Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 38, 1-17; http://www.math.u-szeged.hu/ejqtde/

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Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 38, 1-17; http://www.math.u-szeged.hu/ejqtde/

Asymptoti stability of two dimensional

systems of linear dierene equations and of

seond order half-linear dierential equations

with step funtion oeients

L. Hatvani

1 ∗

and L. Székely

2

1

Corresponding author

University of Szeged, BolyaiInstitute

Aradivértanúk tere 1, Szeged, H-6720 Hungary

E-mail: hatvanimath.u-szeged.hu

2

Szent István University

Institute of Mathematis and Informatis

E-mail: szekely.laszlogek.szie.hu

Abstrat

We give a suient ondition guaranteeing asymptoti stability

with respet to

x

^for ^the ^zero ^solution ^of ^the half-linear dierential equation

x ^′′ | x ^′ | ⁿ⁻ ¹ + q(t) | x | ⁿ⁻ ¹ x = 0, 1 ≤ n ∈ R ,

withstepfuntionoeient

q

^. ^The^geometri^method^of^the^proof^an

be applied alsoto two dimensional systems oflinear non-autonomous

dierene equations. The appliation gives a new simple proof for a

sharpenedversionofÁ.Elbert'sasymptotistabilitytheoremsforsuh

diereneequationsandlinearseondorderdierentialequationswith

step funtionoeients.

∗

Supported by the Hungarian National Foundation for Sienti Researh (OTKA

K75517), Analysis and Stohastis Researh Group of the Hungarian Aademy of Si-

enes, andTÁMOP4.2.2.

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funtion oeients; half-lineardierentialequation.

2000 Mathematis Subjet Classiation: Primary 34D20, 39A30.

1 Introdution

Consider the diereneequation

x _n+1 = M n x n , n = 0, 1, 2, . . . ,

⁽¹⁾

where

x n ∈ R ²

and

M n ∈ R ² ^× ²

. We do not onsider the trivial ase when all the entries of

M n

^are ^equal ^to

0

^for ^some

n

^. ^Let

k M k

^be ^the ^spetral

norm, i.e.,

k M k

îs^the ^square ^rootôf ^the^largest êigenvalue ôf ^the ^symmetri

positive semi-denite matrix

M ^T M

^. Ît îs ^well-known ^[3, ^p. ^232℄ ^that îf

Q ^∞

n=0 k M n k = 0

^, ^then âll ^solutions ôf êquation ⁽¹⁾^tend ^to ^zero âs

n → ∞

^,

i.e., thezerosolutionisasymptotiallystable. Á.Elbert[10℄gaveasuient

ondition for the asymptoti stability under the assumptions

(i) Q ^∞

n=0 max {k M n k , 1 } < ∞

^,

(ii) 0 < Q ^∞

n=0 k M n k

^,

(iii) Q ∞

n=0 max {| det M n | , 1 } < ∞

^.

His proofwasbased onestimation ofthe normof some speial matriesand

a triky deomposition of matries

M n

^. ^He ^applied ^this ^result ^to ^dedue

anArmellini-Tonelli-Sansone-typetheorem(abbreviated asA-T-S theorem),

i.e., a theorem guaranteeing asymptoti stability with respet to

x

^for ^the

zero solution of the linear seondorder dierentialequation

x ^′′ + a(t)x = 0 (a(t) ր ∞ , t → ∞ )

⁽²⁾

with step funtion oeient

a

^[11, ^12℄.

I.Bihari[5℄ andElbert [9℄introduedthehalf-lineardierentialequation

x ^′′ | x ^′ | ^m−1 + q(t) | x | ^m−1 x = 0, m ∈ R ⁺ ,

⁽³⁾

whihhasattratedattention,andithasanextensiveliterature(see,e.g.,[7℄,

[8℄and thereferenes therein). Bihari[6℄has generalizedthe A-T-S theorem

tothisequationintheaseofsmoothoeient

q

^,^requiring^regular^growth

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of

q

^. ^Roughly ^speaking, ^this ^ondition ^means ^that ^the ^growth ^of

q

^annot

be loated to a set with small measure (see Setion 3). Of ourse, a step

funtion

q

^does^not^satisfy^thisôndition. Êlbert's ^method,ûsingâ^wideând

deep mahinery from linear analysis,does not apply tothe half-linear ase.

InthispaperweestablishanA-T-Stheoremforthehalf-lineardierential

equationwithstepfuntionoeient

q

^. ^The^proofîs^basedûponâ^geometri

method. This method applies also to the linear ase, so we an give a new

simple proof for Elbert's result,assuming only

lim sup _n→∞ Q n

k=0 k M k k < ∞

instead of

(i)

(iii)

^.

2 Dierene equation

To investigateequation(1), wewilldeneadiereneequationonthe plane

whih has the same stability properties as equation (1). Let us introdue

the following notations for the matriesof the reetion with respet to the

x

^-axis, ând ôf^the ^rotationâround^the ôrigin ounterlokwise with

ϕ

ⁱⁿ

R ²

:

R =

1 0 0 − 1

, E(ϕ) =

cos ϕ − sin ϕ sin ϕ cos ϕ

.

