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# Periodic solutions for a class of higher-order difference equations

- Huantao Zhu
^{1}and - Weibing Wang
^{2}Email author

**2011**:66

https://doi.org/10.1186/1687-1847-2011-66

© Zhu and Wang; licensee Springer. 2011

**Received:**16 September 2011**Accepted:**23 December 2011**Published:**23 December 2011

## Abstract

In this article, we discuss the existence of periodic solutions for the higher-order difference equation

We show the existence of periodic solutions by using Schauder's fixed point theorem, and illustrate three examples.

**MSC 2010:** 39A10; 39A12.

## Keywords

- functional difference equation
- periodic solution
- fixed point theorem

## 1 Introduction and main results

where *k* ∈ N, *τ* : ℤ → ℤ and *τ* (*n* + *ω*) = *τ* (*n*), *f* (*n* + *ω*, *u*) = *f* (*n*, *u*) for any (*n*, *u*) ∈ ℤ *×* ℝ, *ω* ∈ ℕ.

where *a* ≠ 1, *b* ≠ 1 are positive constants, *τ* : ℤ → ℤ and *τ*(*n*+*ω*) = *τ*(*n*), *ω*, *m*, *k* ∈ ℕ, and obtained existence theorem for single and multiple positive periodic solutions of (1.3).

Our aim of this article is to study the existence of periodic solutions for the higher-order difference equations (1.1) using the well-known Schauder's fixed point theorem. Our results extend the known results in the literature.

The main results of this article are following sufficient conditions which guarantee the existence of a periodic solution for (1.1).

**Theorem 1.1**.

*Assume that there exist constants m*<

*M*,

*r*> 0

*such that g*∈

*C*

^{1}[

*m*,

*M*]

*with r*≤

*g*'(

*u*) ≤ 1

*for any u*∈ [

*m*,

*M*]

*and f*(

*n*,

*u*): ℤ

*×*[

*m*,

*M*] → ℝ

*is continuous in u*,

*for any* (*n*, *u*) ∈ ℤ *×* [*m*, *M*]*, then* (1.1) *has at least one ω-periodic solution x with m* ≤ *x* ≤ *M*.

**Theorem 1.2**.

*Assume that there exist constants*

*m < M*

*such that*

*g*∈

*C*

^{1}[

*m*,

*M*]

*with g*'(

*u*) ≥ 1

*for any*

*u*∈ [

*m*,

*M*]

*and*

*f*(

*n*,

*u*): ℤ × [

*m*,

*M*] → ℝ

*is continuous in u*,

*for any* (*n*, *u*) ∈ ℤ *×* [*m*, *M*]*, then* (1.1) *has at least one ω-periodic solution x with m* ≤ *x* ≤ *M*.

## 2 Some examples

In this section, we present three examples to illustrate our conclusions.

**Example 2.1**. Consider the difference equation

where *k* ∈ ℕ, 0 *< a <* 1, *b >* 1, *q* is one *ω*-periodic function with *q*(*n*) *>* 0 for all *n* ∈ [1, *ω*] and *τ* : ℤ → ℤ and *τ* (*n* + *ω*) = *τ* (*n*).

*m >*0 be sufficiently small and

*M >*0 sufficiently large. It is easy to check that

for *n* ∈ ℤ and *u* ∈ [*m*.*M*]. By Theorem 1.1 (Theorem 1.2), Equation (2.1) (or (2.2)) has at least one positive *ω*-periodic solution *x* with *m* *≤* *x* *≤* *M*. When *k* = 1, this conclusion about (2.1) and (2.2) can been obtained from the results in [15]. Our result holds for all *k* ∈ ℕ.

**Remark 1**Consider the difference equations

where *k* ∈ ℕ, 0 *< a <* 1, *b >* 1, *q* is one *ω*-periodic function with *q*(*n*) *>* 0 for all *n* ∈ [1, *ω*], *τ* : ℤ → ℤ and *τ*(*n* + *ω*) = *τ* (*n*) and *f* : (0, *+* ∞) → (0, *+* ∞) is continuous.

