On the asymptotic stability for

On the asymptotic stability for

intermittently damped nonlinear oscillators

László Hatvani

University of Szeged, Bolyai Institute, Aradi vértanúk tere 1, H-6720 Szeged, Hungary

Received 11 April 2018, appeared 30 July 2018 Communicated by Ferenc Hartung

Abstract. The second order nonlinear differential equation

x⁰⁰+h(t,x,x⁰)x⁰+f(x) =0 x ∈R, t∈R+:= [0,∞), ()⁰:= _dt^d(),

and a sequence {In}^∞_n=1 of non-overlapping intervals are given, where the damping coefficienthadmits an estimate

a(t)|y|^αw(x,y)≤h(t,x,y)≤b(t)W(x,y) (t∈ I:=∪^∞_n=1In;x,y∈_R).

It is known that if the equation is linear (f(x)≡x,h(t,x,x⁰)≡h(t),a(t)≤h(t)≤b(t)), a(t) ≥ a = const. > 0 and b(t) ≤ b = const. < ∞, then ∑^∞n=1|In|³ = ∞ is sufficient for the asymptotic stability, and the exponent 3 is the best possible. (Here|In|denotes the length of In.) We give sufficient conditions for the asymptotic stability of the zero solution via the control functionsa,bon the control setIconsidering cases when a>0 and/orb<∞do not exist.

Keywords: intermittent damping, asymptotic stability, total mechanical energy, dissi- pation, differential inequalities.

2010 Mathematics Subject Classification: Primary 34D20, Secondary 70K20.

1 Introduction

We consider the model

x⁰⁰+h(t,x,x⁰)x⁰+ f(x) =0 x ∈_R, t∈_R+:= [0,∞), ()⁰ := _dt^d(), (1.1) of a nonlinear damped oscillator, where −f(x) is the restoring force (f : (−M,M) → _R is continuous, 0< M≤_∞is a fixed constant,x f(x)>0 ifx6=0);h:R+×(−M,M)×_R→_R+ is the damping coefficient, which allows an estimate

a(t)|y|^αw(x,y)≤h(t,x,y)≤b(t)W(x,y). (1.2)

BEmail: hatvani@math.u-szeged.hu

Hereαis a nonnegative real number;w,W :(−M,M)×_R→_R+are continuous,w(x,y)>0 for all x,y. Functions a,b : R+ → _R+ are piecewise continuous, they are called the lower and the upper control function respectively: we can control the damping via these functions.

Intermittent dampingmeans that we are given a sequence {I_n= (α_n,β_n)}^∞_n=1 (lim_n→_∞α_n =_∞) of non-overlapping intervals, and it is supposed thata,bcan be controlled only over these in- tervals in time; between the intervals nothing but the nonnegativity of a(t),b(t)is supposed.

I := ∪^∞_n=1I_n will be called the control set. The problem is to find conditions on a,b,{I_n}guar- anteeing the asymptotic stability for the equilibrium statex= x⁰ =0 in Lyapunov’s sense [3].

This is the problem of intermittent damping [5,6,8,10–13].

Let us introduce the following notations:

H(x,y):= ¹

2y²+F(x)(total mechanical energy), F(x):=

Z _x

0 f ; H_M := {(x,y)∈(−M,M)×_R:H(x,y)<min{F(M),F(−M)}}; 0<K_M :=

sup

f(x)

x : 0<|x| ≤ M 1/2

≤_∞ (0< M< M); (1.3) A(t):=

Z _t

0 a(s)ds, B(t):=

Z _t

0 b(s)ds;

a_n :=inf{a(t): α_n<t< β_n} a_n:=sup{a(t): α_n <t< β_n}; A_n:=

Z _βn

αn

a(t)dt, B_n:=

Z _βn

αn

b(t)dt (n ∈_N);

b_n, b_n are defined analogously with a_n, a_n. It is easy to see that for every M ∈ (0,M) the closure ofHM is compact, and there is a constantc(M)>0 such that

a(t)≤ c(M)b(t) (t∈ _R+). (1.4) The derivative ofHwith respect to (1.1) [3] is

H⁰(t,x,x⁰) =−h(t,x,x⁰)(x⁰)²≤0, (1.5) so H is non-increasing along any solution of (1.1) (the energy is dissipated). Therefore HM

is an invariant neighborhood of(0, 0), so the equilibrium is stable. We always suppose that (x(0),x⁰(0)) ∈ H_M, so |x(t)| < M is also satisfied for all t ∈ _R+. It has remained to find conditions for the control functions a,b guaranteeing limt→∞H(t,x(t),x⁰(t)) = 0 for every solutiont7→ x(t)with(x(0),x⁰(0))∈H_M.

