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There are three questions that this work tries to answer. First, we investigate some spectral of fine properties of a 3 ž 3 block operator matrix with unbounded entries and with domain consisting of vectors which satisfy certain relations between their components. An application to transport equations that describes the neutron transport in a plane-parallel domain with width 2a, or the transfer of unpolarized light in a plane-parallel atmosphere of optical thickness 2a is given. Second, we discuss under which conditions a 2 ž 2 operator matrix with nonlinear entries, acting on a product of convex closed subsets of Banach spaces have a fixed point. As application, we give some existence results for a mixed stationary problem on Lp-spaces (1 p 1) inspired from the Rotenberg s model. Finally, we study some algebraic and topological properties of a new set defined by Ghyp(T) := { a : T aI is hypercyclic} for a given bounded linear operator acting on separable Banach space.

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I am a doctor in applied mathematics at the preparatory engineering institute of Sfax, Tunisia.

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