The following result generalizes the conclusion of Example 2.1.

**Proposition 2.1**Assume that

*f*

_{0}=

*+*∞ and

*f*

_{∞}, = 0, here

then (2.3) or (2.4) has at least one positive *ω*-periodic solution.

**Proof**Here, we only consider (2.3). From

*f*

_{0}=

*+*∞ and

*f*

_{∞}= 0, we obtain that there exist 0

*< ρ*

_{1}

*< ρ*

_{2}such that

*A*= min

*q*(

*n*) min {

*f*(

*u*):

*ρ*

_{1}

*≤ u ≤ ρ*

_{2}} and

*B*= max

*q*(

*n*) max{

*f*(

*u*):

*ρ*

_{1}

*≤ u ≤ ρ*

_{2}}. Choosing

*θ*∈ (0, 1) such that

By Theorem 1.1, Equation (2.3) has at least one positive *ω*-periodic solution *x* with *θ ρ*
_{1}
*≤ x ≤ θ*
^{-1}
*ρ*
_{2}. □

**Example 2.2**. Consider the difference equation

where *k* ∈ ℕ, *α >* 0, *q* is one *ω*-periodic function.

We claim that there is a *λ >* 0 such that (2.5) has at least two positive *ω*-periodic solutions for min *q*(*n*) *> λ*.

*g*(

*x*) = -

*x*

^{-α }. Let $0<a<\sqrt[\alpha +1]{\alpha}$ be sufficiently small and $b>\sqrt[\alpha +1]{\alpha}$ be sufficiently large, then

then (2.5) has at least one periodic solution $\left[a,\sqrt[\alpha +1]{\alpha}\right]$ and $\left[\sqrt[\alpha +1]{\alpha},b\right]$ respectively. When min *q*(*n*) is sufficiently large, the conditions (2.6) and (2.7) are satisfied.

**Example 2.3**. Consider the difference equation

where *k* ∈ ℕ, *q* is one *ω*-periodic function with *q*(*n*) *>* 0 for all *n* ∈ [1, *ω*], *τ* : ℤ → ℤ and *τ* (*n* + *ω*) = *τ* (*n*).

*m*= 1,

*M >*3 + max

*q*(

*n*) and

*g*(

*u*) =

*u*

^{3}- 2

*u*,

*f*(

*n*,

*u*) =

*q*(

*n*)

*u*

^{2}. It is easy to check that

*g*'(

*u*) ≥ 1 for

*u*∈ [

*m*,

*M*], and

By Theorem 1.2, Equation (2.8) has at least one positive *ω*-periodic solution *x* with *m* ≤ *x* ≤ *M*.

**Remark 2**Consider the difference equation

where *k* ∈ ℕ, *q* is one *ω*-periodic function with *q*(*n*) > 0 for all *n* ∈ [1, *ω*], *τ* : ' → ' and *τ*(*n* + *ω*) = *τ*(*n*) and *f* : (0, +∞) → (0, +∞) is continuous.

**Proposition 2.2**Assume that there exists

*a*> 0 such that

*g*∈

*C*

^{1}([

*a*, +∞),

*R*) with

*g*'(

*u*) ≥ 1 for

*u*>

*a*,

*f*(

*u*) ≥ (

*g*(

*a*) -

*a*)/min

*q*(

*n*) for

*u*≥

*a*. Further suppose that

Then (2.9) has at least one positive *ω*-periodic solution.

**Proof**There exist

*ρ*> 0 such that

*A*= min

*q*(

*n*) min{

*f*(

*u*):

*a*≤

*u*≤

*ρ*} and

*B*= max

*q*(

*n*) max{

*f*(

*u*):

*a*≤

*u*≤

*ρ*}. Since lim

_{ u→+∞}(

*g*(

*u*) -

*u*) = +∞ and

*g*'(

*u*) ≥ 1 for

*u*>

*a*, there is

*M*>

*ρ*such that

*g*(

*M*) -

*M*>

*B*and

Thus, (2.9) has at least one *ω*-periodic solution *x* with *a* ≤ *x* ≤ *M*. □

**3 Proof**

Let *X* be the set of all real *ω*-periodic sequences. When endowed with the maximum norm ||*x*|| = max_{
n∈[0, ω-1]}|*x*(*n*)|, *X* is a Banach space.

