Marachkov Type Stability Conditions for Non-autonomous Functional Differential Equations with Unbounded Right-hand Sides

Sufficient conditions for uniform equi-asymptotic stability and uniform asymptotic stability of the zero solution of the retarded equation x (t) = f (t, x t), (x t (s) := x(t + s), −h ≤ s ≤ 0) are given. In the stability theory of non-autonomous differential equations a result is of Marachkov type if it contains some kind of boundedness or growth condition on the right-hand side of the equation with respect to t. Using Lyapunov's direct method and the annulus argument we prove theorems for equations whose right-hand sides may be unbounded with respect to t. The derivative of the Lyapunov function is not supposed to be negative definite, it may be negative semi-definite. The results are applied to the retarded scalar differential equation with distributed delay x (t) = −a(t)x(t) + b(t) t t−h x(s) ds, (a(t) > 0), where a and b may be unbounded on [0, ∞). The growth conditions do not concern function a, they contain only function b. In addition, the function t → a(t) − t+h t |b(u)| du, measuring the dominance of the negative instantaneous feedback over the delayed feedback, is not supposed to remain above a positive constant, even it may vanish on long intervals.


Introduction
We consider the system where f : R + × C H → R m is continuous and takes bounded sets into bounded sets and f (t, 0) ≡ 0; R + := [0, ∞), C is the Banach space of continuous functions ϕ : [−h, 0] → R m with the maximum norm ϕ := max −h≤s≤0 |ϕ(s)|, | • | denotes an arbitrary norm in R m , h is a nonnegative constant, C H is the open ball of radius H in C around ϕ = 0.As is usual, if x : [−h, β) → R m (β > 0), then x t (s) := x(t + s) for −h ≤ s ≤ 0, 0 ≤ t < β.Let x(•; t 0 , ϕ) : [t 0 − h, t 0 + α) → R m denote a solution of (1.1) satisfying the initial condition x t 0 (•; t 0 , ϕ) = ϕ.It is known [5] that for each t 0 ∈ R + and each ϕ ∈ C there is at least one solution x(•; t 0 , ϕ) : [t 0 − h, t 0 + α) → R m with some α > 0, and if this solution remains bounded on every bounded subinterval of [t 0 , t 0 + α), then α = ∞.We will use Lyapunov's direct method [2,5].A continuous functional V : R + × C → R + which is locally Lipschitz in ϕ is called a Lyapunov functional if its right-hand side derivative with respect to system (1.1) is non-positive: A Lyapunov functional is called positive definite if there exists a wedge (i.e., a continuous, strictly increasing function The following stability concepts are standard [2,5]. Definition 1.1.The zero solution of (1.1) is: (a) stable if for every ε > 0 and t 0 ≥ 0 there is a δ(ε, t 0 ) > 0 such that ϕ < δ, t ≥ t 0 imply that |x(t; t 0 , ϕ)| < ε; (b) uniformly stable if for every ε > 0 there is a δ(ε) > 0 such that ϕ < δ, t 0 ≥ 0, t ≥ t 0 imply that |x(t; t 0 , ϕ)| < ε; (c) asymptotically stable if it is stable and for every t 0 ≥ 0 there is a σ(t 0 ) > 0 such that ϕ < σ implies lim t→∞ x(t; t 0 , ϕ) = 0; (d) uniformly equi-asymptotically stable (UEAS) if it is uniformly stable and there is a D > 0 and for each µ > 0, t 0 ≥ 0 there is a T(µ, t 0 ) such that ϕ < D, t ≥ t 0 + T imply that |x(t; t 0 , ϕ| < µ; (e) uniformly asymptotically stable (UAS) if it is uniformly stable and there is a D > 0 and for each µ > 0 there is a T(µ) such that In stability theory of non-autonomous differential equations a result is of Marachkov's type if it contains some kind of boundedness or growth condition on the right-hand side of the equation with respect of t [9].One of the most classical results in stability theory of functional differential equations is the following theorem.
Theorem A ( [2,5]).Suppose that there are a Lyapunov functional V, wedges W 1 , W 2 , and constants M, H > 0 such that the following conditions are satisfied: Then the zero solution of (1.1) is asymptotically stable.
Condition (iii) is very restrictive, it often raises difficulties in applications of the theorem.T. A. Burton and G. Makay [4] have taken an important step to overcome these difficulties.
Theorem B (T. A. Burton and G. Makay).Suppose there are H > 0, V : R + × C H → R + , wedges W 1 , W 2 , W 3 , and a continuous increasing function F : Then the zero solution of (1.1) is uniformly equi-asymptotically stable.
Throughout this paper we will illustrate abstract results with applications to the retarded scalar differential equation with distributed delay where a, b : R + → R are continuous and a(t) ≥ 0 (t ∈ R + ).This is an important model equation: it describes a process in which there are an instantaneous and a delayed feedback.Define the Lyapunov functional It can be seen that so we can apply the Burton-Makay Theorem B to equation (1.2).

