zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A new contraction principle in Menger spaces. (English) Zbl 1155.54026
Summary: In the present work we introduce a new type of contraction mapping by using a specific function and obtain certain fixed point results in Menger spaces. The work is in line with the research for generalizing the Banach’s contraction principle. We extend the notion of altering distance function to Menger Spaces and obtain fixed point results.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54E70Probabilistic metric spaces
References:
[1]Khan, M. S., Swaleh, M., Sessa, S.: Fixed points theorems by altering distances between the points. Bull. Austral. Math. Soc., 30, 1–9 (1984) · Zbl 0553.54023 · doi:10.1017/S0004972700001659
[2]Arvanitakis, A. D.: A Proof of generalized Banach contraction conjecture. Proc. Amer. Math. Soc., 131(12), 3647–3656 (2003) · Zbl 1053.54047 · doi:10.1090/S0002-9939-03-06937-5
[3]Merryfield, J., Rothschild, B., Stein, J. D.: An application of Ramsey’s Theorem to the Banach Contraction Principle. Proc. Amer. Math. Soc., 130, 927–933 (2002) · Zbl 1001.47042 · doi:10.1090/S0002-9939-01-06169-X
[4]Kirk, W. A.: Fixed points of asymptotic contraction. J. Math. Anal. Appl., 277, 645–650 (2003) · Zbl 1022.47036 · doi:10.1016/S0022-247X(02)00612-1
[5]Rhoades, B. E.: A comparison of various definitions of contractive mappings., Trans. Amer. Math. Soc., 226–257, 1977
[6]Meszaros, J.: A comparison of various definitions of contractive type mappings. Bull. Cal. Math. Soc., 84, 167–194 (1992)
[7]Schweizer, B., Sklar, V.: Probabilistic Metric Space, North-Holland, Amsterdam, 1983
[8]Hadzic, O., Pap, E.: Fixed Point Theory In Probabilistic Metric Spaces, Kluwer Academic Publishers, 2001
[9]Sehgal, V. M., Bharucha-Reid, A. T.: Fixed points of contraction mappings on PM space. Math. Sys. Theory, 6(2), 97–100 (1972) · Zbl 0244.60004 · doi:10.1007/BF01706080
[10]Mihet, D.: Aclass of Sehgal’s Contractions in Probabilistic Metric Spaces, Analele Univ. din Timisoara, Vol. XXXVII, fasc. 1, 1999, Seria Matematica-Informatica, 105–108
[11]Hadzic, O., Pap, E., Radu, V.: Generalized contraction mapping principle in probabilistic metric space. Acta Math. Hungar., 101(1–2), 131–148 (2003) · Zbl 1050.47052 · doi:10.1023/B:AMHU.0000003897.39440.d8
[12]Chang, S. S., Lee, B. S., Cho, Y. S., Chen, Y. Q., Kang, S. M., Jung, J. S: Generalised contraction mapping principle and differential equation in probabilistic metric spaces. Proc. Amer. Math. Soc., 124(8), 2367–2376 (1996) · Zbl 0857.47042 · doi:10.1090/S0002-9939-96-03289-3
[13]Choudhury, B. S.: A unique common fixed point theorems for a sequence of self mappings in Menger spaces. Bull. Kor. Math. Soc., 37(3), 569–575 (2000)
[14]Chang, S. S., Cho, Y. J., Kang, S. M.: Nonlinear Operator Theory In Probabilistic Metric Spaces, Huntington, NY: Nova Science Publishers. X, 338, 2001
[15]Naidu, S. V. R.: Fixed point theorems by altering distances. Adv. Math. Sci. Appl., 11, 1–16 (2001)
[16]Naidu, S. V. R.: Some fixed point theorems in Metric spaces by altering distances. Czechoslovak Mathematical Journal, 53(128), 205–212 (2003) · Zbl 1013.54011 · doi:10.1023/A:1022991929004
[17]Pathak, H. K., Sharma, R.: A note on fixed point theorems of Khan, Swaleh and Sessa. Math. Edn., 28, 151–157 (1994)
[18]Sastry, K. P. R., Babu, G. V. R.: Fixed point theorems in metric space by altering distances. Bull. Cal. Math. Soc., 90, 175–182 (1998)
[19]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)
[20]Sastry, K. P. R., Babu, G. V. R.: A common fixed point theorem in complete metric spaces by altering distances. Proc. Nat. Acad. Sci. India, 71(A), III 237–242 (2001)
[21]Sastry, K. P. R., Naidu, S. V. R., Babu, G. V. R., Naidu, G. A.: Generalisation of fixed point theorems for weekly communting maps by altering distances. Tamkong Journal of Mathematics, 31(3), 243–250 (2000)
[22]Choudhury, B. S., Dutta, P. N.: A unified fixed point result in metric spaces involving a two variable function. FILOMAT, 14, 43–48 (2000)
[23]Choudhury, B. S., Updahyay, A.: An unique common fixed point for a sequence of multivalued mappings on metric spaces, Bulletin of Pure and Applied Science, 19E (2000), 529–533
[24]Singh, B., Jain, S.: A fixed point theorem in Menger space through weak compatibility. J. Math. Anal. Appl., 301, 439–448 (2005) · Zbl 1068.54044 · doi:10.1016/j.jmaa.2004.07.036