of some classes of higher order differential operators and weighted nth order differential inequalities

2021, No.3, 1–20; https://doi.org/10.14232/ejqtde.2021.1.3 www.math.u-szeged.hu/ejqtde/

Oscillation and spectral properties

of some classes of higher order differential operators and weighted nth order differential inequalities

Aigerim Kalybay

, Ryskul Oinarov

and Yaudat Sultanaev

1KIMEP University, 4 Abay Avenue, Almaty, 050010, Kazakhstan

2L. N. Gumilyov Eurasian National University, 5 Munaytpasov Street, Nur-Sultan, 010008, Kazakhstan

3Akmulla Bashkir State Pedagogical University, 3a Oktyabrskaya revolution Street, Ufa, 450000, Russia

Received 22 May 2020, appeared 10 January 2021 Communicated by John R. Graef

Abstract. In this paper, we obtain strong oscillation and non-oscillation conditions for a class of higher order differential equations in dependence on an integral behavior of its coefficients in a neighborhood of infinity. Moreover, we establish some spectral properties of the corresponding higher order differential operator. In order to prove these we establish a certain weighted differential inequality of independent interest.

Keywords: higher order differential operator, oscillation, non-oscillation, variational principle, weighted inequality, eigenvalues, spectrum discreteness, spectrum positive definiteness, nuclear operator.

2020 Mathematics Subject Classification: 34C10, 47B25, 26D10.

1 Introduction

Let I = (0,∞)andu be a continuous and nonnegative function. Suppose that vis a positive function such that it is sufficiently times continuously differentiable on the interval I and for anya >0 the functionv⁻¹is integrable on the interval(0,a).

Let T ≥ 0, I_T = (T,∞) and W_2,vⁿ ≡ W_2,vⁿ (I_T) be the space of functions f : I_T → _R having generalized derivatives up to nth order on the interval I_T, for which kf⁽ⁿ⁾k_2,v < _∞, where kgk_2,v = R_∞

T v(t)|g(t)|²dt¹₂

is the standard norm of the weighted space L2,v(I) ≡ L_2,v. From the conditions on the function v it easily follows the existence of the finite limit lim_t→T⁺ f⁽ⁱ⁾(t) ≡ f⁽ⁱ⁾(T)_, i = 0, 1, . . . ,n−_{1, for any} f ∈ W_2,vⁿ . Therefore, the spaceW_2,vⁿ is a normalized space with the norm

kfk_Wⁿ

2,v =kf⁽ⁿ⁾k_2,v+

n−1 i

∑

|f⁽ⁱ⁾(T)|. (1.1)

Let ˚M₂(I_T) ={f ∈W₂ⁿ(I_T): suppf ⊂ I_T and supp f is compact}.

BCorresponding author. Email: o_ryskul@mail.ru

By the assumptions on the function v we have that ˚M₂(I_T) ⊂ W_2,vⁿ . Denote by ˚W_2,vⁿ = W˚_2,vⁿ (IT)the closure of the set ˚M2 with respect to norm (1.1).

In the paper we investigate three related problems.

Problem 1. Establish criteria of strong oscillation and non-oscillation of the 2nth order differ- ential equation

(−1)ⁿ(v(t)y⁽ⁿ⁾(t))⁽ⁿ⁾−λu(t)y(t) =0, t ∈ I, (1.2) wheren>1 andλ>0.

A solution of equation (1.2) is a functiony : I →_Rthat is ntimes differentiable together with the functionv(_t)_y⁽ⁿ⁾(_t)on the interval I, satisfying equation (1.2) for allt ∈ I.

Equation (1.2) is called [9, p. 6] oscillatory, if for any T > 0 there exists a (non-trivial) solution of this equation, having more than one zero with multiplicity n to the right of T.

Otherwise equation (1.2) is called non-oscillatory. In the sequel, the expression “solution of equation” will mean “non-trivial solution of equation” unless the opposite is specified.

Equation (1.2) is called strong non-oscillatory (oscillatory), if it is non-oscillatory (oscilla- tory) for all valuesλ>0.

