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Solvability of semilinear equations in hilbert spaces

Abstract:

In this paper, we study the existence of solutions for a broad class of semilinear equations in Hilbert spaces. This is done by applying the Rothe’s fixed point theorem and a characterization of dense range linear operators in Hilbert spaces. As an application, we study the approximate controllability of a semilinear control system governed by a semilinear evolution equation, and a particular case of this is a control system given by a semilinear heat equation with interior control.