- Research
- Open access
- Published:
Existence of periodic solutions for a higher-order neutral difference equation
Advances in Difference Equations volume 2018, Article number: 449 (2018)
Abstract
Based on a continuation theorem of Mawhin, the existence of a periodic solution for a higher-order nonlinear neutral difference equation is studied. Our conclusion is new and interesting.
1 Introduction
The periodic solution theory of differential equation and difference equation has important academic value and application background. It has aroused people’s great concern, and many good results have been achieved. For example, see articles [1,2,3,4,5,6,7,8] and the references therein. However, as far as we know, the results of the periodic solutions of neutral difference equations are relatively few (see [7, 8]).
In this paper, we study the periodic solutions of a higher-order nonlinear neutral difference equation of the form
where k is a positive integer, c is a real number different from −1 and 1, σ and \(\tau _{i}\) are integers for \(i\in \{ 1,2,\ldots,l \} \), \(g_{n}\in C(R^{l},R)\) for \(n\in Z\) and \(g_{n}=g_{n + \omega }\), where ω is a positive integer which satisfies \(\omega \geq 2\). We use a continuity theorem to give some criteria for the existence of a periodic solution of (1), and our conclusion is new and interesting.
A solution of (1) is a real sequence of the form \(x= \{ x_{n} \} _{n\in Z}\) which renders (1) into an identity after substitution. As usual, a solution of (1) of the form \(x= \{ x_{n} \} _{n\in Z}\) is said to be ω-periodic if \(x_{n+\omega }=\) \(x_{n}\) for \(n\in Z\).
We also state Mawhin’s continuation theorem (see [1]). Let X and Y be two Banach spaces, and \(L : \operatorname{Dom} L\subset X \rightarrow Y\) is a linear mapping and \(N:X \rightarrow Y\) is a continuous mapping. The mapping L is called a Fredholm mapping of index zero if \(\operatorname{dim} \operatorname{Ker} L = \operatorname{codim} \operatorname{Im} L <+\infty \), and ImL is closed in Y. If L is a Fredholm mapping of index zero, then there exist continuous projectors \(P : X\rightarrow X\) and \(Q:Y\rightarrow Y\) such that \(\operatorname{Im} P = \operatorname{Ker} L\) and \(\operatorname{Im} L = \operatorname{Ker} Q =\operatorname{Im} (I-Q)\). It follows that \(L_{|\operatorname{Dom} L \cap \operatorname{Ker} P} : (I-P) X\rightarrow \operatorname{Im} L\) has an inverse which is denoted by \(K_{p}\). If Ω is an open and bounded subset of X, then the mapping N is called L-compact on Ω̅ when \(QN(\overline{\varOmega })\) is bounded and \(K_{P} (I-Q) N:\overline{\varOmega } \rightarrow X\) is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism \(J:\operatorname{Im}Q\rightarrow \operatorname{Ker} L\).
Theorem A
(Mawhin’s continuation theorem [1])
Let L be a Fredholm mapping of index zero, and let N be L-compact on Ω̅. Suppose
-
(I)
for each \(\lambda \in (0,1)\), \(x\in \partial \varOmega \), \(Lx\neq \lambda Nx \); and
-
(II)
for each \(x \in \partial \varOmega \cap \operatorname{Ker} L\), \(QNx\neq 0\) and \(\deg (JQN,\varOmega \cap \operatorname{Ker} L,0)\neq 0\).
Then the equation \(Lx=Nx\) has at least one solution in \(\overline{\varOmega }\cap \operatorname{Dom} L\).
2 Main result
The main result of this paper is as follows:
Theorem 2.1
Let \(|c|\neq 1\). Assume that there exist constants \(D>0\), \(\alpha \geq 0\), and \(\beta \geq 0\) such that
-
(I)
\(|g_{n}(x_{1},x_{2},\ldots,x_{l})|\leq \beta \max_{1\leq i\leq l}|x_{i}|+\alpha \) for \(n\in Z\) and \((x_{1},x_{2},\ldots,x_{l})^{T}\in R^{l}\),
-
(II)
\(g_{n}(x_{1},x_{2},\ldots,x_{l})>0\) for \(n\in Z\) and \(x_{1},x_{2},\ldots,x_{l}\geq D\),
-
(III)
\(g_{n}(x_{1},x_{2},\ldots,x_{l})<0\) for \(n\in Z\) and \(x_{1},x_{2},\ldots,x_{l}\leq -D\).
Then the higher-order neutral difference equation (1) has an ω-periodic solution when \(\omega ^{k}\beta <2^{k}|1-|c||\).
Remark 2.1
When \(g_{n}\) in (1) is replaced by \(-g_{n}\),the result of Theorem 2.1 still holds.
