Existence of uncountably many bounded positive solutions to higher order nonlinear neutral delay difference equations

R E S E A R C H

Open Access

Existence of uncountably many bounded

positive solutions to higher-order nonlinear

neutral delay difference equations

Magdalena Nockowska-Rosiak

*_{Correspondence:}

[email protected] Institute of Mathematics, Lodz University of Technology, ul. Wólcza ´nska 215, Łód´z, 90-924, Poland

Abstract

This paper studies higher-order nonlinear neutral delay difference equations of the form

Δ(

(Δ(

(

(Δ(

(Δ

)

))

)

))

)

=f(n,xn–τ1,. . .,xn–τs).

Using Krasnoselskii’s fixed point theorem, we obtain the existence of uncountably many bounded positive solutions to the considered problem.

Keywords: nonlinear difference equation; neutral type; Krasnoselskii’s fixed point theorem

1 Introduction and preliminaries

In mathematical models in diverse areas such as economy, biology, computer science, dif-ference equations appear in a natural way; see, for example, [, ]. In the past thirty years, oscillation, nonoscillation, the asymptotic behavior and existence of bounded solutions to many types of difference equation have been widely examined. For the second order, see, for example, [–], and for higher orders, [–], and references therein.

Liuet al.[] discussed the existence of uncountably many bounded positive solutions to

ΔrnΔ(xn+bnxn–τ–cn)

+fn,xf(n), . . . ,xfk(n)=dn

with respect (bn). Using techniques of the measures of noncompactness, Galewskiet al.

[] considered

Δrn

Δ(xn+pnxn–τ)

γ

+qnxαn+anf(xn+) = .

Migda and Schmeidel [] studied the following equation:

Δr_nm–Δrm–_n · · ·Δr_nΔ(xn+pnxn–τ)

· · ·=anf(xn–σ) +bn.

They established sufficient conditions under which for every real constant, there exists a solution to the studied problem convergent to this constant.

In this paper, we study higher-order nonlinear neutral delay difference equations of the form

Δr_nm–Δr_nm–· · ·Δr_nΔ(xn+pnxn–τ)

γγ_{· · ·}γm–γm–

=f(n,xn–τ, . . . ,xn–τs) ()

under the following general settings:

(H) m≥,γ, . . . ,γm–≤are ratios of odd positive integers,τ∈N,τ, . . . ,τs∈Z,(pn)⊂

R,ri= (r_ni)⊂R\ {},i= , . . . ,m– , andf :N×Rs_→R.

Additional conditions will be added to obtain the existence of uncountably many positive (nonoscillatory) solutions to equation (). Krasnoselskii’s fixed point theorem will be used to prove our results. To illustrate them, three examples are included.

Throughout this paper, we assume thatΔis the forward difference operator. By a so-lution to equation () we mean a sequencex:N→Rthat satisfies () for everyn≥kfor somek≥max{τ,τ, . . . ,τs}.

We consider the Banach spacel∞ of all real bounded sequencesx:N→Requipped with the standard supremum norm, that is, forx= (xn)∈l∞,

x=sup n∈N

|xn|.

Definition ([]) A subsetAofl∞is said to be uniformly Cauchy if for everyε> , there existsn∈Nsuch that|xi–xj|<εfor anyi,j≥nandx= (xn)∈A.

Theorem ([]) A bounded,uniformly Cauchy subset of l∞is relatively compact.

We shall use Krasnoselskii’s fixed point theorem in the following form.

Theorem  ([], .B, p.) Let X be a Banach space,B be a bounded closed convex subset of X,and S,G:B→X be mappings such that Sx+Gy∈B for any x,y∈B.If S is a contraction and G is a compact,then the equation

Sx+Gx=x

has a solution in B.

2 Main results

For any nonnegative sequencey= (yn) andn∈N, we use the notation

W(n,y) =

∞

l=n yl;

W(n,y) =

∞

l=n

_rm–

l

∞

l=l yl

γ_m–_–

=

∞

l=n

W(l,y) |rm–_l |

γ_m–_–

;

Wk(n,y) =

∞

lk=n

Wk–(lk,y) |rm–k+

lk |

γ_m–_–k+

, k= , . . . ,m.

