Generalized contractions and fixed point theorems over bipolar \(\mathrm{cone}_{tvs}\) \(b\)-metric spaces with an application to homotopy theory. (English) Zbl 1488.54170

Summary: In this paper, we introduce the concept of bipolar \(\mathrm{cone}_{tvs}\) \(b\)-metric space and prove some generalized fixed point theorems on it. These theorems extend and generalize some recent results obtained by other authors for mappings on a bipolar metric space. Also, a brief study on topological properties of this newly introduced space has been made and in support of our theorems, we give some examples. Moreover, our fixed point result is applied to homotopy theory on such spaces.


54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
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[1] M. Akram, A. A. Siddiqui,A fixed point theorem forA-contractions on a class of generalised metric spaces, Korean J. Math.,10(2)(2003), 1-5.
[2] A. Azam, I. Beg, M. Arshad,Fixed point in topological vector space-valued cone metric spaces, Fixed Point Theory Appl., Article ID 604084, doi:10.1155/2010/604084, (2010). · Zbl 1197.54057
[3] I.A. Bakhtin,The contraction principle in quasimetric spaces,Func. An., Ulianowsk, Gos. Ped. Ins.,30(1989), 26-37. · Zbl 0748.47048
[4] K. Deimling,Nonlinear Functional Analysis, Springler-Verlag, Berlin,105(1985). · Zbl 0559.47040
[5] L.G. Huang, X. Zhang,Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl.,341(2008), 876-882.
[6] N. Hussain, M.H. Shah,KKM mappings in coneb-metric spaces,Comput. Math. Appl.,62 (2011), 1677-1684. · Zbl 1231.54022
[7] Z. Kadelburg, S. Radenovi´c and V. Rakoˇcevi´c,Topological vector space-valued cone metric spaces and fixed point theorems,Fixed Point Theory Appl.,Article ID 170253, doi:10.1155/2010/170253, (2010).
[8] R. Kannan,Some Results on Fixed Points, Bull. Calcutta Math. Soc.10(1968), 71-76. · Zbl 0209.27104
[9] G.N.V. Kishore, R.P. Agarwal, B.S. Rao, R.V.N. S. Rao,Caristi type cyclic contraction and common fixed point theorems in bipolar metric spaces with applications,Fixed Point Theory Appl. (2018), doi:10.1186/s13663-018-0646-z. · Zbl 1462.54073
[10] A. Mutlu, U. G¨urdal,Bipolar metric spaces and some fixed point theorems,J. Nonlinear Sci. Appl.,9(2016), 5362-5373. · Zbl 1378.54029
[11] A. Mutlu, K. ¨Ozkan and, U. G¨urdal,Coupled fixed point theorems on bipolar metric spaces, European J. Pure Appl. Math.,10(4)(2017), 655-667. · Zbl 1370.54030
[12] B. Srinuvasa Rao, G.N.V. Kishore, S. Ramalingeswara Rao,Fixed point theorems under new Caristi type contraction in bipolar metric space with applications,Inter. J. Engineering & Technology (UAE),7(3.31)(2018), 106-110.
[13] B. Srinuvasa Rao, G.N.V. Kishore,Common fixed point theorems in bipolar metric spaces with applications to integral equations,ibid,7(4.10)(2018), 1022-1026.
[14] S. Reich,Kannan’s fixed point theorem,Boll. Un. Math. Ital.,4(1971), 1-11. · Zbl 0219.54042
[15] M. Saha, D. Dey,Fixed point theorems forA-contraction mappings of integral type,J. Nonlinear Sci. Appl.,5(2012), 84-92. · Zbl 1295.54082
[16] C. Vetro, F. Vetro,A homotopy fixed point theorem in 0-complete partial metric space, Filomat, 29(9)(2015), 2037-2048. · Zbl 1461.54110
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