On the class of relatively weakly demicompact nonlinear operators

Abstract

In this paper we discuss some topological properties of the set F (S 0 , T, z) := {x ∈ X : S 0 x ∈ T x + z} , where T is a nonlinear multi-valued mappings and S 0 is a single-valued mappings acting on a Banach space X. This study is based on a new concept, the so called weakly relative demicompactness for nonlinear operators.

Weakly demicompact linear operators and axiomatic measures of weak noncompactness

“…For an example of a S 0 -demicompact operator, we mention operators T such that (S 0 − T ) −1 exists and is continuous on its range R(S 0 − T ). In 2016, B. Krichen and D. O'Regan [19] discussed some topological properties of the set F(S 0 , T, z) := {x ∈ X : S 0 x ∈ T x + z}, where T is a nonlinear multi-valued mappings, S 0 is a single-valued mappings acting on a Banach space X and z is a given element of X. This study is based on a new concept, the so called weakly relative demicompactness for nonlinear operators.…”

“…Throughout this paper, we denote → for the strong convergence (with respect to the norm of X) and for the weak convergence (with respect to the weak topology of X). Recall the following definition introduced in [19].…”

Essential pseudospectra involving demicompact and pseudodemicompact operators and some perturbation results

“…The family of demicompact operators on X will be denoted by DC(X). For more results and applications of the concept of demicompactness, the reader may refer to [11,12].…”

Asymptotic regularity, fixed points and successive approximations

“…From the large list of papers which attest the impact of the asymptotic regularity property in the fixed point theory of operators, mainly in Hilbert and Banach spaces, we mention the following [32], [33], [135], [166], [58], [19], [112], [137], [89], [130], [83], [118], [157], [129], [126], [156], [92], [9], [141], [101], [67], [56], [119], [53], [47], [60], [61], [17], [72], [10].…”