The first result on the intermittent damping was published by R. A. Smith [13] for the linear case

h(t,x,x⁰)≡h(t), f(x)≡x, a(t)≡b(t)≡ h(t). (1.6) Theorem A. Suppose that

∑

h_n|I_n|

min

|I_n|; 1 1+hn

2

=_{∞, (1.7)}

where|In|denotes the length of In. Then the zero solution of (1.1)is asymptotically stable.

This theorem was improved in the linear case (1.6) in [10]. P. Pucci and J. Serrin [12] proved theorems for very general quasi-variational ordinary differential systems of many degrees of freedom. Here we cite the consequences of their two main theorems for (1.1) using the notation

d_n:= ¹

|In|

Z

In

b

end the concept of integral positivity.

Definition 1.1. A locally integrable function g : R+ → _R+ is called integrally positive with parameterδ >0 if

lim inf

t→_∞ Z _t+δ

t g>0. (1.8)

If (1.8) is satisfied for allδ>0, then gis calledintegrally positive.

Theorem B. Suppose that there exist positive constants c1,c2,c3such that

∑

a_n|I_n|_min

|I_n|^2+α_; ^c1 c2+c3a_ndn

=_{∞. (1.9)}

Then the zero solution of (1.1)is asymptotically stable.

Theorem C. Suppose that the lower control function a is integrally positive and there exists a positive constant c₄such that

∑

min

|In|²; c₄ d_n|In|

=_∞. (1.10)

Then the zero solution of (1.1)is asymptotically stable.

These theorems yield the following rather unexpected corollaries for simpler control func- tions.

Corollary D. I. If

a(t)≥ a>0, b(t)≤b<_∞ (t ∈ I) (1.11) with some constants a, b, and

∑

|I_n|^3+α =_∞, (1.12)

then the zero solution of (1.1)is asymptotically stable.

II. If

a(t)≥a >0 (t ∈_R+), b(t)≤b<_∞ (t∈ I),

and ∞

n

∑

|In|²=_∞, (1.13)

then the zero solution of (1.1)is asymptotically stable.

Pucci and Serrin [12] showed that the exponents 3+α and 2 in (1.12) and (1.13), respec- tively are the best possible ones in the sense that without further restrictions no smaller ex- ponents can yield the general conclusion. For this reason we call Theorem B, respectively TheoremC, a result of “type exponent 3+α”, respectively of “type exponent 2”.

In this paper we prove theorems of types 3+α and 2, which do not use the infimum a_n, therefore they will be applicable when the lower control function a often vanishes even on intervals of a fixed length. Pucci and Serrin based their results on the method of quasi- variational inequalities. We use an entirely different approach, the method of differential inequalities.

The paper is organized as follows. In Section 2 and 3 we formulate the basic theorems and their corollaries and discuss the results. In Section 4 we give the proofs.

2 Results of type “exponent 3 + _α”

Equation (1.1) is equivalent to the system

x⁰ =y, y⁰ = −f(x)−h(t,x,y)y. (2.1) By the use of the polar coordinatesr,ϕwith

x=rcosϕ, y=rsinϕ (r>0,−_∞< ϕ<_∞), (2.2) this system can be rewritten into the form

r⁰ =rsinϕcosϕ− f(rcosϕ)_sinϕ−h(t,rcosϕ,rsinϕ)_sin²_{ϕ, (2.3)} ϕ⁰ =−

sin²ϕ+ ^f(rcosϕ) rcosϕ cos²ϕ

−h(t,rcosϕ,rsinϕ)sinϕcosϕ. (2.4) In what follows, ift 7→(x(t)_,y(t))is a solution of (2.1), then we mention the same solution as

“solutiont7→ (r(t),ϕ(t))”, provided thatx(t),y(t)andr(t),ϕ(t)are connected via (2.2).

For a solutiont 7→(x(t),y(t))starting fromH_M (M>0) introduce the notations

h∗(t):=h(t,x(t),y(t)), H∗(t):= H(x(t),y(t)). (2.5) The proof of the main result will be based upon the method of contradiction. We will sup- pose that the equilibrium is not asymptotically stable, i.e., there exists a point (x(0),y(0))∈ H_M such that for the solution (x(t),y(t)) starting from this point there holds H∗(_∞) := lim_t→_∞H∗(t)>0. Then it can be seen that

lim inf

t→_∞ r(t) =:r₀ >0. (2.6)

Integrating (1.5) we get a contradiction if we have the divergence H∗(0)−H∗(_∞)≥r²₀

Z _∞

0 h∗(t)sin²ϕ(t)dt

≥r²₀^+α

infHM

w(x,y) _{Z ∞}

0 a(t)|sinϕ(t)|^2+αdt= _∞.