In the mathematical literature, the most number of works is devoted to the oscillatory properties of linear, semilinear and nonlinear second-order differential equations (see, e.g., [5]

and references given there). However, such studies for a higher order equation are relatively rare due to the fact that not all methods of studying a second order equation are extended to a higher order equation (see [6]). One of the more universal methods to study the oscillatory properties of symmetric differential equations is the variational method. However, in the variational method, the problem is reduced to solving Problem 3, which has not yet been completely studied. Another method of studying an equation in the form (1.2) is to transfer from equation (1.2) to the system of Hamilton’s equations, but even here it is difficult to find the fundamental solutions of the Hamiltonian system, especially when the coefficients of equation (1.2) are arbitrary functions. Therefore, in the works devoted to the problem of oscillation or strong oscillation of higher order equations in the form (1.2), all or one of the coefficients are power functions (see, [6–8] and references given there). In a more general case, in terms of the coefficients of the equation, criteria for its strong oscillation and non-oscillation are given in [20].

The oscillatory and non-oscillatory properties of higher order differential equations and their relation to the spectral characteristics of the corresponding differential operators are well presented in monograph [9].

Problem 2. Investigate the spectral properties of the self-adjoint differential operatorLgener- ated by the differential expression

l(y) = (−1)^{n 1}

u(t)(v(t)y⁽ⁿ⁾)⁽ⁿ⁾, (1.3) in the Hilbert space L_2,u ≡ L_2,u(I) with inner product (f,g)_2,u = R_∞

0 f(t)g(t)u(t)dt, where u>0.

The investigation of the spectral characteristics of the operator L is the subject of many works (see, e.g., [2,3], [9, Chapters 29 and 34] , [10,14,21] and references given there).

Problem 3. Find necessary and sufficient conditions for the validity of the inequality Z _∞

T u(t)|f(t)|²dt≤C_T Z _∞

T v(t)|f⁽ⁿ⁾(t)|²dt, f ∈W^˚_2,vⁿ (1.4) and the sharp estimate of the constantC_T.

The inequality of the type (1.4) was considered in many works (see, e.g., [1,11,17,18] and references given there). The history of the problem and the main achievements are shortly presented in monographs [12] and [13]. Let us note that in [13, Chapter 4] the corresponding comments are given wider than in [12].

We study all these three problems depending on an integral behavior of the functionvin a neighborhood of infinity. Problems 1 and 2 have been already investigated in the strong singular case

Z _∞

T v⁻¹(t)dt= _∞. (1.5)

Here we assume that Z _∞

T v⁻¹(t)dt<_∞ and Z _∞

T v⁻¹(t)t²dt=_∞ (1.6) for any T≥_0.

The work is organized as follows. In Section 2 we give necessary and sufficient conditions for the validity of inequality (1.4). In Section 3 on the basis of the results on inequality (1.4) we find necessary and sufficient conditions for the functionsu and v, under which equation (1.2) is strong oscillatory or non-oscillatory. In Section 4, some spectral characteristics of the operator Lare obtained.

The symbol A B means A≤ CBwith some constantC. If A B A, then we write A≈ B. Moreover,χ_M stands for the characteristic function of the set M.

2 Validity of inequality (1.4)

We investigate (1.4) under condition (1.6). First, we present the known results required for the proof of the validity of inequality (1.4).

Let 0≤a<b≤∞. From the paper [13, p. 6 and 7], the following theorem follows.

Theorem A.

(i) The inequality Z _b

a u(x) _{Z x}

a f(t)dt 2

dx

!¹₂

≤C _{Z b}

a v(t)f²(t)dt ¹₂

, f ≥0, (2.1)

holds if and only if

A⁺ = sup

a<z<b

Z _b

z u(x)dx

¹₂ Z _z

a v⁻¹(t)dt ¹₂

<_∞.

Moreover, A⁺ ≤C≤2A⁺,where C is the best constant in(2.1).

(ii) The inequality Z _b

a u(x) _{Z b}

x f(t)dt 2

dx

!¹₂

≤C _{Z b}

a v(t)f²(t)dt ¹₂

, f ≥0, (2.2)

holds if and only if

A⁻= sup

a<z<b

_{Z z}

a u(x)dx

¹_{2 Z b}

z v⁻¹(t)dt ¹₂

< _∞. Moreover, A⁻≤C≤2A⁻,where C is the best constant in(2.2).

