Trichotomy of nonoscillatory solutions to second order neutral difference equation with quasi difference

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R E S E A R C H

Open Access

Trichotomy of nonoscillatory solutions to

second-order neutral difference equation

with quasi-difference

Agata Bezubik

1

_{, Małgorzata Migda}

2

_{, Magdalena Nockowska-Rosiak}

3

_{and Ewa Schmeidel}

1*

*_{Correspondence:}

[email protected] 1_{Institute of Mathematics, University} of Bialystok, Ciołkowskiego 1M, Białystok, 15-245, Poland Full list of author information is available at the end of the article

Abstract

In this paper the nonlinear second-order neutral diﬀerence equation of the following form:

(a

n

(x

n–pnxn–1)) +qnf(xn–τ) = 0 is considered. By suitable substitution the

above equation is transformed into a new one, which is a third-order non-neutral difference equation. Using results obtained for the new equation, the asymptotic properties of the neutral difference equation are studied. Some classification of nonoscillatory solutions is presented, as well as an estimation of the solutions. Finally, we present necessary and sufficient conditions for the existence of solutions to both considered equations being asymptotically equivalent to the given sequences.

MSC: 39A10; 39A22

Keywords: diﬀerence equation; second order; neutral type; nonoscillatory solutions; estimation of solutions

1 Introduction

In this paper we consider the diﬀerence equation in the following form:

an(xn–pnxn–)

+qnf(xn–τ) = , n∈Nmax{,τ}, ()

whereis the forward diﬀerence operator deﬁned byyn=yn+–yn, (an), (pn), (qn) are sequences of positive real numbers,τis a nonnegative integer, and the functionf:N→R. HereRis the set of real numbersN={, , . . .}, andNk={k,k+ ,k+ , . . .},k∈N.

By a solution to () we mean a sequence (xn) which satisﬁes () fornsuﬃciently large. We consider only solutions which are nontrivial for all largen. A solution to () is called nonoscillatory if it is eventually positive or eventually negative. Otherwise it is called os-cillatory.

Let us denote

yn+=xn n

i=



pi

. ()

This implies thatxn–pnxn–= (yn)

n

i=pi. Substitution of () transforms () into the following:

an

(yn) n

i=

pi

+qnf

yn–τ+

n–τ

i=

pi

= . ()

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Setting

bn= n

i=

pi ()

and assuming that

f(bn–τz) =b∗ng(z), ()

in (), we get the third-order nonlinear diﬀerence equation of the following form:

an(bnyn)

+q∗_ng(yn+–τ) = , n∈Nmax{,τ}, ()

where

q∗_n=qnb∗n. ()

By virtue of (), the positivity of terms of the sequence (pn) implies the positivity of terms of the sequence (bn). Note thatf(xy) =f(x)f(y) is satisﬁed for all power functions. Hence, by () and (), iff(x) =xγ_{, where}_γ _{is a positive constant, then}_g₌_f _and_b∗

n=b γ n–τ

for all n∈Nτ. If f is not a power function, in some cases we can ﬁnd the function g

assumed by (). For example, for f(x) =xx– andbn≡b∈R we have b∗n= b  _and

g(x) = (b₎x_x_.

Neutral type difference equations have been widely studied in the literature. Some recent results on the asymptotic behavior of second-order neutral difference equations can be found, for example, in [–]. The higher-order neutral difference equations were studied in [–].

For results concerning the oscillatory and asymptotic behavior of the third-order dif-ference equation we refer to [, ], for equations with quasi-differences to [–], and to the references cited therein. Many results on the oscillation of second- and third-order functional differential and difference equations can also be found in [].

The purpose of this paper is to study the asymptotic properties of the neutral difference equation (). Transforming the considered equation into a new one, which is a third-order difference equation of type (), we get various results concerning the asymptotic behavior of solutions to this equation. These results are then used to establish some properties of the solutions to (). In particular, we obtain necessary and sufficient conditions for the existence of solutions asymptotically equivalent to the given sequences.

Fourth-order non-neutral difference equations with one quasi-difference, by the tech-niques here used, were studied in [–]. Some generalizations of the results presented in these papers were published in [, ]. Even so, there is not a full analogy to the results since the Kneser type classification of the nonoscillatory solution is different for odd- or even-order equations, and of neutral or non-neutral type as well.

