J. Fixed Point Theory Appl. (2018) 20:57
Springer International Publishing AG,
part of Springer Nature 2018
Journal of Fixed Point Theory
Well-posedness of ﬁxed point problems
Debashis Dey, Ra´ul Fierro and Mantu Saha
Abstract. We state conditions for well-posedness of a ﬁxed point problem
for single and multivalued operators, where these operators are deter-
ministic or random. These results are applied to families of diﬀerent
Mathematics Subject Classiﬁcation. 47H10, 47H04, 47H40.
Keywords. Well-posedness, ﬁxed point, set valued operators, random
operators, Chatterjea, Kannan, Nadler and Reich contractions.
A number of authors have paid attention to the well-posedness of a ﬁxed
point problem for single and multivalued operators. For example, Akkouchi
and Popa , Lahiri and Das , Popa  and, Reich and Zaslavski [18,19]
have studied this issue for a single valued operator, while Petru¸sel and Rus
andPetru¸seletal. have stated diﬀerent deﬁnitions of well-posedness
of ﬁxed point problem for multivalued operators. Of course, some deﬁnitions
related with this subject could be given for random operators. To the best of
our knowledge, no work has been published to this respect.
The main aim of this work is to state conditions for well-posedness
of a ﬁxed point problem for a family of single or multivalued operators,
where these operators are deterministic or random. In our framework, well-
posedness of a ﬁxed point problem for a family of operators requires this
family has a unique ﬁxed point, even in the multivalued case.
Our results are applied to some quasi contraction families [2,6], which
includes a number of particular contractions (see [3,5,10,13,17]). Indeed, we
provide suitable deﬁnitions for these families to their associated ﬁxed point
problem to be well-posed.
The paper is structured as follows. In Sect. 2 some notations and pre-
liminary facts are stated, while in Sects. 3–6, we deal with the well-posedness
of ﬁxed point problem for families belonging to the following types of opera-
tors: single valued and deterministic, single valued and random, multivalued
and deterministic, and multivalued and random, respectively. Examples are
included in the corresponding sections. Finally, Sect. 7 deals with concluding
remarks and the possibility of extension of the results of this paper.