Document Type : Research Paper


1 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

2 Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.


In this paper, via $C$-class functions, as a new class of functions, a fixed theorem in complete $b$-metric spaces is presented. Moreover, we study some results, which are direct consequences of the main results. In addition, as an application, the existence of a solution of an integral equation is given.


Main Subjects

[1] A. Aghajani, M. Abbas, and J.R. Roshan,  Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 4 (2014), pp. 941-960.

[2] A.H. Ansari,  Note on φ-ψ-contractive type mappings and related fixed point, The 2nd regional conference on mathematics and applications, PNU, 2014, pp. 377-380.

[3] A.H. Ansari, S. Chandok, and C. Ionescu,  Fixed point theorems on b-metric spaces for weak contractions with auxiliary functions, J. Inequal. Appl., 429 (2014), pp. 1-17.

[4] H. Aydi, M. Bota, E. Karapinar, and S. Mitrovic,  A fixed point theorem for set-valued quasicontractions in b-metric spaces,  Fixed Point Theory Appl., 88 (2012).

[5] I.A. Bakhtin,  The contraction mapping principle in quasimetric spaces (Russian), Func. An., Gos. Ped. Inst. Unianowsk, 30 (1989), pp. 26-37.

[6] M. Boriceanu,  Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Modern Math., 4 (2009), pp. 285-301.

[7] S. Banach,  Sur les operations dans les ensembles abstraits et leur application aux equations int egrales, Fund. Math., 3 (1922), pp. 133-181.

[8] S. Czerwik,  Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostrav., 1 (1993), pp. 5-11.

[9] S. Czerwik,  Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), pp. 263-276.

[10] Z.M. Fadail, A.G.B. Ahmad, A.H. Ansari, S. Radenovic, and M. Rajovic,  Some common fixed point results of mappings in 0-σ-complete metric-like spaces via new function, Appl. Math. Sci., 9 (2015), pp. 4109-4127.

[11] R.H. Haghi, Sh. Rezapour, and N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal., 74 (2011), pp. 1799-1803.

[12] N. Hussain, V. Parvaneh, J.R. Roshan, and Z. Kadelburg,  Fixed points of cyclic weakly (ψ, φ, L, A, B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl., 256 (2013), pp. 1-18.

[13] M.S. Khan, M. Swaleh, and S. Sessa,  Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), pp. 1-9.