Throughout the rest of our investigations, one or several of the following assumptions will be imposed:

(H)

∞

i=



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(H) i=

pi=O(n);

(H) zf(z) >  for allz= ;

(H) f:R→Ris a continuous function.

Notice that, by virtue of (), the positivity of the sequence (bn) implies that conditions (H) and (H) hold also for the functiong.

The following deﬁnitions and theorems will be used in the sequel.

We say that the sequence (un) is asymptotically constant if this sequence has a nonzero limit, and we say that it is an asymptotically zero sequence if the limit of this sequence equals zero. We say that the sequence (un) is asymptotically equivalent to (vn) if (un_vn) has a nonzero limit. In the present paper, we study the three types of solutions: asymptotically zero solutions, asymptotically constant solutions, and unbounded solutions. It is called a trichotomy of nonoscillatory solutions.

Deﬁnition (Uniformly Cauchy subset []) A subsetSof the Banach spaceBis said to be uniformly Cauchy if for everyε>  there exists a positive integerNsuch that|xi–xj|<ε wheneveri,j>Nfor any (xn)∈B.

Lemma (Arzela-Ascoli’s theorem []) Each bounded and uniformly Cauchy subset of B is relatively compact.

Theorem (Schauder theorem []) Let S be a nonempty,closed,and convex subset of

a Banach space B and T: S→S be a continuous mapping such that T(S)is a relatively compact subset of B.Then T has at least one ﬁxed point in S.

The following theorem of Stolz-Cesáro is a discrete analog of l’Hospital’s rule.

Theorem (Stolz-Cesáro theorem []) Let(un), (vn)be two sequences of real numbers.

Assume that(vn)is a strictly monotone and divergent sequence,and the following limit

exists:limn→∞unvn =g.Then

lim n→∞

un

vn =g.

We introduce the following notation:

Qn= n–

k=



bk k–

j=



aj =

n–

j=



aj n–

k=j+



bk

. ()

2 Existence of nonoscillatory solutions

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Lemma  Condition(H)implies that

∞

i=



bi

=∞, ()

where(bn)is deﬁned by().

Proof Condition (H) implies thatn_i_=pi≤Cn, whereCis a positive constant. It follows

thatn_i_=p–

i ≥Cn. Using the notation of (), the above inequality takes the form



bn ≥



Cn. Since the series

∞

n=n diverges, condition () is satisﬁed.

Remark  Condition (H) and () imply that

lim

n→∞Qn=∞, ()

where (Qn) is deﬁned by ().

Lemma  Assume that(H), (H),and the following conditions:

(H∗) zg(z) >  for all z= ;

(H∗) g: R→Ris a continuous function;

are satisﬁed.Let(yn)be an eventually positive solution to().Then exactly one of the

fol-lowing statements holds:

(i) yn> , yn> , (bnyn) > ,

(ii) yn> , yn< , (bnyn) > 

for all suﬃciently large n.

Proof The proof is obvious and hence omitted.

Lemma  Assume that(H)-(H)hold.If(xn)is an eventually positive solution to(),then

exactly one of the following cases holds: (I)

lim n→∞

xn

bn = ;

(II) there exist positive constantsC,C,and a positive integernsuch that

Cbn≤xn≤CbnQn+ forn≥n, ()

where(bn)is deﬁned by()and(Qn)is deﬁned by().

Proof Let (yn) be an eventually positive solution to (). Then, by Lemma , we have two possibilities:

lim n→∞yn= ,

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Iflimn→∞yn= , then condition (I) is satisﬁed.

Assuming thatyn+≥Cand using substitution (), we obtain

xn

is satisﬁed for the functiong, from the point of view of (), there existsnsuch that

an(bnyn)

Summing again, we have

yn<A

By (), it is easy to see that each term on the right side of inequality () is less than

max{A,A,A}

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whereC= max{A,A,A}. Hence

yn+≤C

n

j=n



bj j–

i=n



ai

for suﬃciently largen.

Using the substitutions () and (), we obtain

xn

bn≤

C

n

j=n



bj j–

i=n



ai .

By (), we see that the required inequality is proved.