By [,M]s_{we denote the set [,M]× · · · ×[,M]⊂R}s_.

Now we are in position to formulate and prove the main theorem.

Theorem  Suppose that(H)is satisfied.Assume further that

(H) supn∈N|pn|=p< /;

(H) there existsM> such that for anyn∈N,the functionf(n,·)is a Lipschitz function on[, M]s_{with Lipschitz constantP(n,M)satisfying}

∞

lm= _r

lm γ

–  ∞

lm–=lm _r

lm–

γ

– 

· · ·

∞

l=l

_rm–

l

γ

–

m–∞

l=l

P(l,M) <∞;

(H) Wm(,|f(·, Rs)|) <∞.

Then, equation () possesses uncountably many bounded positive solutions lying in

[M/, M].

Proof LetM>  be a constant fulfilling assumption (H). It is easy to see that (H) implies that

∞

l=

P(l,M) <∞ ()

and

∞

lk= _rm–k+

lk γ

–

m–k+

· · ·

∞

l=l

_rm–

l

γ

–

m–∞

l=l

P(l,M) <∞, k= , . . . ,m– . ()

From (H) it is clear that

∞

n=

f(n, Rs)<∞,W_k,f(·, _Rs)<∞, k= , . . . ,m. ()

Now we claim that (H) and () imply that

Wk

,P(·,M)<∞, k= , . . . ,m. ()

Indeed, from () we get that there existsnsuch that for anyn≥n, we have ∞

l=nP(l, M) < ; hence, sinceγm–≤, we get that for anyn≥n,

_∞

l=n P(l,M)

γ_m–_–

≤

∞

l=n

Thus, for anyn≥n, we have

In an analogous way, we prove the remaining conditions in (). We now claim that

∞ are left to the reader. Indeed, using (), we have

Once the claim proved, observe that we may findn≥max{τ,τ, . . . ,τs}such that

Wm

n, M√sP(·,M) +f(·, Rs)< M

 –p

. ()

We consider a subset ofl∞of the form

An=

x= (xn)∈l∞:xn= M/,n<n∧ |xn– M/| ≤M/,n≥n

.

Observe thatAnis a nonempty, bounded, convex, and closed subset ofl∞.

Let us denote

ux_(n) =

∞

l=n

f(l,xl–τ, . . . ,xl–τs), x= (xn),xn∈[, M],n≥max{τ, . . . ,τs}.

The following takes care of showing thatux_ is well defined and bounded above. By (H), for any x = (x, . . . ,xs)∈[, M]s_{and for anyn∈N, we have}

f(n, x)≤P(n,M)x_Rs+f(n, _Rs)≤M√sP(n,M) +f(n, _Rs), ()

where · Rs denotes the Euclidean norm inRs. Thus, for anyx= (xn)∈A_n

 andn≥

max{τ, . . . ,τs}, ux_(n)≤W

n, M√sP(·,M) +f(·, Rs). ()

Denote, for anyx= (xn)∈Anandn≥max{τ, . . . ,τs},

ux_(n) =

∞

l=n

ux_(l)

rm–_l

γ_m–_–

.

Thus, for anyx= (xn)∈Anandn≥max{τ, . . . ,τs},

ux_(n)≤W

n, M√sP(·,M) +f(·, Rs). ()

In an analogous way, for anyx= (xn)∈Anandn≥max{τ, . . . ,τs}, we denote

ux_k(n) =

∞

l=n

ux k–(l)

rm–k+_l

γ_m–_–k+

, k= , . . . ,m.

Thus, for anyk= , . . . ,m,x= (xn)∈An, andn≥max{τ, . . . ,τs}, ux_k(n)≤Wk

n, M√sP(·,M) +f(·, Rs). ()

Define two mappingsT,T:An→l∞as follows:

(Tx)n= ⎧ ⎨ ⎩

 for ≤n<n, –pnxn–τ forn≥n;

(Tx)n=

Our next goal is to check the assumptions of Theorem  (Krasnoselskii’s fixed point the-orem). Firstly, we show that Tx+Ty∈An forx,y∈An. Letx,y∈An. For n<n,

so thatTis a contraction.

