### Exact controllability of a non-linear generalized damped wave equation: Application to the sine-Gordon equation.

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Evading fixed point theorems we prove the interior approximate controllability of the following semilinear strongly damped wave equation with impulses and delay [...] in the space Z1/2 = D((−Δ)1/2)×L2(Ω),where r > 0 is the delay, Γ = (0, τ)×Ω, ∂Γ = (0, τ) × ∂Ω, Γr = [−r, 0] × Ω, (ϕ,ψ) ∈ C([−r, 0]; Z1/2), k = 1, 2, . . . , p, Ω is a bounded domain in ℝℕ(ℕ ≥ 1), ω is an open nonempty subset of , 1 ω denotes the characteristic function of the set ω, the distributed control u ∈ L2(0, τ; U), with...

In this paper we prove the interior approximate controllability of the following Semilinear Heat Equation with Impulses and Delay [...] where Ω is a bounded domain in RN(N ≥ 1), φ : [−r, 0] × Ω → ℝ is a continuous function, ! is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ! and the distributed control u be- longs to L2([0, τ]; L2(Ω; )). Here r ≥ 0 is the delay and the nonlinear functions f , Ik : [0, τ] × ℝ × ℝ → ℝ are smooth enough, such that [...] Under this...

In this paper we extend the well known Riesz lemma to the class of bounded $\varphi $-variation functions in the sense of Riesz defined on a rectangle ${I}_{a}^{b}\subset {\mathbb{R}}^{2}$. This concept was introduced in [2], where the authors proved that the space $B{V}_{\varphi}^{R}({I}_{a}^{b};\mathbb{R}$ of such functions is a Banach Algebra. Moreover, they characterized also the Nemytskii operator acting in this space. Thus our result creates a continuation of the paper [2].

In this paper we study existence and uniqueness of solutions for the Hammerstein equation $$u\left(x\right)=v\left(x\right)+\lambda {\int}_{{I}_{a}^{b}}K(x,y)f(y,u\left(y\right))\phantom{\rule{0.166667em}{0ex}}dy\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}x\in {I}_{a}^{b}:=[{a}_{1},{b}_{1}]\times [{a}_{2},{b}_{2}]\phantom{\rule{0.166667em}{0ex}},$$ in the space $B{V}_{\varphi}^{\mathbb{R}}\left({I}_{a}^{b}\right)$ of function of bounded total $\varphi -$variation in the sense of Riesz, where $\lambda \in \mathbb{R}$, $K:{I}_{a}^{b}\times {I}_{a}^{b}\to \mathbb{R}$ and $f:{I}_{a}^{b}\times \mathbb{R}\to \mathbb{R}$ are suitable functions.

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