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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....

Continuous Dependence of Solutions of Quasidifferential Equations with Non-Fixed Time of Impulses

Plotnikov, V., Kitanov, P. (1998)

Serdica Mathematical Journal

In this article on quasidifferential equation with non-fixed time of impulses we consider the continuous dependence of the solutions on the initial conditions as well as the mappings defined by these equations. We prove general theorems for quasidifferential equations from which follows corresponding results for differential equations, differential inclusion and equations with Hukuhara derivative.

Convergence estimate for second order Cauchy problems with a small parameter

Branko Najman (1998)

Czechoslovak Mathematical Journal

We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter ε . The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem.

Currently displaying 81 – 100 of 421