Non - Autonomous Inhomogeneous Boundary Cauchy Problems and Retarded Equations

Abstract

In this paper we prove the existence and the uniqueness of the classical solution of non-autonomous inhomogeneous boundary Cauchy problems, and that this solution is given by a variation of constants formula. This result is applied to show the existence of solutions of a retarded equation.

Approximate solution to abstract differential equations with variable domain

“…Further, we accept the following hypotheses: (B1) We suppose the operator A (2) (s) to be strongly positive, i.e. there exists a positive constant M R independent of s such that on the rays and outside a sector Σ θ = {z ∈ C : 0 ≤ arg(z) ≤ θ, θ ∈ (0, π/2)} the following resolvent estimate holds…”

“…Here L(t) and Φ(t) are some linear operators defined on the boundary of the spatial domain and the second equation represent an abstract model of the time-dependent boundary condition. An existence and uniqueness result for this problem was proved in [2].…”

“…t k t k−1 e −A(2) k (t k −λ) λ s dλ, which can be found by a simple recurrence algorithm:I l = −l A (2) k −1 I l−1 + A (2) k −1 t l k I − t l k−1 e −A (2) k τ k , l = 1, 2, ..., s, I 0 = A (2) k −1…”

Exponentially Convergent Duhamel-Like Algorithms for Differential Equations with an Operator Coefficient Possessing a Variable Domain in a Banach Space

“…Here L(t) and Φ(t) are some linear operators defined on the boundary of the spatial domain, and the second equation represents an abstract model of the time-dependent boundary condition. An existence and uniqueness result for this problem was proven in [5]. Incorporating the boundary condition into the definition of the operator coefficient of the first equation, one obtains problem (1.1).…”