Linear Functional Differential Systems Research Articles

Almost Sure Asymptotic Stability of Stochastic Volterra Integro-Differential Equations with Fading Perturbations

In this note, we address the question of how large a stochastic perturbation an asymptotically stable linear functional differential system can tolerate without losing the property of being pathwise asymptotically stable. In particular, we investigate noise perturbations that are either independent of the state or influenced by the current and past states. For perturbations independent of the state, we prove that the assumed rate of fading for the noise is optimal.

On a Boundary Value Problem for 𝑛-th Order Linear Functional Differential Systems

Abstract In this paper, theorems on the Fredholm alternative and wellposedness of the linear boundary value problem 𝑢′(𝑡) = ℓ(𝑢)(𝑡) + 𝑞(𝑡), ℎ(𝑢) = 𝑐, where ℓ : 𝐶([𝑎, 𝑏]; 𝑅𝑛) → 𝐿([𝑎, 𝑏]; 𝑅𝑛) and ℎ : 𝐶([𝑎, 𝑏]; 𝑅𝑛) → 𝑅𝑛 are linear bounded operators, 𝑞 ∈ 𝐿([𝑎, 𝑏]; 𝑅𝑛), and 𝑐 ∈ 𝑅𝑛, are established even when ℓ is not a strongly bounded operator.

Holomorphic solutions of functional differential systems near singular points

Functional analysis techniques are used to prove a theorem, analogous to the Harris-Sibuya-Weinberg theorem for ordinary differential equations, which yields as corollaries a number of existence theorems for holomorphic solutions of linear functional differential systems of the form z D y ′ ( z ) = A ( z ) y ( z ) + B ( z ) y ( α z ) + C ( z ) y ′ ( α z ) {z^D}y’(z) = A(z)y(z) + B(z)y(\alpha z) + C(z)y’(\alpha z) in the neighborhood of the singularity at z = 0 z = 0 .