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Common fixed point theorems for some new generalized contractive type mappings. (English) Zbl 1133.54028

In this paper the author proves a common fixed point theorem for a pair of mappings \(T,\;S:X\to X\), where \((X,d)\) is a complete metric space, which satisfy a generalized contractive type condition: \[ F(d(Tx,Sy))\leq\psi(F(M(x,y))\quad\text{for all }\; x,y\in X, \] where \(M(x,y)=\max\{d(x,y),d(Tx,x), d(Sy,y),\frac{1}{2}(d(Tx,y)+d(Sy,x))\}\) and \(F\), \(\psi\) satisfy suitable assumptions. Some special cases and an example of a mapping which satisfies the above condition but does not satisfy the general contractive condition are also provided.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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References:

[1] Branciari, A., A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. math. math. sci., 29, 9, 531-536, (2002) · Zbl 0993.54040
[2] Vijayaraju, P.; Rhoades, B.E.; Mohanraj, R., A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int. J. math. math. sci., 2005, 15, 2359-2364, (2005) · Zbl 1113.54027
[3] Rhoades, B.E., A comparison of various definitions of contractive mappings, Trans. amer. math. soc., 226, 257-290, (1977) · Zbl 0365.54023
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