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On the boundedness of the solutions in nonlinear discrete Volterra difference equations
Advances in Difference Equations volume 2012, Article number: 2 (2012)
Abstract
In this article, we investigate the boundedness property of the solutions of linear and nonlinear discrete Volterra equations in both convolution and nonconvolution case. Strong interest in these kind of discrete equations is motivated as because they represent a discrete analogue of some integral equations. The most important result of this article is a simple new criterion, which unifies and extends several earlier results in both discrete and continuous cases. Examples are also given to illustrate our main theorem.
1 Introduction
We consider the nonlinear system of Volterra difference equations
with the initial condition
where
(A) The function f(n,j,.) : ℝ^{d}→ ℝ^{d}is a mapping for any fixed 0 ≤ j ≤ n, x_{0} ∈ ℝ^{d}and h(n) ∈ ℝ^{d}, n ≥ 0.
(B) For any 0 ≤ j ≤ n, there exists an a(n, j) ∈ ℝ_{+}, such that
and
hold. Here ϕ : ℝ_{+} → ℝ_{+} is a monotone nondecreasing mapping such that ϕ (v) > 0, v > 0, and ϕ(0) = 0 where . is any fixed norm on ℝ^{d}.
In recent years, there has been an increasing interest in the study of the asymptotic behavior of the solutions of both convolution and nonconvolutiontype linear and nonlinear Volterra difference equations (see [1–17] and references therein). Appleby et al. [2], under appropriate assumptions, have proved that the solutions of the discrete linear Volterra equation converge to a finite limit, which in general is nontrivial. The main result on the boundedness of solutions of a linear Volterra difference system in [2] was improved by Györi and Horváth [8]. In terms of the kernel of a linear system Györi and Reynolds [10] found necessary conditions for the solutions to be bounded. Also Györi and Reynolds [9] studied some connections between results obtained in [2, 8]. Elaydi et al. [6] have shown that under certain conditions there is a onetoone correspondence between bounded solutions of linear Volterra difference equations with infinite delay and its perturbation. Also Cuevas and Pinto [4] have shown that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear Volterra difference equation with unbounded delay and its perturbation. In most of our references linear and perturbed linear equations are investigated, moreover the boundedness and estimation of the solutions are founded by using the resolvent of the equations.
This article studies the boundedness of the solution of (1.1) under initial condition (1.2). As an illustration, we formulate the following statement which is an interesting consequence of our Corollary 5.7.
Consider the linear convolutiontype Volterra equation
with the initial condition
Here A(n) ∈ ℝ^{d × d}are given matrices, h(n) ∈ ℝ^{d}are given vectors and x_{0} ∈ ℝ^{d}.
Proposition 1.1. Assume that one of the following conditions is satisfied:
Then the solution of (1.3) and (1.4) is bounded for any x_{0} ∈ ℝ^{d}.
We remark that the above proposition gives sufficient conditions for the boundedness, but they are not necessary in general, see Remark 6.5 below.
To the best of our knowledge, this is the first article dealing with the boundedness property of the solutions of a linear inhomogeneous Volterra difference system with the critical case ${\sum}_{i=0}^{\mathrm{\infty}}\left\rightA\left(i\right)\left\right=1$. For some recent literature on the boundedness of the solutions of linear Volterra difference equations, we refer the readers to [10–12]. We give some applications of our main result for sublinear, linear, and superlinear Volterra difference equations. We study the boundedness of solutions of convolution cases and we get a result parallel to the corresponding result of Lipovan [18] for integral equation. Also we give some examples to illustrate our main results.
The rest of the article is organized as follows. In Section 2, we briefly explain some notation and two definitions which are used to state and to prove our results. In Section 3, we sate our main result with its proof. In Section 4, we give three applications based on our main result. In Section 5, some corollaries with convolution estimations and boundedness of convolution infinite delay equation are given. Examples are also given to illustrate our main theorem in Section 6.
2 Preliminaries
In this section, we give some notation and some definitions which are used in this article.
