For the two-term equation

Vol. 19 (2018), No. 2, pp. 1137–1161 DOI: 10.18514/MMN.2018.2250

NONOSCILLATION OF EVEN ORDER EULER TYPE HALF-LINEAR DIFFERENCE EQUATIONS

VOJT ˇECH R ˚U ˇZI ˇCKA Received 24 February, 2017

Abstract. We establish nonoscillation criteria for the even order half-linear difference equation of Euler type

n

X

lD0

. 1/^{n l}ˇn l^{n l}

k^{.˛ lp/}˚

^{n l}xkCl

D0; ˇnWD1;

where˚.t /WD jtj^{p 1}sgnt, p2.1;1/, n2N,k^{.ˇ /} denotes the falling factorial power (for ˇ2R) and˛; ˇ0; ˇ1; : : : ; ˇn 1are real constants. For the two-term equation

. 1/ⁿⁿ

k^.˛/˚ ⁿx_k

Cˇ0k^{.˛ np/}˚.x_kCn/D0

we establish the constant n;p;˛ such that the two-term equation is nonoscillatory if ˇ0> n;p;˛. The criteria are derived using the variational technique and they are further extended via the theory of regularly varying sequences.

2010Mathematics Subject Classification: 39A12; 39A21

Keywords: Euler half-linear difference equation, higher order half-linear difference equation, nonoscillation criterion, variational principle, energy functional

1. INTRODUCTION

We consider the2n-th order half-linear difference equation

n

X

lD0

. 1/^{n ln l}

r_k^{Œn l}˚

^{n l}xkCl

D0; (1.1)

wheren2N, ˚.t /WD jtj^{p 1}sgnt is the odd power function, the real numberp is such that p > 1, n

r_k^{Œj }o₁

kD1 is real-valued sequence for everyj 2 f0; 1; : : : ; ngand r_k^Œn¤0fork2N. The phrase “half-linear” reflects the fact that the solution space

The author was supported by the Grant GA16-00611S of the Czech Grant Foundation and by the Research Project MUNI/A/1103/2016 of Masaryk University.

c 2018 Miskolc University Press

is homogenous, but not additive. Further we consider the energy functional Fn.fy_kgIN;1/WD

1

X

kDN

" _n X

lD0

r_k^{Œn l}ˇ ˇ

ˇ^{n l}y_kClˇ ˇ ˇ

p#

associated with equation (1.1), where N 2N and a sequencefy_kgis from the set Dn.N / (definition of this set will be recalled later). We focus on a special cases of (1.1), namely on Euler type equation (1.2) and its extension (3.27). Consider the Euler type half-linear difference equation

n

X

lD0

. 1/^{n l}ˇ_{n l}^{n l}

k^{.˛ lp/}˚

^{n l}x_kCl

D0; ˇnWD1; (1.2) whereˇ0; ˇ1; : : : ; ˇn 1are real numbers and˛2Rn fp 1; 2p 1; : : : ; np 1g. For k 2N and ˇ2 R the symbol k^{.ˇ /} denotes so-called falling factorial power (see [14, Definition 2.3]), which can be expressed as

k^{.ˇ /}D .kC1/

.kC1 ˇ/

fork2Nn fˇ iji2Ngandˇ2R, where denotes the Gamma function defined fort2Rn f0; 1; 2; : : :g. Recall that fort2.0;1/we have

.t /WD Z ₁

0

e ^ss^{t 1}ds:

Furthermore, recall that for sequencesfa_kgandfb_kgof non-zero real numbers, we writeakÏbkask! 1and say that the sequencesfakgandfbkgareasymptotically equivalent, if limk!1a_k=b_kD1. Now, from Stirling’s formula

tlim!1

.tC1/

t e

tp

2 t D1

we get the known relation ^.k_.k/^{Cˇ /}Ïk^ˇ ask! 1(forˇ2R), hence,

k^ˇ Ïk^{.ˇ /} as k! 1: (1.3)

In this article, we focus on getting conditions (for the coeficientsˇ0; ˇ1; : : : ; ˇn 1) which guarantee the nonoscillation of equation (1.2). We use the variational tech- nique which is at disposal for general equation (1.1) by the results of article [3]. The main result of [3] is formulated (in a slightly different form) in Theorem 3 in this paper.

Our motivation comes mainly from the results for the continuous version of equa- tion (1.2), i.e., for the differential equation

n

X

lD0

. 1/^{n l}ˇn l

t^{˛ lp}˚

x^{.n l/}.n l/

D0; ˇnWD1; (1.4)

where˛2Rn fp 1; 2p 1; : : : ; np 1g. Denote n;p;˛WD

n

Y

iD1

jip 1 ˛j p

p

: Consider also the special cases

p;˛WD1;p;˛D

jp 1 ˛j p

p

and n;2;˛D 1 4ⁿ

n

Y

iD1

.2i 1 ˛/²: The following results are known criteria for special cases of equation (1.4). In [7, Theorem 1.4.4] it is shown that the second order equation

t^˛˚ x⁰₀ C

t^{p ˛}˚.x/D0 (1.5)

is nonoscillatory if and only ifC1;p;˛0(for˛D0see the older result in [9]).

For equation (1.5) the number 1;p;˛is thecritical constant, i.e., the constant which is the “borderline” (as for the parameter) between oscillation and nonoscillation of equation (1.5). For the two-term2n-th order equation

. 1/ⁿ t^˛˚

x^.n/.n/

C t^{˛ np}˚ .x/D0 (1.6)

we have so far only the following implication. Equation (1.6) is nonoscillatory if Cn;p;˛> 0(for general˛see [4, Theorem 3.2], for˛D0see the older result in [7, Theorem 9.4.5]).

