Sciences, Community College Al-Riyadh, King Saud University, Riyadh 4545, Saudi ArabiaIn 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.https://www.mdpi.com/2227-7390/6/12/320fixed pointa double controlled metric typeextended <i>b</i>-metric space |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Thabet Abdeljawad Nabil Mlaiki Hassen Aydi Nizar Souayah |
spellingShingle |
Thabet Abdeljawad Nabil Mlaiki Hassen Aydi Nizar Souayah Double Controlled Metric Type Spaces and Some Fixed Point Results Mathematics fixed point a double controlled metric type extended <i>b</i>-metric space |
author_facet |
Thabet Abdeljawad Nabil Mlaiki Hassen Aydi Nizar Souayah |
author_sort |
Thabet Abdeljawad |
title |
Double Controlled Metric Type Spaces and Some Fixed Point Results |
title_short |
Double Controlled Metric Type Spaces and Some Fixed Point Results |
title_full |
Double Controlled Metric Type Spaces and Some Fixed Point Results |
title_fullStr |
Double Controlled Metric Type Spaces and Some Fixed Point Results |
title_full_unstemmed |
Double Controlled Metric Type Spaces and Some Fixed Point Results |
title_sort |
double controlled metric type spaces and some fixed point results |
publisher |
MDPI AG |
series |
Mathematics |
issn |
2227-7390 |
publishDate |
2018-12-01 |
description |
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. |
topic |
fixed point a double controlled metric type extended <i>b</i>-metric space |
url |
https://www.mdpi.com/2227-7390/6/12/320 |
work_keys_str_mv |
AT thabetabdeljawad doublecontrolledmetrictypespacesandsomefixedpointresults AT nabilmlaiki doublecontrolledmetrictypespacesandsomefixedpointresults AT hassenaydi doublecontrolledmetrictypespacesandsomefixedpointresults AT nizarsouayah doublecontrolledmetrictypespacesandsomefixedpointresults |
_version_ |
1716816825328599040 |