Let ℝ be the set of real numbers, ℝ_{+} the set of nonnegative real numbers, ℤ is the set of integer numbers, and ℤ_{+} = {n ∈ ℤ : n ≥ 0}. Let d be a positive integer, ℝ^{d}is the space of ddimensional real column vectors with convenient norm .. Let ℝ^{d × d}be the space of all d × d real matrices. By the norm of a matrix A ∈ ℝ^{d × d}, we mean its induced norm A = sup{Ax x ∈ ℝ^{d}, x = 1}. The zero matrix in ℝ^{d × d}is denoted by 0 and the identity matrix by I. The vector x and the matrix A are nonnegative if x_{ i }≥ 0 and A_{ ij }≥ 0,1 ≤ i,j ≤ d, respectively. Sequence (x(n))_{n ≥ 0}in ℝ^{d}is denoted by x :ℤ_{+} → ℝ^{d}.
The following definitions will be useful to prove the main results.
Definition 2.1. Let the function ϕ and the sequence a(n,j) ∈ ℝ_{+}, 0 ≤ j ≤ n, be given in condition (B). We say that the nonnegative constant u has property (P_{ N }) with an integer N ≥ 0 if there is v >0, such that
and
hold.
Definition 2.2. We say that the vector x_{0} ∈ ℝ^{d}belongs to the set S if there exist a nonnegative constant u and an integer N ≥ 0 such that u has property (P_{ N }) and the solution x(n;x_{0}), n ≥ 0, of (1.1) and (1.2) satisfies
Remark 2.3. x_{0} ∈ ℝ^{d}belongs to the set S if
and
hold. In this case N = 0 and u = x_{0} have property (P_{0}).
Remark 2.4. If there exists an N ≥ 0 and two positive constants u and v such that (2.2) holds, then
Conditions (2.6) and (2.7) are equivalent to
3 Main result
Our main goal in this section is to establish the following result with the proof.
Theorem 3.1. Let (A) and (B) be satisfied and assume that the initial vector x_{0}belongs to the set S. Then the solution x(n;x_{0}), n ≥ 0, of (1.1) and (1.2) is bounded. More exactly the solution satisfies (2.3) with suitable u and N, such that
where v is defined in (2.1) and (2.2).
Proof. Let x_{0} ∈ S and consider the solution x(n) = x(n; x_{0}), n ≥ 0, of (1.1) with the condition (1.2) and let u and N be defined in (2.3). Then
where we used the monotonicity of ϕ, and the definition of v. Thus (3.1) holds for n = N + 1.
Now we show that (3.1) holds for any n ≥ N + 1. Assume, for the sake of contradiction, that (3.1) is not satisfied for all n ≥ N + 1. Then there exists n_{0} ≥ N + 1 such that
and
Hence, from Equation (1.1), we get
Since ϕ is a monotone nondecreasing mapping, (2.3) and (3.3) yield
But x_{0} ∈ S and u has property (P_{ N }), and hence x(n_{0} + 1) ≤ v. This contradicts the hypothesis that (3.1) does not hold for n_{0} ≥ N + 1. So inequality (3.1) holds.
4 Applications
In this section, we give some applications of our main result. Throughout this section we take ϕ (t) = t^{p}, t > 0 with p > 0. There are three cases:

1.
Sublinear case when 0 < p < 1;

2.
Linear case when p = 1;

3.
Superlinear case when p > 1.
4.1 Sublinear case
Our aim in this section is to establish a sufficient, as well as a necessary and sufficient, condition for the boundedness of all solutions of (1.1) and the scalar case of (1.1), respectively.
The next result provides a sufficient condition for the boundedness of solutions of (1.1).
Theorem 4.1. Let (A), (B) be satisfied and ϕ (t) = t^{p}, t > 0, with fixed p ∈ (0,1). If (2.8) and (2.9) hold, then for any x_{0} ∈ ℝ^{d}the solution x(n; x_{0}), n ≥ 0 of (1.1) and (1.2) is bounded.