⁽⁴⁾

Obviously,

E(ϕ 1 )E(ϕ 2 ) = E(ϕ 1 + ϕ 2 ), E(ϕ)R = RE( − ϕ).

⁽⁵⁾

We willneed the following theorem (see, e.g., [16, p. 188℄):

Theorem (polar fatorization). Every

M ∈ R ^n×n

an be represented as a produt

M = SQ

^where

S

^is ^symmetri, ^positive semi-denite, and

Q

^is

orthogonal.

S

^is ^uniquely ^determined ^while

Q

îs ûnique îf ând ônly îf

M

^is

non-singular.

In this theorem

S

^is ^the ^square ^root ^of ^the ^symmetri ^positive ^semi-

denitematrix

M ^T M

^. ^If

M ∈ R ⁿ ^× ⁿ

isnon-singular,thentheprodut

M ^T M

ispositivedenite,thusitan bediagonalized:

M ^T M = PD ² P ⁻ ¹

^,^where

D ²

isthediagonalmatrixontainingtheeigenvaluesof

M ^T M

^and^the^orthogonal

matrix

P

^has ^the ^propereigenvetorsinitsolumns. Then

S = PDP ⁻ ¹

^and

M = PDP ⁻ ¹ Q.

⁽⁶⁾

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Denote by

Λ

^and

λ

^the eigenvalues of

M ^T M ( k M k = Λ ≥ λ > 0)

^. ^Suppose

that the diagonalelements in

D

âre ⁱⁿ ^dereasing ôrder. Îf

det M = 0

^, ^then

S

^is ^positive semi-denite and the symmetri matrix

S ˜ := k M k ⁻ ¹ S

^an ^be

represented as

S ˜ = P DP ˜ ⁻¹

^, ^where

P

îs ôrthogonalând

D ˜ =

1 0 0 0

.

Applying the aboveargument tothe oeientmatries of (1), we have

M n = k M n k P n D ˆ n P ⁻ _n ¹ Q n ,

⁽⁷⁾

where

D ˆ n :=

1 0 0 d n

, d n :=

( q λ n

Λ n > 0,

^if

det M n 6 = 0

^;

0,

^if

det M n = 0

^.

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Letusexaminetheow

F n := Q n

k=0 M k

ôfêquation^(1). Ûsing^the^fat,^that

the produt of orthogonal matriesare alsoorthogonal,

F n

^has ^the ^form

F n =

n

Y

k=0

P k D ˆ k P ⁻ _k ¹ Q k =

n

Y

k=0

k M k k

! P n

n

Y

k=0

D ˆ k O k

!

,

⁽⁹⁾

where the orthogonal matries

O k (k = 0, . . . , n + 1)

^are ^dened ^by

O 0 := P ⁻¹ ₀ Q 0 , O k = P ⁻¹ _k Q k P k− 1 , k = 1, . . . , n,

⁽¹⁰⁾

and the produt

Q n

k=0 N k

^is^meantⁱⁿ ^the ^order

N n · · · N 0

^. ^It ^is^known ^from

the elementary geometry that in the plane every orthogonal transformation

is a rotation or a produt of a rotation and a reetion with respet to the

x

^-axis. ^Thus, ^if

O k

^is ^not ^a ^rotation, ^then ^let

O k = E(ϑ k )R

^for ^some

ϑ k

^.

Sine

R

îs ômmutable ^with êvery ^diagonal^matries, ^from ⁽⁵⁾^we ôbtain

F n =

n

Y

k=0

k M k k

!

R ^m E(α n )

n

Y

k=0

D ˆ k E(ω k )

!

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for some

m ∈ N ₀

(

m ≤ n + 1

⁾^and^some

ω k

^'s,^where

α k

^,

ω k

^an ^be^alulated

from

M 0 , . . . , M k

^.

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x _n+1 = k M n k

1 0 0 d n

cos ω n − sin ω n

sin ω n cos ω n

x n ,

0 ≤ d n ≤ 1, n = 0, 1, 2, . . .

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The equilibrium

(0, 0)

^of ⁽¹⁾ ^is ^stable (asymptotiallystable) if and only if the equilibrium

(0, 0)

^of ⁽¹²⁾ ^is ^stable (asymptotiallystable). Now, we an state the main theorem of this setion:

Theorem 1. Suppose that

lim sup _n→∞ Q n

k=0 k M k k < ∞

^. ^If

∞

X

n=0

min { 1 − d n , 1 − d n+1 } sin ² ω n+1 = ∞ ,

⁽¹³⁾

then the zero solution of dierene equation (12) is asymptotially stable.

Proof. Obviously, itisenoughto dealwith the ase

k M k k = 1

⁽

k = 0, 1, . . .