(2.7)

However, we cannot require directly divergence (2.7) ofa(t)because we do not knowϕ(t)from the solution (r,ϕ). The main idea is that we estimate |sinϕ(t)| from below, the estimation defines an appropriate family of test functions on the control set I and in the theorem we require the divergence of the integral of the products ofa(t)and the test functions.

Theorem 2.1. I.Suppose that for everyγ>0, for every sequence of non-overlapping intervals{I_n= (α_n,β_n)}^∞_n=1of the propertyβ_n−α_n≤γ, and for arbitraryξ_n∈ I_nthe divergence

∑

Z _ξn

αn

a(t)

min Z _t

αn

exp[−q(B(t)−B(s))]ds;ξ_n−t 2+α

dt +

Z _βn

ξ_n

a(_t)

min Z _t

ξ_n

exp[−q(_B(_t)−B(_s))]_ds;β_n−t 2+α

dt

!

=_∞ (q=q(M):=sup_H

MW(x,y))

(2.8)

holds. Then the equilibrium of (1.1)is asymptotically stable.

II.Suppose K_M < _{∞. If} (2.8)holds for some γ< π/K_M, then the equilibrium of (1.1)is asymptoti- cally stable.

If we estimateRt

α_nexp[−q(B(t)−B(s))]ds in (2.8) in different ways, then we get different corollaries. At first we useb_n=sup_Inb.

Corollary 2.2. Suppose that there are a sequence {I_n} of non-overlapping intervals and a number κ ∈(0, 1)such that

∑

min

|I_n|; 1 1+bn

2+αZ

En

a =_∞ (2.9)

holds for every sequence{E_n}of sets E_n ⊂ I_nsuch that E_nis the union of finite subintervals of I_nand mes(En)≥κ|In|, wheremes(En)denotes the Lebesgue measure of En. Then the equilibrium of (1.1) is asymptotically stable.

This corollary is a generalization of Smith’s Theorem A to the nonlinear equation (1.1).

What is more, in the special case (1.6) it implies a sharpened version of Theorem Aworking also in the case h_n=0. We can get a more general result if we use Bn=R

Inbinstead ofbn. Corollary 2.3. Suppose that for everyγ > 0 there are a sequence {In}of non-overlapping intervals with|I_n| ≤γand a numberκ∈(0, 1)such that

∑

exp

−q Z

In

b

|I_n| 2+αZ

En

a= _∞ (2.10)

holds for every sequence{E_n}of sets E_n ⊂ I_nsuch that E_nis the union of finite subintervals of I_nand mes(E_n)≥κ|I_n|. Then the equilibrium of (1.1)is asymptotically stable.

It is worth noticing that the condition κ ∈ (0, 1)in Corollaries 2.2 and2.3 is sharp in the sense that if we require (2.9) and (2.10) with En = In (the case ofκ = 1), then the corollaries become false. In fact, e.g., if Corollary 2.2 were true withκ = 1, then R∞

0 a = _∞ (this is (2.9) with bounded band I_n = (n,n+1)) would imply asymptotic stability for the zero solution of (1.1) providedbis bounded, but this is not true (see, e.g., [10,12]). The following corollary gives a condition containingR

Inainstead ofR

Ena.

Corollary 2.4. Suppose that for every γ > 0 there is a sequence {I_n} of non-overlapping intervals with|I_n| ≤γsuch that

∑

1 anexp

−q Z

In

b

2+αZ

In

a 3+α

=_∞ (2.11)

holds. Then the equilibrium of (1.1)is asymptotically stable.