As a consequence of Lemma  we obtain the following result.

Lemma  Assume that(H), (H), (H∗),and(H∗)hold.If(yn)is an eventually positive

solution to(),then

(I)

lim n→∞yn= ;

(II) there exist positive constantsCandCsuch that

C≤yn≤CQn for largen.

Before we derive a necessary and suﬃcient condition for the existence of a solution to () that is asymptotically equivalent to (bn), the following theorem needs to be proved.

Theorem  Let conditions(H), (H), (H∗), (H∗)be satisﬁed.Then a necessary condi-tion for()to have an asymptotically constant solution is that

∞

i=



bi

∞

j=i 

aj

∞

k=j

q∗_k<∞. ()

Proof Let (yn) be an asymptotically constant solution to (). Then (yn) is a nonoscillatory sequence. Without loss of generality, we assume that (yn) is an eventually positive solution. By Lemma  it is of type (i) or type (ii). Each solution to type (i) tends to inﬁnity. This implies that (yn) is of type (ii).

Let us denote

lim

n→∞yn=α> . ()

Then there exist positive constantsCandCsuch that

C≤yn+–τ≤C for largen.

By (H∗) and (H∗), we see that there exists a positive constant

C= min

z∈[C,C]

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which means that, foryn+–τ∈[C,C], we have

bk and summing both sides of it from

i=n–  ton–  we obtain

By (), the following inequality holds:

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This means that

∞

i=

q∗_i

i

j=



aj j

k=



bk <∞.

The above condition is equivalent to condition ().

The next example shows that the condition () in Theorem  is not a necessary condi-tion for () to have an asymptotically zero solucondi-tion.

Example  Let us consider the following equation of the form ():

yn+  yn= .

Herean≡,bn≡,qn∗≡ ,g(x) =x, andτ = . It is easy to see that condition () is not

satisﬁed, but the above equation has an asymptotically zero solutionyn=_n.

Sufficient conditions, under which, for every real constant, there exists a solution to the higher-order difference equation with quasi-differences convergent to this constant are obtained in Theorem . in []. Hence, for (), we have the following.

Theorem  Assume that(H), (H), (H∗), (H∗)hold and condition()is satisﬁed.Then for every c∈Rthere exists a solution x to()such thatlimn→∞x(n) =c.

Corollary  Let conditions(H), (H), (H∗), (H∗)be satisﬁed.Then the condition

∞

i=



bi

∞

j=i 

aj

∞

k=j

q∗_k=∞ ()

implies that()has no asymptotically constant solution.

Proof This corollary follows directly from Theorem .

Theorem  If conditions(H)-(H)are satisﬁed,then a necessary and suﬃcient condition for()to have a solution(xn)asymptotically equivalent to the sequence (

n

i=pi)is the

condition

∞

i=

i

l=



pl

∞

j=i 

aj

∞

k=j

qkb∗k<∞. ()

Proof Using the notation of (), (), and () in condition () the conclusion of this

theo-rem follows directly from Theotheo-rem  and Theotheo-rem .

Remark  Let the assumptions of Theorem  be satisﬁed. If

lim n→∞

n

i=

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then condition () is a necessary and suﬃcient condition for () to have an asymptotically zero solution such that (xn)∼(

n i=pi).

As a consequence of Theorem  we get the following result for the Emden-Fowler type equation

an(xn–pnxn–)

+qnxγn–τ = , n∈Nmax{,τ}, ()

where (an), (pn), (qn) are sequences of positive real numbers,τ is a nonnegative integer, andγ is the ratio of odd positive integers.

Corollary  Let conditions(H)and(H)be satisﬁed.A necessary and suﬃcient condition for()to have a solution(xn)asymptotically equivalent to the sequence(

n

Example  Consider the following equation:

of Corollary  are satisﬁed. Hence () has at least one solution asymptotically equivalent to the sequence (n_i_=pi) =

√

n+ . In factxn=

√

n+  +  is such solution.

Example  Consider the following equation:

(xn–xn–) +

sumptions of Corollary  are satisﬁed. Hence, () has at least one solution asymptotically equivalent to the sequence (n_i_=pi)≡. This means that () has an asymptotically con-stant solution. In factxn=  +_n+ is one such solution.

