Previous |  Up |  Next

Article

MSC: 54E40, 54H25
Keywords:
2-Menger space; Cauchy sequence; fixed point; control function; $t$-norm
Summary:
In this paper we introduce generalized cyclic contractions through $r$ number of subsets of a probabilistic 2-metric space and establish two fixed point results for such contractions. In our first theorem we use the Hadzic type $t$-norm. In another theorem we use a control function with minimum $t$-norm. Our results generalizes some existing fixed point theorem in 2-Menger spaces. The results are supported with some examples.
References:
[1] Banach, S.: Sur les Operations dans les Ensembles Abstraits et leur Application aux Equations Integrales. Fundamenta Mathematicae 3 (1922), 133–181.
[2] Choudhury, B. S., Das, K. P.: A new contraction principle in Menger spaces. Acta Mathematica Sinica, English Series 24 (2008), 1379–1386. DOI 10.1007/s10114-007-6509-x | MR 2438308 | Zbl 1155.54026
[3] Choudhury, B. S., Dutta, P. N., Das, K. P.: A fixed point result in Menger spaces using a real function. Acta. Math. Hungar. 122 (2008), 203–216. DOI 10.1007/s10474-008-7242-3 | MR 2480861
[4] Choudhury, B. S., Das, K. P.: A coincidence point result in Menger spaces using a control function. Chaos, Solitons and Fractals 42 (2009), 3058–3063. DOI 10.1016/j.chaos.2009.04.020 | MR 2560014 | Zbl 1198.54072
[5] 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. Bulletin of Mathematical Analysis and Applications 3 (2011), 141–148. MR 2955353
[6] Choudhury, B. S., Das, K. P., Bhandari, S. K.: Fixed point theorem for mappings with cyclic contraction in Menger spaces. Int. J. Pure Appl. Sci. Technol. 4 (2011), 1–9.
[7] Choudhury, B. S., Das, K. P., Bhandari, S. K.: A Generalized cyclic C-contraction priniple in Menger spaces using a control function. Int. J. Appl. Math. 24, 5 (2011), 663–673. MR 2931524
[8] 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 (2012), 21–30. MR 3088824
[9] Choudhury, B. S., Das, K. P., Bhandari, S. K.: Two Ciric type probabilistic fixed point theorems for discontinuous mappings. International Electronic Journal of Pure and Applied Mathematics 5, 3 (2012), 111–126. MR 3016126
[10] Choudhury, B. S., Das, K. P., Bhandari, S. K.: Cyclic contraction result in 2-Menger space. Bull. Int. Math. Virtual Inst. 2 (2012), 223–234. MR 3159041
[11] Choudhury, B. S., Das, K. P., Bhandari, S. K.: Generalized cyclic contraction in Menger spaces using a control function. Rev. Bull. Cal. Math. Soc. 20, 1 (2012), 35–42.
[12] Choudhury, B. S., Das, K. P., Bhandari, S. K.: Cyclic contraction of Kannan type mappings in generalized Menger space using a control function. Azerbaijan Journal of Mathematics 2, 2 (2012), 43–55. MR 2967294
[13] Choudhury, B. S., Das, K. P., Bhandari, S. K.: Fixed points of p-cyclic Kannan type contractions in probabilistic spaces. J. Math. Comput. Sci. 2 (2012), 565–583. MR 2929240
[14] Dutta, P. N., Choudhury, B. S., Das, K. P.: Some fixed point results in Menger spaces using a control function. Surveys in Mathematics and its Applications 4 (2009), 41–52. MR 2485791 | Zbl 1180.54054
[15] Gähler, S.: 2-metrische R$\ddot{a}$ume and ihre topologische strucktur. Math. Nachr. 26 (1963), 115–148. DOI 10.1002/mana.19630260109 | MR 0162224
[16] Gähler, S.: Uber die unifromisieberkeit 2-metrischer Raume. Math. Nachr. 28 (1965), 235–244. DOI 10.1002/mana.19640280309
[17] Fernandez-Leon, A.: Existence and uniqueness of best proximity points in geodesic metric spaces. Nonlinear Analysis 73 (2010), 915–921. DOI 10.1016/j.na.2010.04.005 | MR 2653759 | Zbl 1196.54050
[18] Golet, I.: A fixed point theorems in probabilistic 2-metric spaces. Sem. Math. Phys. Inst. Polit. Timisoara (1988), 21–26.
