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Closed semistable operators and singular differential equations

Jaromír J. Koliha, Trung Dinh Tran (2003)

Czechoslovak Mathematical Journal

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 0 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 C 0 -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...

Compact perturbations of linear differential equations in locally convex spaces

S. A. Shkarin (2006)

Studia Mathematica

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,...

Cone invariance and squeezing properties for inertial manifolds for nonautonomous evolution equations

Norbert Koksch, Stefan Siegmund (2003)

Banach Center Publications

In this paper we summarize an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and some additional technical assumptions like boundedness or smoothing properties. We apply this result to processes (two-parameter semiflows) generated by nonautonomous semilinear parabolic evolution equations.

Cone-type constrained relative controllability of semilinear fractional systems with delays

Beata Sikora, Jerzy Klamka (2017)

Kybernetika

The paper presents fractional-order semilinear, continuous, finite-dimensional dynamical systems with multiple delays both in controls and nonlinear function f . The constrained relative controllability of the presented semilinear system and corresponding linear one are discussed. New criteria of constrained relative controllability for the fractional semilinear systems with delays under assumptions put on the control values are established and proved. The conical type constraints are considered....

Conley index in Hilbert spaces and a problem of Angenent and van der Vorst

Marek Izydorek, Krzysztof P. Rybakowski (2002)

Fundamenta Mathematicae

In a recent paper [9] we presented a Galerkin-type Conley index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indefinite elliptic systems. More specifically, we study the elliptic system - Δ u = ∂ v H ( u , v , x ) in Ω, - Δ v = ∂ u H ( u , v , x ) in Ω, u = 0, v = 0 in ∂Ω, (A1) on a smooth bounded domain Ω in ℝ N for "-"-type Hamiltonians H of class C² satisfying subcritical growth assumptions on their first order derivatives....

Currently displaying 161 – 180 of 966