Document Type : Research Paper


1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

2 Faculty of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran.


In this paper, we give a fixed point theorem for $(\psi,\varphi)$-weakly contractive mappings in complete $b$-metric spaces. We also give a common fixed point theorem for such mappings in complete $b$-metric spaces via altering functions. The given results generalize two known results in the setting of metric spaces. Two examples are given to verify the given results.


Main Subjects

[1] Ya.I. Alber and S. Guerre-Delabrere, Principle of weakly contractive maps in Hilbert spaces, New results in operator theory and its applications, Oper. Theory Adv. Appl., 98, Birkhauser, Basel, (1997) 7-22.
[2] I.A. Bakhtin, The contraction mapping principle in almost metric space, Functional analysis, (Russian), Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, (1989) 26-37.
[3] M. Bota, A. Molnar, and C. Varga, On Ekeland's variational principle in b-metric spaces, Fixed Point Theory, 12 (2) (2011), 21-28.  
[4] S. Chandok, A common fixed point result for (μ,Ψ)-weakly contractive mappings, Gulf J. Math. 1 (2013), 65-71.  
[5] S.K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 727-730.  
[6] G. Cortelazzo, G. Mian, G. Vezzi, and P. Zamperoni, Trademark shapes description by string matching techniques, Pattern Recognit. 27 (8) (1994), 1005-1018.  
[7] S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993), 5-11.
[8] S. Czerwik, K. Dlutek, and S.L. Singh, Round-off stability of iteration procedures for operators in b-metric spaces, J. Natur. Phys. Sci. 11 (1997), 87-94.  
[9] P.N. Dutta and B.S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl., Vol. 2008, (2008), 1-8. Article ID 406368.
[10] R. Fagin and L. Stockmeyer, Relaxing the triangle inequality in pattern matching, Int. J. Comput. Vis. 30 (3) (1998), 219-231.  
[11] H. Faraji, K. Nourouzi, and D. O'Regan, A fixed point theorem in uniform spaces generated by a family of b-pseudometrics, Fixed Point Theory, (to appear).  
[12] N. Hussain and M.H. Shah, KKM mappings in cone $b$-metric spaces, Comput. Math. Appl. 62 (4)  (2011), 1677-1684.
[13] M.A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73 (9) (2010),  3123-3129.
[14] M.S. Khan, M. Swaleh, and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30 (1) (1984), 1-9.  
[15] W. Kirk and N. Shahzad, Fixed point theory in distance spaces, Springer, 2014.  
[16] R. McConnell, R. Kwok, J. Curlander, W. Kober, and S. Pang, Ψ-S correlation and dynamic time warping: two methods for tracking ice floes, IEEE Trans. Geosci. Remote Sens. {29}(6), (1991), 1004-1012.  
[17] Z. Mustafa, J.R. Roshan, V. Parvaneh, and Z. Kadelburg, Fixed point theorems for weakly $T$-Chatterjea and weakly T-Kannan contractions in b-metric spaces, J. Inequal. Appl. 2014 (46) (2014), 14 pp.
[18] A. Petrusel, G. Petrusel, B. Samet, and J.-C Yao, Coupled fixed point theorems for symmetric multi-valued contractions in b-metric space with applications to systems of integral inclusions, J. Nonlinear Convex Anal., 17 (7) (2016), 1265-1282.  
[19] A. Petrusel, G. Petrusel, B. Samet, and J.-C. Yao, Coupled fixed point theorems for symmetric contractions in b-metric spaces with applications to operator equation systems, Fixed Point Theory, 17 (2) (2016),  459-478.  
[20] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (4) (2001), 2683-2693.  
[21] W. Sintunavarat, Nonlinear integral equations with new admissibility types in b-metric spaces, J. Fixed Point Theory Appl., 18 (2) (2016),  397-416.  
[22] Q. Xia, The geodesic problem in quasimetric spaces, J. Geom. Anal. 19 (2) (2009), 452-479.