On Diviccaro, Fisher and Sessa open questions

Ćirić, Ljubomir B.

Archivum mathematicum, Tome 29 (1993) no. 3-4, pp. 145-152.

Résumé

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.

Détail

Classification : 47H10, 54H25
Keywords: convex metric space; Cauchy sequence; fixed point