Document Type : Research Paper

Authors

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.

Abstract

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.

Keywords

Main Subjects

References
[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.