Abstract
Asymptotic properties of solutions of a difference equation of the form
are studied. We present sufficient conditions under which, for any polynomial of degree at most and for any real , there exists a solution x of the above equation such that . We give also sufficient conditions under which, for given real , all solutions x of the equation satisfy the condition for some polynomial of degree at most .
MSC:39A10.
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1 Introduction
Let ℕ, ℤ, ℝ denote the set of positive integers, the set of all integers and the set of real numbers, respectively. For , let , .
Let . In this paper we consider the difference equation of the form
By a solution of (E) we mean a sequence satisfying (E) for all large n. We denote the space of all sequences by SQ. We denote the Banach space of all bounded sequences with the norm by BS. If x, y in SQ, then xy denotes the sequence defined by pointwise multiplication . Moreover, denotes the sequence defined by for every n. We use the symbols ‘big O’ and ‘small o’ in the usual sense, but for , we also regard and as subspaces of SQ. More precisely, let , , and for , let
For , let denote the space , i.e., the space of all polynomial sequences of degree at most m. Now we can define asymptotically polynomial sequences. We say a sequence is asymptotically polynomial of degree at most m if
for some . Note that if , then the condition is equivalent to . Hence if . In particular if , then and the inclusion is proper. Note also that if and , then if and only if
for some fixed constants . For , we use the factorial notation
The integer part of real number t is denoted by .
The purpose of this paper is to study the asymptotic behavior of solutions of equation (E). In Section 2 we present some preliminary results. Main results appear in Sections 3 and 4. We establish sufficient conditions under which, for some natural k and for any such that , there exists a solution x of (E) such that . We also give sufficient conditions under which all solutions are asymptotically polynomial. The proofs of main theorems are based on the Schauder fixed point theorem (Theorem 3.1) and on the discrete Bihari-type inequality (Theorem 4.1).
Asymptotically polynomial solutions appear in the theory of both differential and difference equations. Especially in the theory of second-order equations, the so-called asymptotically linear solutions, i.e., asymptotically polynomial solutions of degree at most one, are considered. Asymptotically linear solutions of differential equations are considered, for example, in papers [1–11]. A historical survey of this topic can be found in [12]. The asymptotic linearity of a solution x, called in some papers ‘property (L)’, usually means (passing over some additional properties of a derivative) one of the following two conditions:
In [9] and [10] the condition of the form for some is also considered.
Asymptotically polynomial solutions of differential equations of higher order appear, for example, in papers [13–17]. In [14], Naito presented necessary and sufficient conditions under which some neutral differential equation of order m possesses a solution x such that
Note that the condition can be written in the form . In [17], Hasanbulli and Rogovchenko obtained sufficient conditions under which every nonoscillatory solution x of some neutral differential equation of order m has the property
In [15], Philos, Purnaras and Tsamatos presented sufficient conditions under which, for given and , every solution of the equation fulfills the condition . Moreover, they obtained sufficient conditions under which, for every polynomial function φ of degree at most , there exists a solution x of this equation such that .
Asymptotically linear solutions of difference equations are studied, for example, in papers [18–22]. Asymptotic linearity, similarly as in the continuous case, usually means one of the following two conditions:
Asymptotically polynomial solutions of difference equations of higher order appear, for example, in papers [23–30]. In [25], Popenda and Drozdowicz presented necessary and sufficient conditions under which the equation has a convergent solution (i.e., a solution that is asymptotically polynomial of degree zero). In [26], Zafer obtained sufficient conditions under which the equation has a solution x such that , i.e., . It is easy to see that the last condition is equivalent to . In [28] sufficient conditions under which, for any , there exists a solution x of the equation such that are presented. In [30] sufficient conditions under which every solution x of the equation has the property for some are presented. Moreover, sufficient conditions under which, for every there exists a solution x of this equation such that , are presented.
2 Asymptotically polynomial sequences
In this section we obtain some technical results which will be used in the next sections. The solutions of ‘the simplest’ difference equation are the polynomial sequences. In Theorem 2.1, which is the main result of this section, we show that if is sufficiently ‘small’ then the solutions of the ‘equation’ are asymptotically polynomial sequences. This result will be used in the proofs of our main Theorems 3.1 and 4.1. Lemmas 2.1, 2.2 and 2.3 are used in the proof of Theorem 2.1, Lemma 2.4 is used in the proof of Theorem 2.2 and Lemma 2.5 is used to justify an important example (see Remark 4.1 and Example 4.2).
Lemma 2.1 Assume that , and . Then .
Proof Induction on m. Let . Using de l’Hospital theorem, we obtain
So, by the assumption , we obtain
Since , the sequence is increasing to infinity. By the Stolz-Cesaro theorem, we obtain . Hence the assertion is true for . Assume that it is true for some , and let . Then , and by an inductive hypothesis, we get . Hence, by the first part of the proof, we obtain . The proof is complete. □
For , , let
Lemma 2.2 Assume that , and let the series be absolutely convergent. Then there exists exactly one sequence z such that and . The sequence z is defined by
Moreover,
Proof The first assertion is an immediate consequence of Lemma 4 in [31]. The second assertion follows from the inequality . □
Lemma 2.3 Assume that u is a positive and nondecreasing sequence, and
Then there exists a sequence such that .
