The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy.
This paper is devoted to the study of the approximation problem for the abstract hyperbolic differential equation u'(t) = A(t)u(t) for t ∈ [0,T], where A(t):t ∈ [0,T] is a family of closed linear operators, without assuming the density of their domains.
We show that the result of Kato on the existence of a semigroup solving the Kolmogorov system of equations in l₁ can be generalized to a larger class of the so-called Kantorovich-Banach spaces. We also present a number of related generation results that can be proved using positivity methods, as well as some examples.
We develop an elementary theory of Fourier and Laplace transformations for exponentially decreasing hyperfunctions. Since any hyperfunction can be extended to an exponentially decreasing hyperfunction, this provides simple notions of asymptotic Fourier and Laplace transformations for hyperfunctions, improving the existing models. This is used to prove criteria for the uniqueness and solvability of the abstract Cauchy problem in Fréchet spaces.
Sono dati nuovi teoremi di esistenza per soluzioni regolari di equazioni di evoluzione paraboliche astratte con applicazioni all'equazione del calore in spazi di funzioni holderiane e alle equazioni semilineari.
We study a class of closed linear operators on a Banach space whose nonzero spectrum lies in the open left half plane, and for which is at most a simple pole of the operator resolvent. Our spectral theory based methods enable us to give a simple proof of the characterization of -semigroups of bounded linear operators with asynchronous exponential growth, and recover results of Thieme, Webb and van Neerven. The results are applied to the study of the asymptotic behavior of the solutions to a singularly...
Herzog and Lemmert have proven that if E is a Fréchet space and T: E → E is a continuous linear operator, then solvability (in [0,1]) of the Cauchy problem ẋ = Tx, x(0) = x₀ for any x₀ ∈ E implies solvability of the problem ẋ(t) = Tx(t) + f(t,x(t)), x(0) = x₀ for any x₀ ∈ E and any continuous map f: [0,1] × E → E with relatively compact image. We prove the same theorem for a large class of locally convex spaces including: • DFS-spaces, i.e., strong duals of Fréchet-Schwartz spaces,...
Let denote the generator of the rotation group in the space , where denotes the unit circle. We show that the stochastic Cauchy problem where is a standard Brownian motion and is fixed, has a weak solution if and only if the stochastic convolution process has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all...