The next Lemma provides a necessary and sufficient condition for the condition (2.2) be satisfied, and will be useful in the proof of Theorem 4.1.
Lemma 4.2. Assume ϕ (t) = t^{p}, t > 0 and 0 < p < 1. Any positive constant u has property (P_{0}) if and only if (2.8) and (2.9) are satisfied.
Proof. Let the nonnegative constant u have property (P_{0}) (N = 0 in Definition 2.1). Then the condition (2.2) is satisfied for some positive v and for all n ≥ 1, so
this imply that conditions (2.8) and (2.9) are satisfied.
Conversely, we assume (2.8), (2.9) and we prove that any positive constant u has property (P_{0}). Clearly, (2.9) is equivalent to α_{0} < ∞, β_{0} < ∞ and (2.8) implies γ_{0} < ∞.
Since p ∈ (0,1), it is clear that for an arbitrarily fixed u > 0, (2.1) and
are satisfied for any v large enough. From (4.3) we get
that (2.5) is satisfied for x_{0} = u and all n ≥ 1. Then by Definition 2.1, u has property (P_{0}).
Now we prove Theorem 4.1.
Proof. Let (A) and (B) be satisfied. By Lemma 4.2, we have that for any x_{0} ∈ ℝ^{d}, u = x_{0} has property (P_{0}) (see Remark 2.3). Thus, the conditions of Theorem 3.1 hold, and the initial vector x_{0} belongs to S, and hence the solution x(n; x_{0}) of (1.1) and (1.2) is bounded.
We consider the scalar case of Volterra difference equation
where x_{0} ∈ ℝ_{+}, a(n,j) ∈ ℝ_{+}, h(n) ∈ ℝ_{+}, 0 ≤ j ≤ n and p ∈ (0,1).
The following result provides a necessary and sufficient condition for the boundedness of the solution of (4.4) and (4.5). The necessary part of the next theorem was motivated by a similar result of Lipovan [18] proved for convolutiontype integral equation.
Theorem 4.3. Assume
moreover for any n ≥ 0, there exists an index j_{ n }such that
For any x_{0} ∈ (0, ∞), the solution of (4.4) is bounded, if and only if (2.8) and (2.9) are satisfied.
Proof. Assume (2.8) and (2.9) are satisfied. Clearly, by Theorem 4.1 the solution of (4.4) is bounded. Conversely, let the solution x(n) = x(n;x_{0}) of (4.4) be bounded on ℝ_{+}, with x_{0} > 0. Under condition (4.7), by mathematical induction we show that x(n) > 0, n ≥ 0. For n = 0 this is clear. Suppose that required inequality is not satisfied for all n ≥ 0. Then there exists index ℓ ≥ 0 such that x(0) > 0, ..., x(ℓ) > 0 and x(ℓ+1) ≤ 0. But by condition (4.7), we get
which is a contradiction. So x(n) > 0 for all n ≥ 0. On the other hand, for any n ≥ N* ≥ 1
hence
Since x(n + 1) ≥ h(n) for all n ≥ 0, clearly sup_{n≥0}h(n) is finite.
Define now
which is finite. First we show that m > 0.
Assume for the sake of contradiction that m = 0. In this case we can find a strictly increasing sequence (N_{ k })_{k≥1}, such that
From (4.4) if n = N_{ k } 1, we deduce
Since x(N_{ k }) > 0, we have that
Since p ∈ (0,1) and x(N_{ k }) → 0, an k → ∞, we get
which contradicts (4.6). So m > 0 and for $\frac{1}{2}m$, there exists N* ≥ 0 such that
Hence,
But the solution x(n) is a bounded sequence, and hence
This and (4.8) imply condition (2.9).
Remark 4.4. In general, without condition (4.7) the necessary part of Theorem 4.3 is not true. In fact if a(0, 0) = 0, and h(0) = 0, that is (4.7) does not hold for n = 0, then for any x_{0} ∈ (0, ∞) the solution of (4.4) satisfies x(1;x_{0}) = 0, and hence
Thus, the solution x(n; x_{0}) does not depend on the choice of the sequence (a(n, 1))_{n≥1}. This shows that the boundedness of the solutions does not imply (2.9), in general.