If pD2 then ˚.t /D jtjsgnt Dt. Therefore, equation (1.4) with pD2is the linear differential equation. As a special case we get the two-term linear equation

. 1/ⁿ

t^˛x^.n/.n/

C t^{˛ np}xD0: (1.7)

In [13, page 132] (for˛D0see [11, pages 97-98]) it is shown that equation (1.7) is nonoscillatory if and only ifCn;2;˛0, i.e., the number n;2;˛ is the critical constant for equation (1.7).

The variational technique is at disposal (see [7, Theorem 9.4.4]) also for the con- tinuous version of equation (1.1), i.e., for the differential equation

n

X

lD0

. 1/^{n l}

r_{n l}.t /˚

x^{.n l/}.n l/

D0: (1.8)

In [4], we use the variational principle together with the Wirtinger inequality, which enables us to show positivity of the energy functional associated with equation (1.8).

In the discrete case (in this article), to show positivity of the energy functional as- sociated with equation (1.2) we use inequalities obtained by using Lemma5. This approach is different from the one in the continuous case, because we do not use any discrete Wirtinger type inequality.

In the linear discrete case it is known the following nonoscillation criterion (see [8, Theorem 9]). The two-term linear difference equation

. 1/ⁿⁿ

k^.˛/nx_k

C

k^{.2n ˛/}x_kCnD0 (1.9) is nonoscillatory ifn;2;˛C > 0. In (1.9) if we takek^{.˛ 2n/}instead of1.

k^{.2n ˛/}

then the proof from [8] of this criterion still works (by using (1.3)) with the same result. Note that the constant n;2;˛is optimal (critical) for a slightly different type of Euler linear difference equation, namely for equation (4.5) with instead of

n;2;˛(see [10, Corollary 4.2]).

This paper is organized as follows. In the second section we rewrite equation (1.1) into a difference system and then we define the concept of generalized zero for the difference system and for equation (1.1) respectively. Further we define the concept of nonoscillation of equation (1.1) and we give two variational lemmas. The second section also contains two nonoscillation criteria, which plays important role in our later proofs. The end of the section is devoted to recalling basic concepts from the theory of regularly varying sequences. Section 3 presents two new nonoscillation criteria for equation (1.2) and is supplemented by remarks on a generalization via the concept of regularly varying sequence.

2. PRELIMINARIES

In order to define the concept of nonoscillation for general half-linear equation (1.1), we need to define the concept of generalized zero for this equation. Further, in order to define the concept of generalized zero for equation (1.1), we transform equation (1.1) into a Hamiltonian type difference system.

Similar observations as in the previous paragraph hold also for the continuous case, i.e. for equation (1.8) (instead of the concept of generalized zero we get the concept of zero point of multiplicity nfrom the transformation of (1.8) into a Hamiltonian type differential system; see [5]).

The following paragraphs (which lead as to the definition of generalized zero) are modeled according to the article [3]. Letfx_kgbe a solution of equation (1.1). Set

u^{Œi }_k D^{i 1}x_{kCn i}; v_k^ŒnDr_k^Œn˚ ⁿx_k

; (2.1)

v_k^{Œn j }D v^{Œn j}_k ^C1Cr_k^{Œn j }˚

^{n j}x_kCj

fork2N,iD1; 2; : : : ; nandj D1; 2; : : : ; n 1. Denote the column vectors u_k D

u^{Œi }_k n

iD1 and v_kD v^{Œi }_k n

iD1

fork2N, and note that the numberv_k^{Œi }can be expressed as v_k^{Œi }D

n i

X

lD0

. 1/^{n i ln i l}

r_k^{Œn l}˚

^{n l}xkCl

for k2N and i D1; 2; : : : ; n. Then the sequence f.uk; vk/g is a solution of the Hamiltonian type difference system

u_k DAu_kC1CB_k˚ ¹.v_k/ ; v_kDC_k˚ .u_kC1/ A^Tv_k; (2.2) wherefBkgandfCkgare square matrix sequences of ordernsuch that

B_k Ddiag n

0; 0; : : : ; 0; 1.

˚ ¹ r_k^Œn o

and C_kDdiag n

r_k^Œ0; r_k^Œ1; : : : ; r_k^{Œn 1}o I and the matrix

AD aij

n

i;jD1 with aij D

(1 forj DiC1;

0 elsewhere.

For vectoraD.ai/ⁿ_iD1, denote˚.a/WD.˚.ai//ⁿ_iD1and˚ ¹.a/WD ˚ ¹.ai/n iD1, where˚ ¹.t /WD jtj^{q 2}t is the inverse function of˚.t /. The constantqis the con- jugate number ofp, i.e.,qWDp 1^p .

Now, we consider the general matrix difference system

u_kDA_ku_kC1CB_k˚ ¹.v_k/; v_k DC_k˚.u_kC1/ A^T_kv_k; (2.3) wherefB_kgandfC_kgare symmetric matrix sequences andfI A_kgis an invertible matrix sequence (symbol I denotes the identity matrix). Let m2N, then we say that an interval.m; mC1contains the generalized zero of a solutionf.u_k; v_k/gof system (2.3) if

um¤0; umC12Im.I Am/ ¹Bm and u^T_mB_m^Ž.I Am/umC10;

whereBm^Ž denotes the Moore-Penrose pseudoinverse of matrixBmand Im denotes the image.

In order to define generalized zero for equation (1.1) we proceed as follows. Let m2N and m n. We say that a nontrivial solution fx_kg of equation (1.1) has a generalized zero in the interval .m; mC1if the solution f.u_k; v_k/g¹_kDn of cor- responding system (2.2) has a generalized zero in .m; mC1, where .u_k; v_k/WD .u_{k nC1}; v_{k nC1}/ for everyk 2Nsuch that kn; and the sequences fu_kgand fv_kgare given by relations (2.1). The shift ensures that our definition will be the same as the one in article [3] (the same shift is used for the linear case in [1, Re- mark 5 (ii)]), where it is considered equation (1.1) withx_{kC1 nCl} instead ofx_kCl (such equation is equivalent to our equation (1.1)).