Let

A₁ = sup

a<z<b

Z _b

z u(x)dx

¹₂ Z _z

a

(z−t)²⁽ⁿ⁻¹⁾v⁻¹(t)dt ¹₂

,

A2= sup

a<z<b

Z _b

z

(x−z)²⁽ⁿ⁻¹⁾u(x)dx

¹₂ Z _z

a v⁻¹(t)dt ¹₂

. The next statement follows from the results in the work [21].

Theorem B. The inequality Z _b

a u(z) _{Z z}

a

(z−t)ⁿ⁻¹f(t)dt 2

dz≤C Z _b

a v(t)f²(t)dt, f ≥0, (2.3) holds if and only ifmax{A₁,A2}<∞. Moreover,

C≈max{A₁,A₂}, (2.4)

where C is the best constant in(2.3).

Assume that lim_t→_∞ f⁽ⁿ⁻¹⁾(t)≡ f⁽ⁿ⁻¹⁾(_∞)and

LR⁽ⁿ⁻¹⁾W_2,vⁿ =f ∈W_2,vⁿ : f⁽ⁱ⁾(T) =0, i=0, 1, . . . ,n−1; f⁽ⁿ⁻¹⁾(_∞) =0 , LW_2,vⁿ =_f ∈W_2,vⁿ : f⁽ⁱ⁾(_T) =_0,i=0, 1, . . . ,n−1 .

From Theorems 1 and 2 in [15] in view of the conditions onv⁻¹ in a neighborhood of zero, it follows the next statement.

Theorem C.

(i) If (1.5)holds, then

W˚_2,vⁿ ≡ LW_2,vⁿ ; (2.5)

(ii) if (1.6)holds, then

W˚_2,vⁿ ≡ LR⁽ⁿ⁻¹⁾W_2,vⁿ and LW_2,vⁿ (I_T+1)≡ LR⁽ⁿ⁻¹⁾W_2,vⁿ (I_T+1)⊕P_∞, (2.6) where P_∞= {P(t) =cχ_IT+1(t)tⁿ⁻¹:c∈ R}.

Assume thatJ(f) =

R_∞

T u(t)|f(t)|²dt R_∞

T v(t)|f⁽ⁿ⁾(t)|²dt,C_L=_supf∈LWn

2,vJ(f)_andC_LR≡C_T=_supf∈LR(n−1)W_2,vⁿ J(f)_. It is obvious thatC_LR ≤C_L. We investigate the estimate of the valueC_LRunder the assumption C_L=∞, that in view of (2.6) is equivalent to the condition

Z _∞

α

u(x)x²⁽ⁿ⁻¹⁾dx= _{∞ (2.7)}

for anyα>T.

Letτbe an arbitrary point of the intervalI_T. Assume A_1,1(T,τ) = sup

T<z<τ

Z _τ

z u(x)dx Z _z

T

(z−t)²⁽ⁿ⁻¹⁾v⁻¹(t)dt, A_1,2(T,τ) = sup

T<z<τ

Z _τ

z u(x)(x−z)²⁽ⁿ⁻¹⁾dx Z _z

T v⁻¹(t)dt, A_1,3(T,τ) =

Z _∞

τ

u(x)(x−τ)²⁽ⁿ⁻²⁾dx Z _τ

T

(τ−t)²v⁻¹(t)dt, A_1,4(T,τ) =

Z _∞

τ

u(x)dx Z _τ

T

(τ−t)²⁽ⁿ⁻¹⁾v⁻¹(t)dt, A_2,1(T,τ) =sup

z>τ

Z _∞

z u(x)(x−τ)²⁽ⁿ⁻²⁾dx Z _z

τ

(t−τ)²v⁻¹(t)dt, A2,2(T,τ) =sup

z>τ

Z _z

τ

u(x)(x−τ)²⁽ⁿ⁻¹⁾dx Z _∞

z v⁻¹(t)dt, A(T,τ) =max

A_1,1(T,τ),A_1,2(T,τ),A_1,3(T,τ),A_1,4(T,τ),A_2,1(T,τ),A_2,2(T,τ) . Due to (2.6) inequality (1.4) can be written in the form

Z _∞

T u(t)|f(t)|²dt≤C_T Z _∞

T v(t)|f⁽ⁿ⁾(t)|²dt, f ∈ LR⁽ⁿ⁻¹⁾W_2,vⁿ .

