Abstract
This paper addresses the max-type difference equation , , where , , and is a positive sequence with a finite limit. We prove that every positive solution to the equation converges to under some conditions. Explicit positive solutions to two particular cases are also presented.
1. Introduction
The study of difference equations, which usually depicts the evolution of certain phenomena over the course of time, has a long history. Many experts recently pay some attention to so-called max-type difference equations which stem from certain models in control theory, see, for example, [1–23] and the references therein.
The study of the following family of max-type difference equations where such that and are real sequences, was proposed by S. Stevic at numerous conferences, for example, [10, 11]. For some results in this direction, see [1, 2, 4, 12–23].
In the beginning of the investigation the following equation was studied: where are real sequences and the initial values are nonzero (see, e.g., [3, 5, 6, 9] and the related references therein).
In [22], Sun studied the second-order difference equation with , and proved that each positive solution to (1.3) converges to the equilibrium point , by considering several subcases. However, the method used there is a bit complicated and difficult for extending. Hence in [14] Stevic extended this, as well as the main result in [13], by presenting a more concise and elegant proof of the next theorem.
Theorem 1.1 (see [14, Theorem ?1]). Every positive solution to the difference equation where are natural numbers such that and , converges to .
Definition 1.2. Let be a function of variables, then the difference equation is called nonautonomous or time variant.
Note that the following nonautonomous difference equation where , and are real sequences (not all constant), is a natural generalization of (1.2), (1.3), and (1.4). It is a special case of (1.1) of particular interest.
The aforementioned works are mainly devoted to the study of (1.6) with constant or periodic numerators.
This paper is devoted to the study of the following nonautonomous max-type difference equation with two delays: where are fixed and is a positive sequence with a finite limit. Inspired by the methods and proofs of the above-mentioned papers, here we try to find some sufficient conditions such that every positive solution to (1.7) converges to .
This paper proceeds as follows. Several useful lemmas are given in Section 2. In Section 3 we establish three main results about the global attractivity of (1.7) under some conditions. Finally motivated by a recent theorem in [21], explicit solutions to two particular cases of (1.7) are presented in Section 4.
2. Auxiliary Results
To establish the main results in Section 3, here we present several lemmas. First we extend Lemma??2.4 in [21] by proving the following result.
Lemma 2.1. Consider the nonautonomous difference equation where and are sequences. If s are nonnegative sequences and there always exists such that for each fixed , then
Proof. Suppose that is fixed, and denote by the set of all indices for which the terms in (2.1) are negative.
If , which means all terms in the right-hand side of (2.1) are nonnegative, then apparently
which implies
Otherwise, , which means that there exist indices such that the corresponding terms in (2.1) are negative, then we derive
Since must be positive for , it follows from (2.5) that
Inequality (2.2) follows easily from (2.4) and (2.6).
The following lemma is widely used in the literature.
Lemma 2.2 (see [24]). Let be a sequence of nonnegative numbers which satisfies the inequality where and are fixed. Then there exists an such that which implies as if .
Lemma 2.3. Assume that is a sequence of nonnegative numbers satisfying the difference inequality where , and are nonnegative sequences. If there exists at least one positive , then the sequence converges to zero as .
Proof. This lemma follows directly from Lemma 2.2 since where .
Remark 2.4. If in Lemma 2.3, we assume , , then the statement also holds, since in this case, if such a sequence exists, then the solution must be trivial, that is, , (for some results on the existence of nontrivial solutions, see, e.g., [25–27] and the references therein).
Through some simple calculations, we have the following result.
Lemma 2.5. Every positive solution to the first-order difference equation with , has the form
Note that Lemma 2.5 leads to the following corollary.
Corollary 2.6. Each positive solution to the k th-order difference equation where and the initial values are positive, has the following form: where represents the integer part function and .
Remark 2.7. By Corollary 2.6 we have that for any positive solution to (2.13) the following three statements hold true if : (1); (2)if for every , then the subsequences are all strictly increasing;(3)if for every , then the subsequences are all strictly decreasing.
3. Main Results
In this section, we prove the main results of this paper, which concern the global attractivity of positive solutions to (1.7) under some conditions. In the sequel, we assume that there is a finite limit of the positive sequence in (1.7).
Theorem 3.1. Consider (1.7), where is a positive monotone sequence with finite limit . If , then every positive solution to (1.7) converges to .
Proof. By the change , (1.7) is transformed into
with . Note that the sequence is also monotone and .
According to the assumption the sequence is nondecreasing or nonincreasing. If is nonincreasing, then for some fixed , there exists a natural number such that for every we have , which implies
On the other hand, if is nondecreasing then obviously for each , hence (3.2) also holds for this case.