Now we rewrite this procedure explicitly in terms of equation (1.1). For a real vectord D.di/ⁿ_iD1, we haved 2Im.I Am/ ¹Bm if and only if there exists the vectorcD.ci/ⁿ_iD1such that

dD.I A/ ¹Bmcor equivalently (by a direct computation)diDcn

.

˚ ¹ r_m^Œn

foriD1; 2; : : : ; n, i.e., the vectord has equal components. Letfxkgbe a nontrivial solution of (1.1). From the definition offu_kgand from the definition offu_kgwe have

u_mC1Du_{.mC1/ nC1}D

^{i 1}x_{.mC1 nC1/Cn i}n iD1D

^{i 1}x_{m .i 2/}n iD1: If

u_mC12Im.I Am/ ¹Bm; thenu_mC1has equal components, i.e., we have

xmC1; xm; ²xm 1; : : : ; ^{n 1}x_{m .n 2/}^T

D.xmC1; xmC1; xmC1; : : : ; xmC1/^T: Hence,xm .n 2/Dxm .n 3/D: : :DxmD0. Next, for an arbitraryi2 f1; 2; : : : ; ng we have

^{i 1}x_{m .i 1/}D

i

X

lD1

. 1/^{i l} i 1 l 1

!

x_{m .i l/};

which is the i-th component of u_m. Fori 2 f1; 2; : : : ; n 1g we havex_{m .i 1/}D xm .i 2/D: : :DxmD0, hence,

u_mD 0; 0; : : : ; 0; . 1/^{n 1}x_{m .n 1/}T

:

Therefore the relationu_m¤0is equivalent with the relationx_{m .n 1/}¤0. Finally, it can be shown that

B_m^Ž.I A/Ddiag n

0; 0; : : : ; 0; ˚ ¹ r_m^Œno

: Hence,

u_m^TB_m^Ž.I A/u_mC1D. 1/^{n 1}˚ ¹ r_m^Œn

x_{m .n 1/}xmC1:

Definition 1. Letm2Nandmn. We say that a nontrivial solutionfxkgof equa- tion (1.1) has ageneralized zeroin the interval.m; mC1ifx_{m .n 2/}Dx_{m .n 3/}D : : :DxmD0(forn > 1),

x_{m .n 1/}¤0 and . 1/^{n 1}r_m^Œnx_{m .n 1/}xmC10: (2.4) Note that in relation (2.4) the constant rm^Œn appears, but only the sign of rm^Œn is important. This definition agrees with Hartman’s one in [12] and it also matches the definition in [1] for the linear case.

Definition 2. We say that equation (1.1) is nonoscillatory (at infinity) if there existsN2Nsuch thatN nand no nontrivial solution of equation (1.1) has two or more generalized zeros in.N;1/. Otherwise, equation (1.1) is calledoscillatory.

Note that Definition1 and Definition2 agree with the definitions in [15] for the second order equation (equation (1.1) withnD1).

Before we formulate variational lemmas, we make another note on the linear case (equation (1.1) with pD2). System (2.3) with pD2 reduces to the general lin- ear Hamiltonian system. For linear Hamiltonian systems we have the Reid type roundabout theorem (see [1]) which guarantees equivalence between nonoscillation of equation (1.1) withpD2, positivity of the energy functional associated with equa- tion (1.1) withpD2and solvability of the so-called Riccati matrix equation associ- ated with equation (1.1) withpD2.

Similar remarks as in the previous paragraph hold also in the continuous case, i.e., for the equation

n

X

lD0

. 1/^{n l}

r_{n l}.t /x^{.n l/}.n l/

D0 (see [18]).

Nonoscillation of an equation is equivalent to positivity of its energy functional also for equations (1.1) and (1.8) ifnD1, i.e., for the second order half-linear equa- tions (the proof for (1.1) withnD1is given in [15]).

Next, we formulate the variational lemma for the second order equation

r_k^Œ1˚ .x_k/

Cr_k^Œ0˚ .x_kC1/D0; (2.5) which is a special case of equation (1.1). Denote

Dn.N /WD ffy_kg¹kD1jy_k D0forkNCn 1;

9m2Nsuch thatm > NCn 1andy_k D0forkmg forN 2N. Note thatDn.N /DD1.NCn 1/forN 2N, andDn.N2/Dn.N1/ forN1; N22Nsuch thatN1N2.

Lemma 1([15]). Equation(2.5)is nonoscillatory if and only if there existsN2N such that

F1.fy_kgIN;1/D

1

X

kDN

h

r_k^Œ1jy_kj^pCr_k^Œ0jy_kC1j^pi is positive for every nontrivial sequencefy_kg 2D1.N /.

For second order equation (2.5) we have the following two nonoscillation criteria, which will be applied to equation (1.2) withnD1.

Theorem 1(O. Doˇsl´y, P. ˇReh´ak [6]). Suppose thatP₁

r_k^Œ0is convergent,r_k^Œ1> 0 for largek,P₁

r_k^Œ11 q

D 1and

klim!1

r_k^Œ11 q

Pk 1

r_j^Œ11 q D0: (2.6)

Denote

A_kWD 0

@

k 1

X r_j^Œ11 q

1 A

p 10

@

1

X

jDk

r_j^Œ0

1 A:

If

lim inf

k!1 A_k > 1 p

p 1 p

p 1

and lim sup

k!1

A_k< 2p 1 p

p 1 p

p 1

; then equation(2.5)is nonoscillatory.

Note that ifr_k^Œ1> 0for largek,P₁ r_k^Œ1

1 q

D 1 andP₁

r_k^Œ0D 1, then equation (2.5) is oscillatory (see [15, Theorem 4] or [7, Theorem 8.2.14]).