L. Hatvani
Corollary C. Suppose that there are constants c 1 , Then the zero solution of (1.2) is uniformly equi-asymptotically stable.
If a, b are constants, i.e., a(t) ≡ a 0 > 0, b(t) ≡ b 0 , then Corollary C says that the zero solution of (1.2) is UEAS, provided that a 0 > h|b 0 |.In other words, the dominance of the negative instantaneous feedback over the delayed one suffices UEAS.Conditions (i) and (ii) are in accordance with this experience in the case of the more general nonautonomous equation (1.2), but condition (iii) contradicts "the larger a(t) is the better" principle.The following problem arises: is the zero solution UEAS if a(t) is large enough and |b(t)| is bounded (e.g., a(t) = t 2 , b(t) ≡ sin t), which is excluded by the growth condition (iii)?We can also ask a question regarding condition (ii) in Theorem B (and in Corollary C).One can expect that the dominance of a over b is not necessarily as uniform as condition (ii) requires.For example, can the zero solution of (1.2) be UEAS if η vanishes on intervals of the same length infinitely many times in R + ?
In this paper we develop further Theorem B essentially weakening both conditions (ii) and (iii).For example, the corollary of the main result for equation (1.2) will imply that the answers to both of the questions above are affirmative.
The paper is organized as follows.Section 2 contains the main theorem and its corollaries.Section 3 is the proof of the main Theorem 3.1 based upon an annulus argument [6].In Section 4 we formulate some applications to equation (1.2).

Main results
To weaken the uniformity of conditions (ii) in Theorem B and Corollary C we need the following concepts, which have played an important role in the stability theory of non-autonomous systems [7,3,8] for a long time.

Definition 2.1. A locally integrable function
holds.
(b) weakly integrally positive (WIP) if for any sequences {t i } ∞ i=1 , {t i } ∞ i=1 satisfying conditions with some δ > 0, ∆ > 0, we have To control the growth of an integral function we introduce a notation.For a locally integrable function M : R + → R + and numbers t ∈ R + , ε > 0 define (see [6]).
Theorem 2.2.Suppose that for system (1.1) there are a Lyapunov function V : R + × C → R + and wedges W 1 , W 2 , W 3 such that the following conditions are satisfied: (ii) there is a locally integrable function η : we have Then the zero solution of (1.1) is uniformly equi-asymptotically stable.
The Burton-Makay theorem is a special case of Theorem 2.2.
Proposition 2.3.Theorem B is a corollary of Theorem 2.2.
Proof.Suppose that conditions (i)-(iv) in Theorem B are fulfilled, and set η(t) ≡ 1.Then conditions (ii) and (iii) in Theorem 2.2 are satisfied.We show that condition (iv) is also satisfied.
Condition (iv) in Theorem 2.2 has a simple form also in the case, when the integral function of G is uniformly continuous.Corollary 2.4.Assume that conditions (i)-(iii) in Theorem 2.2 are satisfied.Suppose, in addition, that (iv ) function G in condition (iv) of Theorem 2.2 has the additional property that t → t 0 G(u) du is uniformly continuous and, instead of (2.5)-(2.6),function η is weakly integrally positive.
Then the zero solution of (1.1) is uniformly equi-asymptotically stable.

