Workshop on Functional Analysis Alhucemas 2014
September, 12th-13th, 2014
(f, g)-Boyd-Wong Contraction Mappings in
Probabilistic Metric Space
Abderrahim Mbarki1 , Abedelmalek Ouahab2 , Tahiri Ismail3
ABSTRACT
The purpose of this talk is to present some results on common fixed point theory in
probabilistic metric space proved by A. Mbarki, A. Ouahab and I. Tahiri [A. Mbarki,
A. Ouahab, I. Tahiri. (f, g)-Boyd-Wong Contraction Mappings in Probabilistic
Metric Space, Applied Mathematical Sciences, Volume (7), no. 13, 623-632, 2013].
Let f , g and h are three self maps on a probabilistic metric space, in this paper we
introduce the notions of (f, g)-Boyd-Wong contraction map, (f, g)-orbit of h starting
at a point and we give some conditions of which f , g and h have a coincidence point
and a unique common fixed point.
Our main result is
Theorem 0.1 Let K be a subset of a complete probabilistic metric space (X, F, τ )
where RanF ⊂ D+ and let h, f , g are three self maps on K which the following
conditions (i), (ii) and (iii) are satisfied
(i) h(K) or f (K) or g(K) is complete;
(ii) There exist x0 ∈ K such that an (f, g)-orbit of h starting at x0 is bounded;
(iii) h is (f, g)-Boyd-Wong contraction.
Then, there exist u, v, z ∈ K such that f u = hu = z = hv = gv.
If in addition (h, f ) and (h, g) are weakly compatible, then z is the unique common
fixed point of h, f and g.
References
[1] A. Mbarki. Quelques aspects de la th´eorie du point fixe pour les semigroupes,
Thèse de Doctorat en Sciences, Faculté des Sciences, Oujda, Maroc, 2001.
[2] B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, New
York. (1983).
[3] M. Elamrani, A. Mbarki and B. Mehdaoui. Nonlinear contractions and semigroups in general complete probabilistic metric spaces, Panam. Math. J. Volume
(11), no.4, 79-87, 2001.
[4] A. Ouhab, S. Lahrech, S. Rais, A. Mbarki and A. Jaddar. Fixed Point
Theorems in General Probabilistic Metric Spaces, Applied Mathematical Sciences.
Volume (1), no. 46, 2277-2286, 2007.
[5] H. Sherwood. Complete probabilistic metric spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. Volume (20), 117-128, 1971/72.
[6] S. Zhang, Q. Zhu and Y. Song. Alternating Picard itearates for hybird BoydWong contractions, Int. Journal of Math. Analysis. Volume (2), no. 12, 563-568,
2008.
1
National school of Applied Sciences P.O. Box 669, Oujda University, Morocco
MATSI Laboratory.
email [email protected]
2
Department of Mathematics Oujda University, 60000 Oujda, Morocco
MATSI Laboratory
email [email protected]
2
Department of Mathematics Oujda University, 60000 Oujda, Morocco
MATSI Laboratory
email [email protected]