Uniformly bounded superposition operators in the space of functions of bounded n-dimensional Φ-variation

We prove that if a superposition operator maps a subset of the space of all functions of n-dimensional bounded Φ-variation in the sense of Riesz, into another such space, and is uniformly bounded, then the non-linear generator h(x, y) of this operator must be of the form h(x, y) = A(x)y + B(x) where, for every x, A(x) is a linear map.peerReviewe

Uniformly bounded superposition operators in the space of functions of bounded n-dimensional Φ-variation

We prove that if a superposition operator maps a subset of the space of all functions of n-dimensional bounded Φ-variation in the sense of Riesz, into another such space, and is uniformly bounded, then the non-linear generator h(x, y) of this operator must be of the form h(x, y) = A(x)y + B(x) where, for every x, A(x) is a linear map.peerReviewe

Remark on globally Lipschitzian composition operators

Let I C R be an interval, f : I x R —> R a fixed two-place function, and J’(Z) the linear space of all the functions u : I —> R. The function F : F(I) —> F{I} given by the formula (F(u))(x) := /(x,u(x)), x G I, u G F(Z), is said to be a composition operator. Let a G I be fixed. Denote by Lip(I) the Banach space of all the functions « E 7(f) with the norm (1) IhllLip(l) := lu(°)l + sup | I Xi — X2 xi,x2 e I-, In [2] it is proved that if a composition operator F mapping Lip(I) into itself is globally Lipschitzian with respect to the Lip(I)-norm, then /(x, y) = g(x)y + h(x), (x G I;y 6 R), for some g,h GLip(I) (Fragment tekstu)

Approximate controllability of the impulsive semilinear heat equation

In this paper we apply Rothe's Fixed Point Theorem to prove the interior approximate controllability of the following semilinear impulsive Heat Equation $$\begin{cases} z_{t} = \Delta z + 1_{\omega}u(t,x) + f(t,z,u(t,x)), & \text{in} \quad (0,\tau] \times \Omega, t \neq t_{k}) \\ z = 0, & \text{on} \quad (0, \tau) \times \delta\Omega,\\ z(0,x) = z_{0}(x), & x \in \Omega, \\ z(t_{k}^{+}, x) = z(t_{k}^{-}, x) + I_{k}(t_{k},z(t_{k},x)u(t_{k},x)), & x \in \Omega, \end{cases}$$ where k = 1, 2, . . . , p, $\Omega$ is a bounded domain in $\mathbb{R}^{N}(N \geq 1), z_{0} \in L_{2}(\Omega), \omega$ is an open nonempty subset of $\Omega$, $1_{\omega}$ denotes the characteristic function of the set $\omega$, the distributed control $u$ belongs to $C\left([0, \tau]; L_{2}\left(\Omega\right)\right)$ and $f,I_{k} \in C([0, \tau] \times \mathbb{R} \times \mathbb{R}; \mathbb{R}), k = 1, 2, 3, \ldots, p$, such that $$|f(t,z,u)| \leq a_{0}|z|^{\alpha_{0}} + b_{0}|u|^{\beta_{0}} +c_{0}, \quad u \in \mathbb{R}, z \in \mathbb{R}.$$ $$|I_{k}(t,z,u)| \leq a_{k}|z|^{\alpha_{k}} + b_{k}|u|^{\beta_{k}} +c_{k}, k=1,2,3 \ldots, pu \in \mathbb{R}, z \in \mathbb{R}$$ with $\frac{1}{2} \leq \alpha_{k} < 1, \frac{1}{2} \leq \beta_{k} < 1, k= 0,1,2,3, \ldots, p$ Under this condition we prove the following statement: For all open nonempty subsets $\omega$ of $\Omega$ the system is approximately controllable on $[0, \tau]$. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state $z_{0}$ to an $\epsilon$ neighborhood of the nal state $z_{1}$ at time $\tau > 0$

Meeting Minutes

Meeting to discuss Lost River Cave internships, Big Red email, provide-a-ride and SGA website

Remarks on uniformly bounded composition operator acting between Banach spaces of functions of two variables of bounded schramm Φ-variation

In this paper we prove that if the composition operator H of generator h : Ib a × C → Y (X is a real normed space, Y is a real Banach space, C is a convex cone in X and Ib a ⸦ R2) maps Φ1 BV (Ib a, C) into Φ2 BV (Ib a, Y) and is uniformly bounded, then the left-left regularization h* of h is an affine function in the third variable

Solutions of Hammerstein equations in the space BV(Iba)

Quaestiones Mathematicae 37(2014), 359-37