Open Access

Periodic solutions for a class of higher-order difference equations

Advances in Difference Equations20112011:66

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

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

x ( n + k ) = g ( x ( n ) ) f ( n , x ( n τ ( n ) ) .

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 equationperiodic solutionfixed point theorem

1 Introduction and main results

Let denote the set of the real numbers, the integers and the positive integers. In this article, we investigate the existence of periodic solutions of the following high-order functional difference equation
x ( n + k ) = g ( x ( n ) ) - f ( n , x ( n - τ ( n ) ) , n ,
(1.1)

where k N, τ : and τ (n + ω) = τ (n), f (n + ω, u) = f (n, u) for any (n, u) × , ω .

Difference equations have attracted the interest of many researchers in the last 20 years since they provided a natural description of several discrete models, in which the periodic solution problem is always a important topic, and the reader can consult [17] and the references therein. There are many good results about existence of periodic solutions for first-order functional difference equations [812]. Only a few article have been published on the same problem for higher-order functional difference equations. Recently, using coincidence degree theory, Liu [13] studied the second-order nonlinear functional difference equation
Δ 2 x ( n - 1 ) = f ( n , x ( n - τ 1 ( n ) ) , x ( n - τ 2 ( n ) ) , , x ( n - τ m ( n ) ) ) ,
(1.2)
and obtain sufficient conditions for the existence of at least one periodic solution of equation (1.2). By using fixed point theorem in a cone, Wang and Chen [14] discussed the following higher-order functional difference equation
x ( n + m + k ) - a x ( n + m ) - b x ( n + k ) + a b x ( n ) = f ( n , x ( n - τ ( n ) ) ) ,
(1.3)

where a ≠ 1, b ≠ 1 are positive constants, τ : and τ(n+ω) = τ(n), ω, m, k , 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 rg'(u) ≤ 1 for any u [m, M] and f (n, u): × [m, M] → is continuous in u,
g ( M ) - M f ( n , u ) g ( m ) - m
(1.4)

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

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): × [m, M] → is continuous in u,
g ( m ) - m f ( n , u ) g ( M ) - M
(1.5)

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

2 Some examples

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

Example 2.1. Consider the difference equation
x ( n + k ) = a x ( n ) + q ( n ) x ( n - τ ( n ) ) 3 ,
(2.1)
x ( n + k ) = b x ( n ) - q ( n ) x ( n - τ ( n ) ) 3 ,
(2.2)

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

Let m > 0 be sufficiently small and M > 0 sufficiently large. It is easy to check that
( a - 1 ) M - q ( n ) u 3 ( a - 1 ) m , ( b - 1 ) m q ( n ) u 3 ( b - 1 ) M

for n 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 .

Remark 1 Consider the difference equations
x ( n + k ) = a x ( n ) + q ( n ) f ( x ( n - τ ( n ) ) ) ,
(2.3)
x ( n + k ) = b x ( n ) - q ( n ) f ( x ( n - τ ( n ) ) ) ,
(2.4)

where k , 0 < a < 1, b > 1, 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.

The following result generalizes the conclusion of Example 2.1.

Proposition 2.1 Assume that f 0 = + ∞ and f , = 0, here
f 0 = lim u 0 + f ( u ) u , f = lim u f ( u ) u ,

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
f ( u ) 1 - a min q ( n ) u , 0 < u ρ 1 , f ( u ) 1 - a max q ( n ) u , u ρ 2 .
Let 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
A 1 - a θ ρ 1 , B 1 - a θ - 1 ρ 2 ,
we obtain that
f ( u ) 1 - a min q ( n ) u θ ( 1 - a ) ρ 1 min q ( n ) , θ ρ 1 u ρ 1 , f ( u ) θ - 1 ( 1 - a ) ρ 2 max q ( n ) , ρ 2 u θ - 1 ρ 2 , A q ( n ) f ( u ) B , n , ρ 1 u ρ 2 .
Using the above three inequalities, we have
( 1 - a ) θ ρ 1 q ( n ) f ( u ) ( 1 - a ) θ - 1 ρ 2 , n , θ ρ 1 u θ - 1 ρ 2 .

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
x ( n + k ) = - 1 x α ( n ) + q ( n ) ,
(2.5)

where k , 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.

