On Diviccaro, Fisher and Sessa open questions
Archivum mathematicum, Tome 29 (1993) no. 3-4, pp. 145-152
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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 $0$ 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.
@article{ARM_1993__29_3-4_a2,
author = {\'Ciri\'c, Ljubomir B.},
title = {On {Diviccaro,} {Fisher} and {Sessa} open questions},
journal = {Archivum mathematicum},
pages = {145--152},
publisher = {mathdoc},
volume = {29},
number = {3-4},
year = {1993},
mrnumber = {1263115},
zbl = {0810.47051},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ARM_1993__29_3-4_a2/}
}
Ćirić, Ljubomir B. On Diviccaro, Fisher and Sessa open questions. Archivum mathematicum, Tome 29 (1993) no. 3-4, pp. 145-152. http://geodesic.mathdoc.fr/item/ARM_1993__29_3-4_a2/