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convex metric space; Cauchy sequence; fixed point
Let $K$ be a closed convex subset of a complete convex metric space $X$ and $T, I: K \rightarrow K$ two compatible mappings satisfying following contraction definition: $Tx, Ty)\le (Ix, Iy)+(1-a)\max \ \lbrace Ix.Tx),\ Iy, Ty)\rbrace $ for all $x,y$ in $K$, where $0<a<1/2^{p-1}$ and $p\ge 1$. If $I$ is continuous and $I(K)$ contains $[T(K)]$ , then $T$ and $I$ have a unique common fixed point in $K$ and at this point $T$ is continuous. This result gives affirmative answers to open questions set forth by Diviccaro, Fisher and Sessa in connection with necessarity of hypotheses of linearity and non-expansivity of $I$ in their Theorem [3] and is a generalisation of that Theorem. Also this result generalizes theorems of Delbosco, Ferrero and Rossati [2], Fisher and Sessa [4], Gregus [5], G. Jungck [7] and Mukherjee and Verma [8]. Two examples are presented, one of which shows the generality of this result.
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[3] Diviccaro, M. L., Fisher, B., Sessa, S.: A common fixed point theorem of Greguš type. Publ. Math. Debrecen 34 (1987), No. 1-2. MR 0901008
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[7] Jungck, G.: On a fixed point theorem of Fisher and Sessa. Internat. J. Math. Math. Sci 13 (1988), 497-500. MR 1068012
[8] Mukherjee, R. N., Verma, V.: A note on a fixed point theorem of Greguš. Math. Japon. 33 (1988), 745-749. MR 0972387
[9] Sessa, S.: On a week commutativity condition in fixed point considerations. Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 149-153. MR 0710984
[10] Takahashi, W.: A convexity in metric space and nonexpansive mappings $I$. Kodai Math. Sem. Rep. 22 (1970), 142-149. MR 0267565 | Zbl 0268.54048
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