On Diviccaro, Fisher and Sessa open questions

Ljubomir B. Ćirić

Archivum Mathematicum (1993)

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

Abstract

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Let K be a closed convex subset of a complete convex metric space X and T , I : K → K two compatible mappings satisfying following contraction definition: T x , T y ) ≤ ( I x , I y ) + ( 1 - a ) max { I x . T x ) , I y , T y ) } for all x , y in K , where 0 < a < 1 / 2 p - 1 and p ≥ 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.

How to cite

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Ćirić, 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 -

References

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  1. On a common fixed point theorem of a Greguš type, Publ. Inst. Math. 49(63) (1991), Beograd, 174-178. (1991) MR1127395
  2. Teoreme di punto fisso per applicazioni negli spazi di Banach, Boll. Un. Mat. Ital. (6) 2-A (1983), 297-303. (1983) MR0724481
  3. A common fixed point theorem of Greguš type, Publ. Math. Debrecen 34 (1987), No. 1-2. MR0901008
  4. On a fixed point theorem of Greguš, Internat. J. Math. Math. 9 (1986), No. 1, 23-28. (1986) MR0837098
  5. A fixed point theorem in Banach space, Boll. Un. Mat. Ital. (5) 7-A (1980), 193-198. (1980) MR0562137
  6. Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), 771-779. (1986) MR0870534
  7. On a fixed point theorem of Fisher and Sessa, Internat. J. Math. Math. Sci 13 (1988), 497-500. (1988) MR1068012
  8. A note on a fixed point theorem of Greguš, Math. Japon. 33 (1988), 745-749. (1988) MR0972387
  9. On a week commutativity condition in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 149-153. (1982) MR0710984
  10. A convexity in metric space and nonexpansive mappings I , Kodai Math. Sem. Rep. 22 (1970), 142-149. (1970) Zbl0268.54048MR0267565

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