4.2 Linear case
Our aim in this section is to obtain sufficient condition for the boundedness of the solution of (1.1) under the initial condition (1.2), but in the linear case of Volterra difference equation.
The following result gives a sufficient condition for the boundedness.
Theorem 4.5. Assume (A), (B) are satisfied and ϕ (t) = t, t ≥ 0. Then the solution x(n;x_{0}), x_{0} ∈ S, n ≥ 0 of (1.1) and (1.2) is bounded, if (2.9) is satisfied and there exists an N ≥ 0 such that one of the following conditions holds:

(i)
condition (2.8) holds and
$${\beta}_{N}=\underset{n\ge N+1}{\text{sup}}\sum _{j=N+1}^{n}a\left(n,j\right)<1;$$(4.9) 
(ii)
${\beta}_{N}=\underset{n\ge N+1}{\text{sup}}\sum _{j=N+1}^{n}a\left(n,j\right)=1,$
and for any$n\in {\Gamma}_{N}^{\left(1\right)}$,
hold, moreover
and
where
and
For the proof of Theorem 4.4, we need the following lemma.
Lemma 4.6. Assume ϕ (t) = t, t ≥ 0. A positive constant u has property (P_{ N }) with an integer N ≥ 0 if and only if the condition (2.9) and either (i) or (ii) are satisfied.
Proof. Necessity. We show that (P_{ N }) implies (2.9) and either (i) or (ii). Suppose a positive constant u has property (P_{ N }) with an integer N ≥ 0, hence (2.1) and (2.2) are satisfied for v > 0 and for any n ≥ N + 1.
From (2.1) and (2.2), it is clear that (2.8), (2.9) are satisfied and
Therefore
and hence
The latest inequality implies two cases with respect the value of β_{ N }.

The first case β_{ N }< 1. In this case the condition (i) is satisfied.

Consider now the second case where β_{ N }= 1. Clearly if ${\sum}_{j=N+1}^{n}a\left(n,j\right)=1$, then from (4.13), we get
$$\sum _{j=0}^{N}a\left(n,j\right)u+h\left(n\right)=0,\phantom{\rule{1em}{0ex}}n\in {\Gamma}_{N}^{\left(1\right)},$$
or equivalently (4.10).
But if ${\sum}_{j=N+1}^{n}a\left(n,j\right)<1$, then
and (4.13) yields
and hence (4.11) and (4.12) are satisfied. Then condition (ii) holds.
Sufficiency. We show that if (2.9) and one of the conditions (i) and (ii) is satisfied with some u ≥ 0 and N ≥ 0, then u has property (P_{ N }). It is easy to observe that (2.9) yields
Let (i) of Theorem 4.5 be satisfied, that is β_{ N }< 1. Then for u ≥ 0 and n ≥ N + 1, N ≥ 0, there exists v > 0 such that
It implies
i.e. (2.2) is satisfied and (2.1) also is satisfied for all v large enough, hence u has property P_{ N }.
Now suppose β_{ N }= 1, and (4.10) holds. Then, clearly (2.1) and (2.2) are satisfied for any v ≥ 0 and $n\in {\Gamma}_{N}^{\left(1\right)}$.
If $n\in {\Gamma}_{N}^{\left(2\right)}$ and (4.11) and (4.12) are satisfied, then for u ≥ 0, we have
Then there exists v > 0 large enough such that
since $1{\sum}_{j=N+1}^{n}a\left(n,j\right)>0$, for all $n\in {\Gamma}_{N}^{\left(2\right)}$.
Therefore
i.e. for all v large enough the conditions (2.2) and (2.1) are satisfied. Hence, u has property (P_{ N }).
The following lemma is extracted from [2] (Lemma 5.3) and will be needed in this section.