In work [18] it is obtained that A(T,τ) < _∞ is necessary and sufficient condition for the validity of this inequality, whereRτ

T v⁻¹(t)dt=R_∞

τ v⁻¹(t)dt. Here we obtain a simpler criterion that is usable for the application to Problem 1and2.

Theorem 2.1. Let T ≥0. Let(1.6)and(2.7)hold. Inequality(1.4)holds if and only if

zlim→_∞ Z _∞

z u(x)(x−τ)²⁽ⁿ⁻²⁾dx Z _z

τ

(t−τ)²v⁻¹(t)dt<_∞ (2.8) and

zlim→_∞ Z _z

τ

u(x)(x−τ)²⁽ⁿ⁻¹⁾dx Z _∞

z

v⁻¹(t)dt<_{∞. (2.9)} Moreover, there exists a pointτ_T : T<τ_T < _∞such that

C_T ≈ A(T,τ_T) =max{A_2,1(T,τ_T),A_2,2(T,τ_T)}, (2.10) where C_T is the best constant in(1.4).

Proof. Sufficiency. Let (2.8) and (2.9) hold. Then, due to the conditions on the weight functions u and v, we get that A(T,τ) < _∞ for any τ ∈ I_T. Therefore, on the basis of the results in [18], inequality (1.4) holds. Now, let us estimate the constant C_T from above. From (2.6) it follows that f⁽ⁱ⁾(T) = 0, i = 0, 1, . . . ,n−1, f⁽ⁿ⁻¹⁾(_∞) = 0 for any f ∈ W^˚_2,vⁿ . Hence, we present f ∈W^˚_2,vⁿ in the form f(x) = ₍ ¹

n−2)!

Rx

T(x−s)ⁿ⁻²f⁽ⁿ⁻¹⁾(s)ds, x > T, where f⁽ⁿ⁻¹⁾(s) = Rs

T f⁽ⁿ⁾(t)dt = −R_∞

s f⁽ⁿ⁾(t)dt, s > T. Let τ ∈ I_T. Next, for T < s < τ we assume that f⁽ⁿ⁻¹⁾(s) = Rs

T f⁽ⁿ⁾(t)dt, and for s > τ we assume that f⁽ⁿ⁻¹⁾(s) = −R_∞

s f⁽ⁿ⁾(t)dt. Then f(x) = _(n−¹

2)_!

Rx

T(x−s)ⁿ⁻²Rs

T f⁽ⁿ⁾(t)dtds forT< x<τand f(x) = ¹

(n−2)! Z _x

T

(x−s)ⁿ⁻²f⁽ⁿ⁻¹⁾(s)ds

= ¹

(n−2)! Z _τ

T

(x−s)ⁿ⁻²f⁽ⁿ⁻¹⁾(s)ds+

Z _x

τ

(x−s)ⁿ⁻²f⁽ⁿ⁻¹⁾(s)ds

= ¹

(n−2)! _{Z τ}

T

(x−s)ⁿ⁻²

Z _s

T f⁽ⁿ⁾(t)dtds−

Z _x

τ

(x−s)ⁿ⁻²

Z _∞

s f⁽ⁿ⁾(t)dtds

forx>τ. Therefore, we have Z _∞

T u(x)|f(x)|²dx =

Z _τ

T u(x)|f(x)|²dx+

Z _∞

τ

u(x)|f(x)|²dx

= ¹

[(n−2)!]²

Z _τ

T u(x)

Z _x

T

(x−s)ⁿ⁻²

Z _s

T f⁽ⁿ⁾(t)dtds

2

dx

+ ¹

[(n−2)!]²

Z _∞

τ

u(x)

Z _τ

T

(x−s)ⁿ⁻²

Z _s

T f⁽ⁿ⁾(t)dtds−

Z _x

τ

(x−s)ⁿ⁻²

Z _∞

s f⁽ⁿ⁾(t)dtds

2

dx

= ¹

[(n−2)!]²

F₁(f⁽ⁿ⁾) +F₂(f⁽ⁿ⁾), (2.11)

where

F₁(f⁽ⁿ⁾) =

Z _τ

T u(x)