Further note that ifr_k^Œ00for largek, then the constant _p¹_{p 1}

p

p 1

is critical.

Indeed, the condition

lim sup

k!1

A_k< 1 p

p 1 p

p 1

with assumptionsr_k^Œ1> 0for largek,r_k^Œ00for largek,

1

X r_k^Œ11 q

D 1;

1

Xr_k^Œ0is convergent and (2.6) implies that equation (2.5) is oscillatory (see [7, Theorem 8.2.15]).

Theorem 2 (O. Doˇsl´y, P. ˇReh´ak [6]). Suppose that r_k^Œ1 > 0 for large k, P₁

r_k^Œ11 q

<1and

klim!1

r_k^Œ11 q

P₁

jDk

r_j^Œ11 q D0: (2.7)

Denote

B_kWD 0

@

1

X

jDk

r_j^Œ11 q

1 A

p 10

@

k 1

Xr_j^Œ0

1 A:

If

lim inf

k!1B_k > 1 p

p 1 p

p 1

and lim sup

k!1

B_k< 2p 1 p

p 1 p

p 1

; then equation(2.5)is nonoscillatory.

In case of general (even order) equation (1.1), the variational relation presented later in Lemma2is obtained from the following theorem.

Theorem 3(O. Doˇsl´y [3]). LetN02Nbe such that N0nand letN12Nbe such thatN1N0CnC1. If the interval .N0; N1C1 contains two generalized zeros of a solution fxkg of equation (1.1), then there exists a nontrivial sequence fy_kg 2D1.N0/such thaty_kD0forkN1 nC2and

Fn.fy_kgIN0 nC1; N1 nC1/D

N₁ nC1

X

kDN₀ nC1

" _n X

lD0

r_k^{Œn l}

ˇ ˇ

ˇ^{n l}y_kCl ˇ ˇ ˇ

p# 0:

Theorem3can be reformulated as follows. If we setN DN0 nC1andX D N1 nC1, then the conditionN0nmeans thatN 2Nand the conditionN1 N0CnC1is reduced toX NCnC1. If we rewrite Theorem3in terms of such N andX then we can easily obtain the following variational lemma (in Definition2 we takeNCn 1(instead ofN), which is obviously greater than or equal ton).

Lemma 2. Equation(1.1)is nonoscillatory if there existsN 2Nsuch that Fn.fy_kgIN;1/D

1

X

kDN

" _n X

lD0

r_k^{Œn l}

ˇ ˇ

ˇ^{n l}y_kCl ˇ ˇ ˇ

p#

is positive for any nontrivial sequencefy_kg 2Dn.N /.

Next we recall the definition of regularly varying sequences and some of their selected properties (see [2,17]).

Definition 3. Let#2R. A positive sequencefa_kgis said to beregularly varying (at infinity) of index#, if

klim!1

aŒk

a_k D^#

for every positive real, whereŒt denotes the integer part oft. The set of all regularly varying sequences of index# is denoted byRV.#/. FurtherRVWDS

#2RRV.#/

andS VWDRV.0/. Sequences from the setS V are calledslowly varying.

Lemma 3. The following statements hold.

.a/ A sequencefa_kgbelongs to RV.#/ if and only if there existsfL_kg 2S V such thata_kDk^#L_k fork2N.

.b/ IffLkg 2S V andfKkgis such thatKkÏLkask! 1, thenfKkg 2S V.

.c/ A sequencefakg 2RV.#/ if and only if there existsfLkg 2S V such that a_kDk^.#/L_k fork2N.

.d/ Iffa_kg 2RV.#/, thenfb_kg 2RV.#ˇ/for everyˇ2R, whereb_kDa^ˇ_k for k2N.

.e/ Letfa_kg 2RV.#1/andfb_kg 2RV.#2/. Thenfa_kb_kg 2RV.#1C#2/.

Now, we give a few examples of the slowly varying sequences. Trivial examples are positive constant sequences. A typical example is the sequence flnkg. Further the sequencesfKkg,fLkgandfMkgare slowly varying, where

K_kD

n

Y

iD1

.lnik/ⁱ fork2N; where ln1kWDlnk;lniC1kWDln.lnik/

andi 2Rfori2NI L_kDexp

( _n Y

iD1

.lnik/ⁱ )

fork2N;wherei 2.0; 1/fori 2 f1; 2; : : : ; ngI M_kDln .k/

k fork2N:

The next statement is very important for the extension of our results (Theorem5 and Theorem6) from the next section via the theory of regularly varying sequences.

Theorem 4(Karamata type theorem [2,17]). LetfL_kg 2S V. Then

k 1

X

jD1

j^#Lj Ïk^#C1

#C1Lk as k! 1 for every real# such that# > 1; and

1

X

jDk

j^#Lj Ï k^#C1

#C1L_k as k! 1 for every real# such that# < 1.

3. NONOSCILLATION CRITERIA

Consider equation (1.2) withnD1andˇ0D, i.e., the equation

k^.˛/˚ .x_k/

Ck^{.˛ p/}˚ .x_kC1/D0: (3.1) Lemma 4. Let˛¤p 1andCp;˛> 0. Then equation(3.1)is nonoscillatory.

Proof. If 0, then nonoscillation of equation (3.1) immediately follows from Lemma1. Alternatively, we can use the Sturm comparison theorem (see [6,15]) for equation (2.5). By this theorem, from nonoscillation of the equation

k^.˛/˚ .x_k/

D0it follows nonoscillation of equation (3.1) for0.