Lemma 4.7. Assume (A), (B) are satisfied and ϕ (t) = t, t ≥ 0. For every integer N > 0, there exists a nonnegative constant K_{1}(N) independent of the sequence (h(n))_{n≥0}and x_{0}, such that the solution (x(n))_{n≥0}of (1.1) and (1.2), satisfies
Now we give the proof of Theorem 4.5.
Proof. Let (A), (B), (2.9) and either (i) or (ii) in Theorem 4.5 be satisfied. By Lemma 4.7 the solution of (1.1) and (1.2) satisfies (4.14) for all 0 ≤ n ≤ N. This means that, there exists a nonnegative constant u such that
By Lemma 4.6 we have u has property (P_{ N }). Then the conditions of Theorem 3.1 hold for the initial vector x_{0} belonging to S, and hence the solution x(n; x_{0}) of (1.1) and (1.2) is bounded.
4.3 Superlinear case
Our aim in this section is to obtain sufficient condition for the boundedness in the superlinear case.
Theorem 4.8. Assume that conditions (A) and (B) are satisfied with the function ϕ(t) = t^{p}, t > 0, where p > 1. Suppose also (2.8) and (2.9) hold. Then the solution x(n;x_{0}) of (1.1) and (1.2) is bounded for some x_{0} ∈ ℝ^{d}if there exists$v\in \left[{\left(\frac{1}{p{\beta}_{0}}\right)}^{\frac{1}{p1}},{\left(\frac{1}{{\beta}_{0}}\right)}^{\frac{1}{p1}}\right]$such that
and
where α_{0}, β_{0}and γ_{0}are defined in (4.1) and (4.2).
Proof. Assume (2.8), (2.9), (4.15), and (4.16) are satisfied. The case β_{0} = 0 is clear. So we assume that β_{0} > 0. In this case, clearly, $v{\beta}_{0}{v}^{p}\ge 0$ if
and the maximum value of the function $g\left(v\right)=v{\beta}_{0}{v}^{p}\mathsf{\text{is}}{\left(\frac{1}{p{\beta}_{0}}\right)}^{1/\left(p1\right)}$.
Then there exists v such that
and the conditions (2.4) and (2.5) hold. By Remark 2.3 we get that under conditions (4.15) and (4.16), u = x_{0} has property (P_{0}) and x_{0} belongs to S. Then the conditions of Theorem 3.1 hold, so the solution of (1.1) with the initial condition (1.2) is bounded.
5 Some corollaries with convolution estimations
In this section we give some corollaries on the boundedness of the solutions of (1.1) and (1.2) but in the convolutiontype. Through out in this section we take a(n, i) = α(n  i), n ≥ 0, and 0 ≤ i ≤ n, and the following condition

(C)
For any n ≥ 0, there exists an α(n) ∈ ℝ_{+}, such that
$$\parallel f\left(n,j,x\right)\parallel \le \alpha \left(nj\right)\varphi \left(\parallel x\parallel \right),$$
with a monotone nondecreasing mapping ϕ : ℝ_{+} → ℝ_{+} and . is any norm on ℝ^{d}.
Remark 5.1. If a(n,j) = α(n  j), α(n) ∈ ℝ_{+}, n ≥ 0, and ϕ(t) > 0, t > 0, then the nonnegative constant u has property (P_{ N }) with an integer N ≥ 0 if and only if
and
are satisfied.
Remark 5.2. x_{0} ∈ ℝ^{d}belongs to the set S if
and
hold. In this case N = 0 and u = x_{0} has property (P_{0}).
By our main result, we have the following corollary.
Corollary 5.3. Let (A), (C) and sup_{n≥0}h(n) < ∞ be satisfied, and assume that the initial vector x_{0}belongs to the set S. Then the solution x(n;x_{0}), n ≥ 0, of (1.1) and (1.2) is bounded.
Proof. Assume (A), (C) are satisfied. By Theorem 3.1 and Remark 5.1, it is easy to prove that the solution of (1.1) and (1.2) is bounded.
The following two corollaries are immediate consequence of Theorems 4.1 and 4.3 of the sublinear convolution case, respectively.