⁾

and to showthat

Q ^∞

n=0 D ˆ n E(ω n )

^=0. Geometrially,the dynamis of (12) is omposed of onseutive rotationsand ontrations along the

y

^-axis. ^Let

us introdue polaroordinates

r

^,

ϕ

^so ^that

x :=

x y

, x = r sin ϕ, y = r cos ϕ.

Inthese oordinatesthe phasespaeforsystem(12) is

r ≥ 0

^,

−∞ < ϕ < ∞

^.

Using the notations

˜

x n = E(ω n )x n , κ n := ϕ n+1 − (ϕ n + ω n ), ∆r n := r n+1 − r n , n = 0, 1, . . .

we have

p x ² _n + y _n ² = p

˜

x ² _n + ˜ y _n ² , x n+1 = ˜ x n , y n+1 = d n y ˜ n

ϕ _n+1 = ϕ ₀ +

n

X

i=0

(ω i + κ i ), r _n+1 = r ₀ +

n

X

i=0

∆r i ,

and

∆r i ≤ 0

^beause ^of ^the ^ontration. ^Therefore, ^the ^sequene

{ r n } ^∞ n=0

^is

monotonously dereasing.

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Supposethatthestatementofthetheoremisnottrue,i.e.,

¯ r := lim n→∞ r n

> 0

^. ^Then

− ∆r i = r i − r i+1 = q

x ² _i + y _i ² − q

x ² _i+1 + y _i+1 ²

= q

˜

x ² _i + ˜ y ² _i − q

˜

x ² _i + d ² _i y ˜ ² _i = (1 − d ² _i )˜ y _i ² p x ˜ ² _i + ˜ y _i ² + p

˜

x ² _i + d ² _i y ˜ ² _i

≥ (1 − d ² _i )r ² _i cos ² (ϕ i + ω i ) 2r i ≥ ¯ r

2 (1 − d i ) cos ² (ϕ i + ω i ).

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Wewanttogetthe ontraditionthat thesum ofthe lowerestimatingterms

in(14)diverges. Theproblemisthatthesetermsontain

ϕ i

^'s,^whih^depend

on solutions,so they are unknown; we haveto get rid of them. Obviously,

| cos(ϕ i + ω i ) | = | cos ϕ i cos ω i − sin ϕ i sin ω i |

≥ | sin ϕ i || sin ω i | − | cos ϕ i || cos ω i | .

⁽¹⁵⁾

For arbitrarily xed

0 < γ < ε < 1

^, ^dene

µ(ε, γ) := p

1 − γ ² − εγ

^. ^Sine

lim ε→0,γ→0 µ(ε, γ) = 1

^, ^we ^may ^assume ^that

µ(ε, γ) ≥ 1/2

^. ^W^e distinguish three ases:

a)

γ| sin ω _i | ≥ | cos ϕ _i |

^and

| cos ω _i | ≥ ε.

^Then

| sin ϕ i | ≥ | cos ω i |

^, ^and

from (15) weget

| cos(ϕ i + ω i ) | ≥ | sin ω i || cos ω i | (1 − γ) ≥ | sin ω i | (1 − γ)ε.

⁽¹⁶⁾

In this ase, estimate (14) isontinued as

− ∆r i ≥ r ¯

2 (1 − d i ) cos ² (ϕ i + ω i ) ≥ r ¯

2 (1 − γ) ² ε ² (1 − d i ) sin ² ω i .

⁽¹⁷⁾

b)

γ| sin ω _i | ≥ | cos ϕ _i |

^and

| cos ω _i | < ε

^. ^Then

| sin ϕ i | ≥ q

1 − γ ² sin ² ω i ≥ p

1 − γ ² ,

⁽¹⁸⁾

and

| cos(ϕ i + ω i ) | ≥ ( p

1 − γ ² − εγ) | sin ω i | = µ(ε, γ ) | sin ω i | ≥ 1

2 | sin ω i | .

Then

− ∆r i ≥ r ¯

2 (1 − d i ) cos ² (ϕ i + ω i ) ≥ r ¯

8 (1 − d i ) sin ² ω i .

⁽¹⁹⁾

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)

γ| sin ω _i | < | cos ϕ _i |

^. În ^this âse ^we ân êstimate

− ∆r i− 1

^(instead

of

− ∆r i )

^from ^below ^by

| sin ω i |

^. În ^fat, ûsingâlso ^the înequality

| cos ϕ i | = | y i |

p x ² _i + y ² _i = d i−1 | y ˜ i−1 | p x ˜ ² _i− ₁ + d ² _i− ₁ y ˜ _i− ² ₁

≤ | y ˜ i − 1 |

p x ˜ ² _i− ₁ + ˜ y _i− ² ₁ = | cos(ϕ i−1 + ω i−1 ) | ,

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from (14) weobtain

− ∆r i−1 ≥ ¯ r

2 (1 − d i−1 ) cos ² (ϕ i−1 + ω i−1 ) ≥ r ¯

2 (1 − d i−1 ) cos ² ϕ i

≥ ¯ r

2 γ ² (1 − d i− 1 ) sin ² ω i ≥ r ¯

2 γ ² min { 1 − d i− 1 , 1 − d i } sin ² ω i .

