Document Type : Research Paper
Authors
- Roghaye Jalal Shahkoohi ^{1}
- Zohreh Bagheri ^{} ^{2}
^{1} Department of Mathematics, Aliabad katoul Branch, Islamic Azad University, Aliabad katoul, Iran.
^{2} Department of Mathematics, Azadshahr Branch, Islamic Azad University, Azadshahr, Iran.
Abstract
In 2014, Zead Mustafa introduced $b_2$-metric spaces, as a generalization of both $2$-metric and $b$-metric spaces. Then new fixed point results for the classes of rational Geraghty contractive mappings of type I,II and III in the setup of $b_2$-metric spaces are investigated. Then, we prove some fixed point theorems under various contractive conditions in partially ordered $b_2$-metric spaces. These include Geraghty-type conditions, conditions that use comparison functions and almost generalized weakly contractive conditions. Berinde in [17-20] initiated the concept of almost contractions and obtained many interesting fixed point theorems. Results with similar conditions were obtained, \textit{e.g.}, in [21] and [22]. In the last section of the paper, we define the notion of almost generalized $(\psi ,\varphi )_{s,a}$-contractive mappings and prove some new results. In particular, we extend Theorems 2.1, 2.2 and 2.3 of Ciric et.al. in [23] to the setting of $b_{2}$-metric spaces. Also, some examples are provided to illustrate the results presented herein and several interesting consequences of our theorems are also provided. The findings of the paper are based on generalization and modification of some recently reported theorems in the literature.
Keywords
Main Subjects
[2] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena., 46 (1998), pp. 263-276.
[3] V.S. Gahler, 2-metrische Raume und ihre topologische Struktur, Math. Nachr., 26 (1963/64), pp. 115-118.
[4] N. Hussain, V. Parvaneh, J.R. Roshan, and Z. Kadelburg, Fixed points of cyclic weakly $(psi,varphi,L,A,B)$-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl., (2013):256 (2013).
[5] N.V. Dung and V.T. Le Hang, Fixed point theorems for weak C-contractions in partially ordered 2-metric spaces, Fixed Point Theory Appl., (2013):161 (2013).
[6] S.V.R. Naidu and J.R. Prasad, Fixed point theorems in 2-metric spaces, Indian J. Pure Appl. Math., 17 (1986), pp. 974-993.
[7] A. Aliouche and C. Simpson, Fixed points and lines in 2-metric spaces, Adv. Math., 229 (2012), pp. 668-690.
[8] B. Deshpande and S. Chouhan, Common fixed point theorems for hybrid pairs of mappings with some weaker conditions in 2-metric spaces, Fasc. Math., 46 (2011), pp. 37-55.
[9] R.W. Freese, Y.J. Cho, and S.S. Kim, Strictly 2-convex linear 2-normed spaces, J. Korean Math. Soc., 29 (1992), pp. 391-400.
[10] K. Iseki, Fixed point theorems in 2-metric spaces, Math. Semin. Notes., 3 (1975), pp. 133-136.
[11] K. Iseki, Mathematics on 2-normed spaces, Bull. Korean Math. Soc., 13 (1976), pp. 127-135.
[12] B.K. Lahiri, P. Das, and L.K. Dey, Cantor's theorem in 2-metric spaces and its applications to fixed point problems, Taiwan. J. Math., 15 (2011), pp. 337-352.
[13] S.N. Lai and A.K. Singh, An analogue of Banach's contraction principle in 2-metric spaces, Bull. Aust. Math. Soc., 18 (1978), pp. 137-143.
[14] V. Popa, M. Imdad, and J. Ali, Using implicit relations to prove infied fixed point theorems in metric and 2-metric spaces, Bull. Malaysian Math. Sci. Soc., 33 (2010), pp. 105-120.
[15] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), pp. 604-608.
[16] D. Dukic, Z. Kadelburg, and S. Radenovic, Fixed points of Geraghty-type mappings in various generalized metric spaces, Abstr. Appl. Anal., (2011), Article ID 561245, 13 pages.
[17] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math., 19 (2003), pp. 7-22.
[18] V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nolinear Anal. Forum., 9 (2004), pp. 43-53.
[19] V. Berinde, General contractive fixed point theorems for Ciric-type almost contraction in metric spaces, Carpathian J. Math., 24 (2008), pp. 10-19.
[20] V. Berinde, Some remarks on a fixed point theorem for Ciric-type almost contractions, Carpathian J. Math., 25 (2009), pp. 157-162.
[21] G.V.R. Babu, M.L. Sandhya, and M.V.R. Kameswari, A note on a fixed point theorem of Berinde on weak contractions, Carpathian J. Math., 24 (2008), pp. 8-12.
[22] J.R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei, and W. Shatanawi, Common fixed points of almost generalized $(psi,varphi)_s$-contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl., 2013:159 (2013).
[23] Lj. Ciric, M. Abbas, R. Saadati, and N. Hussain, Common fixed points of almost generalized contractive mappings in ordered metric spaces, Applied Math. Comput., 217 (2011), pp. 5784-5789.
[24] M.S. Khan, M. Swaleh, and S. Sessa, Fixed point theorems by altering distancces between the points, Bull. Aust. Math. Soc., 30 (1984), pp. 1-9.
[25] Sh. Fathollahi, N. Hussain, and L.A. Khan, Fixed point results for modified weak and rational $alpha$-$psi$-contractions in ordered 2-metric spaces, Fixed Point Theory Appl., 2014.
[26] Z. Mustafa, V. Parvaneh, J.R. Roshan, and Z. Kadelburg, $b_2$-Metric spaces and some fixed point theorems, Fixed Point Theory Appl, 2014.
[27] R.J. Shahkoohi and A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, J. Inequal. Appl., 2014, 2014:373.
[28] F.Zabihi and A.Razani, Fixed point theorems for Hybrid Rational Geraghty contractive mappings in ordered b-metric spaces, Journal of Applied Mathematics, (2014), Article ID 929821, 9 pages.