Corollary 5.4. Assume (A), (C) are satisfied and ϕ(t) = t^{p}, t > 0, with fixed p ∈ (0,1). If
then for any x_{0} ∈ ℝ^{d}the solution x(n; x_{0}), n ≥ 0 of (1.1) and (1.2) is bounded.
Corollary 5.5. Consider Equation (4.4) with p ∈ (0, 1) and nonnegative coefficients. Assume
and for any n ≥ 0 one has h(n) > 0 if α(j) = 0, 0 ≤ j ≤ n. Then the solution of (4.4) and (4.5) is bounded if and only if the condition (5.5) is satisfied.
Proof. The proof is immediate consequence from proof of Theorem 4.3 with Remark 5.1.
Remark 5.6. The Corollary 5.5 is analogous to the corresponding result of Lipovan (Theorem 3.1, [18]) for integral equation.
In the next corollary we assume that
Under condition (5.6) we have three cases
Case 1. $\sum _{n=0}^{\mathrm{\infty}}\alpha \left(n\right)<1$;
Case 2. $\sum _{j=0}^{\mathrm{\infty}}\alpha \left(j\right)=1$ and for any n ≥ 1,
Case 3. There exists an index M ≥ 0 such that $\sum _{n=0}^{M}\alpha \left(n\right)=1$, moreover α(M) ≠ 0 and α(n) = 0, n ≥ M + 1.
Corollary 5.7. Assume (A), (C) and (5.6), and let ϕ(t) = t, t ≥ 0. Then the solution of (1.1) and (1.2) is bounded for any x_{0} ∈ ℝ, if one of the following conditions holds

(a)
Case 1. holds and sup_{n≥0}h(n) < ∞.

(b)
Case 2. holds and
$$\underset{n\ge 1}{\text{sup}}\frac{\parallel h\left(n\right)\parallel}{\sum _{j=n}^{\mathrm{\infty}}\alpha \left(j\right)}<\mathrm{\infty}.$$ 
(c)
Case 3. holds and h(n) = 0, n ≥ M + 1.
Proof. (a) This part is an immediate consequence of (i) from Theorem 4.5 with Remark 5.1.

(b)
Assume that (b) is satisfied. For a fixed N ≥ 0 and u ∈ ℝ_{+}, there exists a positive constant v such that
$$\frac{\sum _{j=nN}^{n}\alpha \left(j\right)}{\sum _{j=nN}^{\mathrm{\infty}}\alpha \left(j\right)}u+\frac{\parallel h\left(n\right)\parallel}{\sum _{j=nN}^{\mathrm{\infty}}\alpha \left(j\right)}\le u+\frac{\parallel h\left(n\right)\parallel}{\sum _{j=n}^{\mathrm{\infty}}\alpha \left(j\right)}\le v,\phantom{\rule{1em}{0ex}}n\ge N+1,$$
and
hold. Thus
i.e.
By Remark 5.1, u has property (P_{ N }). For the initial value x_{0} ∈ ℝ^{d}applying Corollary 5.3, we get the boundedness of the solution of (1.1) and (1.2).

(c)
Assume the condition (c). Clearly, for all v ≥ 0 the conditions (2.1) and (2.2) are satisfied and u has property (P _{ N }). Then for any x _{0} ∈ S, the solution of (1.1) is bounded according to Corollary 5.3.
The proof of the following corollary is an immediate consequence of Theorem 4.8 and Remark 5.1 and it is therefore omitted.
Corollary 5.8. Assume that conditions (A) and (C) are satisfied with the function ϕ (t) = t^{p}, t > 0,p > 1, and (5.5) holds. Then for an x_{0} ∈ ℝ^{d}the solution x(n;x_{0}) of (1.1) and (1.2) is bounded, if there exists$v\in \left[{\left(\frac{1}{p{\beta}_{c}}\right)}^{\frac{1}{p1}},{\left(\frac{1}{{\beta}_{c}}\right)}^{\frac{1}{p1}}\right]$such that
and
where
Now we show the application of Corollary 5.7 to the linear convolution Volterra difference equation with infinite delay
with initial condition
where Q(n) ∈ ℝ^{d × d}, n ≥ 0 and φ(m) ∈ ℝ^{d}, m ≤ 0.
