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Controlled Metric Type Spaces and the Related Contraction Principle

Author

Listed:
  • Nabil Mlaiki

    (Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia)

  • Hassen Aydi

    (Department of Mathematics, College of Education in Jubail, Imam Abdulrahman Bin Faisal University, P. O. 12020, Industrial Jubail 31961, Saudi Arabia)

  • Nizar Souayah

    (Department of Natural Sciences, Community College Al-Riyadh, King Saud University, Riyadh 11451, Saudi Arabia
    ESSEC Tunis, University of Tunis, Tunis 2058, Tunisia)

  • Thabet Abdeljawad

    (Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia)

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 ) , for all 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.

Suggested Citation

  • Nabil Mlaiki & Hassen Aydi & Nizar Souayah & Thabet Abdeljawad, 2018. "Controlled Metric Type Spaces and the Related Contraction Principle," Mathematics, MDPI, vol. 6(10), pages 1-7, October.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:10:p:194-:d:174224
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    Citations

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    Cited by:

    1. Nayab Alamgir & Quanita Kiran & Hassen Aydi & Aiman Mukheimer, 2019. "A Mizoguchi–Takahashi Type Fixed Point Theorem in Complete Extended b -Metric Spaces," Mathematics, MDPI, vol. 7(5), pages 1-15, May.
    2. Naeem Saleem & Salman Furqan & Kinda Abuasbeh & Muath Awadalla, 2023. "Fuzzy Triple Controlled Metric like Spaces with Applications," Mathematics, MDPI, vol. 11(6), pages 1-30, March.
    3. Abdullah Eqal Al-Mazrooei & Jamshaid Ahmad, 2022. "Fixed Point Results in Controlled Metric Spaces with Applications," Mathematics, MDPI, vol. 10(3), pages 1-15, February.
    4. Gunaseelan Mani & Salma Haque & Arul Joseph Gnanaprakasam & Ozgur Ege & Nabil Mlaiki, 2023. "The Study of Bicomplex-Valued Controlled Metric Spaces with Applications to Fractional Differential Equations," Mathematics, MDPI, vol. 11(12), pages 1-19, June.
    5. Amiri, Pari & Afshari, Hojjat, 2022. "Common fixed point results for multi-valued mappings in complex-valued double controlled metric spaces and their applications to the existence of solution of fractional integral inclusion systems," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    6. Hasanen A. Hammad & Mohra Zayed, 2022. "New Generalized Contractions by Employing Two Control Functions and Coupled Fixed-Point Theorems with Applications," Mathematics, MDPI, vol. 10(17), pages 1-18, September.
    7. Umar Ishtiaq & Doha A. Kattan & Khaleel Ahmad & Salvatore Sessa & Farhan Ali, 2023. "Fixed Point Results in Controlled Fuzzy Metric Spaces with an Application to the Transformation of Solar Energy to Electric Power," Mathematics, MDPI, vol. 11(15), pages 1-17, August.
    8. Irshad Ayoob & Ng Zhen Chuan & Nabil Mlaiki, 2023. "Double-Composed Metric Spaces," Mathematics, MDPI, vol. 11(8), pages 1-12, April.

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