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Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. (English) Zbl 1185.54037
Let \(X\neq\emptyset\). 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 \(x\neq y\).
(c) \(G(x,x,y)\leq G(x,y,z)\) for all \(x,y\in X\), with \(z\neq y\),
(d) \(G(x,y,z)= G(x,z,y)= G(y,z,x)=\cdots\) (symmetry in all three variables),
(e) \(G(x,y,z)\leq G(x,a,a)+ G(a,y,z)\) 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:
\[ G(fx,fy,fy)\leq k\max\{G(gx,fy,fy), G(gy,fx, fx), G(gy,fy,fy)\} \] and
\[ G(fx,fy,fy)\leq k\max\{G(gx,gx,fy), G(gy, gy,fx), G(gy, gy, fy)\} \] for all \(x,y\in X\), where \(0\leq k< 1\). 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.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
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[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
[2] Hicks, T.L.; Rhoades, B.E., A Banach type fixed point theorem, Math. japonica, 24, 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
[4] Jungck, G., Compatible mappings and common fixed points, Int. J. math. sci., 9, 4, 771-779, (1986) · Zbl 0613.54029
[5] Jungck, G., Common fixed points for commuting and compatible maps on compacta, Proc. am. math. soc., 103, 977-983, (1988) · Zbl 0661.54043
[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, 2, 1003-1012, (2007) · Zbl 1110.54024
[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, 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
[11] Pant, R.P., Common fixed points of noncommuting mappings, J. math. anal. appl., 188, 436-440, (1994) · Zbl 0830.54031
[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
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