To prove the continuity ofT, notice that from () we get

ux_m–(n)≤Wm–

k _{is Lipschitz continuous, say, with constant}

SinceT(An) is uniformly Cauchy and bounded, by Theorem ,T(An) is relatively

com-pact inl∞, which means thatTis a compact operator.

From Theorem  we get that there exists a fixed pointx= (xn) ofT+TonAn. Hence,

xn+pnxn–τ= (–)muxm(n) + M/

forn≥n. Applying the operatorΔto both sides of the last equation, raising to the power γ(recalling that it is the ratio of odd positive integers), and multiplying byrn, we get

r_nΔ(xn+pnxn–τ)

γ

= (–)m–ux_m–(n)

forn≥n. Repeating this procedurem–  times, we get thatx= (xn) is a solution to equation () forn≥nwithxn∈[M, M].

Now we prove the existence of uncountably many solutions to () lying in [M/, M]. LetM,Mbe such thatM/ <M<M<M. It is easy to see that the assumptions of the theorem are fulfilled forM,M. So there existn,n≥max{τ,τ, . . . ,τs}andx= (xn) and

x_{= (x}

n), each a fixed point of the operatorTi+Ti inAni, respectively, where

T_ix_n= ⎧ ⎨ ⎩

 for ≤n<ni,

–pnxn–τ forn≥ni;

T_ix_n= ⎧ ⎨ ⎩

Mi/ for ≤n<ni,

Mi/ + (–)mux_m(n) forn≥ni.

Thus, xi _{are solutions to () for n≥ max{n}

,n}. By () there exists n ∈N, n ≥ max{n,n}, such that

ux_m(n)+ux_m(n)≤/(M–M) forn≥n.

From this we get that, forn≥n,

x_n–x_n+pn

x_n–τ–xn–τ≥/(M–M) –ux



m(n)+ux



m(n)> ,

which means thatx_andx_{are different solutions to () lying in [M/, M].}

Remark  It is obvious that condition (H) in Theorem  can be replaced by the condition

(H_) Wm(,|f(·, x)|) <∞for somex∈[, M]s.

3 Examples

Now, we present examples of equations for which our method can be applied.

Example  Let us consider the second-order nonlinear neutral delay difference equation

Δ√nΔ(xn+pnxn–τ)

= x

 n–τ

whereτ∈N,τ∈Z,γ= , and (pn) is any sequence of real numbers such thatsupn∈N|pn|<

so that assumption (H) of Theorem  is satisfied. To see that assumption (H) of The-orem  is fulfilled, notice thatf(n, ) = ,n∈N. Hence, there exist uncountably many solutions to () in any interval [M/, M] for anyM> . On the other hand, Theorem . in [] is inapplicable because

∞

Example  Consider the third-order nonlinear neutral delay difference equation

Δ

sufficiently largenand

∞ fied. Hence, there exist uncountably many solutions to () in any interval [M/, M] for anyM> .

Example  Let us consider a nonlinear neutral delay difference equation of the form

wherem∈N,τ∈N,τ∈Z, andγ, . . . ,γm–are the ratios of odd positive integers. More-over, (pn)n∈Nis any sequence of real numbers such thatsupn∈N|pn|< /,rn=· · ·=rnm–= , andf(n,u) =_un for anyn∈Nandu∈R.

In an analogous way to Example , we have to check only assumption (H) of Theorem . We have thatf(n,·)∈C_{(R) for anyn∈N, and it is easy to calculate thatP(n,M) =}M

n and

∞

lm=

∞

lm–=lm · · ·

∞

l=l ∞

l=l

M l = M



 

m <∞.

Hence, there exist uncountably many solutions to () in any interval [M/, M] for any

M> .

Competing interests

The author declares that she has no competing interests.

Author’s contributions

The author solely contributed in this article.

Acknowledgements

The author wishes to express her thanks to the referee for insightful remarks improving quality of the paper. The research was supported by Lodz University of Technology.

Received: 5 May 2016 Accepted: 20 July 2016 References

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