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Setting

c := r ¯

2 min { (1 − γ) ² ε ² ; 1

4 ; γ ² } > 0,

for every

i

^we ^have

c min { 1 − d i − 1 ; 1 − d i } sin ² ω i ≤ − ∆r i − 1 − ∆r i = r i − 1 − r i+1 .

Summarizing these inequalitieswe obtain

c

∞

X

i=1

min { 1 − d i− 1 ; 1 − d i } sin ² ω i ≤ r ₀ − r < ¯ ∞ ,

whih ontradits assumption (13).

3 The half-linear equation

In this setion we onsider the half-linearseond orderdierentialequation

x ^′′ | x ^′ | ⁿ⁻ ¹ + q(t) | x | ⁿ⁻ ¹ x = 0, n ∈ R ⁺ ,

⁽²²⁾

whihwas introdued by Bihari[5℄ and Elbert [9℄. They alled it half-linear

beauseitssolutionsetishomogeneous, butitisnotadditive. Thisequation

is a generalizationof the seond order lineardierentialequation

x ^′′ + q(t)x = 0

⁽²³⁾

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500℄, we alla non-trivialsolution

x 0 (t)

^of ⁽²²⁾ ^small ^if

t lim →∞ x 0 (t) = 0.

⁽²⁴⁾

H. Milloux [18℄ proved, that if

q

^is dierentiable, monotonously inreasing andtends toinnityas

t → ∞

^,^then^the ^linearêquation⁽²³⁾ ^hasât^leastône

smallsolution. Healsoonstrutedanequationwithsuhaoeient

q

^hav-

ingnotsmallsolutions,too. The famousArmellini-Tonelli-SansoneTheorem

(see, e.g., [17℄) gave a suient ondition guaranteeing that all solutions of

(23) were small. Many papers examined andsharpened theabove theorems,

even for nonlinear dierential equations or dierene equations (see, e.g.,

[15,17℄ and the referenes therein).

F. V. Atkinson and Elbert [4℄ extended the theorem of H. Milloux to

the half-lineardierentialequation(22). Anextensionof theA-T-S theorem

to (22) was given by Bihari with the following onept. A nondereasing

funtion

f : [0, ∞ ) → (0, ∞ )

^with

lim t→∞ f(t) = ∞

îs âlled ^to ^grow în-

termittently if for every

ε > 0

^there ^is ^a ^sequene

{ (a i , b i ) } ^∞ i=0

^of ^disjoint

intervals suh that

a i → ∞

^as

i → ∞

^,^and

lim sup

i →∞

i

X

k=1

b k − a k

b i ≤ ε,

∞

X

i=1

(f (a i+1 ) − f (b i )) < ∞

are satised. If suh a sequene does not exist, then

f

^is ^alled ^to ^grow

regularly.

Theorem B (Bihari[6℄). If

q

^is ontinuously dierentiable and it grows to innity regularly as

t → ∞

^, ^then ^all non-trivial solutions of equation (22) are small.

Thesimplestaseoftheintermittentgrowthiswhen

q

^is^amonotonously inreasing step funtion. In this setion we will examine this ase, i.e., the

equation

x ^′′ | x ^′ | ⁿ⁻ ¹ + q k | x | ⁿ⁻ ¹ x = 0 (t k ≤ t < t k+1 , k = 0, 1, . . .),

⁽²⁵⁾

where

t 0 = 0, lim

k→∞ t k = ∞ , 0 < q 0 ≤ q 1 ≤ . . . ≤ q k ≤ q k+1 ≤ . . . , lim

k→∞ q k = ∞ .

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equation (25)has a smallsolution. Elbert [11, 12℄proved anA-T-S theorem

for the linear (

n = 1

⁾ âse ôf êquation ⁽²⁵⁾ âs â ^diret âppliation ôf ^his

theorem onthe asymptotistability of the trivialsolution of (1).

Theorem C (Elbert [11℄). Let

n = 1

^. ^If

∞

X

k=0

min

1 − q k

q k+1

, 1 − q k+1

q k+2

sin ² ( √ q _k+1 (t _k+2 − t _k+1 )) = ∞ ,

⁽²⁶⁾

then allnon-trivial solutions of equation (25) are small.

Our main goal is to extend Theorem C to the ase

n > 1

^of half-linear equation(25). To thisend, weneed the so-alledgeneralizedsineand osine

funtions introdued by Elbert [9℄. Consider the solution

S = S n (Φ)

^of ^the

initialvalue problem

( S ^′′ | S ^′ | ⁿ⁻ ¹ + S | S | ⁿ⁻ ¹ = 0

S(0) = 0, S ^′ (0) = 1.

⁽²⁷⁾

Multiplying the dierential equation by

S ^′

^and integrating it over

[0, Φ]

^we

obtain the relation

| S ^′ | ⁿ⁺¹ + | S | ⁿ⁺¹ = 1 ( −∞ < Φ < ∞ ),

⁽²⁸⁾

whihan be onsideredas ageneralizationof the lassialidentity

cos ² ϕ + sin ² ϕ = 1

^(the ^ase

n = 1

^).

