Common fixed points for $\psi$-Geraghty-Jungck contraction type mappings in Branciari b-metric spaces

Abstract

The main purpose of this paper is to define a class of contraction-type pair of mappings, called psi-Geraghty-Jungck contraction pair, which consists of a Jungck pair of mappings satisfying the Geraghty condition and, furthermore, its contractive inequality is controlled by an altering distance function. For this class of mappings, we discuss the existence and uniqueness of its common fixed points under the weakly compatibility property. These mappings are defined in the setting of the so-called Branciari b-metric spaces.

Keywords

• contraction-type mapping
• common fixed point
• weakly compatible mappings.

Kaynakça

1. [1] Z.I. Al-Muhiameed, Z. Mostefaqui and M. Bousselsal, Coincidence and common fixed point thorems for (psi,phi)-weakly contractive mapping in rectangular b-metric spaces, Elec. J. Math. Anal. Appl., 6(2) (2018), 211–220.
2. [2] I.A. Bakhtin, The contraction mapping Principle in almost metrics spaces, Functional Analysis, 30, (1989), 26–37.
3. [3] A. Branciari, A fixed point theorem of Banach–Caccioppoly type on a class of generalized metric space, Publ. Math. Debrecen, 57, (2000), 31–37.
4. [4] H-S. Ding, M. Imdad, S. Radenovic and J. Vujakovic, On some fixed point results in b-metric, rectangular and b-rectangular metric spaces, Arab J. Math. Sci., 22, (2016), 151–164
5. [5] H.S. Ding, V. Ozturk and S. Radenović, On some new fixed point results in b-rectangular metric spaces, J. Nonlinear Sci. Appl., 8, (2015), 378–386.
6. [6] R. George, S. Radenovic, S. Reshma and S. Shukla, Rectangular b-metric spaces and contraction principles, J. Nonlinear Sci. Appl. 8, (2015), 1005-1013.
7. [7] M.A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40, (1973), 604–608.
8. [8] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly, 83(4) (1976), 261–263.

9. [9] G. Jungck and B.E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory, 7(2) (2006) 287–296.
10. [10] M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc., vol. 30, (1984), 1–9.
11. [11] J.R. Morales and E.M. Rojas, Contractive mappings of rational type controlled by minimal requirements functions, Afr. Mat., vol. 27, no. 1-2, (2016), 65–77.
12. [12] J.R. Morales, E.M. Rojas and R.K. Bisht, Common fixed points for pairs of mappings with variable contractive parameters, Abstract and Applied Analysis, Volume 2014, Article ID 209234, 7 pages.
13. [13] J.R. Roshan, V. Parvaneh, Z. Kadelburg and N. Hussain, New fixed point results in b-rectangular metric spaces, Nonlinear Anal. Model. Control, 21(5) (2016), 614–634.
14. [14] T. Suzuki, Generalized metric spaces do not have the compatible topology, Abstract and Applied Analysis Volume 2014, Article ID 458098, 5 pages.