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 provide some existence and uniqueness results for a .p; q/-quasi-contraction mapping acting on an orbitally-complete cone metric space. These results generalize several fixed point theorems, in particular those due to Ilić and Rakočević's for quasi-contraction mappings (Ilić and Rakočević, Appl Math Lett 22(5):728-731, 2009), convex contraction mapping, and two-sided convex contraction of order 2.