Mediterr. J. Math. (2018) 15:41
published online February 21, 2018
Springer International Publishing AG,
part of Springer Nature 2018
Demicompact and k-D-Set-contractive
Multivalued Linear Operators
Aymen Ammar, Houcem Daoud and Aref Jeribi
Abstract. In this paper, we extend the concept of demicompactness and
k-set-contractive linear operators on multivalued linear operators and
we develop some properties. Finally, we establish a rapport with the
theory of ﬁxed points.
Mathematics Subject Classiﬁcation. 47A06.
Keywords. Linear relation, multivalued linear operator, demicompact,
semi-Fredholm relation, k-D-set-contractive, D-set-condensing, ﬁxed point.
The concept of demicompactness was introduced into functional analysis by
Petryshyn , to discuss ﬁxed points and it has been studied in a large
number of papers (see, for example, [7,9,13,14,16,18]). A linear operator
T : D(T ) ⊂ X → X is said to be demicompact if for every bounded sequence
in D(T ) such that x
→ x ∈ X, there is a convergent subsequence of
}. The aim of this paper is to extend the concept of demicompactness and
k-set-contractive linear operators on linear relations, study some properties
and show a connection with the theory of ﬁxed points. The paper is organized
in the following way. In Sect. 2, we recall some deﬁnitions and results needed
in the rest of the paper. In the next section, we will deﬁne a demicompact
linear relation and we give few properties and results. In Sect. 4,weintro-
duce the deﬁnition of k-D- set-contractive and D-condensing linear relations
and cite few results. Finally, we establish a rapport between k-D-Lipschitz
relations and the ﬁxed point theory.
2. Preliminary Results
The concept of a linear relation in a linear space generalizes one of a linear
operator to that of a multivalued linear operator. A systematic treatment was
given by Arens  and by Coddington . This concept has been studied