Part-metric and Its Applications to Cyclic Discrete Dynamic Systems

We adapt the part metric and use it in studying positive solutions of a certain family of discrete dynamic systems. Some examples are presented, and we also compare some results in the literature.


Introduction
There has been an increasing interest in studying discrete dynamic systems recently see, e.g., 1-37 .For some recent papers on the systems of difference equations which are not derived from differential equations, see, for example, 4, 12, 14, 15 , and the related references therein.In particular, in 4 , were considered some cyclic systems of difference equations for the first time.Motivated by 4 , in 12 , the global attractivity of four k-dimensional systems of higherorder difference equations with two or three delays was investigated.The results in 12 can be easily extended to the corresponding systems with arbitrary number of delays by using the main results in 28 .In 9 , the authors used Thompson's part-metric 32 to investigate the behaviour of positive solutions to a difference equation from the William Lowell Putman Mathematical Competition 33 by applying a result on discrete dynamic systems in finite dimensional complete metric spaces.Further investigations devoted to applying various part-metricrelated inequalities and some asymptotic methods in order to study scalar difference equations related to the equation in 33 can be found, for example, in 1, 3, 5, 18-22, 34-37 see also the related references therein .
In this paper, we adapt the part-metric and apply it in studying of the behaviour of positive solutions to the following family of discrete dynamic systems Y −k q , Y −k q 1 , . . . ,Y −1 are positive initial vectors and Φ : Ê q×q → Ê q is a continuous mapping which will be specified later.
In Section 2, we present some preliminary results which will be applied in the proofs of main results, given in Section 3. Some applications of the main result are given in Sections 4 and 5.In Section 6, we show that some recent results follow from a result in 9 .

Auxiliary Results
Let Ê be the whole set of reals and let Ê 0, ∞ .Denote by Ê n the set of all positive ndimensional vectors and by Ê m×n the set of all m × n matrices with positive entries, that is, The following theorem was proved in 9, Theorem 1 .

Suppose that for the discrete dynamic system
x n 1 Tx n , n ∈ AE 0 ,

2.1
there is a k ∈ AE such that for the kth iterate of T, the next inequality holds for all x / x * .Then, x * is globally asymptotically stable with respect to metric d.Based on these properties and Theorem 2.1, Kruse and Nesemann in 9 obtained the following result.Lemma 2.2 see 9, Corollary 2 .Let T : Ê n → Ê n be a continuous mapping with a unique equilibrium x * ∈ Ê n .Suppose that for the discrete dynamic system 2.1 there is some k ∈ AE such that for the part-metric p inequality p T k x, x * < p x, x * holds for all x / x * .Then, x * is globally asymptotically stable.
Our idea is to adapt the part-metric to matrices.For any two matrices with positive entries A a ij m×n ∈ Ê m×n and B b ij m×n ∈ Ê m×n , we define the part-metric in the following natural way: Note that an m × n matrix a ij m×n is equivalent to a vector with mn elements, such as

2.5
Thus, for the above matrices A and B, we have that where From this and the above-mentioned properties for the part-metric we have the following: 1 the part-metric P is a continuous metric on Ê m×n , 2 Ê m×n , P is a complete metric space, 3 the distances induced by the part-metric P and the Euclidean norm are equivalent on Ê m×n .From this and by Lemma 2.2, we have that the next result holds.Theorem 2.3.Let T : Ê m×n → Ê m×n be a continuous mapping with the unique equilibrium C ∈ Ê m×n .Suppose that for the discrete dynamic system there is a k ∈ AE such that for metric P, the inequality P T k X, C < P X, C holds for each X / C.

Main Result
Let Y Y 1 , Y 2 , . . ., Y q be a square q × q matrix, where Y i y 1 i , y 2 i , . . ., y q i T , i 1, 2, . . ., q, and Φ is defined by where φ : Ê q → Ê is a continuous mapping.Clearly, Φ is a continuous mapping and our system becomes As an application of Theorem 2.3, we will establish a theorem regarding the global asymptotic stability of cyclic system of difference equations in 3.2 , as follows.
Proof.Define a matrix mapping T : where X i x 1i , x 2i , . . ., x qi T , i 1, 2, . . ., k q .Then, 3.2 can be converted into the first-order recursive q × k q matrix equation with M 0 initial matrix, with positive entries.