Now we cite a result showing that Corollary2.4, and consequently Theorem2.1are sharp enough. It is known that in the case of “small damping” (0 ≤ h(t) ≡ h(t,x,y) ≤ h < _∞, t∈ _R+) there is no necessary and sufficient condition for the asymptotic stability even in the linear case except linear equations with step function coefficients. Á. Elbert [1] dealt with the linear equation

x⁰⁰+h(t)x⁰+x=0, h(t):=

(h_n>0, fort ∈ I_n = [ω_n,ω_n+1), n∈_N

0, otherwise. (2.12)

He solved (2.12) on intervalsIn, then, to get the global solution of an initial value problem, he

“glued together” the pieces of the solution at the endpoints of I_n’s so that the solution might be continuously differentiable on [_0,∞). He proved the following

Theorem E. Suppose that the sequences{|I_n|}and{h_n|I_n|}are bounded. Then the zero solution of (2.12)is asymptotically stableif and only if

∑

h_n|I_n|³ =_∞. (2.13)

It is easy to see that under the conditions of Theorem2.4and TheoremEon{I_n}_and{h_n} conditions (2.11) and (2.13) are equivalent. Therefore we can say that condition (2.11) is not only sufficient but alsonecessaryin some sense.

CorollaryDwas unexpected because the effect of the damping was controlled from below via lengths |In| in condition (1.12). Now we already see that this was made possible by the condition a(t) ≥ a > 0 in (1.11). One expects that in the general case without this condition it is R

Ina that can be used for estimating the effect of damping from below (see (1.5)). Corollary2.4 gives the possibility of dropping conditiona(t)≥ a > 0 from (1.11), and the result verifies this expectation.

Corollary 2.5. Suppose that for every γ > 0 there is a sequence {I_n}of non-overlapping intervals with|In| ≤γsuch that the sequence{bn}is bounded and

∑

_Z

In

a 3+α

=_∞ (2.14)

holds. Then the equilibrium of (1.1)is asymptotically stable.

Remark 2.6. For the sake of brevity, in Corollaries2.3–2.5we did not mention that in the case of K_M < _∞ the boundedness condition on |I_n| can be weakened in the following way: it is enough to require that there exists a constantγ∈ (0,π/K_M)such that condition (2.10), (2.11), or (2.14) is satisfied for every sequence{I_n}of the property|I_n| ≤γ.

In general, the boundedness condition on |I_n| cannot be dropped from the corollaries.

This can be seen very easily in the case of Corollary2.5. Suppose that (2.14) alone guaranties asymptotic stability provided that {b_n} is bounded. However if there is no boundedness condition on |I_n|, then R_∞

0 a = _∞ implies (2.14). In fact, it is enough to choose I_n so that R

Ina≥1. This would mean thatR_∞

0 a=_∞guarantees asymptotic stability provided that{b_n} is bounded, but it is well-known that this is not true (see, e.g., [10,12]).

Fortunately, by the aid of a new method of proof we can strengthen condition (2.14) so that the boundedness condition on|I_n|may be omitted.

Theorem 2.7. Suppose that there is a sequence{I_k}of non-overlapping intervals such that the sequence {b_k}is bounded and

∑

1 1+|I_k|^2+α

Z

Ik

a 3+α

=_∞ (2.15)

holds. Then the equilibrium of (1.1)is asymptotically stable.

3 Results of type “exponent 2”

In contrast with the previous one, in this section we always suppose that the damping is

“large” in the sense that the lower control function is integrally positive with a suitable pa- rameter not only on the control set I but on the whole half-line [_0,∞). It can be proved [4]

that under this condition there exists no oscillatory solution of the property H∗(_∞) > 0, so we have to deal with only non-oscillatory solutions. As is known, they are monotonous; we have to exclude the so-calledoverdamping x(_∞)6=0. R. A. Smith gave the first necessary and sufficient condition for the linear case (1.6) requiring

Z _∞

0

Z _t

0 exp

−

Z _t

s h

dsdt =_∞ (3.1)

provided that the oscillator is controlled on the whole half-line[0,∞).

In this section we generalize (3.1) to the intermittent damping of nonlinear systems.

Theorem 3.1. Suppose that

(a is integrally positive if K_M =_∞,

a is integrally positive with parameterπ/K_M if K_M <_∞. ^(3.2) If, in addition,

∑

Z _βn

αn

Z _t

αn

exp[−(B(t)−B(s))]ds

dt=_∞, (3.3)

then the zero solution of the equilibrium of (1.1)is asymptotically stable.

We give two more explicit corollaries usingL_∞ andL₁norm ofb.

Corollary 3.2. Suppose(3.2). If, in addition,

∑

1

b_n|I_n| − ¹ b²_n

1−exp[−b_n|I_n|]

!

=_∞, (3.4)

then the zero solution of (1.1)is asymptotically stable. Especially,(3.2)and

∑

1 bn

|I_n|=_∞ and

∑

1 b²_n

<_∞ (3.5)

imply that the zero solution of (1.1)is asymptotically stable.