Finally, we present a necessary and suﬃcient condition for the existence of an asymp-totically (Qn) solution to (). We start with the following theorem.

Theorem  If conditions(H), (H), (H∗), (H∗)are satisﬁed and

g is a monotonic function, ()

then a necessary and suﬃcient condition for()to have a solution(yn)satisfying

lim n→∞

yn

Qn

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is that

∞

i=

q∗_ig(CQi+–τ)<∞, ()

where C is some nonzero constant.

Proof Necessity. Let (yn) be a nonoscillatory solution to () which satisﬁes (). Without loss of generality, we may assume that

lim n→∞

yn

Qn

=β> .

Then there exist positive constantsCandCsuch that

CQn≤yn≤CQn for largen.

Hence

CQn+–τ≤yn+–τ ≤CQn+–τ for largen, sayn≥n.

Thus, by (), we get

g(yn+–τ)≥g(CQn+–τ), ()

whereC=Cif the functiongis nondecreasing andC=C if the functiongis

nonin-creasing.

By (H∗), we see thatg(CQn+–τ) is positive. On the other hand, summing () from

n=n+τton– , by Lemma , we obtain

 <an(bnyn) =an(bnyn) –

n–

i=n

q∗_ig(yi+–τ) forn≥n.

This implies that

n–

i=n

q∗_ig(yi+–τ)≤an(bnyn) <∞.

So, by (), we have

∞

i=n

q∗_ig(CQn+–τ) <∞.

Suﬃciency. LetCbe a given positive constant. Set

In=

C

 Qn,CQn

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From the above, () and (H∗), there exists a maximum of the functiongon intervalIn, which we denote as the pointCQnwithC=C_ if the functiongis nonincreasing and

C=Cif the functiongis nondecreasing. Thus we get

g(yn)≤g(CQn) forn∈In. ()

Assume that () holds forC=C. Then there exists a positive integernsuch that

∞

Consider the Banach spaceBof all real sequencesy= (yn) such that

y=sup

It is easy to see thatSis a bounded, convex and closed subset ofB. Now we deﬁne an operatorT:S→Bin the following way:

ThusTmapsSinto itself.

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and therefore

We see thatTis a continuous mapping.

Finally, we need to show thatT(S) is uniformly Cauchy. To see this, we have to show that, given anyε> , there exists an integernsuch that form>n>n; we have

Furthermore, by Stolz’s theorem (see Theorem ) and (), we have

lim

This completes the proof.

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Now, let_an ≡ and _bn ≡. This means thatan≡ andbn≡. HenceQn=_n–_n+  is a quadratic polynomial. Obviously, by virtue of (), this case holds only ifpn≡.

Theorem  Let conditions(H)-(H)be satisﬁed and

f is a monotonic function.

Then a necessary and suﬃcient condition for()to have a solution(xn)which is

asymptot-ically equivalent to the sequence(Qn+ni=pi)is the convergence of the series

∞

i=

qi

Cf

Qi+–τ

i

j=

pj

, ()

where C is some nonzero constant.

Proof Using the notation of (), (), and () in condition () the conclusion of this

theo-rem follows directly from Theotheo-rem .

Note that for particular cases of (), if (_an) is a polynomial sequence andpn≡, from Theorem  we get the existence of asymptotically polynomial solutions.

Example  In Example  () is considered. In this equationan≡ andpn≡. All as-sumptions of Theorem  are satisﬁed. Hence () has an asymptotically (Qn) solution, whereQn=_n–_n+ . It means that () has an asymptotically polynomial solution.

Some results concerning asymptotically polynomial solutions to diﬀerence equations can be found, for example, in [–].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the ﬁnal manuscript.

Author details

1_{Institute of Mathematics, University of Bialystok, Ciołkowskiego 1M, Białystok, 15-245, Poland.}2_{Institute of Mathematics,} Poznan University of Technology, Piotrowo 3A, Pozna ´n, 60-965, Poland.3_{Institute of Mathematics, Lodz University of} Technology, Wólczanska 215, Łód´z, 90-924, Poland.

Acknowledgements

This work was partially supported by the Ministry of Science and Higher Education of Poland (PB-43-081/14DS).

Received: 13 February 2015 Accepted: 8 June 2015 References

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