Z _x

T

(x−s)ⁿ⁻²

Z _s

T f⁽ⁿ⁾(t)dtds

2

dx

= ¹

(n−1)²

Z _τ

T u(x)

Z _x

T

(x−t)ⁿ⁻¹f⁽ⁿ⁾(t)dt

2

dx,

F₂(f⁽ⁿ⁾) =

Z _∞

τ

u(x)

Z _τ

T

(x−s)ⁿ⁻²

Z _s

T f⁽ⁿ⁾(t)dtds−

Z _x

τ

(x−s)ⁿ⁻²

Z _∞

s f⁽ⁿ⁾(t)dtds

2

dx

=

Z _∞

τ

u(x)

Z _τ

T

(x−s)ⁿ⁻²

Z _s

T f⁽ⁿ⁾(t)dtds−

Z _x

τ

(x−s)ⁿ⁻²

Z _x

s f⁽ⁿ⁾(t)dtds

−

Z _x

τ

(x−s)ⁿ⁻²dx Z _∞

x f⁽ⁿ⁾(t)dtds

2

dx.

Assume that f⁽ⁿ⁾ = g, then R_∞

T g(t)dt = 0 and the condition f ∈ W^˚_2,vⁿ is equivalent to the condition g ∈ eL₂(I_T) ≡ {g ∈ L₂(I_T) : R_∞

T g(t)dt = 0}. Therefore, from (2.11) it follows that inequality (1.4) is equivalent to the inequality

1 [(n−2)!]²

F₁(g) +F₂(g)≤C_T Z _∞

T v(t)|g(t)|²dt, g∈eL₂(I_T). (2.12) Moreover, the best constants in inequalities (1.4) and (2.12) coincide.

On the basis of TheoremBwe have F₁(g) = ¹

(n−1)²

Z _τ

T u(x)

Z _x

T

(x−t)ⁿ⁻¹g(t)dt

2

dx max{A_1,1(T,τ),A_1,2(T,τ)}

Z _τ

T v(t)|g(t)|²dt. (2.13) Now, we estimateF₂(g).

F₂(g)≤

Z _∞

τ

u(x)

Z _τ

T

(x−s)ⁿ⁻²

Z _s

T

|g(t)|dtds+

Z _x

τ

(x−s)ⁿ⁻²

Z _x

s

|g(t)|dtds +

Z _x

τ

(x−s)ⁿ⁻²ds Z _∞

x

|g(t)|dt

2

dx

=

Z _∞

τ

u(x)

Z _τ

T

|g(t)|

Z _τ

t

(x−s)ⁿ⁻²dsdt+

Z _x

τ

|g(t)|

Z _t

τ

(x−s)ⁿ⁻²dsdt

+ ¹

n−1(x−τ)ⁿ⁻¹

Z _∞

x

|g(t)|dt

2

dx

≤3

"

Z _∞

τ

u(x)

Z _τ

T

|g(t)|

Z _τ

t

(x−s)ⁿ⁻²dsdt

2

dx +

Z _∞

τ

u(x)

Z _x

τ

|g(t)|

Z _t

τ

(x−s)ⁿ⁻²dsdt

2

dx

+ ¹

(n−₁)²

Z _∞

τ

u(x)(x−τ)²⁽ⁿ⁻¹⁾ _{Z ∞}

x

|g(t)|dt 2

dx

#

=3

J₀+J₁+ ^J2 (n−1)²

, (2.14)

where

J0=

Z _∞

τ

u(x)

Z _τ

T

|g(t)|

Z _τ

t

(x−s)ⁿ⁻²dsdt

2

dx, J₁=

Z _∞

τ

u(x)

Z _x

τ

|g(t)|

Z _t

τ

(x−s)ⁿ⁻²dsdt

2

dx, J₂=

Z _∞

τ

u(x)(x−τ)²⁽ⁿ⁻¹⁾ _{Z ∞}

x

|g(t)|dt 2

dx.

