Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 4 (2009), 41 -- 52
SOME FIXED POINT RESULTS IN MENGER SPACES USING A CONTROL FUNCTION
P.N. Dutta, Binayak S. Choudhury and Krishnapada Das
Abstract. Here we prove a probabilistic contraction mapping principle in Menger spaces. This is in line with research in fixed point theory using control functions which was initiated by Khan et al. [Bull. Austral. Math. Soc., 30(1984), 1-9] in metric spaces and extended by Choudhury et al. [Acta Mathematica Sinica, 24(8) (2008), 1379-1386] in probabilistic metric spaces. An example has also been constructed.2000 Mathematics Subject Classification: 54H25; 54E70.
Keywords: Menger space; p-convergence; Φ-function; fixed point.
B.S. Choudhury and P.N. Dutta, A unified fixed point result in metric spaces involving a two variable function, FILOMAT, 14(2000), 43-48. MR1953993. Zbl 1012.54047.
B.S. Choudhury, A unique common fixed point theorem for a sequence of self-mappings in Menger spaces, Bull. Korean Math. Soc., 37(2000), No.3, 569-573. MR1779246 Zbl 0959.54026.
B.S. Choudhury, A common unique fixed point result in metric spaces involving generalised altering distances, Math. Communications, 10(2005), 105-110. MR2199098. Zbl 1089.54514.
B.S. Choudhury and P.N. Dutta, Common fixed points for fuzzy mappings using generalised altering distances, Soochow J. Math., 31(2005), 71-81. MR2130497. Zbl 1068.54043.
B.S. Choudhury and K. Das, A new contraction principle in Menger spaces, Acta Mathematica Sinica, English Series, 24(8) (2008), 1379-1386. MR2438308. Zbl pre05497467.
B.S. Choudhury, P.N. Dutta and K. Das , A fixed points result in Menger space using a real function, Acta. Math. Hungar., 122 (2009), 203-216.
P.N. Dutta and B.S. Choudhury A generalization of contraction mapping principle, Fixed Point Theory and Applications, Volume 2008, Article ID 406368, doi: 10.1155/2008/406368.
A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(1994), 395-399. MR1289545. Zbl 0843.54014.
A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90(1997), 365-368. MR1477836. Zbl 0917.54010.
O. Hadzic and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, 2001. MR1896451. Zbl 0994.47077.
O. Hadzic and E. Pap, A fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces, Fuzzy Sets and Systems, 127(2002), 333-344. MR1899066. Zbl 1002.54025.
M.S. Khan, M. Swaleh and S. Sessa , Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30(1984), 1-9. MR0753555. Zbl 0553.54023.
D. Mihet, Multivalued generalisations of probabilistic contractions, J. Math. Anal. Appl., 304(2005), 464-472. MR2126543 Zbl 1072.47066 .
D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems, 158(2007), 915-921. MR2302646. Zbl 1117.54008.
S.V.R. Naidu, Some fixed point theorems in metric spaces by altering distances, Czechoslovak Mathematical Journal, 53(2003), 205-212. MR1962009. Zbl 1013.54011.
K.P.R. Sastry and G.V.R. Babu, Some fixed point theorems by altering distances between the points, Ind. J. Pure. Appl. Math., 30(6),(1999), 641-647. MR1701042. Zbl 0938.47044.
K.P.R. Sastry, S.V.R. Naidu, G.V.R. Babu and G.A. Naidu, Generalisation of common fixed point theorems for weakly commuting maps by altering distances, Tamkang Journal of Mathematics, 31(3),(2000), 243-250. MR1778222. Zbl 0995.47035 .
B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, North-Holland, 1983. MR0790314. Zbl 0546.60010.
V.M. Sehgal and A.T. Bharucha-Reid, Fixed point of contraction mappings on PM space, Math. Sys. Theory, 6(2)(1972), 97-100. MR0310858. Zbl 0244.60004.
B. Singh and S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl., 301(2005), 439-448. MR2105684. Zbl 1068.54044.
R. Vasuki and P. Veeramani, Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy Sets and Systems, 135(2003), 415-417. MR1979610 Zbl 1029.54012.
Tatjana Zikic-Dosenovic, A multivalued generalization of Hick's C-contraction, Fuzzy Sets and Systems, 151(2005), 549-562. MR2126173 Zbl 1069.54025.
P.N. Dutta Department of Mathematics Government College of Engineering and Ceramic Technology, 73 A.C. Banergee Lane , Kolkata - 700010, West Bengal, INDIA. e-mail: email@example.com
Binayak S. Choudhury
Department of Mathematics Bengal Engineering and Science University P.O.- B. Garden, Shibpur, Howrah - 711103, West Bengal, INDIA. e-mail: firstname.lastname@example.org
Department of Mathematics, Bengal Engineering and Science University, Shibpur P.O.- B. Garden, Shibpur, Howrah - 711103, West Bengal, INDIA. e-mail: email@example.com