S

^and

S ^′

^are ^periodi ^funtions^with ^period

2ˆ π

^,

where

π ˆ

^is ^dened ^as

ˆ

π = 2 _n+1 ^π sin _n+1 ^π ,

whih gives bak

π

ⁱⁿ ^the ^ordinary ^ase

n = 1

^(see ^[9℄). ^Furthermore,

S

^is

odd and

S ^′

^is ^even. ^The generalized tangent funtion an be introdued as well:

T (Φ) = S(Φ) S ^′ (Φ) .

Now wean state our main theorem.

Theorem 2. Let

n > 1

^. ^If

∞

X

k=0

min

1 − q k

q k+1

, 1 − q k+1

q k+2

S q

1 n+1

k+1 (t k+2 − t k+1 )

!

n+1

= ∞ ,

⁽²⁹⁾

then allnon-trivial solutions of equation (25) are small.

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Proof. First, using the notation

q(t) := q k (t k ≤ t < t _k+1 , k = 0, 1, 2 . . .)

^we

introduea new time variable

τ = ϕ(t) = Z t

0

q(s) ⁿ⁺¹ ¹ ds, τ k := ϕ(t k ).

⁽³⁰⁾

Let

x(t) = x(ϕ ⁻ ¹ (τ )) =: y(τ )

^, ^where

ϕ ⁻ ¹

îs ^the înverse ^funtionôf

ϕ

^. ^Then

x ^′ (t) = ˙ y(τ)q ⁿ⁺¹ ¹ (t), x ^′′ (t) = ¨ y(τ )q ⁿ⁺¹ ² (t) (t 6 = t k , k = 0, 1, 2, . . .),

where

( · ) ^· = d( · )/dτ

^. ^Thus, ^equation ⁽²⁵⁾ ^is transformed intothe form

¨

y(τ) | y(τ ˙ ) | ⁿ ⁻ ¹ + | y(τ) | ⁿ ⁻ ¹ y(τ ) = 0, (τ 6 = τ k k = 0, 1, . . .).

⁽³¹⁾

Sineany solution

x

^of^equation⁽²⁵⁾ ^has^to^beontinuouslydierentiableon

(0, ∞ )

^,

x ^′ (t k+1 − 0) = x ^′ (t k+1 + 0) = x ^′ (t k+1 )

^must ^hold ^for^every

k ∈ N

, i.e.,

˙

y(τ k+1 ) = ˙ y(τ k+1 + 0) = q k

q _k+1 _n+1 ¹

˙

y(τ k+1 − 0),

where

f (t − 0)

^and

f(t + 0)

^denotes ^the ^left-hand ^side ^and ^the ^right-hand

side limitofafuntion

f

^at

t

^, respetively. Weobtainthat(25) isequivalent to the followingdierentialequation with impulses:







¨

y(τ) | y(τ ˙ ) | ⁿ⁻ ¹ + | y(τ ) | ⁿ⁻ ¹ y(τ ) = 0, τ 6 = τ k

˙

y(τ k+1 ) = q k

q k+1

_n+1 ¹

˙

y(τ k+1 − 0), k = 0, 1, 2, . . .

(32)

Let us introdue the generalized polar oordinates

y ˙ = ρS ^′ (Φ)

^,

y = ρS(Φ)

^,

where

ρ = ( | y ˙ | ⁿ⁺¹ + | y | ⁿ⁺¹ ) ⁿ⁺¹ ¹ , T (Φ) = y

˙

y , −∞ < Φ < ∞ .

Thisistheso-alledgeneralizedPrüfertransformation. Withtheaidofthese

variableswe an rewrite equation(31) into

˙Φ = 1, ρ ˙ = 0, (τ k ≤ τ < τ k+1 , k = 0, 1, . . .).

⁽³³⁾

So the dynamis of system (32) on the Minkowski plane [19℄

( ˙ y, y)

^is ^the

following. It turns any point

( ˙ y 0 , y 0 )

âround ^the ôrigin ôn ^the ^Minkowski

(11)

irle with radius

ρ 0 := ( | y ˙ 0 | ⁿ⁺¹ + | y 0 | ⁿ⁺¹ ) ⁿ⁺¹ ¹

^on

[τ 0 , τ 1 )

^, ^and ^at

τ 1

^the ^point

( ˙ y(τ 1 − 0), y (τ 1 − 0))

^jumps ^to^the ^point

( ˙ y(τ 1 ), y(τ 1 )) :=

q 0

q 1

_n+1 ¹

˙

y(τ 1 − 0), y(τ 1 − 0)

! .

This proess is repeated onseutively for

[τ ₁ , τ ₂ )

^,

[τ ₂ , τ ₃ ), . . .

