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 some results on the common fixed point of self-mappings defined on complete $b$-metric spaces. Our results generalize Kannan and Chatterjea fixed point theorems on complete $b$-metric spaces. In particular, we show that two self-mappings satisfying a contraction type inequality have a unique common fixed point. We also give some examples to illustrate the given results.
 H. Aydi, M.F. Bota, E. Karapinar, and S. Moradi, A common fixed point for weak $phi$-contractions on $b$-metric spaces, Fixed Point Theory, 13 (2012), no. 2, 337-346.
 I.A. Bakhtin, The contraction mapping principle in almost metric space, Functional analysis, (Russian), Ulýanovsk. Gos. Ped. Inst., Ulýanovsk, (1989) 26-37.
 M.F. Bota and E. Karapinar, A note on "Some results on multi-valued weakly Jungck mappings in b-metric space", Cent. Eur. J. Math., 11 (2013), No. 9, 1711-1712.
 M.F. Bota, E. Karapinar, and O. Mlesnite, Ulam-Hyers stability results for fixed point problems via $alpha - psi$-contractive mapping in $b$-metric space, Abstr. Appl. Anal. (2013), Art. ID 825293, 6 pp.
 S.K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci., 25 (1972) 727-730.
 S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993) 5-11.
 S. Czerwik, Nonlinear Set-valued contraction mappings in $b$-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998) 263-276.
 R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968) 71-76.
 M.A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 73 (2010), no. 9, 3123-3129.
 M.S. Khan, M. Swaleh, and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), no. 1, 1-9.
 W. Kirk and N. Shahzad, Fixed point theory in distance spaces, Springer, Cham, 2014.
 M.A. Kutbi, E. Karapinar, J. Ahmad, and A. Azam, Some fixed point results for multi-valued mappings in $b$-metric spaces, J. Inequal. Appl. 2014, (2014:126), 11 pp.
 C.S. Wong, Common fixed points of two mappings, Pacific J. Math., 48 (1973) 299-312.