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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Common Fixed Point Theorem for Hardy-Rogers Contractive Type in Cone 2-Metric Spaces and Its Results

Discontinuity, Nonlinearity, and Complexity 12(1) (2023) 197--206 | DOI:10.5890/DNC.2023.03.014

Basel Hardan$^{1}$, Jayashree Patil$^2$, Ahmed A. Hamoud$^3$, Amol Bachhav$^4$

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In this paper, Hardy-Rogers type common fixed point theorem of self contractive maps in cone 2-metric spaces over Banach algebra is proved. The corresponding conclusions in the literature are improved and generalize by obtained results. Some examples proposed to illustrate our main results.


%The authors would like to thank the referees and the editor %of this journal for their valuable suggestions and comments that improved this paper.


  1. [1]  Gahler, S. (1963), 2-metrische Raume und ihre topologi sche strukuren, Mathematische Nachrichten, 26(1-4), 115-148.
  2. [2]  Gahler, S. (1965), Uberdie Uniformisierbarkeit 2-metric sche Raume, Mathematische Nachrichten 28(3-4), 235-244.
  3. [3]  Debnath, P. and Mohan, H. (2020), New extensions of Kannan's and Reich's fixed point theorems for multivalued maps using Wardowski's technique with application to integral equation, Symmetry, 12(7), 1-9.
  4. [4]  Faragi, H. and Nourouzi, K. (2017), A generalization of Kannan and Chatterjea fixed point theorems on complete b-metric spaces, Sahand Communications in Mathematical Analysis, 6(1), 77-86.
  5. [5]  Goswami, N. and Haokip, N. (2019), Some fixed point theorems for generalized Kannan type mappings in b-metric spaces, Antofagasta, 38(4), 763-782.
  6. [6]  Jaroslaw, G. (2017), Fixed point theorems for Kanna type mappings, Journal of Fixed Point Theory and Applications, 19(3), 2145-2152.
  7. [7]  Kumar, S. and Kumar, H. (2018), A generalized fixed point theorem in 2-metric space, International Journal of applied Engineering Research, 13(9), 6935-6937.
  8. [8]  Oztunc, S. and Mutla, A. (2019), Some Kannan type fixed point results in rectangular soft metric space and an application of fixed point for thermal science problem, Thermal Science, 23(1), 215-225.
  9. [9]  Pourmoslemi, A., Rezaei, S., Nazari, T. and Sailmi, M. (2020), Generalizations of Kannan and Reich fixed point theorems using sequentially convergent mappings and subadditive altering distance functions, Mathematics, 8(9), 1-11.
  10. [10]  Rhoades, B. (1979), Contraction type mappings on a 2-metric space, Mathematische Nachrichten, 91(1), 151-155.
  11. [11]  Abbas, M. and Jungck, G. (2008), Common fixed point results for non-commuting mappings without continuity in cone metric spaces, Journal of Mathematical Analysis and Applications, 341(1), 416-420.
  12. [12]  Asadi, M. and Soleimani, H. (2012), Examples in cone metric spaces: A Survey, Middle-East Journal of Scientific Research, 11(12), 1636-1640.
  13. [13]  Long-Huang, H. and Xian, Z. (2007), Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, 332(2), 1468-1476.
  14. [14]  Rezapour, Sh. and Hamlbarani, R. (2008), Some notes on Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, 345(2), 719-724.
  15. [15]  Liu, H. and Shaoyuan, X. (2013), Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory and Applications, 2013(1), 1-10.
  16. [16]  Rodrigues, H. (2009), A note on the relationship between spectral radius and norms of bounded linear operators, Matematica Contemporanea, 36, 131-137.
  17. [17]  Jin, M. and Piao, Y. (2018), Generalizations of Banach-Kannan-Chatterjea type fixed point theorems on non-normed cone metric spaces with Banach algebras, Advances in Fixed Point Theory, 8(1), 68-82.