^. ^Dene

ρ k := | y(τ ˙ k ) | ⁿ⁺¹ + | y(τ k ) | ⁿ⁺¹ _n+1 ¹

, Φ k := Φ(τ k ), Ω k := τ k+1 − τ k ,

∆ρ k := ρ k+1 − ρ k , κ k := Φ k+1 − (Φ k + Ω k ), k = 0, 1, . . .

Obviously,

Φ k+1 = Φ 0 +

k

X

i=0

(Ω i + κ i ), ρ k+1 = ρ 0 +

k

X

i=0

∆ρ i , k = 0, 1 . . .

Sine

∆ρ i ≤ 0

^,^the^sequene

{ ρ k } ^∞ k=0

^ismonotonouslydereasing,thereforeit has alimit

ρ ¯ := lim k→∞ ρ k

^. Îf^the ^statementôf ^the^theorem îs ^not^true, ^then

there exists a solution

(ρ, Φ)

^suh ^that

ρ > ¯ 0

^. ^Let ^us ^onsider ^this ^solution

and estimate

− ∆ρ i

^:

− ∆ρ i = ρ i − ρ i+1

= ( | y(τ ˙ i ) | ⁿ⁺¹ + | y(τ i ) | ⁿ⁺¹ ) ⁿ⁺¹ ¹ − ( | y(τ ˙ i+1 ) | ⁿ⁺¹ + | y(τ i+1 ) | ⁿ⁺¹ ) ⁿ⁺¹ ¹

= ( | y(τ ˙ i+1 − 0) | ⁿ⁺¹ + | y(τ i+1 − 0) | ⁿ⁺¹ ) ⁿ⁺¹ ¹

− ( | y(τ ˙ i+1 ) | ⁿ⁺¹ + | y(τ i+1 ) | ⁿ⁺¹ ) ⁿ⁺¹ ¹

= ( | y(τ ˙ i+1 − 0) | ⁿ⁺¹ + | y(τ i+1 − 0) | ⁿ⁺¹ ) ⁿ⁺¹ ¹

− q i

q i+1 | y(τ ˙ i+1 − 0) | ⁿ⁺¹ + | y(τ i+1 − 0) | ⁿ⁺¹ _n+1 ¹

= 1

n + 1 ρ ⁿ⁺¹ _i+1 + η i ρ ⁿ⁺¹ _i − ρ ⁿ⁺¹ _i+1 − _n+1 ⁿ

×

1 − q i

q i+1

| y(τ ˙ i+1 − 0) | ⁿ⁺¹

≥ 1

n + 1 (¯ ρ) ⁿ⁺¹ ⁻ _n+1 ⁿ

1 − q i

q i+1

ρ ⁿ⁺¹ _i | S ^′ (Φ i + Ω i ) | ⁿ⁺¹

≥ ρ ¯ n + 1

1 − q i

q _i+1

| S ^′ (Φ i + Ω i ) | ⁿ⁺¹

(34)

(12)

with some

η i ∈ (0, 1)

^for ^all

i ∈ N ₀

. Now we need to estimate

| S ^′ (φ i + Ω i ) |

from belowby either

| S(Ω i ) |

^or

| S(Ω i+1 ) |

^, ^similarly^to^the ^proof ^of ^Theorem

1, where weused the addititonal formulae for the osine funtion. However,

to our best knowledge,the problemof nding exat additionformulaefor

S

and

S ^′

îs ^not ômpletely ^solved, âlthough ^there âre ^some ^papers âbout ^this

topi (see, e.g., [1℄, [2℄). Therefore, to omplete the proof we need a new

method dierent from one we used in the proof of Theorem 1 after formula

(14).

Funtions

| S ^′ (Φ + Ω) |

^and

| S(Ω) |

^are

π ˆ

^-periodi ^with ^respet ^to ^both

variables

Φ, Ω

^,^hene^we^may^restrit^ourselves^to^the^quadrant

[ − π/2, ˆ ˆ π/2] × [ − π/2, ˆ π/2] ˆ

^on ^the

(Φ, Ω)

^plane. ^Thanks ^to ^the ^symmetry ^properties ^of

S

and

S ^′

^,ît îsênough ^to^make^the êstimate ôn

Q := [0, π/2] ˆ × [0, ˆ π/2]

^.

At rst, letus handlethe set

Q ε := { (Φ, Ω) ∈ Q : | S ^′ (Φ) | < ε } ,

where

ε > 0

^is ^small ^enough. ^The omplementer set of

Q ε

^with ^respet ^to

Q

^will ^be ^treated ⁱⁿ ânother ^way. ^The ^same ^way ^will ^be ûsed âlso ^for ^the

omplementer set of

Q ^γ := { (Φ, Ω) ∈ Q : | S ^′ (Φ) | ≤ γ | S(Ω) |} (0 < γ < 1),

so nowweonsider the set

Q ^γ _ε := Q ε ∩ Q ^γ

^(see ^the ^gure).

A part of the boundary of this set is a piee of the urve dened by the

equation

Γ : | S ^′ (Φ) | = γ | S(Ω) | .