Let 2 p;˛; 0

. We often use relation (1.3) without referring to it. We prove the cases˛ < p 1and˛ > p 1separately. In the both cases we have

k^.˛/1 q

Ï k^{˛.1 q/}ask! 1, because

klim!1

k^.˛/1 q

k^{˛.1 q/} D lim

k!1

k^.˛/

k^˛

!1 q

D1:

Furthermore we havek^{˛.1 q/}Ïk^{.˛.1 q//}fork! 1. Hence,

k^.˛/1 q

Ïk^{.˛.1 q//}

fork! 1. It holds that˛.1 q/D _{p 1}^˛ , hence,˛.1 q/ > 1for˛ < p 1and

˛.1 q/ < 1for˛ > p 1.

Let˛ < p 1. We verify the assumptions of Theorem1for equation (3.1). We have

klim!1

k^.˛/1 q

Pk 1

j^.˛/1 q D lim

k!1

k^{.˛.1 q//}

Pk 1

j^{˛.1 q/} D˛.1 q/ lim

k!1

k^{.˛.1 q/ 1/}

k^{˛.1 q/} D0:

(3.2) Indeed, from the discrete l’Hospital rule we getPk 1

j^.˛/1 q

ÏPk 1

j^{˛.1 q/}

ask! 1, because limk!1Pk 1

j^{˛.1 q/}D 1for˛.1 q/ > 1.

Further, from the limit comparison test,

1

X k^.˛/1 q

D 1 and

1

Xk^{.˛ p/} is convergent. (3.3) Now we compute limk!1A_k. We have

1

X

jDk

j^{.˛ p/}D k^{.˛ pC1/}

˛ pC1 and

k 1

X j^.˛/1 q

Ï k^{.˛.1 q/C1/}

˛.1 q/C1 ask! 1; where the latter relation is obtained using the discrete l’Hospital rule. Hence,

klim!1Ak D lim

k!1

2 4

k^{.˛.1 q/C1/}

˛.1 q/C1

!p 13

5. /k^{.˛ pC1/}

˛ pC1 D.p 1/^{p 1}

.p 1 ˛/^p: (3.4) The inequalities

.p 1/^{p 1} .p 1 ˛/^p > 1

p

p 1 p

p 1

and .p 1/^{p 1}

.p 1 ˛/^p < 2p 1 p

p 1 p

p 1

are equivalent (for˛ < p 1) with the inequalities > p;˛and < .2p 1/p;˛, respectively, and the inequality < .2p 1/p;˛ holds for an arbitrary 2. 1; 0/.

Hence, by Theorem1, equation (3.1) is nonoscillatory for 2 p;˛; 0

and˛ <

p 1.

Let ˛ > p 1. Similarly as in the previous case, we verify the assumptions of Theorem2for equation (3.1). By the limit comparison test we get

1

X k^.˛/1 q

<1; and by the l’Hospital rule we have

klim!1

k^.˛/1 q

P₁

jDk j^.˛/1 q D lim

k!1

k^{.˛.1 q//}

P₁

jDkj^{˛.1 q/} D ˛.1 q/ lim

k!1

k^{.˛.1 q/ 1/}

k^{˛.1 q/} D0:

Now we compute limk!1B_k. By the l’Hospital rule, we can verify that

1

X

jDk

j^.˛/1 q

Ï k^{.˛.1 q/C1/}

˛.1 q/C1 and

k 1

Xj^{.˛ p/}Ïk^{.˛ pC1/}

˛ pC1 ask! 1. Hence,

klim!1B_k D lim

k!1

2 4

k^{.˛.1 q/C1/}

˛.1 q/C1

!p 13

5k^{.˛ pC1/}

˛ pC1 D .p 1/^{p 1} .˛ pC1/^p: The inequalities

.p 1/^{p 1} .˛ pC1/^p > 1

p

p 1 p

p 1

and .p 1/^{p 1}

.˛ pC1/^p < 2p 1 p

p 1 p

p 1

are equivalent (for˛ > p 1) with the inequalities > p;˛and < .2p 1/p;˛, respectively. Hence, by Theorem2, equation (3.1) is nonoscillatory for2 p;˛; 0

and˛ > p 1. The proof is completed.

Note that we can prove also the following oscillation complement of Lemma 4 (see the text after Theorem1). If˛¤p 1andCp;˛< 0, then equation (3.1) is oscillatory. We will not present the proof of this result in details since we do not need it.

Remark1. The previous lemma can be generalized in the following sense. Con- sider the equation

.f_k˚ .x_k//Cg_k˚ .x_kC1/D0; (3.5) whereff_kg 2RV.˛/,fg_kg 2RV.˛ p/,˛¤p 1,2Rand the sequencesff_kg andfg_kgtake the forms

f_kDk^˛K_k; k2N and g_kDk^{˛ p}L_k; k2N (3.6) wherefK_kg;fL_kg 2S V (see Lemma3). We show that ifCp;˛> 0andK_kÏL_k as k! 1, then equation (3.5) is nonoscillatory. We use the proof of Lemma 4 with the following modifications. The paragraphs with relations (3.2), (3.3), (3.4) are replaced by the paragraphs with relations (3.7), (3.8), (3.9) below, respectively.

Let2 p;˛; 0

and˛ < p 1. It holds that

klim!1

k^{˛.1 q/}K_k^{1 q} Pk 1

j^{˛.1 q/}K_j^{1 q} D.˛.1 q/C1/ lim

k!1

k^{˛.1 q/}K_k^{1 q}

k^{˛.1 q/C1}K_k^{1 q} D0: (3.7) Indeed, we haven

K_k^{1 q}o

2S V and˛.1 q/ > 1 for˛ < p 1, hence, by The- orem4, we getPk 1

j^{˛.1 q/}K_j^{1 q}Ï.˛.1 q/C1/ ¹k^{˛.1 q/C1}K_k^{1 q} ask! 1. Further, by Theorem4,

1

Xk^{˛.1 q/}K_k^{1 q} D 1;

1

Xk^{˛ p}L_k is convergent (3.8) andP₁

jDkj^{˛ p}Lj Ï ^k_{˛ p}^{˛ p}_C^C₁¹L_k ask! 1for˛ p < 1.