  18. [18]  Singh, B., Jain, S. and Bhagat, P. (2012), Cone 2-metric space and Fixed point theorem of contractive mappings, Commentationes Matematicae, 52(2), 143-151.
  19. [19]  Tiwari, S. (2015), Cone 2-metric spaces and Fixed point theorems for pair of contractive mappings, Journal of Progressive Research in Mathematic, 5(3), 543-552.
  20. [20]  Tiwari, S. (2015), Cone 2-metric spaces and an extension of fixed point theorems for contractive mappings, IOSR Journal of Mathematics, 11(4), 1-8.
  21. [21]  Agarwal, R. and Karapnar, E. (2013), Remarks on some coupled fixed point theorems in G-metric spaces, Fixed Point Theory and Applications 2013(1), 1-33.
  22. [22]  Al-Mezel, A., Alsulami, H., Karapnar, E. and Roldn, A. (2014), Discussion on Multidimensional coincidence points via recent publications, Abstract and Applied Analysis, 2014(2014), 1-13.
  23. [23]  Karapinar, E., Roldn, A., Shahzad, N. and Sintunaravat, W. (2014), Discussion on coupled and tripled coincidence point theorems for f-contractive mappings without the mixed g-monotone property, Fixed Point Theory and Applications, 2014(1), 1-16.
  24. [24]  Roldn, A., Martnez-Moreno, J., Roldn, C. and Cho, Y. (2914), Multidimensional coincidence point results for compatible mappings in partially ordered fuzzy metric spaces, Fuzzy Sets and Systems, 251, 71-82.
  25. [25]  Banach, S. (1922), Sur les operations dans les ensembles abstraits et leur application aux equations int├ęgrales, Fundamental Mathematics, 3, 133-181.
  26. [26]  Hardy, G. and Rogers, T. (1973), A generalization of a fixed point theorem of Reich, Canadian Mathematical Bulletin, 16(2), 201-206.
  27. [27]  Khammahawong, K. and Kumam, P. (2017), Fixed point theorems for generalized Roger Hardy type F-contraction mappings in a metric-like space with an application to second-order differential equations, Cogent Mathematics, 4(1), 1-20.
  28. [28]  Kumari, P. and Panthi, D. (2016), Connecting various types of cyclic contractions and contractive self-mappings with Hardy-Rogers self-mappings, Fixed Point Theory and Applications, 2016(1), 1-19.
  29. [29]  Patil, J., Hardan, B., Abdo, M., Chaudhari, A. and Bachhav, A. (2020), fixed point theorem for Hardy-Rogers type on generalized fractional differential equations, Advances in the Theory of Nonlinear Analysis and its Applications, 4(4), 407-420.
  30. [30]  Rudin, W. (1991), Functinal Analysis, second edition, McGraw Hill, New York.
  31. [31]  Rangamma, M. and Rama, P. (2016), Hardy and Rogers type Contractive condition and common fixed point theorem in Cone 2-metric space for a family of self-maps, Global Journal of Pure and Applied Mathematics, 12(3), 2375-2383.
  32. [32]  Wang, T., Yin, J. and Yan, Q. (2015), Fixed point theorems on cone 2-metric spaces over Banach algebras and an application, Fixed Point Theory and Applications, 2015(1), 1-13.
  33. [33]  Piao, Y.(2016), Unique common fixed points for two mappings with Kannan-Chatterjea type conditions on cone metric spaces over Banach algebras without normality, Advances in Inequalities and Applications, 2016, 1-16.
  34. [34]  Piaoa, Y. and Xu, S. (2018), Unique Common Fixed Points for Mixed Contractive Mappings on Non-Normal Cone Metric Spaces over Banach Algebras, Faculty of Sciences and Mathematics, Filomat, 32(6), 2067-2079.
  35. [35]  Xu, S. and Radenovic, S. (2014), Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory and Applications, 2014(1), 1-12.
  36. [36]  Kadeburg, Z. and Radenovic, S. (2013), A note on various types of cones and fixed point results in cone metric spaces, Asian Journal of Mathematics and Applications, 2013, 1-7.
  37. [37]  Mlaiki, N., Abodayeh, K., Aydi, H., Abdeljawad, T. and Abuloha, M. (2018), Rectangular Metric-Like Type Spaces and Related Fixed Points, Journal of Mathematics, 2018, 1-8.