Corollary 3.3. Suppose(3.2). If, in addition,

∑

exp

−

Z _βn

αn

b

|I_n|²=_{∞, (3.6)}

then the zero solution of (1.1)is asymptotically stable.

Conditions (3.2)–(3.6) are knew; we can say that they extend (1.12)–(1.13) to unbounded control function b.

In comparison, e.g., with (1.10), condition (3.3) is not explicit enough for applications.

Now we make it more explicit and applicable. Let us fix a constant d > 0 and introduce the notations

t_n,k :=inf

t≥ α_n: Z _t

αn

b=kd

, r_n,k :=min{t_n,k;β_n}, (k =0, 1, . . .). (3.7)

Theorem 3.4. For every d>0condition(3.3)is equivalent to

∑

∑

(r_n,k−r_n,k−1)²

!

=_∞. (3.8)

Theorem C follows from Theorems 3.1 and 3.4 with d = c₄, namely condition (1.10) in Theorem Cimplies (3.8). In fact, if B_n < c₄, thenr_n,1−r_n,0 = β_n−α_n = |I_n| ≥ |I_n|/√

2. If Bn ≥ c₄, then let kn ≥ 1 denote the smallest natural number of the propertyr_n,kn = β_n. By Cauchy’s inequality we have

∑

(r_n,k−r_n,k−1)²=

kn

k

∑

(r_n,k−r_n,k−1)²

≥ ¹ kn

kn

k

∑

(r_n,k−r_n,k−1)

!₂

= |I_n|² kn

>|I_n|^{2 1} 1+ ^Bn

c₄

≥ ¹ 2|I_n|^2c4

Bn

,

which completes the proof.

In [8] it is proved by an example that (3.8) does not imply (1.10). This means that Theo- rem3.1does not follow from TheoremC.

4 Proofs

We will need an earlier lemma [4, Lemma 2.2] to estimate the distances of consecutive zeros of sinϕ(t)for an oscillatory solution(r,ϕ).

Lemma 4.1. Suppose KM < _∞ and consider a solution (r,ϕ) ((x(0),x⁰(0)) ∈ HM). Suppose, in addition, that0<ε<π/2, and

ϕ(T) =−kπ−ε, ϕ(S) =−(k+1)π+ε with some k∈_N,0<T< S.

Then there areµ(ε)>0,ν(ε)>0, independent of T,S such that lim

ε→0+0ν(ε) =0, Z _S

T h∗(t)sin²ϕ(t)dt≥µ(ε) π

K_M −(S−T)−ν(ε)

. (4.1)

The following lemma estimates the waste of energy between two zeros of sinϕ(t) _{for an} oscillatory solution.

Lemma 4.2. Let us given an oscillatory solution of (1.1) with H∗(_∞):=lim_t→_∞H∗(t)>0, and let us use the notation r₀ :=_{lim inft→∞}r(t)>0. Denote by{τ_n}^∞_n=1 the increasing sequence of all zeros ofsinϕ(t). Then there are constants c₅ = c₅(r₀,M)>0 and q =q(M)> 0, independent of n such that for every n∈ _Nwe have

H∗(τ_n)−H∗(τ_n+1)≥r²₀ Z _τn+1

τn

h∗(t)sin²ϕ(t)dt

≥c₅ Z _τn+1

τn

a(t)

min _{Z t}

τn

exp[−q(B(t)−B(s))ds];τ_n+1−t 2+α

dt.

(4.2)

Proof. Suppose that ϕ(τ_n) = −kπ and, consequently ϕ(τ_n+1) = −(k+1)π (k ∈ _{N). Let} τ_n⁰ ∈ (τ_n,τ_n+₁) denote the first time point where ϕ(τ_n⁰) = −kπ−π/2 holds. From (2.4) it follows that

(ϕ(t)−ϕ(τ_n))⁰ ≤ −p(r₀,M)−qb(t)(ϕ(t)−ϕ(τ_n)) (τ_n≤ t≤τ_n⁰), (4.3) q=q(M):=sup

HM

W(x,y), p(r0,M):=min

sin²ϕ+ ^f(x)

x cos²ϕ:(x,y)∈ HM,r≥r0

>0.

By the basic theorem of the theory of differential inequalities [3, Theorem III.4.1, p. 26] we have

ϕ(t)≤ −kπ−p(r₀,M)

Z _t

τn

exp[−q(B(t)−B(s))]ds (τ_n ≤t≤τ_n⁰). (4.4) On the other hand, ϕ⁰(t)≤ −p(r₀,M)on the interval[τ_n⁰,τ_n+1], therefore

ϕ(t)≥ −(k+1)π+p(r₀,M)(τ_n+1−t) (τ_n⁰ ≤t ≤τ_n+1). (4.5) Combining (4.4) and (4.5) we obtain

|sinϕ(t)| ≥ ²

πp(r₀,M)min _{Z t}

τn

exp[−q(B(t)−B(s))]ds;τ_n+1−t

(τn≤t ≤τ_n+1).