*k*∈ ℕ and 0 <

*c*≠ 1, and consider the equation

*γ*∈

*X*. Set (

*k*,

*ω*) is the greatest common divisor of

*k*and

*ω*,

*h*=

*ω*/(

*k*,

*ω*). We obtain that if

*x*∈

*X*satisfies (3.1), then

By summing the above equations and using periodicity of *x*, we obtain the following result.

**Lemma 3.1**.

*Assume that*0 <

*c*≠ 1,

*then*(3.1)

*has a unique periodic solution*

The following well-known Schauder's fixed point theorem is crucial in our arguments.

**Lemma 3.2**. [16]*Let X be a Banach space with D* ⊂ *X closed and convex. Assume that T* : *D* → *D is a completely continuous map, then T has a fixed point in D*.

*p*> 0 is a constant which is determined later. By Lemma 3.1, if

*x*is a periodic solution of (1.1),

*x*satisfies

*h*=

*ω*/(

*k*,

*ω*), the mapping

*H*

_{ p }is defined as

*T*

_{ p }in

*X*by

Clearly, the fixed point of *T*
_{
p
} in *X* is a periodic solution of (1.1).

**Proof of Theorem 1.1**Let

*p*=

*r*and Ω = {

*x*∈

*X*:

*m*≤

*x*(

*n*) ≤

*M*for

*n*∈ '}, then Ω is a closed and convex set. If

*r*= 1, then

*g*(

*u*) =

*u*on [

*m*,

*M*]. It is easy to check that any constant

*c*∈ [

*m*,

*M*] is a periodic solution of (1.1). Set

*r*< 1. Now we show that

*T*

_{ r }satisfies all conditions of Lemma 3.2. Noting that the function

*g*(

*u*) -

*ru*is nondecreasing in [

*m*,

*M*], we have for any

*x*∈ Ω,

*x*∈ Ω and

*n*∈ ℤ,

*x*∈ Ω and

*n*∈ ℤ,

Hence, *T*
_{
r
} (Ω) ⊆ Ω.

Since *X* is finite-dimensional and *g*(*u*), *f*(*n*, *u*) are continuous in *u*, one easily show that *T*
_{
r
} is completely continuous in Ω. Therefore, *T*
_{
r
} has a fixed point *x* ∈ Ω by Lemma 3.2, which is a *ω* = periodic solution of (1.1). The proof is complete. □

**Proof of Theorem 1.2**Since

*g*∈

*C*

^{1}[

*m*,

*M*], max{

*g*'(

*u*):

*m*≤

*u*≤

*M*} exists and max{

*g*'(

*u*):

*m*≤

*u*≤

*M*} ≥ 1. Let

*p*= max{

*g*'(

*u*):

*m*≤

*u*≤

*M*}. If

*p*= 1, then

*g*(

*u*) ≡

*u*on [

*m*,

*M*]. It is easy to check that any constant

*c*∈ [

*m*,

*M*] is a periodic solution of (1.1). Next, we assume that

*p*> 1. Set Ω = {

*x*∈

*X*:

*m*≤

*x*(

*n*) ≤

*M*for

*n*∈ ℤ}. Noting that the function

*g*(

*u*) -

*pu*is nonincreasing in [

*m*,

*M*], we have for any

*x*∈ Ω,

*x*∈ Ω and

*n*∈ ℤ,

*x*∈ Ω and

*n*∈ ℤ,

Hence, *T*
_{
p
} (Ω) ⊆ Ω. *T*
_{
p
} has a fixed point *x* ∈ Ω. The proof is complete. □

## Declarations

### Acknowledgements

The authors would like to thank the referee for the comments which help to improve the article. The study was supported by the NNSF of China (10871063) and Scientific Research Fund of Hunan Provincial Education Department (10B017).

## Authors’ Affiliations

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