Some Fixed Point Theorems in Menger Probabilistic Partial Metric Spaces with Application to Volterra Type Integral Equation

Main Article Content

Amir ‎Ghanenia
Mahnaz khanehgir
Reza Allahyari
Mohammad ‎Mehrabinezhad

Abstract

In this paper, we introduce the notion of Menger probabilistic partial metric space and prove some fixed point theorems in the framework of such spaces. Some examples and an application to Volterra type integral equation are given to support the obtained results. Finally, we apply successive approximations method to find a solution for a Volterra type integral equation with high accuracy.

Article Details

References

  1. T. Abdeljawada, E. Karapinar and K. Tas, Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett. (2011), 1-5.
  2. W. Abdul-Majid, Linear and Nonlinear Integral Equations: Methods and Applications, Springer, 2011.
  3. H. Aydi, E. Karapinar and S. Rezapour, A Generalized Meir-Keeler contraction on partial metric spaces, Abstr. Appl. Anal. 2012 (2012), Article ID 287127.
  4. N.A. Babacev, Nonlinear generalized contraction on Menger PM-spaces, Appl. Anal. Discrete Math. 6 (2012), 257-264.
  5. M. Bukatin, R. Kopperman, S. Matthews and H. Pajoohesh, Partial metric spaces, Am. Math. Montly, 116 (2009), 708-718.
  6. S. Chauhan, S. Bhatnagar and S. Radenovic, Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces, Le Mathematiche, LXVIII (2013)-Fasc. I, 87-98.
  7. S. Chauhan, M. Imdad, C. Vetro and W. Sintunavarat, Hybrid coincidence and common fixed point theorems in Menger probabilistic metric spaces under a strict contractive condition with an application, Appl. Math. Comput. 239 (2014), 422-433.
  8. T. Dosenovic, D. Rakic, B. Caric and S. Radenovic, Multivalued generalizations of fixed point results in fuzzy metric spaces, Nonlinear Anal., Model. Control, 21(2) (2016), 211-222.
  9. P.N. Dutta, B.S. Choudhury and K.P. Das, Some fixed point results in Menger spaces using a control function, Surv. Math. Appl. 4 (2009), 41-52.
  10. D. Gopal, M. Abbas and C. Vetro, Some new fixed point theorems in Menger PM-spaces with application to Volterra type integral equation, Appl. Math. Comput. 232 (2014), 955-967.
  11. L. Grammont, Nonlinear integral equations of the second kind: A new version of Nystr ¨om method. Numer. Funct. Anal. Optim. 34(5) (2013), 496-515.
  12. O. Hadzic and E. Pap, Fixed point theory in probabilistic metric spaces, Kluwer Academic Publishers, 2001.
  13. F. Hasanvand and M. Khanehgir, Some fixed point theorems in Menger P bM-spaces with an application, Fixed Point Theory Appl. 2015 (2015), 81.
  14. S.G. Matthews, Partial metric topology, Proc. 8th Summer Conference on General Topology and Applications. Ann. N.Y. Acad. Sci. 728 (1994), 183-197.
  15. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA, 28 (1942), 535-537.
  16. Z. Mustafa, J. Rezaei Roshan, V. Parvaneh and Z. Kadelburg, Some common fixed point results in orderd partial b-metric spaces, J. Inequal. Appl. 2013 (2013), 562.
  17. P. Patle, D. Patel, H, Aydi and S. Radenovi ´c, On H+type multivalued contraction and its applications in symmetric and probabilistic spaces, Mathematics, 7 (2019), 144.
  18. M. Rabbani and R. Arab, Extension of some theorems to find solution of nonlinear integral equation and homotopy perturbation method to solve it, Math. Sci. 11(2) (2017), 87-94.
  19. S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl. 2010 (2010), Article ID 493298, 6 pages.
  20. B. Schweizer and S. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 313-334.
  21. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, North-Holland, New York, 1983.
  22. S. Sedghi, N. Shobkolaei, T. Dosenovi ´c and S. Radenovic, Suzuki-type of common fixed point theorems in fuzzy metric spaces, Math. Slovaca 68(2) (2018), 451-462.
  23. V.M. Sehgal and A.T. Bharucha-Reid, Fixed point of contraction mappings in PM-spaces, Math. Syst. Theory, 6 (1972), 97-102.
  24. Y. Shi, L. Ren and X. Wang, The extension of fixed point theorems for set valued mapping, J. Appl. Math. Comput. 13 (2003), 277-286.
  25. Stevens, Metrically generated PM-spaces, Fund. Math. (1968), 259-269.
  26. Y. Su and J. Zhang, Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces, Fixed Point Theory Appl. 2014 (2014), 170.