In fact, g(x) = - x -α . Let 0 < a < α α + 1 be sufficiently small and b > α α + 1 be sufficiently large, then
α b α + 1 g ( x ) = α x α + 1 1 , for x [ α α + 1 , b ] , g ( x ) = α x α + 1 1 , for x [ a , α α + 1 ] .
If the following conditions are fulfilled
- 1 b α - b - q ( n ) - 1 α α α + 1 - α α + 1 , n ,
(2.6)
- 1 a α - a - q ( n ) - 1 α α α + 1 - α α + 1 , n ,
(2.7)

then (2.5) has at least one periodic solution [ a , α α + 1 ] and [ α α + 1 , b ] respectively. When min q(n) is sufficiently large, the conditions (2.6) and (2.7) are satisfied.

Example 2.3. Consider the difference equation
x ( n + k ) = x 3 ( n ) - 2 x ( n ) - q ( n ) x 2 ( n - τ ( n ) ) ,
(2.8)

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

Let m = 1, M > 3 + max q(n) and g(u) = u 3 - 2u, f (n, u) = q(n)u 2. It is easy to check that g'(u) ≥ 1 for u [m, M], and
g ( m ) - m = - 2 < f ( n , u ) g ( M ) - M = M 3 - 3 M , n , u [ m , M ] .

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

Remark 2 Consider the difference equation
x ( n + k ) = g ( x ( n ) ) - q ( n ) f ( x ( n - τ ( n ) ) ) ,
(2.9)

where k , 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 ua. Further suppose that
lim u + g ( u ) - u f ( u ) > max q ( n ) , lim u + ( g ( u ) - u ) = + .

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

Proof There exist ρ > 0 such that
g ( u ) - u f ( u ) max q ( n ) , u ρ .
Let A = min q(n) min{f(u): auρ} and B = max q(n) max{f(u): auρ}. Since lim u→+∞(g(u) - u) = +∞ and g'(u) ≥ 1 for u > a, there is M > ρ such that g(M) - M > B and
f ( u ) max q ( n ) g ( u ) - u g ( M ) - M , ρ u M .

Thus, (2.9) has at least one ω-periodic solution x with axM. □

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.

Let k and 0 < c ≠ 1, and consider the equation
x ( n + k ) = c x ( n ) + γ ( n ) ,
(3.1)
where 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
c - 1 x ( n + k ) - x ( n ) = c - 1 γ ( n ) , c - 2 x ( n + 2 k ) - c - 1 x ( n + k ) = c - 2 γ ( n + k ) , c - p x ( n + h k ) - c 1 - p x ( n + ( h - 1 ) k ) = c - p γ ( n + ( h - 1 ) k ) .

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
x ( n ) = ( c - h - 1 ) - 1 i = 1 h c - i γ ( n + ( i - 1 ) k ) .

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 : DD is a completely continuous map, then T has a fixed point in D.

Now, we rewrite (1.1) as
x ( n + k ) = p x ( n ) + [ g ( x ( n ) ) - f ( n , x ( n - τ ( n ) ) - p x ( n ) ] ,
(3.2)
where p > 0 is a constant which is determined later. By Lemma 3.1, if x is a periodic solution of (1.1), x satisfies
x ( n ) = ( p - h - 1 ) - 1 i = 1 h p - i ( H p x ) ( n + ( i - 1 ) k ) ,
where h = ω/(k, ω), the mapping H p is defined as
( H p x ) ( n ) = g ( x ( n ) ) - p x ( n ) - f ( n , x ( n - τ ( n ) ) , x X .
Define a mapping T p in X by
( T p x ) ( n ) = ( p - h - 1 ) - 1 i = 1 h p - i ( H p x ) ( n + ( i - 1 ) k ) , x X .

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 : mx(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,
g ( m ) - r m g ( x ( n ) ) - r x ( n ) g ( M ) - r M , n .
Let (1.4) be fulfilled. For any x Ω and n ,
( H r x ) ( n ) = g ( x ( n ) ) p x ( n ) f ( n , x ( n τ ( n ) g ( M ) r M ( g ( M ) M ) = ( 1 r ) M , ( H r x ) ( n ) = g ( x ( n ) ) p x ( n ) f ( n , x ( n τ ( n ) g ( m ) r m ( g ( m ) m ) = ( 1 r ) m .
Hence, for any x Ω and n ,
( T r x ) ( n ) = ( r - h - 1 ) - 1 i = 1 h r - i ( H p x ) ( n + ( i - 1 ) k ) ( r - h - 1 ) - 1 i = 1 h r - i ( 1 - r ) M = M , ( T r x ) ( n ) = ( r - h - 1 ) - 1 i = 1 h r - i ( H p x ) ( n + ( i - 1 ) k ) ( r - h - 1 ) - 1 i = 1 h r - i ( 1 - r ) m = m .