Now we compute limk!1A_k. We have

klim!1A_kD lim

k!1

2 4

k^{˛.1 q/C1}K_k^{1 q}

˛.1 q/C1

!p 13

5. / k^{˛ pC1}

˛ pC1L_kD .p 1/^{p 1} .p 1 ˛/^p:

(3.9) Similarly in the case 2 p;˛; 0

and˛ > p 1.

Now we formulate the lemma, which help us to estimate the summands from the energy functional associated with equation (1.2).

Lemma 5. Letm2N,˛2Rn fp 1; 2p 1; : : : ; mp 1gand"0; "1; : : : ; "m 1be arbitrary positive real numbers. Then there existsN 2Nsuch that

1

X

kDN

h

k^{.˛ jp/}

ˇ ˇ

ˇ^{m j}y_kCj ˇ ˇ ˇ

p

C "j p;˛ jp

k^{.˛ .jC1/p/}

ˇ ˇ

ˇ^{m j 1}y_kCjC1 ˇ ˇ ˇ

pi

(3.10) is positive for every nontrivial sequencefy_kg 2Dm.N /and for everyj 2 f0; 1; : : : ; m 1g.

Proof. First, consider the equation

k^{.ˇ /}˚ .xk/

Ck^{.ˇ p/}˚ .xkC1/D0; (3.11) whereˇ2R,p2.1;1/and2R.

By Lemma 4, equation (3.11) is nonoscillatory if C_p;ˇ > 0and ˇ¤p 1.

Choose an arbitrary"2.0;1/, j 2N[ f0gand˛ 2Rn fp.jC1/ 1g. SetˇD

˛ jpand D" p;˛ jp, then equation (3.11) becomes the equation

k^{.˛ jp/}˚ .x_k/

C." p;˛ jp/k^{.˛ .jC1/p/}˚ .x_kC1/D0 (3.12) and we have

Cp;ˇ D" p;˛ jpCp;˛ jpD" > 0 and ˇD˛ jp¤p 1:

Hence, by Lemma4, equation (3.12) is nonoscillatory.

Now, letm2N,˛2Rn fp 1; 2p 1; : : : ; mp 1gand"0; "1; : : : ; "m 1 be pos- itive real numbers. Equation (3.11) withˇD˛ jpand D"j p;˛ jp, i.e., the equation

k^{.˛ jp/}˚ .x_k/

C."j p;˛ jp/k^{.˛ .jC1/p/}˚ .x_kC1/D0

is nonoscillatory for anyj 2 f0; 1; : : : ; m 1g. By Lemma1, for everyj 2 f0; 1; : : : ; m 1gthere existsNj 2Nsuch that the energy functional

1

X

kDNj

h

k^{.˛ jp/}j´_kj^pC "j p;˛ jp

k^{.˛ .jC1/p/}j´_kC1j^pi is positive for every nontrivialf´kg 2D1.Nj/.

DenoteN DmaxfN0; N1; : : : ; Nm 1g, then we have D1.N /D1.Nj/for j D 0; 1; : : : ; m 1. Note that forf´_kg 2D1.N /we have´_k D0fork2 f1; 2; : : : ; Ng. Hence,

N 1

X

kDNj

h

k^{.˛ jp/}j´_kj^pC "j p;˛ jp

k^{.˛ .jC1/p/}j´_kC1j^pi D0

for everyf´kg 2D1.N /and for everyj 2 f0; 1; : : : ; m 1gsuch thatNj N 1, i.e.,

1

X

kDN

h

k^{.˛ jp/}j´kj^pC "j p;˛ jp

k^{.˛ .jC1/p/}j´kC1j^pi

> 0 for every nontrivialf´_kg 2D1.N /and for everyj2 f0; 1; : : : ; m 1g.

Choose an arbitraryj2 f0; 1; : : : ; m 1gand an arbitrary nontrivialfykg 2Dm.N /.

Note thaty1Dy2D: : :DyNCj D: : :DyNCm 1D0. Then the sequence f´_kg, defined by the relation´_k D^{m j 1}y_kCj fork2N, is nontrivial and it belongs to the setD1.N /. Hence,

1

X

kDN

h

k^{.˛ jp/}

ˇ ˇ

ˇ^{m j}y_kCj ˇ ˇ ˇ

p

C "j p;˛ jp

k^{.˛ .jC1/p/}

ˇ ˇ

ˇ^{m j 1}y_kCjC1 ˇ ˇ ˇ

pi

> 0

for every nontrivialfy_kg 2Dm.N /and for everyj2 f0; 1; : : : ; m 1g. Remark2. Due to Remark1, Lemma5can be generalized in the following way.

Expression (3.10) is replaced by

1

X

kDN

h

k^{.˛ jp/}L^{Œm j }_k ˇ ˇ

ˇ^{m j}ykCj

ˇ ˇ ˇ

p

C "j p;˛ jp

k^{.˛ .jC1/p/}L^{Œm .j}_k ^C1/

ˇ ˇ

ˇ^{m j 1}y_kCjC1 ˇ ˇ ˇ

pi

; where n

L^{Œi }_k o

2S V for i D0; 1; : : : ; m and L^{Œm j }_k ÏL^{Œm .j}_k ^C1/ as k ! 1 for j D0; 1; : : : ; m 1.

In the proof, equation (3.11) is replaced by

k^{.ˇ /}K_k˚ .x_k/

Ck^{.ˇ p/}L_k˚ .x_kC1/D0; (3.13) whereˇ2R,p2.1;1/, 2Rand the sequencesfK_kgandfL_kgare from the set S V such thatK_k ÏL_k ask! 1. Then it is easy to rewrite the rest of the proof.