Proof Since u is positive and nondecreasing, we have . By Lemma 2.2, there exist sequences such that and . Moreover, using Lemma 2.2, we obtain
Hence , and we can take . □
Theorem 2.1 Assume that , , , and
Then .
Proof Assume that . Since , we have . Let . By Lemma 2.3, there exists a sequence such that . Then
Let . Choose such that . Then
and by Lemma 2.3 there exists such that . Choose such that . Since , so by Lemma 2.1, . Moreover,
Hence . The proof is complete. □
Lemma 2.4 Let and . Then
Proof If , then . Hence
and we obtain the result. □
Theorem 2.2 If and , then
Proof First we show, by induction on m, that there exists a sequence c such that
For this assertion follows from Lemma 2.4. Assume it is true for some . By Lemma 2.1 and by an inductive hypothesis, we obtain sequences , b, , , and c such that
Hence we obtain (2). The assertion (1) is an easy consequence of (2). □
Lemma 2.5 Assume that and . The following conditions are equivalent:
Proof If , then . If , then
Assume that and . Obviously, . Hence the assumption implies the existence of a nonzero constant c such that for some . If , then dividing by , we obtain , which is impossible. If , then dividing by , we have , which is impossible too. Hence, in this case, . The proof is complete. □
3 Asymptotically polynomial solutions
In this section we consider the first issue of the Abstract. In Theorem 3.2 we establish sufficient conditions under which, for any and for any real , there exists a solution x of (E) such that . However, the main result of the section is more general Theorem 3.1 in which we establish sufficient conditions under which, for some natural k and for any such that , there exists a solution x of (E) such that . In the second part of the section (Theorems 3.3, 3.4 and Corollary 3.1), we present some consequences of Theorem 3.1.
Theorem 3.1 generalizes Theorem 2 of [28]. The method of the proof of Theorem 3.1 shows certain similarities to the method used in the continuous case in the proof of Theorem 1 of [15].
In this section we regard as a metric subspace of the plane . Moreover, we assume , , and
Theorem 3.1 Assume that , is continuous, f is continuous and
Then for any such that , there exists a solution x of (E) such that
Proof Let x be a solution of equation (E), and let , . Let us denote . Then and equation (E) takes the form
Hence, we have to prove that equation (E*) has a solution y such that . Let . For , we define and by
Let . Since , there exists a constant L such that for . Hence, by the continuity of g, there exists a constant such that for every and every n. Moreover,
for every and every n. Let
Choose p such that and for . Let
Then and S is a convex subset of the Banach space BS. Moreover, as in the proof of Theorem 1 in [30], one can show that S is compact. If , then . Hence, by (3) we have . For , let
Then, for and , by (5) and (6), we get
Hence . Let . Choose and such that . There exists such that for . Choose such that
Let , and let . By the continuity of f at a point , there exists such that implies . Let , and let , . If , then . Hence
and
Moreover, by (5) and (6), we have
Hence
This shows that the map H is continuous. By the Schauder fixed point theorem, there exists a sequence such that . Then for . By Lemma 2.2 we obtain
for . Hence y is a solution of (E*). Moreover, by (5) we have
Hence , and by Theorem 2.1 we obtain . Moreover, and . Hence . The proof is complete. □
Remark 3.1 Let denote the greatest natural number such that for every polynomial , there exists a solution x of (E) such that . Note that if in Theorem 3.1 for some fixed integer p, then the condition takes the form . Hence . But if the sequence σ is of another form, then may be greater than k. In the following example, we have and .
Example 3.1 Let , , , , , ,
Then the conditions of Theorem 3.1 are satisfied and equation (E) takes the form
Note, that for a polynomial φ, the condition is equivalent to the condition . Thus, by Theorem 3.1, for any , equation (7) has a solution such that . One such solution is .
Theorem 3.2 Assume that , is continuous, f is continuous and
Then, for any , there exists a solution x of (E) such that .
Proof Let . Choose M, K such that and . Then
Hence . Now the assertion follows from Theorem 3.1.
We say that a function is locally equibounded if for every , there exists a neighborhood U of t such that f is bounded on . Obviously, every bounded function is locally equibounded. □
Example 3.2 Let and . Then is continuous, unbounded and locally equibounded, is continuous but not locally equibounded.
Example 3.3 Let be continuous, and let . Then f is continuous and locally equibounded. If are continuous and , then f is continuous and locally equibounded.
Example 3.4 Assume that are continuous, are bounded, and let . Then f is continuous and locally equibounded.
Theorem 3.3 If , U is a neighborhood of c and the function is bounded and continuous, then there exists a solution x of (E) such that . Moreover, if f is continuous and locally equibounded, then for any , there exists a solution x of (E) such that .
