Geodesic
On Diviccaro, Fisher and Sessa open questions
Archivum mathematicum, Tome 29 (1993) no. 3-4, pp. 145-152 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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

[1] Ćirić, Lj.B.: On a common fixed point theorem of a Greguš type. Publ. Inst. Math. 49(63) (1991), Beograd, 174-178. | MR

[2] Delbosco, D., Ferrero, O., Rossati, F.: Teoreme di punto fisso per applicazioni negli spazi di Banach. Boll. Un. Mat. Ital. (6) 2-A (1983), 297-303. | MR

[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

[4] Fisher, B., Sessa, S.: On a fixed point theorem of Greguš. Internat. J. Math. Math. 9 (1986), No. 1, 23-28. | MR

[5] Greguš, M.: A fixed point theorem in Banach space. Boll. Un. Mat. Ital. (5) 7-A (1980), 193-198. | MR

[6] Jungck, G.: Compatible mappings and common fixed points. Internat. J. Math. Math. Sci. 9 (1986), 771-779. | MR

[7] Jungck, G.: On a fixed point theorem of Fisher and Sessa. Internat. J. Math. Math. Sci 13 (1988), 497-500. | MR

[8] Mukherjee, R. N., Verma, V.: A note on a fixed point theorem of Greguš. Math. Japon. 33 (1988), 745-749. | MR

[9] Sessa, S.: On a week commutativity condition in fixed point considerations. Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 149-153. | MR

[10] Takahashi, W.: A convexity in metric space and nonexpansive mappings $I$. Kodai Math. Sem. Rep. 22 (1970), 142-149. | MR | Zbl