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): muM} exists and max{g'(u): muM} ≥ 1. Let p = max{g'(u): muM}. 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 : mx(n) ≤ M for n }. Noting that the function g(u) - pu is nonincreasing in [m, M], we have for any x Ω,
g ( M ) - p M g ( x ( n ) ) - p x ( n ) g ( m ) - p m , n .
For any x Ω and n ,
( H p x ) ( n ) = g ( x ( n ) ) p x ( n ) f ( n , x ( n τ ( n ) g ( m ) p m ( g ( m ) m ) = ( 1 p ) m , ( H p x ) ( n ) = g ( x ( n ) ) p x ( n ) f ( n , x ( n τ ( n ) g ( M ) p M ( g ( M ) M ) = ( 1 p ) M .
Hence, for any x Ω and n ,
( T p x ) ( n ) = ( p - h - 1 ) - 1 i = 1 h p - i ( H p x ) ( n + ( i - 1 ) k ) ( p - h - 1 ) - 1 i = 1 h p - i ( 1 - p ) m = m , ( T p x ) ( n ) = ( p - h - 1 ) - 1 i = 1 h p - i ( H p x ) ( n + ( i - 1 ) k ) ( p - h - 1 ) - 1 i = 1 h p - i ( 1 - p ) M = M .

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

(1)
Hunan College of Information
(2)
Department of Mathematics, Hunan University of Science and Technology

References

  1. Agarwal RP: Difference Equations and Inequalities. 2nd edition. Marcel Dekker, New York; 2000.Google Scholar
  2. Antonyuk PN, Stanyukovic KP: Periodic solutions of the logistic difference equation. Rep Acad Sci USSR 1990, 313: 1033-1036.Google Scholar
  3. Berg L: Inclusion theorems for non-linear difference equations with applications. J Differ Equ Appl 2004, 10: 399-408. 10.1080/10236190310001625280View ArticleGoogle Scholar
  4. Cheng S, Zhang G: Positive periodic solutions of a discrete population model. Funct Differ Equ 2000, 7: 223-230.MathSciNetGoogle Scholar
  5. Zheng B: Multiple periodic solutions to nonlinear discrete Hamiltonian systems. Adv Differ Equ 2007. doi: 10.1155/2007/41830Google Scholar
  6. Zhu B, Yu J: Multiple positive solutions for resonant difference equations. Math Comput Model 2009, 49: 1928-1936. 10.1016/j.mcm.2008.09.009MathSciNetView ArticleGoogle Scholar
  7. Zhang X, Wang D: Multiple periodic solutions for difference equations with double resonance at infinity. Adv Differ Equ 2011. doi:10.1155/2011/806458Google Scholar
  8. Chen S: A note on the existence of three positive periodic solutions of functional difference equation. Georg Math J 2011, 18: 39-52.Google Scholar
  9. Gil' MI, Kang S, Zhang G: Positive periodic solutions of abstract difference equations. Appl Math E-Notes 2004, 4: 54-58.MathSciNetGoogle Scholar
  10. Jiang D, Regan DO, Agarwal RP: Optimal existence theory for single and multiple positive periodic solutions to functional difference equations. Appl Math Comput 2005, 161: 441-462. 10.1016/j.amc.2003.12.097MathSciNetView ArticleGoogle Scholar
  11. Padhi S, Pati S, Srivastava S: Multiple positive periodic solutions for nonlinear first order functional difference equations. Int J Dyn Syst Differ Equ 2009, 2: 98-114.MathSciNetGoogle Scholar
  12. Raffoul YN, Tisdell CC: Positive periodic solutions of functional discrete systems and population model. Adv Differ Equ 2005, 2005: 369-380.MathSciNetView ArticleGoogle Scholar
  13. Liu Y: Periodic solutions of second order nonlinear functional difference equations. Archivum Math 2007, 43: 67-74.Google Scholar
  14. Wang W, Chen X: Positive periodic solutions for higher order functional difference equations. Appl Math Lett 2010, 23: 1468-1472. 10.1016/j.aml.2010.08.013MathSciNetView ArticleGoogle Scholar
  15. Raffoul YN: Positive periodic solutions of nonlinear functional difference equations. Electron J Differ Equ 2002, 2002: 1-8.MathSciNetView ArticleGoogle Scholar
  16. Guo D, Lakshmikantham V: Nonlinear Problem in Abstract Cones. Academic Press, New York; 1988.Google Scholar

Copyright

© Zhu and Wang; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.