Let be fixed. Employing the transformation , (3.1) becomes
which implies
Note that for all . From this and by Lemma 2.1 we get
When both and are zero, it is clear that is always zero for . Otherwise, it follows from Lemma 2.3 that , which implies
Finally, from the above two transformations we get
The proof is complete.
Theorem 3.2. Consider (1.7). Let be a positive solution to (2.13) such that (or ), , and denote If , then every positive solution to (1.7) converges to .
Proof. Employing the transformation , (1.7) becomes
where and .
Then by the change , (3.9) is transformed into
In the sequel, we proceed by considering two cases.
Case 1. Let .
By Remark 2.7, we have . From (3.10) we get
for . By the change , (3.11) becomes
where
Claim 1. There exists an integer such that for every . Proof. Since , we easily have that
Hence
On the other hand, for , there exists an such that for each we have , which along with the fact , implies that
The claim follows directly from (3.15) and (3.16), as desired.
Next, from Lemma 2.1 and (3.12) it follows that
From (3.17) and by Lemma 2.3, we derive . Hence
and consequently
Case 2. Let .
By Remark 2.7, we have , and (3.10) is transformed into
for all . Then employing the following change
(3.20) is transformed into
where . In this case, obviously holds. The rest of the proof is similar to that of Case 1 so is omitted.
To illustrate Theorem 3.2, we present the following example.
Example 3.3. Consider the difference equation
where , and .
By Theorem 3.2 and through some calculations, we obtain
Hence if , then every positive solution to (3.23) converges to .
Theorem 3.4. Consider (1.7). If is an increasing sequence converging to and , then every positive solution to (1.7) converges to .
Proof. By the change , (1.7) becomes where . The rest of the proof is analogous to that of Theorem 3.1 and thus is omitted.
4. Explicit Solutions
Recently, Stevic and Iricanin in [21] proved the following theorem.
Theorem 4.1 (see [21, Theorem??2.8]). Consider where . Then every well-defined solution of the equation has the following form: where , and where is equal to one of the initial values .
The result is interesting since (4.2) holds for all real ’s and for all nonzero initial values if one of these exponents is negative. However, (4.2) does not give explicit solutions to (4.1) since ’s and ’s in (4.2) are uncertain. Thus the problem of finding more explicit expressions of solutions to (4.1) is of interest.
In this section we find explicit solutions to the next particular cases of (4.1) with and positive initial values . First we prove a useful lemma.
Lemma 4.2. Let be a positive solution to (4.3) or (4.4). If there exists an such that then for each the following equalities hold:
Proof. We will only consider (4.3), because similar proof can be given to (4.4). The case obviously holds due to (4.5). Next assume that (4.6) holds for for some . Then by (4.3) we derive Thus (4.6) holds for , finishing the inductive proof of the lemma.
Proposition 4.3. Let be a solution to (4.3) with and positive initial values , then for each the following statements hold true. (1)If , then .(2)If , then .(3)If then and (4)If then and
Proof. (1) By the assumption and (4.3) it follows that Then by and (4.3), we have the following equalities: Hence (4.5) is satisfied for . Then by Lemma 4.2 we have that as desired.(2)By similar calculations as in (1), the following equalities hold: Thus (4.5) holds for , and the result follows again by Lemma 4.2.(3)By the assumption and (4.3) it follows that Then from and (4.3), we get Now we will consider two cases and . When , the following equalities hold: On the other hand, the case leads to Hence (4.5) holds for no matter the value of is bigger or less than one. From this the result follows by Lemma 4.2.(4)Through analogous calculations to (3), if then If then Hence (4.5) holds for and any . Hence, the results also follow from Lemma 4.2, finishing the proof of the proposition.
The next proposition can be similarly proved as the proof of Proposition 4.3, hence the proof is omitted here.
Proposition 4.4. Let be a solution to (4.4) with and positive initial values , then for each the following statements hold true. (1)If , then .(2)If , then .(3)If then , and (4)If then , and
Remark 4.5. From the above propositions, we know that any positive solution to (4.3) or (4.4) can be divided into three subsequences which have explicit expressions. If we regard the sequence as a general periodic solution to (4.3), then the solution eventually converges to the general period-three solution .
Acknowledgments
This paper is partially supported by the Fundamental Research Funds for the Central Universities (Project no. CDJXS10181130), the New Century Excellent Talent Project of China (no. NCET-05-0759), the National Natural Science Foundation of China (no. 10771227), and the Serbian Ministry of Science (Projects III41025 and III44006).