Let us estimate J₀, J₁ and J₂ separately. For the estimate of J₀ using (x− s)ⁿ⁻² = (x−τ+τ−s)ⁿ⁻²≈ (x−τ)ⁿ⁻²+ (τ−s)ⁿ⁻²and Hölder’s inequality, we get

J₀≈

Z _∞

τ

u(x)(x−τ)²⁽ⁿ⁻²⁾dx _{Z τ}

T

(τ−t)|g(t)|dt 2

+

Z _∞

τ

u(x)dx _{Z τ}

T

(τ−t)ⁿ⁻¹|g(t)|dt 2

max{A_1,3(T,τ)_,A_1,4(T,τ)}

Z _τ

T

v(t)|g(t)|²dt. (2.15)

For the estimate ofJ₁usingRt

τ(x−s)ⁿ⁻²ds=_n−¹₁ (x−τ)ⁿ⁻¹−(x−t)ⁿ⁻¹≈(x−τ)ⁿ⁻²(t−τ) and Theorem A, we get

J₁ ≈

Z _∞

τ

u(x)(x−τ)²⁽ⁿ⁻²⁾ Z _x

τ

(t−τ)|g(t)|dt 2

dx A_2,1(T,τ)

Z _∞

τ

v(t)|g(t)|²dt. (2.16) By TheoremAwe have

J₂ A_2,2(T,τ)

Z _∞

τ

v(t)|g(t)|²dt. (2.17) From (2.11), (2.12), (2.13), (2.14), (2.15), (2.16) and (2.17) it follows that there exist positive numbersαandβsuch that the estimate

Z _∞

T u(x)|f(x)|²dx ≤βA₀(T,τ)

Z _τ

T v(t)|f⁽ⁿ⁾(t)|²dt+αA(T,τ)

Z _∞

τ

v(t)|f⁽ⁿ⁾(t)|²dt (2.18) holds, where A₀(T,τ) = max{A_1,1(T,τ), A_1,2(T,τ), A_1,3(T,τ),A_1,4(T,τ)} and A(T,τ) = max{A_2,1(T,τ),A_2,2(T,τ)}.

In view of (2.8) and (2.9), we have that the value A₀(T,τ) satisfies the properties lim_τ→TA₀(T,τ) =0 and lim_τ→_∞A₀(T,τ) =∞, and the valueA(T,τ)is non-increasing inτ and limτ→_∞A(T,τ)< ∞. Therefore, the following point

τ_T =_sup{τ∈ I_T :βA₀(T,τ)≤αA(T,τ)} _(2.19)

is defined. Then from (2.18) we have Z _∞

T u(t)|f(t)|²dt A(T,τT)

Z _∞

T v(t)|f⁽ⁿ⁾(t)|²dt, (2.20) i.e., inequality (1.4) holds with the estimate

C_T A(T,τ_T) (2.21)

for the best constantC_T in (1.4).

Necessity. Let us use the technique used in works [17] and [18]. Let inequality (1.4) hold with the best constant C_T > 0. By condition (1.6) we have that R_∞

T v⁻¹(t)dt < ∞. Suppose that γ_τT =γ(τ_T)>0 and the functionρ:(T,τ_T)→(τ_T,∞)is such that

Z _τT

T v⁻¹(t)dt=γτ_T

Z _∞

τT

v⁻¹(t)dt and

Z _s

T

v⁻¹(t)dt= γ_τT Z _∞

ρ(_s)v⁻¹(t)dt, s∈ (T,τ_T)_{. (2.22)} It is obvious that the decreasing function ρ is locally absolutely continuous on the interval (T,τ_T)and lim

s→T⁺ρ(s) =_{∞, lim}

s→τT

ρ(s) =τ_T. The differentiation of the both sides of (2.22) gives v⁻¹(s) =−γτ_Tv⁻¹(ρ(s))ρ⁰(s) =γτ_Tv⁻¹(ρ(s))|ρ⁰(s)|

or

1 γτ_T

= ^v

−1(ρ(s))|ρ⁰(s)|

v⁻¹(s) ^(2.23)

for almost alls ∈(T,τ_T). Let

K⁺(T,τ_T) =g∈ L₁(T,τ_T)∩L_2,v(T,τ_T)_: g≥0,g6≡0 , K⁻(τ_T,∞) =g∈ L₁(τ_T,∞)∩L_2,v(τ_T,∞): g≤0,g6≡0 .

