Double-Composed Metric Spaces
The double-controlled metric-type space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi mathvariant="script">D</mi&g...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/8/1866 |
| _version_ | 1827744605132554240 |
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| author | Irshad Ayoob Ng Zhen Chuan Nabil Mlaiki |
| author_facet | Irshad Ayoob Ng Zhen Chuan Nabil Mlaiki |
| author_sort | Irshad Ayoob |
| collection | DOAJ |
| description | The double-controlled metric-type space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi mathvariant="script">D</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a metric space in which the triangle inequality has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>≤</mo><msub><mi>ζ</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>ζ</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>. The maps <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ζ</mi><mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>≤</mo><mi>α</mi><mfenced separators="" open="(" close=")"><mi mathvariant="script">D</mi><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mfenced><mo>+</mo><mi>β</mi><mfenced separators="" open="(" close=")"><mi mathvariant="script">D</mi><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mfenced></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>. In our new space, the control functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>:</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula> are composed of the metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">D</mi></semantics></math></inline-formula> in the triangle inequality, where the control functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ζ</mi><mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in a double-controlled metric-type space are multiplied with the metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">D</mi></semantics></math></inline-formula>. We establish some fixed-point theorems along with the examples and applications. |
| first_indexed | 2024-03-11T04:47:11Z |
| format | Article |
| id | doaj.art-f04939c9a6634a7e88dea87b21f3b5c6 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:47:11Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f04939c9a6634a7e88dea87b21f3b5c62023-11-17T20:17:41ZengMDPI AGMathematics2227-73902023-04-01118186610.3390/math11081866Double-Composed Metric SpacesIrshad Ayoob0Ng Zhen Chuan1Nabil Mlaiki2Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi ArabiaSchool of Mathematics, Universiti Sains Malaysia, Gelugor 11800, Penang, MalaysiaDepartment of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi ArabiaThe double-controlled metric-type space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi mathvariant="script">D</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a metric space in which the triangle inequality has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>≤</mo><msub><mi>ζ</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>ζ</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>. The maps <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ζ</mi><mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>≤</mo><mi>α</mi><mfenced separators="" open="(" close=")"><mi mathvariant="script">D</mi><mo>(</mo><mi>η</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mfenced><mo>+</mo><mi>β</mi><mfenced separators="" open="(" close=")"><mi mathvariant="script">D</mi><mo>(</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mfenced></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>μ</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>. In our new space, the control functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>:</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula> are composed of the metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">D</mi></semantics></math></inline-formula> in the triangle inequality, where the control functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ζ</mi><mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in a double-controlled metric-type space are multiplied with the metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">D</mi></semantics></math></inline-formula>. We establish some fixed-point theorems along with the examples and applications.https://www.mdpi.com/2227-7390/11/8/1866b-metric spacescontrolled metric spacesdouble-controlled metric-type spacesfixed pointdouble-composed metric spaces |
| spellingShingle | Irshad Ayoob Ng Zhen Chuan Nabil Mlaiki Double-Composed Metric Spaces Mathematics b-metric spaces controlled metric spaces double-controlled metric-type spaces fixed point double-composed metric spaces |
| title | Double-Composed Metric Spaces |
| title_full | Double-Composed Metric Spaces |
| title_fullStr | Double-Composed Metric Spaces |
| title_full_unstemmed | Double-Composed Metric Spaces |
| title_short | Double-Composed Metric Spaces |
| title_sort | double composed metric spaces |
| topic | b-metric spaces controlled metric spaces double-controlled metric-type spaces fixed point double-composed metric spaces |
| url | https://www.mdpi.com/2227-7390/11/8/1866 |
| work_keys_str_mv | AT irshadayoob doublecomposedmetricspaces AT ngzhenchuan doublecomposedmetricspaces AT nabilmlaiki doublecomposedmetricspaces |