zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. (English) Zbl 1185.54037
Let $X\ne\emptyset$. Suppose that a mapping $G: X\times X\times X\to[0,\infty)$ satisfies: (a) $G(x,y,z)= 0$ if and only if $x= y= z$, (b) $0< G(x,y,z)$ for all $x,y\in X$, with $x\ne y$. (c) $G(x,x,y)\le G(x,y,z)$ for all $x,y\in X$, with $z\ne y$, (d) $G(x,y,z)= G(x,z,y)= G(y,z,x)=\cdots$ (symmetry in all three variables), (e) $G(x,y,z)\le G(x,a,a)+ G(a,y,z)$ for all $x,y,z,a\in X$. Then $G$ is called a $G$-metric on $X$ and $(X,G)$ is called a $G$-metric space. In the present paper the authors, using the setting of $G$-metric space, prove a fixed point theorem for one map, and several fixed point theorems for two maps. They prove, for example: Theorem 2.5. Let $(X, G)$ be a $G$-metric space. Suppose that $f,g: X\to X$ satisfy one of the following conditions: $$G(fx,fy,fy)\le k\max\{G(gx,fy,fy), G(gy,fx, fx), G(gy,fy,fy)\}$$ and $$G(fx,fy,fy)\le k\max\{G(gx,gx,fy), G(gy, gy,fx), G(gy, gy, fy)\}$$ for all $x,y\in X$, where $0\le k< 1$. If the range of $g$ contains the range of $f$ and $g(X)$ is a complete subspace of $X$, then $f$ and $g$ have a unique point of coincidence in $X$. Moreover, if $f$ and $g$ are weakly compatible, then $f$ and $g$ have a unique common fixed point.

54H25Fixed-point and coincidence theorems in topological spaces
Full Text: DOI
[1] I. Beg, M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theor. Appl. (2006) 1 -- 7 (Article ID 74503). · Zbl 1133.54024 · doi:10.1155/FPTA/2006/74503
[2] Hicks, T. L.; Rhoades, B. E.: A Banach type fixed point theorem, Math. japonica 24, No. 3, 327-330 (1979) · Zbl 0432.47036
[3] Jungck, G.: Commuting maps and fixed points, Am. math. Monthly 83, 261-263 (1976) · Zbl 0321.54025 · doi:10.2307/2318216
[4] Jungck, G.: Compatible mappings and common fixed points, Int. J. Math. sci. 9, No. 4, 771-779 (1986) · Zbl 0613.54029 · doi:10.1155/S0161171286000935
[5] Jungck, G.: Common fixed points for commuting and compatible maps on compacta, Proc. am. Math. soc. 103, 977-983 (1988) · Zbl 0661.54043 · doi:10.2307/2046888
[6] Jungck, G.: Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far east J. Math. sci. 4, 199-215 (1996) · Zbl 0928.54043
[7] Jungck, G.; Hussain, N.: Compatible maps and invariant approximations, J.m.m.a 325, No. 2, 1003-1012 (2007) · Zbl 1110.54024 · doi:10.1016/j.jmaa.2006.02.058
[8] Z. Mustafa and B. Sims, Some Remarks concerning D-metric spaces, in: Proc. Int. Conf. on Fixed Point Theor. Appl., Valencia (Spain), July 2003, pp. 189 -- 198. · Zbl 1079.54017
[9] Mustafa, Z.; Sims, B.: A new approach to generalized metric spaces, J. nonlinear convex anal. 7, No. 2, 289-297 (2006) · Zbl 1111.54025
[10] Z. Mustafa, H. Obiedat, F. Awawdeh, Some common fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theor Appl. (2008) (Article ID 189870, 12 pages). · Zbl 1148.54336 · doi:10.1155/2008/189870
[11] Pant, R. P.: Common fixed points of noncommuting mappings, J. math. Anal. appl. 188, 436-440 (1994) · Zbl 0830.54031 · doi:10.1006/jmaa.1994.1437
[12] Park, Sehie: A unified approach to fixed points of contractive maps, J. korean math. Soc. 16, 95-105 (1980) · Zbl 0431.54028
[13] Sessa, S.: On a weak commutativity condition of mappings in fixed point consideration, Publ. inst. Math. soc. 32, 149-153 (1982) · Zbl 0523.54030
[14] Kannan, R.: Some results on fixed points, Bull. Calcutta math. Soc. 60, 71-76 (1968) · Zbl 0209.27104