Let us show that for every g₂ ∈ K⁻(τ_T,∞) there exists g_1,2 ∈ K⁺(T,τ_T) such that for the functionsg(t) =g_1,2(t),t∈ (T,τ_T)andg(t) =g₂(t),t∈ (τ_T,∞)we have thatg ∈eL_2,v(T,∞).

For g₂ ∈ K⁻(τ_T,∞) we assume that g_1,2(x) = −γτ_Tg₂(ρ⁻¹(x)) ^v−1(x)

v⁻¹(ρ⁻¹(x)). Then g_1,2 ≥ 0.

Changing the variablesρ⁻¹(x) =t and using (2.23), we have Z _τT

T g_1,2(x)dx= γτ_T

Z _τT

T

g₂(ρ⁻¹(x))

v⁻¹(x)

v⁻¹(ρ⁻¹(x))^dx=−γτ_T

Z _∞

τT

|g₂(t)|^v

−1(ρ(t)) v⁻¹(t) ^ρ

0(t)dt

= γτ_T

Z _∞

τT

|g₂(t)|^v

−1(ρ(t))

v⁻¹(t) |ρ⁰(t)|dt=

Z _∞

τT

|g₂(t)|dt<_∞. (2.24) From (2.24) it follows thatRτT

T g_1,2(x)dx<_∞and Z _τT

T g_1,2(x)dx+

Z _∞

τT

g2(t)dt=

Z _∞

T g(t)dt= 0. (2.25)

Again, changing the variables ρ⁻¹(x) =tand using (2.23), we have Z _τT

T

|g_1,2(t)|²v(t)dt=γ²_τT Z _τT

T

g₂(ρ⁻¹(x)) ^v

−1(x) v⁻¹(ρ⁻¹(x))

2

v(x)dx

=γ²_τT Z _∞

τ_T

|g₂(t)|²v(t)^v

−1(ρ(t))

v⁻¹(t) |ρ⁰(t)|dt

=γ_τT Z _∞

τ_T

|g₂(t)|²v(t)dt<_∞. Hence,

Z _∞

T

|g(t)|²v(t)dt=

Z _τT

T

|g_1,2(t)|²v(t)dt+

Z _∞

τ_T

|g₂(t)|²v(t)dt

= (1+γτ_T)

Z _∞

τT

|g₂(t)|²v(t)dt<_{∞, (2.26)} i.e.,g ∈L2,v(IT). The last and (2.25) give thatg∈ eL2,v(IT).

Let g₂ ∈ K⁻(τ_T,∞)and g_1,2 ∈ K⁺(T,τ_T)be a function defined by g₂. Then g ∈ eL_2,v(I_T), where g(t) =g_1,2(t),t∈ (T,τ_T)andg(t) =g₂(t),t∈(τ_T,∞). Sinceg ∈eL₂(I_T), then replacing the function gin (2.12) forτ=τT and taking into account that g_1,2≥0, g2≤0, we have