From (5.8), we have
If we compare the latest equation with Equation (1.1), we have f(n, i, x) = Q(n  i)x and
If ${M}_{\phi}=\underset{m\le 1}{\text{sup}}\parallel \phi \left(m\right)\parallel $, then we get
Corollary 5.9. Assume Q(n) ∈ ℝ^{d × d}, n ≥ 0, and
Then the solution of (5.8) with the initial condition (5.9) is bounded, if
Proof. To prove that the solution of (5.8) with the initial condition (5.9) is bounded, it is enough to show that one of the hypotheses (a), (b), (c) of Corollary 5.7 holds. Let (5.10) and (5.11) be satisfied. There are three cases.
First consider the case when
and (5.11) are satisfied. This implies (a) of Corollary 5.7.
Second consider the case when
This yields
and hence condition (b) of Corollary 5.7 is satisfied.
The last case is when there exists N ≥ 0, such that
In this case, we get h(n) = 0 for all n ≥ N + 1, hence (c) of Corollary 5.7 holds.
6 Examples
In this section we give some examples to illustrate our results.
Example 6.1. Let us consider the case when
We have $a\left(0,0\right)=\frac{1}{2},a\left(n,0\right)=\frac{2n+1}{\left(n+1\right)\left(n+2\right)},{\text{sup}}_{n\ge 1}a\left(n,0\right)=\frac{1}{2}$,
and
One can easily see that conditions (2.4) and (2.5) are equivalent to the inequalities
and
Consider the scalar equation
with the initial condition
where x_{0} ∈ ℝ_{+}, h(n) ∈ ℝ_{+}, n ≥ 0 and p > 0. In fact there are three cases with respect to the value of p.
(a1) p ∈ (0,1) and ϕ (t) = t^{p}, t > 0. If sup_{n≥0}h(n) < ∞ and (6.1) are satisfied, then for any x_{0} ≥ 0, the solution of (6.4) and (6.5) is bounded by Theorem 4.1.
(a2) p = 1 and ϕ (t) = t, t >0. It is not difficult to see that for any v > 0 large enough the inequalities (6.2) and (6.3) are equivalent to the inequalities
and
Let k = sup_{n≥1}nh(n) < ∞, it is easily to see that the inequality (6.6) is satisfied if
Hence (2.4) and (2.5) are satisfied, by Lemma 4.6, x_{0} has property (P_{0}). It is worth to note that in this case our Theorem 4.5 is applicable for any x_{0} ∈ ℝ_{+}, but the results in [2, 8–12, 14] are not applicable in this case.
(a3) p > 1 and ϕ(t) = t^{p}, t > 0. Assume k = sup_{n≥1}h(n), and for x_{0} ≥ 0, there exists $v\in \left[{\left(\frac{1}{p}\right)}^{\frac{1}{p1}},1\right]$, such that
and
hold. Hence x_{0} has property (P_{0}), and by Theorem 4.8, for small x_{0}, the solution of (6.4) and (6.5) is bounded.
Summarizing the observations and applying Theorem 4.5, we get the next new result.
Proposition 6.2. If Equation (6.4) is linear, that is p = 1, and sup_{n ≥ 1}n h(n) < ∞, then every positive solution of (6.4) with initial condition (6.5) is bounded.
Proposition 6.3. Assume Equation (6.4) is superlinear, that is p > 1. If for some x_{0} ≥ 0 there exists$v\in \left[{\left(\frac{1}{p}\right)}^{\frac{1}{p1}},1\right]$, such that
and
hold, then the positive solution of (6.4) with initial condition (6.5) is bounded.
The following example shows that applicability of our result in the critical case.