We show that the tangent to

Γ

^at

(ˆ π/2, 0)

^is ^the ^line

Φ = ˆ π/2

^, ^i.e.,

Φ → lim ^π ₂ − 0 f ^′ (Φ) = −∞ ; f(Φ) := S ⁻¹ 1

γ S ^′ (Φ)

,

⁽³⁵⁾

provided

n > 1

^. ^The ^statement^of ^the ^theorem ^for ^the ^linear^ase

n = 1

^was

proved in Theorem 1, so proving (35) we an restrit ourselves to the ase

n > 1

^.

It iseasy tosee that

(S ⁻ ¹ ) ^′ (W ) = 1

(1 − W ⁿ⁺¹ ) ⁿ⁺¹ ¹ (0 ≤ W ≤ 1).

(13)

S ^′′ (Φ) = −| S ^′ (Φ) | ⁻ⁿ⁺¹ | S(Φ) | ⁿ⁻¹ S(Φ).

⁽³⁶⁾

Therefore,

d

Φ f(Φ) = f ^′ (Φ) = − _γ ¹ (S ^′ (Φ)) ⁻ⁿ⁺¹ S ⁿ (Φ) 1 − γ ⁿ⁺¹ ¹ (S ^′ (Φ)) ⁿ⁺¹ _n+1 ¹ ,

onsequently, (35) holds, independently of

γ.

⁽³⁵⁾ împlies ^the êxistene ôf â

(14)

δ > 0

^suh ^that

f ^′ (Φ) < − 2

(S ^′ ) ⁻¹ (ε) < π ˆ

2 − δ < Φ < π ˆ 2

,

whene we get

f (Φ) ≥ − 2

Φ − π ˆ 2

,

whihmeansthat

Γ

îs^loatedôn^the ^right-hand^side ôf^the ^line

Ω = − 2(Φ − ˆ

π/2)

^near ^the ^point

(ˆ π/2, 0)

^(see ^the ^gure). ^T^o ^estimate

| S ^′ (Φ i + Ω i ) |

^from

belowby

| S(Ω i ) |

ⁱⁿ⁽³⁴⁾^we^have^to^estimate^the^quotient

| S ^′ (Φ + Ω) | / | S(Ω) |

frombelow. In

Q ^γ _ε

^we^derease ^this ^quotient^exhanging^point

(Φ, Ω)

^for ^the

horizontally orresponding point

(ˆ π/2 − Ω/2, Ω)

^of ^the ^line

Φ = ˆ π/2 − Ω/2

(see the gureagain). Therefore, by the L'Hospital Rule and (36) we get

lim _Φ _→ ˆ ^π

2 − 0,Ω → 0+0,(Φ,Ω) ∈ Q ^γ ε

| S ^′ (Φ + Ω) |

| S(Ω) | ≥ lim

Ω → 0+0

− S ^′ ˆ

π

2 − ¹ ₂ Ω + Ω S(Ω)

= lim

Ω → 0+0

− S ^′ ˆ

π

2 + ¹ ₂ Ω

S(Ω) = lim

Ω → 0+0

− S ^′′

ˆ ^π

2 + ¹ ₂ Ω

1 2

S ^′ (Ω)

= lim

Ω → 0+0

S ^′

ˆ ^π

2 + ^Ω ₂

− n+1 S

ˆ ^π

2 + ^Ω ₂

n − 1

S ˆ

π 2 + ^Ω ₂

2S ^′ (Ω) = ∞ .

This means that there exists a

κ > 0

^suh ^that

| S ^′ (Φ + Ω) | ≥ κ | S(Ω) | ((Φ, Ω) ∈ Q ^γ _ε ).

⁽³⁷⁾

Nowwe are readyto ompleteestimate(34). Wedistinguish threeases:

A)

(Φ i , Ω i ) ∈ Q ^γ _ε .

^Then ^by ⁽³⁴⁾ ^and ⁽³⁷⁾ ^we ^have

− ∆ρ i ≥ ρ n + 1

1 − q i

q i+1

κ ⁿ⁺¹ | S(Ω i ) | ⁿ⁺¹ .

⁽³⁸⁾

In the remaining ases we estimate

− ∆ρ i− 1 .

^By ^the ânalogue ôf ⁽²⁰⁾ ît îs

alwaystrue that

− ∆ρ i − 1 ≥ ρ n + 1

1 − q i − 1

q i

| S ^′ (Φ i − 1 + Ω i − 1 ) | ⁿ⁺¹

≥ ρ n + 1

1 − q i − 1

q i

| S ^′ (Φ i ) | ⁿ⁺¹ .

(15)

B)

(Φ i , Ω i ) ∈ Q _ε \Q ^γ _ε

^. ^Then

| S ^′ (Φ i ) | ≥ γ | S(Ω i ) |

^, ^and

− ∆ρ i−1 ≥ γ ⁿ⁺¹ ρ n + 1

1 − q i − 1

q i

| S(Ω i ) | ⁿ⁺¹ .

