44 research outputs found
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
where k = 1, 2, . . . , p, is a bounded domain in is an open nonempty subset of , denotes the characteristic function of the set , the distributed control belongs to and , such that
with
Under this condition we prove the following statement: For all open nonempty subsets of the system is approximately controllable on . Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state to an neighborhood of the nal state at time
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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