(4.6)

From this estimate and (1.5) we get (4.2) with c₅ :=

infHM

w(x,y) ²

πr₀p(r₀,M) 2+α

. We also need an analogous lemma for non-oscillatory solutions.

Lemma 4.3. For every non-oscillatory solution of (1.1) with H∗(_∞) > 0there exists a T∗ > 0such that for arbitraryτ₁,τ₂(T∗ < τ₁< τ₂)we have

H∗(τ₁)−H∗(τ₂)≥ c₅ Z _τ2

τ₁

a(t) _{Z t}

τ₁

exp[−q(B(t)−B(s))ds] ₂+α

dt. (4.7)

Proof. If (x,y)is a non-oscillatory solution, then it can bee seen that x(t)y(t)< 0 fort large enough, let us say if t≥ T∗, and limt→_∞ϕ(t)≡0 (mod π). Then

−

k+¹ 4

π< ϕ(t)<−kπ for somek∈ _Z+. Similarly to (4.3) we obtain

(ϕ(t) +kπ)⁰ ≤ −p_r0,M−qb(t)(ϕ(t) +kπ) (T∗ ≤ τ¹≤t ≤τ²), and

ϕ(t)≤ −kπ+ (ϕ(τ¹) +kπ)e^{−q(B(t)−B(τ1))}−p_r0,M Z _t

τ¹

e^{−q(B(t)−B(s))}ds

≤ −kπ−p_r0,M Z _t

τ¹

e^{−q(B(t)−B(s))}ds,

|sinϕ(t)| ≥ ²

π|ϕ(t) +kπ| (τ¹ ≤t≤τ²), from which (4.7) follows.

Proof of Theorem2.1. Suppose the contrary and let (x,y) be such a solution that H∗(_∞) > 0.

If this solution is oscillatory, then let {τ_n}^∞_n=1 denote the increasing sequence of all zeros of sinϕ(t). For a fixednthere is ak ∈_Z+ such thatϕ(τ_n) =−kπand ϕ(τ_n+1) =−(k+1)π. Let t_n,s_n∈ (τ_n,τ_n+1)denote the time points when ϕ(t_n) =−kπ−π/8 andϕ(s_n) =−kπ−3π/8 for the first time. Then

ϕ⁰(_t)≥ −p(_r0,M) (_tn ≤t≤ sn)_, consequently,

τ_n+1−τ_n ≥s_n−t_n≥ π/(4p(r₀,M)). (4.8) We state that I_m contains at most one member of {τ_n}provided thatmis large enough.

In fact, at first let us consider the case K_M = ∞, choose a γ from the interval (0,π/(4p(r₀,M))) and an {I_m} with |I_m| ≤ γ and such that (2.8) is satisfied. In view of (4.8) the statement is true for arbitrary m. If K_M < ∞, then choose γ ∈ (_0,π/K_M)_and {I_m} so that|Im| ≤γand (2.8) hold, and suppose that the statement is not true. Since|Im| ≤ γfor allm∈ N, this means that there are infinitely manyn’s withτ_n+1−τ_n ≤ γ< π/K_M. Let us denote by{(τ_k⁰,τ_k⁰₊₁)}^∞_k=1the subsequence of {(τ_n,τ_n+1)}with this property. Choosingε > ₀ applying so small thatν(ε)<((π/K_M)−γ)/2 and applying (2.7) and Lemma4.1we obtain

H∗(0)−H∗(_∞)≥r²₀

∑

Z _τ⁰

m+1

τ_m⁰

h∗(t)sin²ϕ(t)dt

≥r²₀µ(ε)

∑

π KM

−γ

−ν(ε)

= _∞, which is a contradiction.

We have proved that I_m contains at most one member of {τ_n} provided that m is large enough. Without loss of the generality we may drop the finitely manyIm’s containing at least twoτ_n’s.

Now we estimate the integrals over[τ_n,τ_n+₁] in (2.2) by integrals on Im. We will use the simple fact that ifτ<α<t, then

Z _t

τ

exp[−q(B(t)−B(s))ds]≥

Z _t

α

exp[−q(B(t)−B(s))ds].