Next, some preparations are presented to prove our theorems. Let \(X_{\omega } \) be the Banach space of all real ω-periodic sequences of the form \(x=\{x_{n}\}_{n\in Z}\) and endowed with the usual linear structure as well as the norm \(\vert x \vert _{\infty }=\max_{1\leq i\leq \omega } \vert x_{i} \vert \).
Define the mappings \(L:X_{\omega }\rightarrow X_{\omega }\) and \(N:X_{\omega }\rightarrow X_{\omega }\) respectively by
and
It is easy to see that L is a linear mapping. Similar to the paper [8], in case \(\vert c \vert \neq 1\), direct calculation shows that \(\operatorname{Ker} L= \{ x\in X_{\omega } \vert x_{n}=x_{0},n\in Z \} \). Since \(\operatorname{dim} X_{\omega }=\omega \) and \(L:X_{\omega }\rightarrow X_{\omega }\) is a linear mapping, by the knowledge of linear algebra, we know that \(\operatorname{dim}\operatorname{Ker} L\bigoplus \dim \operatorname{Im} L=\dim X_{\omega }\). It is easy to see that \(\operatorname{dim}\operatorname{Ker} L = \operatorname{codim} \operatorname{Im} L=1\) and \(\operatorname{dim} \operatorname{Im} L=\omega -1\). It follows that ImL is closed in \(X_{\omega }\). Thus L is a Fredholm mapping of index zero. Now, we assert that
To do that, we just have to prove that
Indeed, if \(y=\{y_{n}\}_{n\in Z}\in \operatorname{Im} L\), then there is \(x=\{x_{n}\}_{n\in Z}\in X_{\omega }\) such that
Thus
Note that \(x=\{x_{n}\}_{n\in Z} \in X_{\omega }\). It follows that \(\{\Delta ^{k-1}x_{n}\}_{n\in Z} \in X_{\omega }\). Furthermore, direct calculation shows that
We see that for any \(u= \{ u_{n} \} _{n\in Z}\in \operatorname{Ker}L\cap \operatorname{Im}L\), then \(u= \{ u_{n} \} _{n\in Z}\in \operatorname{Ker}L\) and \(u= \{ u_{n} \} _{n\in Z}\in \operatorname{Im}L\). Because of \(u= \{ u_{n} \} _{n\in Z}\in \operatorname{Ker}L\) and \(\operatorname{Ker} L= \{ x\in X_{\omega } \vert x_{n}=x_{0},n\in Z \} \), thus for any \(n\in Z\), we have
On the other hand, since \(u= \{ u_{n} \} _{n\in Z}\in \operatorname{Im}L\), by (9), we have
By (10) and (11), we see that, for any \(n\in Z\), \(u_{n}=0\). This implies that (5) is true, that (4) is true. Now, for any \(u= \{ u_{n} \} _{n\in Z}\in X_{\omega }\), if
where \(x=\{x_{n}\}_{n\in Z}\in \operatorname{Ker} L\) and \(y=\{y_{n}\}_{n\in Z}\in \operatorname{Im} L\), then
and
As in paper [8], we define \(P=Q:X_{\omega }\rightarrow X_{\omega }\) by
The operators P and Q are projections. We have \(\operatorname{Im} P=\operatorname{Ker} L\), \(\operatorname{Ker} Q=\operatorname{Im} L\), and \(X_{\omega }=\operatorname{Ker} P\bigoplus \operatorname{Ker} L=\operatorname{Im} L\bigoplus \operatorname{Im} Q\). It follows that \(L_{|\operatorname{Dom}L \cap \operatorname{Ker} P}: (I-P) X_{\omega }\rightarrow \operatorname{Im} L\) has an inverse which is denoted by \(K_{p}\). By (3) and (13), we see that, for any \(x=\{x_{n}\}_{n\in Z}\in X_{\omega }\),
and
Since the Banach space \(X_{\omega }\) is finite dimensional, \(K_{p }\) is linear. By relations (14) and (15), we see that QN and \(K_{p} ( I-Q ) N\) are continuous on \(X_{\omega }\) and take bounded sets into bounded sets respectively. Thus, we know that if Ω is an open and bounded subset of \(X_{\omega }\), then the mapping N is called L-compact on Ω̅.
Lemma 2.1
(see [7])
Let \(\{ u_{n } \} _{n\in Z}\) be a real ω-periodic sequence, then we have
where the constant factor \(1/2\) is the best possible.