Note that equation (3.13) is equation (3.5), where the sequences (k^.˛/

k^˛ Kk

)1

nD1

and

(k^{.˛ p/}

k^{˛ p} Lk

)1

nD1

are slowly varying components offf_kgandfg_kg(see Lemma3), respectively.

Consider equation (1.2) with ˇ₁Dˇ₂D: : :Dˇ_{n 1} D0 and ˇ₀D, i.e., the two-term equation

. 1/ⁿⁿ

k^.˛/˚ ⁿxk

Ck^{.˛ np/}˚.xkCn/D0: (3.14) Theorem 5. If

Cn;p;˛> 0

and˛2Rn fp 1; 2p 1; : : : ; np 1g, then equation(3.14)is nonoscillatory.

Proof. By Lemma2it is sufficient to prove that there existsN 2Nsuch that

1

X

kDN

h

k^.˛/jⁿy_kj^pCk^{.˛ np/}jy_kCnj^pi

> 0 (3.15)

for every nontrivalfy_kg 2Dn.N /.

To prove inequality (3.15) we use inequalities obtained via Lemma5. Therefore, first we determine a set ofn 1positive real numbers. Recall that we have

p;˛D

jp 1 ˛j p

p

; _{p;˛ lp}D

j.lC1/p 1 ˛j p

p

and n;p;˛D

n 1

Y

lD0

_{p;˛ lp}

forlD0; 1; : : : ; n 1. Let"2.0;1/be such that

" <min˚

n;p;˛; Cn;p;˛ : (3.16)

Define real numbers"1; "2; : : : ; "2n 2by the recurrence relations

"2l 1D "_2lC1

2_{p;˛ lp}; "2l D "_{2lC1p;˛ lp} 2_lC1;p;˛ "_2lC1;

"2.n 1/ 1D "

2_{p;˛ .n 1/p}; "2.n 1/D"_{p;˛ .n 1/p} 2n;p;˛ "

forlD1; 2; : : : ; n 2. Condition (3.16) guarantees that the inequalities p;˛ lp> "2l > 0; l;p;˛> "2l 1> 0

hold forl D1; 2; : : : ; n 1. Indeed, we have

n;p;˛> " > 0 implies _{p;˛ .n 1/p}> "_{2.n 1/}> 0 and n 1;p;˛> "_{2.n 1/ 1}> 0I

n 1;p;˛> "2n 3> 0 implies _{p;˛ .n 2/p}> "_{2.n 2/}> 0 and n 2;p;˛> "2.n 2/ 1> 0I

:::

3;p;˛> "5> 0 implies p;˛ 2p> "4> 0 and 2;p;˛> "3> 0I

2;p;˛> "3> 0 implies p;˛ p> "2> 0 and 1;p;˛Dp;˛> "1> 0:

Now we use Lemma 5. We have ˛ 2Rn fp 1; 2p 1; : : : ; np 1g. Denote H D f"1g [ f"2iji D1; 2; : : : ; n 1g. By Lemma 5, for the elements of the set H there existsN 2Nsuch that for every nontrivialfy_kg 2Dn.N /we have

1

X

kDN

h

k^.˛/jⁿykj^pC "1 p;˛

k^{.˛ p/}ˇ

ˇ^{n 1}ykC1

ˇ ˇ

pi

> 0 (3.17) and

1

X

kDN

h

k^{.˛ jp/}

ˇ ˇ

ˇ^{n j}y_kCj ˇ ˇ ˇ

p

C "2j p;˛ jp

k^{.˛ .jC1/p/}

ˇ ˇ

ˇ^{n j 1}y_kCjC1 ˇ ˇ ˇ

pi

> 0 (3.18)

forj D1; 2; : : : ; n 1.

By a direct computation we can easily verify that forlD1; 2; : : : ; n 2we have ._l;p;˛ "_{2l 1}/._{p;˛ lp} "_2l/D_lC1;p;˛ "_{2.lC1/ 1} (3.19) and

.n 1;p;˛ "2n 3/._{p;˛ .n 1/p} "2n 2/Dn;p;˛ ": (3.20) Now we are ready to prove inequality (3.15). Among others, we use the relations

l;p;˛> "2l 1 and n;p;˛ " > (3.21)

forlD1; 2; : : : ; n 1. We have

1

X

kDN

k^.˛/jⁿykj^{p (3.17)}> p;˛ "1

1

X

kDN

k^{.˛ p/}ˇ

ˇ^{n 1}ykC1

ˇ ˇ

p

(3.18);(3.21)

> p;˛ "1

p;˛ p "2

1

X

kDN

k^{.˛ 2p/}ˇ

ˇ^{n 2}y_kC2ˇ ˇ

p

(3.19)

D 2;p;˛ "3

1

X

kDN

k^{.˛ 2p/}ˇ

ˇ^{n 2}y_kC2ˇ ˇ

p

(3.18);(3.21)

> 2;p;˛ "3

p;˛ 2p "4

1

X

kDN

k^{.˛ 3p/}ˇ

ˇ^{n 3}ykC3

ˇ ˇ

p

(3.19)

D 3;p;˛ "5

1

X

kDN

k^{.˛ 3p/}ˇ

ˇ^{n 3}y_kC3ˇ ˇ

p

:::

(3.18);(3.21)

> n 1;p;˛ "2n 3

_{p;˛ .n 1/p} "2n 2

1

X

kDN

k^{.˛ np/}jy_kCnj^p

(3.20)

D n;p;˛ "

1

X

kDN

k^{.˛ np/}jy_kCnj^p

(3.21)

>

1

X

kDN

k^{.˛ np/}jykCnj^p

for every nontrivialfykg 2Dn.N /.

Note that the constant n;p;˛is optimal in the casenD1of equation (3.14) (see the paragraph below the proof of Lemma4).

