Common Fixed Point Theorem for Hybrid Pair of Mappings in a Generalized $(F,\xi,\eta)$-contraction in weak Partial $b$- Metric Spaces with an Application

Abstract

In the present paper, we proved a common fixed-point theorem for two-hybrid pair of non-self mappings satisfying a generalized $(F, \xi, \eta) $- contraction condition under joint common limit range property in weak partial $b$- metric spaces. Our result is a generalization of many works available in metric space setting. An example and application to the integral equation are given to support the results proved in this paper.

Keywords

• Common fixed point
• weak partial b - metric space
• weak F-contraction
• joint common limit range property
• non-self mappings.

Kaynakça

1. [1] M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270(1) ( 2002) 181-188.
2. [2] E. Ameer, H. Aydi, M. Arshad, H. Alsamir, H. M.S. Noorani, Hybrid multi-valued type contraction mappings in αK- complete partial b-metric spaces and applications, Filomat 31(5) (2019) 1141-1148.
3. [3] A.A. Aserkar, M.P. Gandhi, The Unique Common Fixed Point Theorem for Four Mappings Satisfying Common Limit in the Range Property, Mathematical Analysis I: Approximation Theory: ICRAPAM 2018 New Delhi India 306(161), (2020) 23-25.
4. [4] H. Aydi, M.F. Bota, E. Karapinar, S. Moradi, A common fixed point for weak ϕ-contractions on b-metric spaces, Fixed Point Theory 13(2) (2012) 337-346.
5. [5] H. Aydi, M. Abbas, C. Vetro, Partial Hausdorf and Nadler's fixed point theorem on partial metric space, Topology Appl. 159(14) (2012) 3234-3242.
6. [6] H. Aydi, E. Karapinar, H. Yazidi, Modified F-Contractions via α-Admissible Mappings and Application to Integral Equa- tions, Filomat 31 (5) (2012) 1141-1148.
7. [7] S. Banach, Sur Les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922) 133-181.
8. [8] I. Beg, H.K. Pathak, A variant of Nadler's theorem on weak partial metric spaces with application to homotopy result, Vietnam J. Math. 46 (2018) 693-706.

9. [9] M. Bota, A. Molnar, C.SABA. Varga, On Ekeland's variational principle in b-metric spaces, Fixed Point Theory 12(2) (2011) 21-28.
10. [10] M.F. Bota, E. Karapinar, A note on "Some results on multi-valued weakly Jungck mappings in b-metric space", Cent. Eur. J. Math. 11 (9), (2013) 1711-1712. DOI: 10.2478/s11533-013-0272-2
11. [11] T. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett. 11(1) (1998) 85-88.
12. [12] M. Cosentino, M. Jieli, B. Samet, C. Vetro, Solvability of integrodifferential problems via fixed point theory in b-metric spaces, Fixed Point Theory Appl. (2015) Article ID: 70 1-15.https://doi.org/10.1186/s13663-015-0317-2.
13. [13] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostrav. 1 (1993) 5-11.
14. [14] B.C. Dhage, A non-linear alternative with applications to non-linear perturbed differential equations, Nonlinear Studies 13(4) (2006)343-354.
15. [15] B. C. Dhage, Partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, Tamkang Journal of Mathematics 45(4) (2014) 397-426.
16. [16] K. Goebel, A coincidence theorem, Bull. Acad. Polon. Sci. S6r. Sci. Math. 16 (1968) 733 -735.
17. [17] M. Imdad, A. Ahmad, S. Kumar, On Non-linear non-self hybrid contractions, Rad. Mat. 10(2) (2001) 233-244.
18. [18] M. Imdad, S. Chauhan, P. Kumam, Fixed point theorems for two hybrid pairs of non-self mappings under joint common limit range property in metric spaces, J. Nonlinear Convex Anal. 16 (2) (2015) 243-254.
19. [19] M. Imdad, S. Chauhan, A.H. Soliman, M.A. Ahmed, Hybrid fixed point theorems in symmetric spaces via common limit range property, Demonstratio Mathematica 47(4) ( 2014) 949-962.
20. [20] V. Joshi, D. Singh, A. Petrusel, Existence Results for Integral Equations and Boundary Value Problems via Fixed Point Theorems for Generalized-Contractions in-Metric-Like Spaces, Journal of Function Spaces 2017 (2017) 1-14. [21] T. Kanwal, A. Hussain, P. Kumam, E. Savas, Weak Partial b-Metric Spaces and Nadler's Theorem, Mathematics 7(4) ( 2019) 332.
21. [22] E. Karapinar, A Short Survey on the Recent Fixed Point Results on b-Metric Spaces, Constructive Mathematical Analysis 1(1) ( 2018) 15-44.
22. [23] E. Karapinar, A. Fulga, R.P. Agarwal, A survey: F-contractions with related fixed point results, Journal of Fixed Point Theory and Applications 22(3) ( 2020) 1-58.
23. [24] M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30(1984) 1-9.
24. [25] M.A. Krasnoselski, Topological Methods in the theory of Nonlinear Integral Equations, Pergamon Press Oxford (1964).
25. [26] M.A. Kutbi, E. Karapinar, J. Ahmad, A. Azam, Some fixed point results for multi-valued mappings in b-metric spaces, Journal of Inequalities and Applications 2014(1) 1-11.
26. [27] S. Mathews, Partial metric topology in Papers on General Topology and Applications, Eighth Summer Conference at Queens College, Eds. S. Andima et al., Annals of the New York Academy of Sciences 728 (1994) 183-197.
27. [28] Z. Mustafa, J.R. Roshan, V. Parvanesh, Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, J. Ineq. Appl. 2013(2013) 562.
28. [29] S. A. Naimpally, S.L.J Singh, H.M. Whitfield, Coincidence theorems for hybrid contractions, Math. Nachr. 127 (1986) 177-180.
29. [30] H. K. Nashine, M. Imdad, M. Ahmadullah, Common fixed-point theorems for hybrid generalized (F,ϕ)-contractions under the common limit Range property with applications, Ukrainian Mathematical Journal 69(11) (2018) 1784-1804.
30. [31] H.K. Nashine, M. Imdad, M.D. Ahmadullah, Using (JCLR)-property to prove hybrid fixed point theorems via quasi F-contractions, J. Pure Appl. Math. 11 (1) (2020) 43-56.
31. [32] H.K. Pathak, An Introduction to Nonlinear Analysis and Fixed Point Theory Springer (2018).
32. [33] H. Piri, S. Rahrovi, Generalized multi-valued F-weak contractions on complete metric spaces, Sahand Communications in Mathematical Analysis 2(2) (2015) 1-11.
33. [34] N.A. Secelean, Weak F-contractions and some fixed point results, Bulletin of the Iranian Mathematical Society 42(3) (2016) 779-798.
34. [35] O.G. Selma Gulyaz, On some α-admissible contraction mappings on Branciari b-metric spaces, Advances in the Theory of Nonlinear Analysis and its Applications 1(1) 1-13 (2017) Article Id: 2017.
35. [36] S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math. 11(2014) 703-711.
36. [37] W. Sintunavarat, P. Kumam, P, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, J. Appl. Math. (2011), Article ID 637958, 1-14.
37. [38] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012, 94.
38. [39] D. Wardowski, N. Van Dung, Fixed points of F-weak contractions on complete metric spaces, Demonstratio Mathematica 47(1) ( 2014) 146-155.