Key words Demicompact linear operator, essential spectrum, Fredholm and semi-Fredholm operators MSC (2010) 47A53In this paper, we present some results on Fredholm and upper semi-Fredholm operators involving demicompact operators. Our results generalize many known ones in the literature, in particular those obtained by Petryshyn in [27] and Jeribi et al. in [1], [22]. They are used to establish a fine description of the Schechter essential spectrum of closed densely defined operators, and to investigate the essential spectrum of the sum of two bounded linear operators defined on a Banach space by means of the essential spectrum of each of the two operators.

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In this paper, we study some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras, where the entries are nonlinear operators. Furthermore, we apply the obtained results to a coupled system of nonlinear equations. Copyright © 2012 John Wiley & Sons, Ltd.

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.