We consider only those (τ_n,τ_n+1)’s that have points from the control set I = ∪^∞_m=1I_m: n₁ <

n₂ <· · ·<n_k <· · · are the natural numbers such that(τ_nk,τ_nk+1)∩I 6=_∅. Let us fix ak∈_N and denote by Ip_k,Ip_k+₁, . . . ,Iq_k (1 ≤ p_k ≤ q_k ≤ p_k+1) the control intervals having common points with(τ_nk,τ_nk+1):

I_pk+j∩(τ_nk,τ_nk+1)6= _∅ (j=0, 1, . . . ,q_k−p_k). Then we have the estimate

H∗(τ_nk−1)−H∗(τ_nk+2))

≥

q_k−p_k

∑

c₅



 Z _ξ

pk+j

α_pk+j

a(t) _min (

Z _t

α_pk+j

e^{−q(B(t)−B(s))ds};ξ_pk+j−t )!2+α

dt

+

Z _β

pk+j

ξ_pk+j

a(t) min (

Z _t

ξ_pk+j

e^{−q(B(t)−B(s))ds};β_pk+j−t )!2+α

dt



.

(4.9)

If τ_nk ∈ I_pk (τ_nk+1 ∈ I_qk), then ξ_pk = τ_nk (ξ_qk = τ_nk+1), otherwiseξ_pk+j is arbitrary in I_pk+j. On the other hand

3(H∗(0)−H∗(_∞))≥3

∑

(H∗(τ_n)−H∗(τ_n−1))

!

≥3

∑

(H∗(τ_nk+2)−H∗(τ_nk−1))

! .

(4.10)

It follows from (4.9) and (4.10) that condition (2.8) implies H∗(0)− H∗(_∞) = ∞, that is a contradiction.

If the solution (x,y) is non-oscillatory, then we apply Lemma 4.3. There exists a natural numberm∗such thatm>m∗ impliesα_m > T∗, so from (4.7) we obtain

H∗(T∗)−H∗(_∞)≥ −

Z

[T∗,∞)∩IH⁰_∗(t)dt

≥c₅

∑

Z _βm

αm

a(t) _{Z t}

αm

exp[−q(B(t)−B(s))ds] 2+α

dt

≥c₅

∑

Z _βm

αm

a(t)

min _{Z t}

αm

exp[−q(B(t)−B(s))ds];βm−t 2+α

dt.

Condition (2.8) impliesH∗(₀)−H∗(_∞) =_∞again.

In what follows we will use the notation

g_G:=sup{g(t):t∈ G} (G⊂_R+,g :R+→_R). Lemma 4.4.

Z _t

α

exp[−q(B(t)−B(s))ds]≥ ¹

3(t−α) α≤t ≤ ¹ qb_(α,β)

!

. (4.11)

Proof. Since the functiont7→₁−_exp[−qb_(α,β)(t−α)]is concave we have the estimate Z _t

α

exp[−q(B(t)−B(s))ds]≥

Z _t

α

exp[−qb(_α,β)(t−s)]ds

= ¹

qb(α,β)

(1−exp[−qb_(α,β)(t−α)])≥ ¹

3(t−α) α≤t≤α+ ^{ln 3} qb(α,β)

! ,

from which (4.11) follows.

Proof of Corollary2.2. Suppose that there exist{In}andκ ∈ (0, 1)such that condition (2.9) in Corollary 2.2is satisfied. We show that condition (2.8) in Theorem2.1 is also satisfied for the same {I_n}_.

Starting from (2.8) and using the estimate (4.11) in Lemma4.4 we obtain J_n¹:=

Z _ξn

αn

a(t)

min _{Z t}

αn

exp[−q(B(t)−B(s))]ds;ξ_n−t 2+α

dt

≥

Z _min{αn+1/qbn;ξn} αn

a(t)

min 1

3(t−αn;ξn−t) 2+α

dt +

Z _ξn

min{α_n+1/qbn;ξn}a(t) min ( 1

qb_n(1−exp[−1]);ξ_n−t )!2+α

dt

≥c₆ Z _ξn

αn

a(t)

min

t−α_n;ξ_n−t; 1 1+bn

2+α

dt with a suitably chosen constantc₆>0 independent of n. Similarly,

J_n²:=

Z _βn

ξn

a(t)

min _{Z t}

ξn

exp[−q(B(t)−B(s))]ds;β_n−t 2+α

dt

≥c₇ Z _βn

ξn

a(t)

min

t−ξn;βn−t; 1 1+b_n

2+α

dt

with a constantc₇ >0 independent ofn. Therefore there is a constantc₈ >0 such that

