Abstract
Cyclic contractions have received attention especially due to their uses in problems of global optimizations. In this paper we consider p-cyclic maps in 2-Menger spaces which are cyclic mappings between p-number of subsets of the space. We establish a fixed point result for these maps under the assumption of an inequality which is Kannan type and involves two control functions. Our results extends a previous results and is illustrated with an example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Choudhury, B.S., Das, K.P.: A new contraction principle in Menger spaces. Acta. Math. Sin. Engl. Ser. 24, 1379–1386 (2008)
Choudhury, B.S., Das, K.P.: Fixed points of generalized Kannan type mappings in generalized Menger spaces. Commun. Korean Math. Soc. 24, 529–537 (2009)
Choudhury, B.S., Das, K.P., Bhandari, S.K.: A fixed point theorem for Kannan type mappings in 2-Menger spaces using a control function. Bull. Math. Anal. Appl. 3, 141–148 (2011)
Choudhury, B.S., Dutta, P.N., Das, K.P.: A fixed point result in Menger spaces using a real function. Acta. Math. Hungar. 122, 203–216 (2008)
Choudhury, B.S., Das, K.P.: A coincidence point result in Menger spaces using a control function. Chaos Solitons Fractals 42, 3058–3063 (2009)
Choudhury, B.S., Das, K.P., Bhandari, S.K.: A fixed point theorem in 2-Menger space using a control function. Bull. Cal. Math. Soc. 104(1), 21–30 (2012)
Choudhury, B.S., Das, K.P., Bhandari, S.K.: Cyclic contraction result in 2-Menger space. Bull. Int. Math. Virtual Inst. 2, 223–234 (2012)
Choudhury, B.S., Bhandari, S.K.: Kannan type cyclic contraction results in 2-Menger spaces. Math. Bohem. 141(1), 37–58 (2016)
Connell, E.H.: Properties of fixed point spaces. Proc. Am. Math. Soc. 10, 974–979 (1959)
Gähler, S.: 2-metrische R\(\ddot{a}\)ume and ihre topologische strucktur. Math. Nachr. 26, 115–148 (1963)
Gähler, S.: Uber die unifromisieberkeit 2-metrischer Raume. Math. Nachr. 28, 235–244 (1965)
Gopal, D., Abbas, M., Vetro, C.: Some new fixed point theorems in Menger PM-spaces with application to Volterra type integral equation. Appl. Math. Comput. 232, 955–967 (2014)
Hadzic, O.: A fixed point theorem for multivalued mappings in 2-Menger spaces. Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 24, 1–7 (1994)
Hadzic, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic Publishers, Berlin (2001)
Kannan, R.: Some results on fixed point. Bull. Cal. Math. Soc. 60, 71–76 (1968)
Kannan, R.: Some results on fixed point II. Am. Math. Mon. 76, 405–408 (1969)
Karpagam, S., Agrawal, S.: Best proximity point theorems for cyclic orbital MeirKeeler contraction maps. Nonlinear Anal. 74, 1040–1046 (2011)
Karpagam, S., Agrawal, S.: Best proximity point theorems for \(p\)-cyclic Meirkeeler contractions. Fixed Point Theory Appl. vol. 2009, pp. 9 (Article ID 197308) (2009)
Kirk, W.A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4, 79–89 (2003)
Khan, M.S., Swaleh, M., Sessa, S.: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30, 1–9 (1984)
Lal, S.N., Singh, A.K.: An analogue of Banach’s contraction principle for 2-metric spaces. Bull. Aust. Math. Soc. 18, 137–143 (1978)
Mihet, D.: Altering distances in probabilistic Menger spaces. Nonlinear Anal. 71, 2734–2738 (2009)
Naidu, S.V.R.: Some fixed point theorems in metric and 2-metric spaces. Int. J. Math. Math. Sci. 28(11), 625–638 (2001)
Sastry, K.P.R., Babu, G.V.R.: Some fixed point theorems by altering distances between the points. Ind. J. Pure. Appl. Math. 30(6), 641–647 (1999)
Sastry, K.P.R., Naidu, S.V.R., Babu, G.V.R., Naidu, G.A.: Generalisation of common fixed point theorems for weakly commuting maps by altering distances. Tamkang J. Math. 31(3), 243–250 (2000)
Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Elsevier, North-Holland (1983)
Sehgal, V.M., Bharucha-Reid, A.T.: Fixed point of contraction mappings on PM space. Math. Sys. Theory 6(2), 97–100 (1972)
Shioji, N., Suzuki, T., Takahashi, W.: Contractive mappings, Kannan mappings and metric completeness. Proc. Am. Math. Soc. 126, 3117–3124 (1998)
Chang, S., Huang, N.J.: On generalized 2-metric spaces and probabilistic 2-metric spaces, with applications to fixed point theory. Math. Jap. 34(6), 885–900 (1989)
Shi, Y., Ren, L., Wang, X.: The extension of fixed point theorems for set valued mapping. J. Appl. Math. Comput. 13, 277–286 (2003)
Subrahmanyam, P.V.: Completeness and fixed points. Monatsh. Math. 80, 325–330 (1975)
Vetro, C.: Best proximity points: convergence and existence theorems for p-cyclic mappings. Nonlinear Anal. 73, 2283–2291 (2010)
Zeng, Wen-Zhi: Probabilistic 2-metric spaces. J. Math. Res. Expo. 2, 241–245 (1987)
Wlodarczyk, K., Plebaniak, R., Banach, A.: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 70, 3332–3341 (2009)
Wlodarczyk, K., Plebaniak, R., Obczyski, C.: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 72, 794–805 (2010)
Wlodarczyk, K., Plebaniak, R.: Kannan-type contractions and fixed points in uniform spaces. Fixed Point Theory Appl. 2011(90), 1–24 (2011)
Acknowledgments
We are grateful to learned referees for their valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Choudhury, B.S., Bhandari, S.K. & Saha, P. Unique fixed points of p-cyclic kannan type probabilistic contractions. Boll Unione Mat Ital 10, 179–189 (2017). https://doi.org/10.1007/s40574-016-0073-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40574-016-0073-1