Lemma 2.2
(see [7])
Let \(\{ u_{n } \} _{n\in Z}\) be a real ω-periodic sequence, then
Lemma 2.3
(see [7])
If \(|c|\neq 1\) and \(\{ u_{n } \} _{n\in Z}\) is a real ω-periodic sequence, then
Proof
Note that
It follows that
If \(k=1\), by Lemma 2.1 and (19), then
If \(k\geqslant 2\), by Lemma 2.2, then
In view of (19) and (20), we have
The proof is completed. □
Proof of Theorem 2.1
Consider the system
where \(\lambda \in ( 0,1 ) \) is a parameter. Let \(u\in X_{\omega }\) be a solution of (21). By (2), (3), and (21),
Let \(u_{\xi }= \max_{1\leq n\leq \omega } u_{n }\) and \(u_{\eta }=\min_{1\leq n\leq \omega } u_{n}\). By Lemma 2.3 and (21),
If there exists a constant \(m\in \{ 1,2,\ldots,\omega \} \) such that \(\vert u_{m } \vert < D\), by (23), for any \(n\in Z\), then
Otherwise by (22),
In view of conditions (II), (III) and (25), we know \(u_{\xi }\geq D\) and \(u_{\eta }\leq -D\). By (23),
and
By (26) and (27), for any \(n\in Z\),
From (24) and (28), for any \(n\in Z\), we have
It follows that
By condition (I),
where \(C=D+\frac{\omega ^{k}\alpha }{2^{k}|1-|c||}\) and \(\rho =\frac{\omega ^{k}\beta }{2^{k}|1-|c||}\).
Set
where D̅ is a fixed number which satisfies \(\overline{D}>D+\frac{C}{1-\rho }\). We have that Ω is an open and bounded subset of \(X_{\omega }\). By (31), for each \(\lambda \in ( 0,1 ) \), \(u\in \partial \varOmega \), \(Lu\neq \lambda Nu\). If \(u\in \partial \varOmega \cap \operatorname{Ker} L\), then \(u= \{ \overline{D} \} _{n\in Z}\) or \(u= \{ -\overline{D} \} _{n\in Z}\). By (13),
In particular, we see that if \(u= \{ \overline{D} \} _{n\in Z}\), then
and if \(u= \{ -\overline{D} \} _{n\in Z}\), then
This indicates
By Theorem A, we see that the equation \(Lx=Nx\) has at least one solution in \(\overline{\varOmega }\cap \operatorname{Dom} L\). In other words, (1) has an ω-periodic solution. The proof is completed. □
3 Example
Example 3.1
The difference equation
is one of the form (1), where \(k=4\), \(c=\frac{1}{3}\), \(\sigma =2\), \(l=3\), \(\tau _{1}=2\), \(\tau _{2}=1\), \(\tau _{3}=0\), and
We can prove that (32) has a 3-periodic nontrivial solution. Indeed, let \(D=1\), \(\beta =\frac{10}{81}\), and \(\alpha =1\). Then the conditions of Theorem 2.1 are satisfied. Therefore (32) has a 3-periodic solution. Furthermore, the solution is nontrivial since \(g_{n}(0,0,0)\) is not identically zero.
References
Gaines, R.E., Mawhin, J.L.: Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math., vol. 568 (1977)
Gopalsamy, K., He, X., Wen, L.: On a periodic neutral logistic equation. Glasg. Math. J. 33, 281–286 (1991)
Wang, G.Q., Yan, J.R.: Existence of periodic solutions for nth order neutral differential equations with multiple variable lags. Acta Math. Sci. 26(2A), 306–313 (2006) (In Chinese)
Elaydi, S.: Periodic solutions of difference equations. J. Differ. Equ. Appl. 6(2), 203–232 (2000)
Wang, G.Q., Chen, S.S.: Periodic solutions of higher order nonlinear difference equations via a continuation theorem. Georgian Math. J. 12(3), 539–550 (2005)
Wang, G.Q., Chen, S.S.: Positive periodic solutions for nonlinear difference equations via a continuation theorem. Adv. Differ. Equ. 4, 311–320 (2004)
Wang, G.Q., Chen, S.S.: Periodic solutions of a neutral difference system. Bol. Soc. Parana. Mat. 22(2), 117–126 (2004)
Wang, G.Q.: Steady State Solutions of Neural Networks and Periodic Solutions of Difference Equations. Jinan University Press, Guangzhou (2012) (in Chinese)
Acknowledgements
The authors would like to thank the referees for invaluable comments and insightful suggestions.
Funding
This work was supported by GDSFC project (No. 9151008002000012).
Author information
Authors and Affiliations
Contributions
JLZ and GQW worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhang, JL., Wang, GQ. Existence of periodic solutions for a higher-order neutral difference equation. Adv Differ Equ 2018, 449 (2018). https://doi.org/10.1186/s13662-018-1890-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-018-1890-6