∑

(J_n¹+J_n²)≥ c₈

∑

Z _βn

αn

a(t)

min

t−α_n;|ξ_n−t|;β_n−t; 1 1+bn

₂+α

dt. (4.12) Now we choose a small δ_n > 0 (it will be defined later) and cut out δ_n-neighborhoods of α_n,ξ_n, andβ_n. Then we get

Z _βn

αn

a(t)

min

t−α_n;|ξ_n−t|_;β_n−t; 1 1+bn

2+α

dt

≥

min

δn; 1 1+b_n

2+αZ

En

a,

(4.13)

where

En := In\([αn,αn+δn]∪[ξn−δn,ξn+δn]∪[βn−δn,βn]). (4.14) Letδ_nbe defined by

δn := ((1−κ)/4)|In|. (4.15) Combining (4.12), (4.13), and (2.9) we obtain

∑

(J_n¹+J²_n)≥c₈

1−κ 4

2+α ∞ n

∑

min

|I_n|; 1 1+bn

2+αZ

En

a=_∞, (4.16) which means that divergence (2.8) is also satisfied.

It has remained to prove that for every γ > 0 we may suppose that |I_n| ≤ γ. This means that if (2.9) is satisfied for{In}andκ, then it also holds for another sequence {Lm}^∞_m=1 with the same κ, but |L_m| ≤ γ (m ∈ N). The same fact was proved for linear systems in [7, Corollary 4.2]. To make the present paper self-contained we repeat the short proof here.

Let us observe at first the obvious fact that for arbitrary {un,vn,wn} (0 < un < 1, v_n,w_n>0) andδ >0 the two divergences

∑

min{un;vn}wn= _∞,

∑

min{un;vn;δ}wn=_∞

are equivalent.

If|In|> γ, then we divide the interval Ininto non-overlapping subintervals I_n=∪^l_jⁿ₌₁I_nj, γ/2≤ |I_nj| ≤γ,

and take arbitrary setsE_nj⊂ I_njwhich are finite union of intervals such that mes(E_nj)≥κ|I_nj|. Then

ln

∑

min ( 1

1+b_nj;|I_nj| )!2+α

Z

Enj

a≥

min

|In|; 1 1+b_n;γ

2

2+αZ

En

a,

provided that En:=∪^l_jⁿ₌₁ We obtain{Lm}if we exchange all Inof the properties |In|> γwith {I_nj}^l_jⁿ₌₁.

Proof of Corollary2.3. If we use the lower estimate Z _t

αn

exp[−q(B(t)−B(s))]ds ≥exp

−

Z _βn

αn

q(B(t)−B(s))ds

(t−α_n) and its analogy on [ξn,βn], then for (2.8) we get an inequality analogous with (4.12):

∑

(J¹_n+J_n²)≥c₉

∑

(exp[−qB_n])^2+α

Z _βn

αn

a(t) (min{t−α_n;|ξ_n−t|;β_n−t})^2+α dt. (4.17) Defining δ_n, Enby (4.14), (4.15) we obtain

∑

(J_n¹+J_n²)≥c₉

1−κ 4

2+α ∞ n

∑

(exp[−qB_n]|I_n|)^2+α

Z

En

a =_∞, (4.18)

which means that divergence (2.8) is also satisfied.

Proof of Corollary2.4. We start from (4.17). If we cut out δ_n-neighborhoods of α_n,ξ_n, and β_n, then we have

∑

(J_n¹+J_n²)≥c₉

∑

(exp[−qBn]δn)^2+α

Z

En

a

≥c₉

∑

(exp[−qBn]δn)^2+α _Z

In

a−4δnan

. Defining δ_n:= (1/8)R

Ina/a_n, from condition (2.11) we obtain

∑

(J_n¹+J_n²)≥ ¹ 2

1 8

2+α

c₉

∑

exp[−qBn] a_n

2+α_Z

In

a 3+α

=_∞,

i.e., divergence (2.8) holds.

Proof of Corollary2.5. If we take into account (1.4), then (2.14) implies (2.11).

Lemma 4.5. Suppose that for everyγ>0small enough and for every{τ_n}withγ≤τ_n+₁−τ_n≤2γ we have

∑

Z _τn+1

τn

a(t)

min _{Z t}

τn

exp[−q(B(t)−B(s))ds];τ_n+1−t 2+α

dt =_∞. (4.19) Then the equilibrium of (1.1)is asymptotically stable.