1 [(n−2)!]²

F₁(g_1,2) +

Z _∞

τT

u(x) _{Z τT}

T

(x−s)ⁿ⁻²

Z _s

T g_1,2(t)dtds +

Z _x

τT

(x−s)ⁿ⁻²

Z _∞

s

|g₂(t)|dtds 2

dx

#

≤C_T Z _∞

T v(t)|g(t)|²dt, that together with (2.26) gives

Z _∞

τ_T

u(x) Z _x

τ_T

(x−s)ⁿ⁻²

Z _∞

s

|g₂(t)|dtds 2

dx (₁+γ_τT)C_T

Z _∞

τ_T

|g₂(t)|²v(t)dt, g₂∈K⁻(τ_T,∞)_{. (2.27)} Since

Z _x

τ_T

(x−s)ⁿ⁻²

Z _∞

s

|g₂(t)|dtds≥(x−τ_T)ⁿ⁻²

Z _x

τ_T

(t−τ_T)|g₂(t)|dt+ ¹

n−1(x−τ_T)ⁿ⁻¹

Z _∞

x

|g₂(t)|dt, then from (2.27) we have

Z _∞

τ_T

u(x)(x−τ_T)²⁽ⁿ⁻²⁾ _{Z x}

τ_T

(t−τ_T)|g₂(t)|dt ₂

dx

≤(₁+γ_τT)C_T Z _∞

τ_T

|g₂(t)|²v(t)dt, g₂∈K⁻(τ_T,∞)_{, (2.28)}

Z _∞

τ_T

u(x)(x−τ_T)²⁽ⁿ⁻¹⁾ Z _∞

x

|g₂(t)|dt 2

dx

≤C_T(₁+γ_τT)

Z _∞

τ_T

|g₂(t)|²v(t)dt, g₂∈K⁻(τ_T,∞)_{. (2.29)}

For anyτ_T <z< _∞the functionsg₂⁺(t) =−χ_(τT,z)(t)(t−τ_T)v⁻¹(t), g⁻₂(t) =−χ_(z,∞)(t)v⁻¹(t) belong to the setK⁻(τ_T,∞). Replacing the functionsg⁺₂ andg₂⁻into (2.28) and (2.29), respec- tively, we get

A(T,τ_T)C_T. (2.30)

This relation together with (2.21) gives (2.10). From the finiteness of the value A(T,τ_T) = max{A_2,1(T,τ_T),A_2,2(T,τ_T)}we have (2.8) and (2.9). The proof of Theorem 2.1 is complete.

3 Oscillatory properties of equation (1.2)

The main aim of this Section is the investigation of strong oscillation and non-oscillation of differential equation (1.2) in a neighborhood of infinity. Oscillatory properties of (1.2) we investigate under conditions (1.6) and (2.7). Case (1.5) has been investigated in paper [20].

We consider the inequality Z _∞

T λu(t)|f(t)|²dt≤λC_T Z _∞

T v(t)|f⁽ⁿ⁾(t)|²dt, f ∈W^˚_2,vⁿ , (3.1) with a constantλC_T, whereC_T is the best constant in (1.4).

We investigate the oscillatory properties of equation (1.2) by the variation method, i.e., on the basis of the known variational statement.

Lemma A([9, Theorem 28]). Equation(1.2)is non-oscillatory if and only if there exists T>0such

that Z _∞

T

v(t)|f⁽ⁿ⁾(t)|²−λu(t)|f(t)|²dt≥0 (3.2) for all f ∈ M^˚₂(I_T).

Due to the compactness of the setsupp f for f ∈ M^˚₂(I_T), inequality (3.2) coincide with the inequality

Z _∞

T λu(t)|f(t)|²dt≤

Z _∞

T v(t)|f⁽ⁿ⁾(t)|²dt, ∀f ∈ M^˚₂(I_T). (3.3) Lemma 3.1. Equation(1.2)

(i) is non-oscillatory if and only if there exists T > 0 such that inequality(3.1) holds with the best constantλC_T : 0<λC_T ≤1;

(ii) is oscillatory if and only if for any T>₀the best constant is such thatλC_T >₁in(3.1).

Proof. Let us prove the statement (i), the statement (ii) is the opposite of the statement (i).

If equation (1.2) is non-oscillatory, then for some T > 0 inequality (3.3) holds, which means that inequality (3.1) holds with the best constant 0 < λC_T ≤ 1. Inversely, if for some T > 0 inequality (3.1) holds with the best constant 0 < λC_T ≤ 1, then inequality (3.3) holds and by LemmaAequation (1.2) is non-oscillatory. The proof of Lemma3.1is complete.

On the basis of Lemma3.1and Theorem 2.1, we have the following statement.

Theorem 3.2. Let(1.6)and(2.7)hold. Then equation(1.2)is strong non-oscillatory if and only if

zlim→_∞ Z _∞

z u(x)(x−T)²⁽ⁿ⁻²⁾dx Z _z

T

(t−T)²v⁻¹(t)dt=0 (3.4) and

zlim→_∞ Z _z

T u(x)(x−T)²⁽ⁿ⁻¹⁾dx Z _∞

z v⁻¹(t)dt=0. (3.5)