Quasicontractions on metric spaces
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- by Brian Fisher PDF
- Proc. Amer. Math. Soc. 75 (1979), 321-325 Request permission
Abstract:
It is proved that if T is a continuous mapping on the complete metric space X into itself satisfying the inequality \[ \begin {array}{*{20}{c}} \hfill {d({T^p}x,{T^q}y) \leqslant c\cdot \max \{ d({T^r}x,{T^s}y),d({T^r}x,{T^{r’}}x),d({T^s}y,{T^{s’}}y):} \\ \hfill {0 \leqslant r,r’ \leqslant p\;{\text {and}}\;0 \leqslant s,s’ \leqslant q\} } \\ \end {array} \] for all x,y in X, where $0 \leqslant c < 1$, for some fixed positive integers p and q, then T has a unique fixed point. Further, it is shown that the condition that T be continuous is unnecessary if $q\;({\text {or}}\;p) = 1$.References
- James Merryfield and James D. Stein Jr., A generalization of the Banach contraction principle, J. Math. Anal. Appl. 273 (2002), no. 1, 112–120. MR 1933019, DOI 10.1016/S0022-247X(02)00215-9
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 75 (1979), 321-325
- MSC: Primary 54H25
- DOI: https://doi.org/10.1090/S0002-9939-1979-0532159-9
- MathSciNet review: 532159