3.11
From relations 3.10 and 3.11 , we obtain that

3.12
From the following set of inequalities and since M 0 / C, it follows that there exists at least one index j ∈ {0, 1, . ..,k q − 1} such that the relation " j " is "<", which implies From the definition of the part-metric, we have that Then, we derive

3.16
Because M 0 is arbitrary and M 0 / C, then by Theorem 2.3 see also Remark 2.4 we have that C is a globally asymptotically stable equilibrium of 3.5 , which implies that the equilibrium C of system 3.2 is globally asymptotically stable, as desired.

On Some Symmetric Discrete Dynamic Systems
For the sake of convenience, first we define two continuous mappings f, g : Ê q → Ê , q ≥ 2, as follows: where r is a real parameter belonging to the interval 0, 1 .
Many researchers have studied the symmetric difference equation . ., k q ∈ AE, and φ ∈ {f, g}.
In the following, we mainly investigate the behaviour of positive solutions to the following class of cyclic difference equation systems , and φ ∈ {f, g}.
In order to establish the main result concerning 4.3 , we need some preliminary lemmas.Proof.Let c 1 , c 2 , . . ., c q T be an arbitrary positive equilibrium of system 4.3 .Since the mappings f and g are both symmetric, then we derive that c i φ c 1 , c 2 , . . ., c q , i 1, 2, . . ., q, 4.4 from which it follows that c i c > 0, i 1, 2, . . ., q, and then c φ c, c, . . ., c .

4.5
By Lemma 2.1 in 13 , we obtain c 1, as desired.
Lemma 4.2.Let a 1 and a 2 be positive real numbers with a 1 , a 2 / 1, 1 , and φ ∈ {f, g}.Then, Proof.From the next identities it is easy to see that when a 1 , a 2 / 1, 1 the following inequalities hold: Because r ∈ 0, 1 , then for the case φ f, we easily obtain that min a 1 , a 2 , 4.9 The case φ g follows immediately from the case φ f due to the fact that fg ≡ 1.
Lemma 4.3.Let q ≥ 2 be an integer and φ ∈ {f, g}.Let a 1 , a 2 , . . ., a q be positive real numbers with a 1 , a 2 , . . ., a q / 1, 1, . . ., 1 .Then, Proof.For the case q 2, the assertion follows from Lemma 4.2.Next, we argue by the induction and assume that the assertion is true for q k k ≥ 2 .Then, it suffices to prove that the assertion holds when q k 1.Now, let a 1 , a 2 , . . ., a k 1 be positive real numbers with a 1 , a 2 , . . ., a k 1 / 1, 1, . . ., 1 .Consider the following function h in a variable x where a 1 , a 2 , . . ., a k are arbitrary but fixed positive numbers.Clearly, The first derivative of the function h regarding the variable x is equal to

4.13
In the following, we distinguish three possibilities.

Abstract and Applied Analysis 9
Case 1 k j 1 a 2r j − 1 < 0 .Then, h x; a 1 , a 2 , . . ., a k > 0 holds for all x > 0, which implies that the function h x; a 1 , a 2 , . . ., a k is strictly increasing in variable x.From this, we obtain

4.14
Case 2 k j 1 a 2r j − 1 > 0 .Then, h x; a 1 , a 2 , . . ., a k < 0 holds for all x > 0, implying that the function h x; a 1 , a 2 , . . ., a k is strictly decreasing in variable x.From this, we obtain that

4.15
Case 3 k j 1 a 2r j − 1 0 .This implies h x; a 1 , a 2 , . . ., a k 1 for all x > 0. From this relation and the inspection that the condition a 1 , a 2 , . . ., a k 1 / 1, 1, . . ., 1 implies max 1≤i≤k 1 {a i , 1/a i } > 1 and min 1≤i≤k 1 {a i , 1/a i } < 1, we obtain that min 1≤i≤k 1 Hence, by induction, the assertion immediately holds for φ f, and then the case φ g follows directly from the case φ f because f • g ≡ 1 By Lemmas 4.1 and 4.3 and Theorem 3.1, we obtain the following theorem.
Theorem 4.4.The unique equilibrium of system 4.3 is globally asymptotically stable.Remark 4.5.Note that the following two systems particular cases q 2 and q 3 of the system 4.3 , resp.

On a System of Difference Equations
Let μ : Ê q → Ê , q ≥ 4 be a continuous mapping defined by μ t 1 , t 2 , . . ., t q q−2 i 1 t i t q−1 t q t 1 t 2 q i 3 t i .

