Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

COUPLED FIXED POINT THEOREMS OF SOME CONTRACTION MAPS OF INTEGRAL TYPE ON CONE METRIC SPACES OVER BANACH ALGEBRAS

  • Received : 2021.01.30
  • Accepted : 2021.03.22
  • Published : 2021.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we prove some coupled fixed point theorems satisfying some generalized contractive condition in a cone metric space over a Banach algebra. We also applied the results obtained to show coupled fixed point of some contractive mapping of integral type.

Keywords

Acknowledgement

The second author will like to appreciate the first and third authors for their unique style of mentoring. Also we would like to appreciate the reviewers for their comments and suggestions.

References

  1. A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, International Journal of Mathematics and Mathematical Sciences. 29(9), (2002), 531-536. https://doi.org/10.1155/S0161171202007524
  2. E. Sabetghadam, H. Masiba, A. H. Sanatpour, Some Couple Fixed Point Theorems in Cone Metric Spaces, Fixed Point Theoty And Appl. Vol. (2009), Article ID125426, 8 pages.
  3. F Khojasteh, Z Goodarzi, A Razani, Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Spaces, Fixed Point Theory and Appl. Vol. (2010), Article ID189684, 13 pages.
  4. D Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. TMA 11, (1987) 623-632. https://doi.org/10.1016/0362-546X(87)90077-0
  5. H Liu, S Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory and Appl. (2013), 2013: 320 10 pages. https://doi.org/10.1186/1687-1812-2013-10
  6. H Long-Guang, Z Xian Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332, (2007) 1468-1476. https://doi.org/10.1016/j.jmaa.2005.03.087
  7. J.O. Olaleru, G.A Okeke, H. Akewe, Coupled fixed point Theorems of Integral Type mappings in cone metric spaces, Kragujevac Journal of Mathematics Volume 36 No. 2, (2012), Pages 215-224.
  8. R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60, (1968), 71-76.
  9. S. Xu, S. Radenovic, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl. 2014, 102.
  10. Athasit Wongcharoen, Bashir Ahmad, Sotiris K. Ntouyas, and Jessada Tariboon, Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional, Advances in Mathematical Physics (Hindawi) Volume 2020, Article ID 9606428, 11 pages, https://doi.org/10.1155/2020/9606428
  11. S. C. Lim, M. Li, and L. P. Teo, Langevin equation with two fractional orders, Physics Letters A, vol. 372 no. 42, (2008), pp. 6309-6320. https://doi.org/10.1016/j.physleta.2008.08.045
  12. W. Yukunthorn, S. K. Ntouyas, and J. Tariboon, Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions, Advances in Difference Equations, vol. 2014 no. 1, (2014).
  13. S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3 (1922), 133-181 (French). https://doi.org/10.4064/fm-3-1-133-181
  14. Sh. Rezapour and R. Hamlbarani, Some notes on the paper: "Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, Vol. 345, No. 2, (2008) pp. 719-724. https://doi.org/10.1016/j.jmaa.2008.04.049
  15. S.K Chattterjea "Fixed point Theorems", C.R Acad. Bulgare Sci. 25, (1972), 727-730.
  16. T. G. Bhaskar and V. Lakshmikantham, "Fixed point theorems in partially ordered metric spaces and applications", Nonlinear Analysis: Theory, Methods & Applications, Vol. 65, No. 7, (2006) pp. 1379-1393. https://doi.org/10.1016/j.na.2005.10.017
  17. V. Lakshmikantham and L. Ciric, "Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces," Nonlinear Analysis: Theory, Methods & Applications, Vol. 70, No. 12, (2009) pp. 4341-4349. https://doi.org/10.1016/j.na.2008.09.020
  18. W Rudin Functional Analysis, 2nd edn. McGraw-Hill, New York (1991).
  19. Afif Ben Amar, Aref Jeribi, and Maher Mnif, Some Fixed Point Theorems and Application to Biological Model, Numerical Functional Analysis and Optimization, 29(1-2):1-23, 2008. https://doi.org/10.1080/01630560701749482
  20. Jonathan Eckstein, Dimitri P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming 55 (1992) 293-318 North-Holland. https://doi.org/10.1007/bf01581204
  21. M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat Vesn 66, (2014), 223-234.
  22. H, Akewe, J. Olilima, A. Adeniran, On modified Picard-S-AK Hybrid Iterative algorithm for approximating fixed point of Banach contraction map, MatLab Journal, vol 4(2019), ISSN: 2582-0389
  23. S. Ishikawa. Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44, (1974), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
  24. F. Gursoy, V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, (2014) arXiv:1403.2546v2.
  25. I. Karahan, M. Ozdemir, A general iterative method for approximation of fixed points and their applications. Adv. Fixed Point Theory 3(2013), 510-526.
  26. G.A Okeke, Convergence analysis of the Picard-Ishikawa hybrid iterative process with applications, Afrika Matematika 2019.
  27. G.A Okeke, M. Abbas, A solution of delay differential equations via Picard-Krasnoselskii hybrid iterative process, Arab. J. Math. 6, 21-29 (2017) https://doi.org/10.1007/s40065-017-0162-8
  28. Yekini Shehu, Olaniyi Iyiola. On a Modified Extragradient Method for Variational Inequality Problem with Application to Industrial Production, J. of Ind. and Management Optimization, vol 15 Num. 1, (2019) 319-342 https://doi.org/10.3934/jimo.2018045
  29. A.A Mogbademu, New Iteration process for a general class of contractive mappings, Acta et Commentationes Universitatis Tartuensis de Mathematical, vol 20(2), (2016, 117-122. https://doi.org/10.12697/ACUTM.2016.20.10
  30. S.H Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl (2013) doi:10.1186/1687-1812-2013-69
  31. G. Viglialoro and J. Murcia, A singular elliptic problem related to the membrane equilibrium equations, Int. J. Comput. Math., 90(10) (2013), 2185-2196. https://doi.org/10.1080/00207160.2013.793317
  32. T. Li, N. Pintus, and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys., 70(3) (2019), Art. 86, pp. 1-18. https://doi.org/10.1007/s00033-018-1046-2
  33. T. Li and G. Viglialoro, Analysis and explicit solvability of degenerate tensorial problems, Bound. Value Probl., 2018 (2018), Art. 2, pp. 1-13. https://doi.org/10.1186/s13661-017-0920-8
  34. J. Olilima, A. Mogbademu, A. Adeniran, Strong convergence theorem for uniformly L-Lipschitzian mapping of Gregus type in Banach spaces, Facta Universitatis (NIS) Ser. Math. Inform. Vol. 35, No 5 (2020), 1259-1271