⁽³⁹⁾

C)

(Φ i , Ω i ) ∈ Q \ Q _ε

^. ^Then

| S ^′ (Φ i ) | ≥ ε | S(Ω i ) |

^and

− ∆ρ i− 1 ≥ ε ⁿ⁺¹ ρ n + 1

1 − q i−1

q i

| S(Ω i ) | ⁿ⁺¹ .

⁽⁴⁰⁾

Setting

C := ρ

n + 1 min { κ ⁿ⁺¹ ; γ ⁿ⁺¹ ; ε ⁿ⁺¹ } > 0,

and taking intoaount (38), (39), (40), for every

i

^we ^have

C min

1 − q i − 1

q i

; 1 − q i

q _i+1

| S(Ω i ) | ⁿ⁺¹ ≤ ∆ρ i−1 − ∆ρ i = ρ i−1 − ρ i+1 .

Summarizing these inequalitieswe obtain

C

∞

X

n=1

min

1 − q i − 1

q i

; 1 − q i

q _i+1

| S(Ω i ) | ⁿ⁺¹ ≤ ρ 0 − ρ < ∞ ,

whih ontradits the assumption of the theorem.

Theorem2extends Elbert'sTheorem Ctohalf-linearequationsprovided

n > 1

^. ^It ^would ^bee interesting to nd anextension tothe ase

n < 1

^,^too.

Referenes

[1℄ I. Adamaszek, On generalized sine and osine funtions,Demonstratio

Math. 28(1995),263270.

[2℄ I. Adamaszek-Fidytek, On generalized sine and osine funtions, II.

Demonstratio Math.37(2004), 571578.

[3℄ R.P.Agarwal,Dierene EquationsandInequalities,Monographsand

TextbooksinPureandAppliedMathematis,vol.115,MarelDekker,

In., New York, 1992.

(16)

linear dierential equations, Pro. 6'th Colloq. Qual. Theory Dier.

Equ., No.8, Eletron. J. Qual. Theory Dier. Equ., Szeged, 2000.

[5℄ I.Bihari,Ausdehnung derSturmshenOsillationsundVergleihssätze

aufdieLösungen gewissernihtlinearenDierentialgleihungenzweiter

Ordnung, Publ. Math. Inst. Hungar. Aad. Si.,2(1957), 154165.

[6℄ I. Bihari, Asymptoti result onerning equation

x ^′′ | x ^′ | ⁿ ⁻ ¹ + a(t)x ^∗ ⁿ

^.

ExtensionofatheorembyArmellini-Tonelli-Sansone,StudiaSi.Math.

Hungar. 19(1984), 151157.

[7℄ O. Do²lý, Half-linear dierential equations, Handbook of Dierential

Equations, 161357, Elsevier/North-Holland,Amsterdam, 2004.

[8℄ O.Do²lý,P.ehák, Half-linearDierentialEquations, Elsevier/North-

Holland, Amsterdam, 2005.

[9℄ Á. Elbert, A half-linear seondorder dierentialequation, Qualitative

Theory of Dierential Equations, Vol. I, II (Szeged, 1979), 153180,

Colloq.Math. So.János Bolyai,30, North-Holland,Amsterdam-New

York,1981.

[10℄ Á. Elbert, Stability of some differene equations, Advanes in Die-

rene Equations: Proeedings of the Seond International Conferene

on Dierene Equations and Appliations, Veszprém, Hungary, 7-11

August 1995,Gordon and Breah Siene Publishers, 1997, 155178.

[11℄ Á.Elbert,On asymptotistabilityof someSturm-Liouvilledierential

equations,General Seminar in Mathematis, University ofPatras,22-

23(1996/97), 5766.

[12℄ Á. Elbert,On dampingoflinearosillators,Studia Si.Math. Hungar.

38(2001), 191208.

[13℄ P. Hartman, Ordinary Dierential Equations, Birkhäuser, Boston,

1982.

[14℄ L. Hatvani, On stability properties of solutionsof seond order dier-

entialequations, Pro.6'th Colloq.Qual. Theory Dier. Equ.,No. 11,

Eletron. J. Qual. Theory Dier. Equ., Szeged,2000.

(17)

near osillatorwithaninreasing elastiityoeient,Georgian Math.

J. 14(2007),no. 2,269278.

[16℄ N.Jaobson, Letures inAbstrat Algebra,II. Linear Algebra,Springer

Verlag (New York- Heidelberg - Berlin),1953.

[17℄ J. W. Maki, Regular growth and zero tending solutions, Ordinary

Dierential Equations and Operators (Dundee, 1982), Leture Notes

in Math., vol. 1032, 358399.

[18℄ H.Milloux,Surl'equationdierentielle

x ^′′ +A(t)x = 0

^,^Prae^Mat.-Fiz.,

41(1934), 3954.

[19℄ H.Rund,TheDierentialGeometryofFinslerSpaes,Springer,Berlin,

1959.

(Reeived April 2,2011)

Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 38, 1-17; http://www.math.u-szeged.hu/ejqtde/