L. Hatvani
Proof.We have to prove that (iv) in Theorem 2.2 is satisfied.
i=1 are arbitrary with because η is weakly integrally positive.
Burton and Makay gave a sophisticated counterexample showing that it is impossible to strengthen the conclusion of Theorem B to uniform asymptotic stability.The following theorem says that if we strengthen condition (iv ) in Corollary 2.4 to "integral positivity", then we get UAS.Therefore, if F is bounded in Theorem B, then the conclusion of the theorem can already be strengthened to UAS, and we get Theorem 5.2.1 in [5].So the following theorem can be considered as a generalization of this theorem.Theorem 2.5.Assume that conditions (i)-(ii) in Corollary 2.4 (i.e., conditions (i)-(ii) in Theorem 2.2) are satisfied.Suppose, in addition, that (iv ) function η in condition (iv ) is not only weakly integrally positive but integrally positive.
Then the zero solution of (1.1) is uniformly asymptotically stable.

Proofs of the theorems
The proof of Theorem 2.2 is based upon the annulus argument [1,6].This is a method of the proof for the existence of a limit, which can detect that a trajectory x : R + → R m crosses the annulus ε 1 ≤ |x| ≤ ε 2 infinitely many times.

Proof of Theorem 2.2
Suppose that conditions (i)-(iv) in Theorem 2.2 are satisfied.(i) and (ii) guarantee uniform stability for the zero solution of (1.1) [2,5]; take δ(ε) from the definition of this property.Define D := δ(H), where H is from condition (iv).We always suppose throughout this proof that initial functions ϕ satisfy ϕ < D, i.e., we have |x(t)| < H for all solutions x and for all t 0 , t with t 0 ≤ t.Since t → v(t) := V(t, x t ) is nonincreasing, we also have In the first step we prove that (ii) and (iii) imply the following property: for every ε > 0 there exists a ∆(ε i.e., If T > ∆(ε) were possible, then by the choice of ∆(ε) the reversed strict inequality would also hold, what is impossible.
To prove UEAS we have to show the existence of T(µ, t 0 ) in the definition of the property.Thanks to the US, it is enough to guarantee the existence of T(µ, t 0 ) such that for every ϕ with ϕ < D there is a t ∈ [t 0 , t 0 + T(µ, t 0 )] such that x t (•; t 0 , ϕ) < δ(µ).Suppose the contrary, i.e., there are µ, t 0 such that for each T > 0 there exists ϕ = ϕ(•; T) with ϕ < D such that Let us fix T arbitrarily, take a corresponding ϕ and denote x(t) = x(t; t 0 , ϕ).Then there are sequences {t i } Here N 0 and N(T) are defined by where [α] denotes the integer part of a real number α.That is, N 0 is independent of T, but N(T) does depend on T and lim T→∞ N(T) = ∞.
Let us observe that For the function t → v(t) := V(t, x t (•; t 0 , ϕ)) we have To get a contradiction we want to apply condition (iv) taking T → ∞.However, the problem is that the initial function ϕ and, consequently, sequences {t i }, {t i } depend on T. We overcome this difficulty by the use of a universal sequence {t i } ∞ i=N 0 .Since G is locally integrable, the integral of G is absolute continuous [10] and Γ G (u, ε) is continuous in u, so we can define {t i } by On the other hand, by the definition (3.5) of {t i }, from (3.4) we obtain for all T > 0. Now we can already take the limit T → ∞ and write which contradicts (3.6).

Proof of Theorem 2.5
We will construct an upper bound for T of properties (3.2)-(3.3)independent of ϕ and also of t 0 .
R + ); in addition, for every ε > 0 there is L such that for every {t i } ∞ i=1 satisfying the inequalities t i > ih, t i+1 < t i + L iii) lim T→∞ t+T t η(u) du = ∞ uniformly with respect to t ∈ R + ; (iv) there are H ∈ R + and a locally integrable function G : R + → R + such that if a solution is bounded by H, then |x(t)| ≤ G(t) (t ∈