# On Diviccaro, Fisher and Sessa open questions

Archivum Mathematicum (1993)

- Volume: 029, Issue: 3-4, page 145-152
- ISSN: 0044-8753

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topĆirić, Ljubomir B.. "On Diviccaro, Fisher and Sessa open questions." Archivum Mathematicum 029.3-4 (1993): 145-152. <http://eudml.org/doc/247433>.

@article{Ćirić1993,

abstract = {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.},

author = {Ćirić, Ljubomir B.},

journal = {Archivum Mathematicum},

keywords = {convex metric space; Cauchy sequence; fixed point; contraction type condition; unique common fixed point in a closed convex set},

language = {eng},

number = {3-4},

pages = {145-152},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {On Diviccaro, Fisher and Sessa open questions},

url = {http://eudml.org/doc/247433},

volume = {029},

year = {1993},

}

TY - JOUR

AU - Ćirić, Ljubomir B.

TI - On Diviccaro, Fisher and Sessa open questions

JO - Archivum Mathematicum

PY - 1993

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 029

IS - 3-4

SP - 145

EP - 152

AB - 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.

LA - eng

KW - convex metric space; Cauchy sequence; fixed point; contraction type condition; unique common fixed point in a closed convex set

UR - http://eudml.org/doc/247433

ER -

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