5.1
Then, the following difference equation is an extension of the difference equation in 33 , which was studied in 9 .First, we consider the next four-dimensional system of difference equations: Applying 5.5 , the system in 5.4 is reduced to

5.6
If a 1, then it follows from the second identity that b 1.Now, assume a ∈ Ê \ {1}.By solving the first equation in 5.6 with respect to variable b, we get that the discriminant which implies the first equation in 5.6 has no real roots.This contradicts b > 0. Hence, a b 1, which along with 5.5 implies c d 1, finishing the proof.
The following lemma follows directly from Lemma 3.3 in 34 or the proof of Lemma 4 in 35 .
In the following, we consider the next q-dimensional q ≥ 5 generalization of system 5.3 It is easy to see that 1, 1, . . ., 1 T is a positive equilibrium of system 5.9 , but it is not so easy to confirm its uniqueness as in the proof of Lemma 5.1.However, we have the following lemma which follows directly from Lemmas 3.4 and 3.5 in 34 .

An Application of a Kruse-Nesseman Result
Numerous papers studied particular cases of 4.2 by using semi-cycle analysis of their solutions.It was shown by Berg and Stević in 1 that this analysis is unnecessarily complicated and useful only for lower-order difference equations.They also described some methods for determining rules of semi-cycles which can be used in many classes of difference equations.On the other hand, it has been noticed in several papers see, e.g., 18 that the stability results in many of these papers follow from the following result by Kruse and Nesemann in 9 .

6.1
where F : Ê m → Ê is a continuous function with a unique equilibrium x * ∈ Ê .Suppose that there is a k 0 ∈ AE such that for each solution y n of 6.1 , with equality if and only if y n x * .Then, x * is globally asymptotically stable.
Motivated by 18 , in recent paper 2 , Berg and Stević also applied Theorem 6.1 by proving the next result, which covers numerous particular cases appearing in the literature.We formulate the proposition here as a useful information to the reader.Before we formulate it we need some notation.Let S j {1, 2, . . ., j}, j 1, . . ., k, let or reversed, T k r is the sum over the even, and T k r is the sum over the odd r.Another proof of the previous result, in the case χ 0, can be also find in recent paper 28 by Stević.
Recently Sun and Xi in 31 gave an interesting proof of the following result.At first sight their result looked new and not so closely related to Theorem 6.1.However, we prove here that it is also a consequence of Theorem 6.1.
and satisfy the following two conditions: Then, x 1 is the unique positive equilibrium of the difference equation which is globally asymptotically stable (here u * max{u, 1/u}). Proof.Let We should determine the sign of the product of the next expressions

6.10
From 6.9 and 6.10 , we see if we show that x n−r 1 −1 f n − 1 and x n−r 1 −1 /f n − 1 have the same sign for n ∈ AE, then P n Q n will be nonpositive.
There are four cases to be considered.x n−r 1 −1 .
Assume that P n Q n 0, then, P n 0 or Q n 0. Using 6.9 or 6.10 along with 6.12 in any of these two cases, we have that f n 1 x n−r 1 −1 x n−r 1 −1 , n ∈ AE.where y * j y * , . . ., y * denotes the vector consisting of j copies of y * .Then according to the considerations in Cases 1-4 it follows that f y * k y * 1/y * , so that y * 1. Hence y * 1 is a unique positive equilibrium of 6.7 .
From all above mentioned and by Theorem 6.1, we get the result.

Theorem 2 . 1 .
Let M, d be a complete metric space, where d denotes a metric and M is an open subset of Ê n , and let T : M → M be a continuous mapping with the unique equilibrium x * ∈ M.

3 . 6 then
we get a matrix sequence M n ∞ n 0 .Apparently, the matrix sequence M n ∞ n 0 is a solution to 3.5 .When M 0 C, it is clear that M n C holds for n ∈ AE 0 .Hence, in what follows, we assume that M 0 / C.

Lemma 5 . 1 .
System 5.3 has unique positive equilibrium 1, 1, 1, 1 T .Proof.Let a, b, c, d T be an arbitrary positive equilibrium of the system 5.3 .Then, we get

Finally, let y * be a solution of
Theorem 6.2.Suppose χ is a nonnegative continuous function on Ê k , k ∈ AE, and1 ≤ i 1 < i 2 < • • • < i k .If a sequence y i satisfies the difference equation with y −i k , . . ., y −1 ∈ Ê , then it converges to the unique positive equilibrium 1.