Delay-Dependent Stability Conditions for Non-autonomous Functional Differential Equations with Several Delays in a Banach Space

: Let B j ( t ) ( j = 1, ..., m ) and B ( t , τ ) ( t ≥ 0, 0 ≤ τ ≤ 1) be bounded variable operators in a Banach space. We consider the equation where h k ( t ) ( t ≥ 0; k = 1, ..., m ) and h 0 ( τ ) are continuous nonnegative bounded functions. Explicit delay-dependent exponential stability conditions for that equation are established. Applications to integrodifferential equations with delay are also discussed.


Introduction and statement of the main result
This paper is devoted to a class of linear nonautonomous functional di erential equations in a Banach space with several variable delays, whose coe cients are bounded operators. Such equations include integrodi erential equations with delay.
The basic method for the stability analysis of functional di erential equations is the Lyapunov-Krasovskij method [5]. By that method, many great results have been obtained. Recently, that method has been extended to functional di erential equations in a Hilbert space, cf. [10,11,14,16,18] and references given therein. Besides, mainly equations with one delay have been considered. To the best of our knowledge, the stability of nonautonomous equations in a Banach space with several delays are not investigated in the available literature. Below we obtain delay-dependent exponential stability conditions for nonautonomous functionaldi erential equations in a Banach space with several delays.
It should be noted that nding the Lyapunov-Krasovskij type functionals or solving the corresponding operator inequalities are often connected with serious mathematical di culties, especially regarding nonautonomous equations with many delays. To the contrary, the stability conditions presented in this paper are explicitly formulated in terms of the coe cients and delays. The literature on the delay-dependent stability criteria is rather rich, but mainly equations in a nite dimensional space are considered, cf. [7,8,15].
Introduce the notations. Everywhere below, X is a complex Banach space with a norm · X = · and the unit operator I X = I. By B(X), we denote the set of all bounded linear operators in X. For an A ∈ B(X), σ(A) is the spectrum and A is the operator norm.
For real a < b < ∞, C([a, b], X) is the space of X-valued functions f de ned and continuous on [a, b], and equipped with the nite norm where h k (t) (k = , ..., m) and h (τ) are continuous nonnegative functions de ned on R+ and [ , ], respectively, such that The present paper is devoted to the equation where ϕ is given. Throughout the paper it is a assumed that and therefore, The proof of this theorem is presented in the next section. As it follows from Theorem 1.1 our stability conditions are based, in particular, on the norm estimates for V M . Below we consider such estimates under various conditions. In particular, in Section 3 we assume that M(t) is dissipative. In Section 4 it is supposed that M(t) satis ed the so called generalized Lipschitz condition. Section 5 is devoted to equations with di erentiable operators a Hilbert space.
In Section 6 Theorem 1.1 is applied to integro-di erential equations.

Proof of Theorem 1.1
Consider the non-homogeneous equation corresponding to (1.1): with the zero initial condition A solution of problem (2.1), (2.2) is a strongly di erentiable function w : Rη → X satisfying (2.1) for all nite t > and (2.2). Integrating (2.1), we have is a Volterra one and therefore, problemg (2.1), (2.2) has a unique solution representable by Furthermore, for a nite T > for the brevity put |w| T = w C( ,T) . Due to (1.3) for a solution of (2.1), (2.2) we have.
Thus (2.1) can be written as By the Variation of Constants formula problem (2.5), (2.2) takes the form Observe that From (2.6) and (2.8) it follows Hence, letting T → ∞, we get We thus have proved the following result. Let v(t) be a solution of problem (1.1), (1.2). Put where f = Eφ. Besides, the zero initial condition holds. Under conditions (1.3), (1.5) Lemma 2.1 implies Note that φ C(R+) ≤ ϕ C(−η, ) and due to (2.3) we arrive at the following result. For an ϵ > put where Uϵ(t, s) is the evolution operator of the di erential equation z ′ (t) = Mϵ(t)z(t). into (1.1), we obtain the equation where

Equations with dissipative operators
We will say that A ∈ B(X) is dissipative if I+Aδ < for all su ciently small δ > . In this section we consider (1.1), assuming that M(t) is dissipative for su ciently large t. To this end introduce the operator products This equality is proved in [9, Chapter 1].

Lemma 3.1. Let there be a real Riemann-integrable function function ν(t), such that
for all su ciently small δ > . Then The passage to the limit as m → ∞ and representation (3.1) give the required estimate. Let X = H be a Hilbert space and Λ(M R (t)) = sup σ(M R (t)), where M R (t) = (M(t) + M * (t)) and the asterisk means the adjointness. Since

Equations in a Banach space with the generalized Lipschitz property
In this section we consider equation (1.1), assuming that M(t) satis es the generalized Lipschitz condition

hold. Then a solution u(t) to the non-homogeneous problem
satis es the inequality Proof. Rewrite (4.4) as with an arbitrary xed τ ≥ . So problem (4.4) is equivalent to the equation
From this lemma it follows. and With Re A = (A + A * )/ and λ inf (Re A ) = inf σ(Re A ), we conclude that d(y(t), y(t)) dt ≥ λ inf (Re A )(y(t), y(t)) and therefore e A t z ≥ e tλ inf (Re A ) z (t ≥ ).

Integro-di erential equations with delays
Our main object in this section is the equation About other approaches to integro-di erential equations see [1-4, 17, 19]. Con ict of interest: The author states that there is no con ict of interest.
Data Availability Statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.