On the boundedness of the solutions in nonlinear discrete Volterra difference equations
 István Györi^{1}Email author and
 Essam Awwad^{1, 2}
https://doi.org/10.1186/1687184720122
© Györi and Awwad; licensee Springer. 2012
Received: 20 April 2011
Accepted: 16 January 2012
Published: 16 January 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
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.
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.
Here A(n) ∈ ℝ^{d × d}are given matrices, h(n) ∈ ℝ^{ d }are given vectors and x_{0} ∈ ℝ^{ d }.
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.
hold.
hold. In this case N = 0 and u = x_{0} have property (P_{0}).
3 Main result
Our main goal in this section is to establish the following result with the proof.
where v is defined in (2.1) and (2.2).
where we used the monotonicity of ϕ, and the definition of v. Thus (3.1) holds for n = N + 1.
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
 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.
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} < ∞.
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.
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.
For any x_{0} ∈ (0, ∞), the solution of (4.4) is bounded, if and only if (2.8) and (2.9) are satisfied.
Since x(n + 1) ≥ h(n) for all n ≥ 0, clearly sup_{n≥0}h(n) is finite.
which is finite. First we show that m > 0.
This and (4.8) imply condition (2.9).
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.
 (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)
${\beta}_{N}=\underset{n\ge N+1}{\text{sup}}\sum _{j=N+1}^{n}a\left(n,j\right)=1,$
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.
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).
and hence (4.11) and (4.12) are satisfied. Then condition (ii) holds.
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 $n\in {\Gamma}_{N}^{\left(1\right)}$.
since $1{\sum}_{j=N+1}^{n}a\left(n,j\right)>0$, for all $n\in {\Gamma}_{N}^{\left(2\right)}$.
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.
Now we give the proof of Theorem 4.5.
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.
where α_{0}, β_{0}and γ_{0}are defined in (4.1) and (4.2).
and the maximum value of the function $g\left(v\right)=v{\beta}_{0}{v}^{p}\mathsf{\text{is}}{\left(\frac{1}{p{\beta}_{0}}\right)}^{1/\left(p1\right)}$.
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
 (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 }.
are satisfied.
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.
then for any x_{0} ∈ ℝ^{ d }the solution x(n; x_{0}), n ≥ 0 of (1.1) and (1.2) is bounded.
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.
Under condition (5.6) we have three cases
Case 1. $\sum _{n=0}^{\mathrm{\infty}}\alpha \left(n\right)<1$;
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.
 (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.
 (b)Assume that (b) is satisfied. For a fixed N ≥ 0 and u ∈ ℝ_{+}, there exists a positive constant v such that$\frac{\sum _{j=nN}^{n}\alpha \left(j\right)}{\sum _{j=nN}^{\mathrm{\infty}}\alpha \left(j\right)}u+\frac{\parallel h\left(n\right)\parallel}{\sum _{j=nN}^{\mathrm{\infty}}\alpha \left(j\right)}\le u+\frac{\parallel h\left(n\right)\parallel}{\sum _{j=n}^{\mathrm{\infty}}\alpha \left(j\right)}\le v,\phantom{\rule{1em}{0ex}}n\ge N+1,$
 (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.
where Q(n) ∈ ℝ^{d × d}, n ≥ 0 and φ(m) ∈ ℝ^{ d }, m ≤ 0.
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.
and (5.11) are satisfied. This implies (a) of Corollary 5.7.
and hence condition (b) of Corollary 5.7 is satisfied.
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.
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.
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.
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.
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.
where x_{0} ∈ ℝ_{+}, q ∈ (0,1), c ∈ (0,1) and h(n) ∈ ℝ_{+}, n ≥ 0.
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.
The next example shows the sharpness of our Corollary 5.8.
where x(0) = x_{0} > 0, p > 1, q ∈ (0,1).
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.
Declarations
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.
Authors’ Affiliations
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