Demicompact and k-D-Set-contractive Multivalued Linear Operators

Demicompactness, selection of linear relation and application to multivalued matrix

“…Lemma 2.3. [29] Let D be a compact linear subspace of a space X. Let {x n } in X be a sequence such that {Q D x n } is a convergent sequence, then {x n } has a convergent subsequence.…”

“…The concept of Fredholm operators is one of the attempts to understand the classical Fredholm theory of integral equations. Further important contributions were due to A. Jeribi [33] who gave a simple and unified treatment of this theory which covered all the basic points while avoiding some of the involved concepts (see also [2]- [22]). Recently, W. Chaker, A. Jeribi and B. Krichen [32] have utilized demicompact operators in order to investigate the essential spectra of closed linear operators.…”

“…In [34] B. Krichen introduced the relative demicompactness class with respect to a given closed linear operator as a generalization of the demicompactness notion. Lately, in [29] A. Ammar, H. Daoud and A. Jeribi defined the demicompact of a linear relation by T : D(T) ⊆ X → X is said to be demicompact if for every bounded sequence {x n } in D(T) such that Q I−T (I − T)x n → y ∈ X/(I − T)(0), there is a convergent subsequence of {Q T x n }. Then, is to generalize some results given in [32] to multivalued linear operators.…”

Fredholm theory for demicompact linear relations

“…In what follows, we shall present two definitions set forward by A. Ammar, H. Daoud and A. Jeribi in 2017 [4], who extended the concept of demicompact and k-set-contraction of linear operators on multivalued linear operators and developed some pertinent properties.…”

“…[4, Definition 4.1]). T : D(T ) ⊆ X → Y is a linear relation, while δ 1 and δ 2 are respectively Kuratowski measures of noncompactness in X/D and Y , where D is a closed subspace of N (T ).…”