Consider equation (1.2), i.e., the full-term equation . 1/ⁿⁿ

k^.˛/˚ ⁿx_k

C. 1/^{n 1}ˇ_{n 1}^{n 1}

k^{.˛ p/}˚ ^{n 1}x_kC1 C: : : : : : ˇ1

k^{.˛ .n 1/p/}˚ .x_{kCn 1}/

Cˇ0k^{.˛ np/}˚ .x_kCn/D0:

The technique of the previous proof can be also used to obtain a criterion for equa- tion (1.2). In fact, the criterion for the two-term equation is a special case of the criterion for the full-term equation.

The following notation greatly simplifies the formulation of the next theorem. De- note

.1/WDp;˛Cˇn 1 and .kC1/WD.k/p;˛ kpCˇn 1 k

forkD1; 2; : : : ; n 1. Then

.2/Dp;˛p;˛ pCˇn 1p;˛ pCˇn 2;

.3/Dp;˛p;˛ pp;˛ 2pCˇn 1p;˛ pp;˛ 2pCˇn 2p;˛ 2pCˇn 3; :::

.n/Dn;p;˛C

n 1

X

kD1

"_{n 1} Y

lDk

_{p;˛ lp}

#

ˇ_{n k}Cˇ0: Theorem 6. If

.k/ > 0

for everyk2 f1; 2; : : : ; ngand˛2Rn fp 1; 2p 1; : : : ; np 1g, then equation(1.2) is nonoscillatory.

Proof. By Lemma2it is sufficient to prove that there existsN 2Nsuch that the energy functional

Fen.fykg; N;1/ WD

1

X

kDN

h

k^.˛/jⁿykj^pCˇn 1k^{.˛ p/}ˇ

ˇ^{n 1}ykC1

ˇ ˇ

pC: : :Cˇ0k^{.˛ np/}jykCnj^pi (3.22) associated with equation (1.2) is positive for every nontrivialfykg 2Dn.N /.

Similarly as in the proof of Theorem5we use Lemma5here. Let"2.0;1/be such that

" < .n/ and " < 2^{n l 1}.l/

n 1

Y

jDl

p;˛ jp (3.23)

for everylD1; 2; : : : ; n 1. Define real numbers"1; "2; : : : ; "2n 2by the relations

"2l 1D "

2^{n l}Qn 1

jDl p;˛ jp

and "2lD "_{p;˛ lp} 2^{n l}.l/h

Qn 1

jDl p;˛ jp

i

"

for lD1; 2; : : : ; n 1. Note that if ˇ1 Dˇ2 D: : :Dˇn 1D0, then the constants

"_{2l 1} and "_2l are the same as in the previous proof for each l 2 f1; 2; : : : ; n 1g. From conditions (3.23) we have the inequalities

p;˛ lp> "2l > 0 and .l/ > "2l 1> 0 forlD1; 2; : : : ; n 1.

Denote H D f"1g [ f"2iji D1; 2; : : : ; n 1g. By Lemma5, for the elements of the setH there existsN2Nsuch that the relations (3.17) and (3.18) hold for every nontrivialfykg 2Dn.N /and for everyj D1; 2; : : : ; n 1.

By direct computation we can easily verify that forl D1; 2; : : : ; n 2we have ..l/ "_{2l 1}/ _{p;˛ lp} "_2l

DŒ.lC1/ ˇ_{n 1 l} "_{2.lC1/ 1} (3.24) and

..n 1/ "2n 3/ p;˛ .n 1/p "2n 2

DŒ.n/ ˇ0 ": (3.25) Now we prove positivity of functional (3.22). Among others, we use the relations

.l/ > "_{2l 1} and .n/ " > 0 (3.26) forlD1; 2; : : : ; n 1. We have

1

X

kDN

h

k^.˛/jⁿy_kj^pCˇn 1k^{.˛ p/}ˇ

ˇ^{n 1}y_kC1ˇ ˇ

pi

(3.17)

>

p;˛ "1

Cˇn 1

1

X

kDN

k^{.˛ p/}ˇ

ˇ^{n 1}y_kC1ˇ ˇ

p

D..1/ "1/

1

X

kDN

k^{.˛ p/}ˇ

ˇ^{n 1}ykC1

ˇ ˇ

p

(3.18);(3.26)

> ..1/ "1/ p;˛ p "2

1

X

kDN

k^{.˛ 2p/}ˇ

ˇ^{n 2}y_kC2ˇ ˇ

p

(3.24)

D ..2/ ˇn 2 "3/

1

X

kDN

k^{.˛ 2p/}ˇ

ˇ^{n 2}y_kC2ˇ ˇ

p:

for every nontrivialfy_kg 2Dn.N /. Therefore,

1

X

kDN

h

k^.˛/jⁿy_kj^pCˇ_{n 1}k^{.˛ p/}ˇ

ˇ^{n 1}y_kC1ˇ ˇ

pCˇ_{n 2}k^{.˛ 2p/}ˇ

ˇ^{n 2}y_kC2ˇ ˇ

pi

> Œ..2/ ˇn 2 "3/Cˇn 2

1

X

kDN

k^{.˛ 2p/}ˇ

ˇ^{n 2}y_kC2ˇ ˇ

p

(3.18);(3.26)

> ..2/ "3/ p;˛ 2p "4

1

X

kDN

k^{.˛ 3p/}ˇ

ˇ^{n 3}y_kC3ˇ ˇ

p

(3.24)

D ..3/ ˇn 3 "5/

1

X

kDN

k^{.˛ 3p/}ˇ

ˇ^{n 3}ykC3

ˇ ˇ

p:

for every nontrivialfykg 2Dn.N /. Continuing similarly step by step, we obtain

1

X

kDN

h

k^.˛/jⁿykj^pC: : :Cˇ2k^{.˛ .n 2/p/}ˇ

ˇ²ykCn 2

ˇ ˇ

p