Proof Choose such that and . Let
Then is continuous and bounded. Choose such that for any . Let , , and let . Then
for any . Hence, by Theorem 3.1, there exists a solution x of the equation
such that . Since , for large n. Hence for large n. Therefore x is a solution of (E). The second assertion is an easy consequence of the first one. □
Theorem 3.4 Assume that for some , , and that at least one of the following conditions is satisfied:
-
(1)
f is continuous and bounded on and ,
-
(2)
f is continuous and bounded on and ,
-
(3)
f is continuous and bounded on and φ is nonconstant.
Then there exists a solution x of (E) such that .
Proof
Assume (1) is satisfied. Let
Then is continuous and bounded. Choose such that for any . Let , , and let . Then and
for any . Then, by Theorem 3.1, there exists a solution x of the equation
such that . Moreover, for large n. Therefore x is a solution of (E). The assertion (1) is proved. (2) is analogous to (1) and (3) is a consequence of (1) and (2). □
Corollary 3.1 Assume that for some , , , , and that at least one of the following conditions is satisfied:
-
(1)
g is continuous on some neighborhood of c and ,
-
(2)
g is continuous and bounded on and ,
-
(3)
g is continuous and bounded on and ,
-
(4)
g is continuous and bounded on and φ is nonconstant.
Then there exists a solution x of the equation
such that .
Proof This is a consequence of Theorems 3.3 and 3.4. □
Example 3.5 Assume that , , , and
By Corollary 3.1, for every nonconstant polynomial , there exists a solution x of (E1) such that . Moreover, by Corollary 3.1, for every real , there exists a solution x of (E1) such that . We will show that a solution of (E1), which is convergent to c, does not exist. Assume that m is even. Then . Let x be a solution of (E1) such that . Then
for large n. Moreover, . Hence for large n. Thus for large n and so on. After -steps, we obtain for large n. Choose p such that for . Then for . If for some , then for every and for . On the other hand, if , we obtain , which is impossible. Hence for . Choose such that for every . Then for . Hence for , and there exists a polynomial sequence such that for . But the only polynomial which satisfies the condition is the constant polynomial . Hence we again obtain for large n, which is impossible. Similarly, if m is odd, one can show that a solution of (E1), which is convergent to c, does not exist.
4 Approximations of solutions
In this section we consider the second issue of the Abstract. In Theorem 4.1 we establish sufficient conditions under which, for given real , all solutions x of (E) satisfy the condition for some . In the second part of the section, we present some consequences of Theorem 4.1. Moreover, in Example 4.2 we show that the assertion of Theorem 4.1 is in some sense optimal.
Theorem 4.1 generalizes Theorem 4 of [26]. The way Theorem 4.1 is proved partially resembles the methods used in the continuous case in the proofs of Theorem 1 in [5] and Theorem 2 in [15].
In this section we assume , , and
Lemma 4.1 Assume that is a continuous and nondecreasing function such that for and . Let and let a, u be sequences of nonnegative real numbers such that
Then the sequence u is bounded.
Proof The assertion is an easy consequence of Theorem 1 in [32]. □
The next lemma can be found in [33].
Lemma 4.2 Let be a sequence of real numbers, and let , . Then
Theorem 4.1 Assume that , is nondecreasing and continuous, f is continuous, and assume that
and x is a solution of (E). Then .
Proof Assume that x is a solution of (E). Then, by (9) for large n, we have
From the identity , we get . Similarly, . Hence we have
Analogously, and then
and so on. After steps, we get
Let . By Lemma 4.2 we obtain
Using the inequality
we have . Hence
where . By Lemma 4.1, the sequence is bounded. Hence, by (10) and the continuity of g, there exists a constant such that
for all large n. Hence , and from Theorem 2.1, we obtain . The proof is complete. □
Corollary 4.1 Assume that , , and x is a solution of the equation
Then .
Proof Let , and . Then
Hence the assertion follows from Theorem 4.1. □
Corollary 4.2 Assume that , , ,
and x is a solution of the equation . Then .
Proof If , then and x is a solution of the equation
Hence by Corollary 4.1. □
Example 4.1 Let , ,
Then the conditions of Theorem 4.1 are satisfied and equation (E) takes the form
The general solution x of (11) can be written in the form
Using the formula , we obtain . Note that .
Remark 4.1 If the assumptions of Theorem 4.1 are satisfied, then every solution of (E) is an element of the space
We will show that for every , there exist sequences a, b and a function f such that the assumptions of Theorem 4.1 are satisfied and equation (E) has a solution x such that
Example 4.2 Let , , , , , , , , , , ,
Since , we have . By Lemma 2.5,
By Theorem 2.2,
Moreover, . Hence for some . Therefore . Moreover,
Hence x is a solution of the equation .
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Migda, J. Asymptotically polynomial solutions of difference equations. Adv Differ Equ 2013, 92 (2013). https://doi.org/10.1186/1687-1847-2013-92
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DOI: https://doi.org/10.1186/1687-1847-2013-92