Example 6.4. Consider the equation
where x_{0} ∈ ℝ_{+}, q ∈ (0,1), c ∈ (0,1) and h(n) ∈ ℝ_{+}, n ≥ 0.
If q + c < 1 and sup_{n≥0}h(n) < ∞, then our condition (a) in Corollary 5.7 holds. Relation q + c = 1 implies
Note that the results in [2, 8–12] are not applicable. But our Corollary 5.7 is applicable under condition
since it implies condition (b) in Corollary 5.7:
Remark 6.5. By mathematical induction, it is easy to see the solution of (6.7) and (6.8) is given in the form
where x(0) = x_{0}and x(1) = cx_{0} + h(0). Let${\sum}_{j=0}^{\mathrm{\infty}}c{q}^{j}=1$or equivalently q + c = 1 moreover h(n) = k^{n+1}, 0 < q < k < 1. In this case the above solution is bounded for any x_{0} ∈ ℝ_{+}. At the same time condition (b) in Corollary 5.7 does not hold, and hence condition (β) in Proposition 1.1 is not necessary.
Therefore by statements (a) and (b) of Corollary 5.7 we get the following.
Proposition 6.6. The solution of (6.7) and (6.8) is bounded if either q + c < 1 and sup_{n≥0}h(n) < ∞, or q + c = 1, and
The next example shows the sharpness of our Corollary 5.8.
Example 6.7. Consider the equation
where x(0) = x_{0} > 0, p > 1, q ∈ (0,1).
Since x_{0} > 0, implies x(n) > 0 for all n ≥ 0, therefore from (6.9), we have x(n + 1) ≥ x^{p}(n), n ≥ 0. For fixed p > 1, inequality ${x}_{0}^{p}>{x}_{0}$ holds if and only if x_{0} > 1. In the latest, by mathematical induction it is easy to prove that the sequence (x(n))_{n≥0}is strictly increasing. Now for x_{0} > 1, we prove that x(n) → ∞ as n → ∞. Assume, for the sake of contradiction, that the sequence (x(n))_{n≥0}is bounded. Since it is strictly increasing, x* = lim_{n→∞}x(n) is finite and x* > x_{0}. On the other hand x(n+ 1) ≥ x^{p}(n), and hence we get x* ≥ (x*) p > x*, which is a contradiction. Hence for all x_{0} > 1 the solution of (6.9) is unbounded. Now applying Corollary 5.8 to (6.9), there exists
such that ${x}_{0}^{p}\le v$ and
Hence
i.e.
Then the solution of (6.9) with initial value ${x}_{0}\le {\left(1q\right)}^{\frac{2}{p\left(p1\right)}}$ is bounded, but the solution of (6.9) with initial value x_{0} > 1 is unbounded. Our results do not give any information about the boundedness of the solutions, whenever ${x}_{0}\in \left({\left(1q\right)}^{\frac{2}{p\left(p1\right)}},1\right]$ but this gap tends to zero if either p is large enough or q is very close to zero. Hence, our results for the superlinear case are sharp in some sense. As a special case let p = 2. In this case, the solution of (6.9) with initial condition x(0) = x_{0} is bounded whenever x_{0} ∈ [0,1  q] and it is unbounded whenever x_{0} > 1.
Based on our results we state the following conjecture as an open problem.
Conjecture 6.8. Let p > 1 and 0 < q <1. Then there exists a constant κ > 1 such that the solution of (6.9) with initial condition x(0) = x_{0}is bounded whenever x_{0} ∈ [0, κ) and it is unbounded whenever x_{0} > κ.
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Acknowledgements
The authors thank to a referee for valuable comments. This study was supported by Hungarian National Foundations for Scientific Research Grant no. K73274, and also the project TÁMOB4.2.2/B10/120100025.
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Györi, I., Awwad, E. On the boundedness of the solutions in nonlinear discrete Volterra difference equations. Adv Differ Equ 2012, 2 (2012). https://doi.org/10.1186/1687184720122
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Keywords
 Linear Case
 Initial Vector
 Mathematical Induction
 Volterra Equation
 Boundedness Property