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A generalisation of contraction principle in metric spaces. (English) Zbl 1177.54024
In this paper, the main result states that a self-mapping \(T: X \to X\) defined on a complete metric space \((X,d)\) has a unique fixed point if it satisfies the following inequality
\[ \psi(d(Tx,Ty)) \leq \psi(d(x,y))-\phi(d(x,y)), \]
where \(\psi, \phi: [0,\infty) \to [0,\infty)\) are two monotone nondecreasing continuous functions with \(\psi(t)=0= \phi(t)\) if and only if \(t=0\). When \(\psi\) is the identity function on \([0,\infty)\), this reduces to a result of B. E. Rhoades [Nonlinear Anal., Theory Methods Appl. 47, No. 4, 2683–2693 (2001; Zbl 1042.47521)]. Moreover, the main result is illustrated by an example.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
Keywords:
fixed point
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References:
[1] doi:10.2307/2035677 · Zbl 0175.44903
[2] doi:10.1090/S0002-9939-03-06937-5 · Zbl 1053.54047
[3] doi:10.1090/S0002-9939-01-06169-X · Zbl 1001.47042
[4] doi:10.1090/S0002-9939-07-09055-7 · Zbl 1145.54026
[9] doi:10.1023/A:1022991929004 · Zbl 1013.54011
[16] doi:10.1007/s10114-007-6509-x · Zbl 1155.54026
[17] doi:10.1016/S0362-546X(01)00388-1 · Zbl 1042.47521
[18] doi:10.1155/FPTA/2006/74503 · Zbl 1133.54024
[19] doi:10.1016/S0022-247X(02)00063-X · Zbl 1005.47053
[20] doi:10.1016/j.aml.2008.02.007 · Zbl 1163.47304
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