[19] Hadžić, O.: A fixed point theorem for multivalued mappings in 2-menger spaces. Univ. Novi Sad, Zb. Rad. Prirod., Mat. Fak., Ser. Mat. 24 (1994), 1–7. MR 1413932 | Zbl 0897.54036
[20] Hadžić, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Spaces. Mathematics and Its Applications 536, Springer Netherlands, New York–Heidelberg–Berlin, 2001. MR 1896451
[21] Iseki, K.: Fixed point theorems in 2-metric space. Math. Sem. Notes, Kobe Univ. 3 (1975), 133–136. MR 0415596
[22] Khan, M. S.: On the convergence of sequences of fixed points in 2-metric spaces. Indian J. Pure Appl. Math. 10 (1979), 1062–1067. MR 0547888 | Zbl 0417.54020
[23] Karpagam, S., Agrawal, S.: Best proximity point theorems for cyclic orbital Meir–Keeler contraction maps. Nonlinear Analysis 74 (2011), 1040–1046. DOI 10.1016/j.na.2010.07.026 | MR 2746787 | Zbl 1206.54047
[24] Khan, M. S., Swaleh, M., Sessa, S.: Fixed point theorems by altering distances between the points. Bull. Austral. Math. Soc. 30 (1984), 1–9. DOI 10.1017/S0004972700001659 | MR 0753555 | Zbl 0553.54023
[25] Kirk, W. A., Srinivasan, P. S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theorys 4 (2003), 79–89. MR 2031823 | Zbl 1052.54032
[26] Lal, S. N., Singh, A. K.: An analogue of Banach’s contraction principle for 2-metric spaces. Bull. Austral. Math. Soc. 18 (1978), 137–143. DOI 10.1017/S0004972700007887 | MR 0645161 | Zbl 0385.54028
[27] Mihet, D.: Altering distances in probabilistic Menger spaces. Nonlinear Analysis 71 (2009), 2734–2738. DOI 10.1016/j.na.2009.01.107 | MR 2532798 | Zbl 1176.54034
[28] Naidu, S. V. R., Prasad, J. R.: Fixed point theorems in metric, 2-metric and normed linear spaces. Indian J. Pure Appl. Math 17 (1986), 602–612. MR 0844195 | Zbl 0584.54042
[29] Naidu, S. V. R., Prasad, J. R.: Fixed point theorems in 2-metric spaces. Indian J. Pure Appl. Math 17 (1986), 974–993. MR 0856334 | Zbl 0592.54049
[30] Naidu, S. V. R.: Some fixed point theorems in metric and 2-metric spaces. Int. J. Math. Math. Sci. 28, 11 (2001), 625–638. DOI 10.1155/S016117120101064X | MR 1892319 | Zbl 1001.47037
[31] Naidu, S. V. R.: Some fixed point theorems in metric spaces by altering distances. Czechoslovak Math. J. 53 (2003), 205–212. DOI 10.1023/A:1022991929004 | MR 1962009 | Zbl 1013.54011
[32] Rhoades, B. E.: Contraction type mapping on a 2-metric spaces. Math. Nachr. 91 (1979), 151–155. DOI 10.1002/mana.19790910112 | MR 0563606
[33] Sastry, K. P. R., Babu, G. V. R.: Some fixed point theorems by altering distances between the points. Indian J. Pure. Appl. Math. 30, 6 (1999), 641–647. MR 1701042 | Zbl 0938.47044
[34] 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 Journal of Mathematics 31, 3 (2000), 243–250. MR 1778222
[35] Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Elsevier, North-Holland, New York, 1983. MR 0790314 | Zbl 0546.60010
[36] Sehgal, V. M., Bharucha-Reid, A. T.: Fixed point of contraction mappings on PM space. Math. Sys. Theory 6, 2 (1972), 97–100. DOI 10.1007/BF01706080 | MR 0310858
[37] Sharma, A. K.: A note on fixed points in 2-metric spaces. Indian J. Pure Appl. Math. 11 (1980), 1580–1583. MR 0617834 | Zbl 0448.54049
[38] Chang, S.-S., Huang, N.-J.: On generalized 2-metric spaces and probabilistic 2-metric spaces, with applications to fixed point theory. Math. Jap. 34, 6 (1989), 885–900. MR 1025044
[39] Shi, Y., Ren, L., Wang, X.: The extension of fixed point theorems for set valued mapping. J. Appl. Math. Computing 13 (2003), 277–286. DOI 10.1007/BF02936092 | MR 2000215 | Zbl 1060.47057
[40] Singh, S. L., Talwar, R., Zeng, W.-Z.: Common fixed point theorems in 2-menger spaces and applications. Math. Student 63 (1994), 74–80. MR 1292372 | Zbl 0878.54040
[41] Vetro, C.: Best proximity points: Convergence and existence theorems for p-cyclic mappings. Nonlinear Analysis 73 (2010), 2283–2291. DOI 10.1016/j.na.2010.06.008 | MR 2674204 | Zbl 1229.54066
[42] Zeng, W.-Z.: Probabilistic 2-metric spaces. J. Math. Research Expo. 2 (1987), 241–245. MR 0929343
[43] Wlodarczyk, K., Plebaniak, R., Banach, A.: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Analysis 70 (2009), 3332–3341. DOI 10.1016/j.na.2008.04.037 | MR 2503079 | Zbl 1171.54311
[44] Wlodarczyk, K., Plebaniak, R., Obczyński, C.: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Analysis 72 (2010), 794–805. DOI 10.1016/j.na.2009.07.024 | MR 2579346 | Zbl 1185.54020
Partner of
EuDML logo