Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations

Abstract

We consider nonlinear difference equations of unbounded order of the form 𝑥_𝑖=𝑏_𝑖−∑^𝑖_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗(𝑥_𝑗),𝑖=0,1,2,…, where 𝑓_𝑗(𝑥)(𝑗=0,…,𝑖) are suitable functions. We establish sufficient conditions for the boundedness and the convergence of 𝑥_𝑖 as 𝑖→+∞. Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.

1. Introduction

We consider the following nonlinear Volterra discrete equation of nonconvolution type: 𝑥_𝑖=𝑏_𝑖−∑^𝑖_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗(𝑥_𝑗𝑥),𝑖=0,1,2,…,_𝑖,𝑏_𝑖,𝑎_𝑖,𝑗∈𝐼𝑅,𝑏_𝑖≠0,∀𝑖=0,1,2,…. (1.1) The existence problem for solution of Volterra discrete equations arises in the nonlinear implicit case. For linear implicit equations and nonlinear explicit equations, the problem is easily solved. Recently, some local and global existence theorems for Volterra discrete equations in the general case are given in [1, 2].

From now on, we assume that there exists a strictly increasing function 𝑓(𝑥) such that 0≤𝑓_𝑖(𝑥)≤𝑓(𝑥)for𝑥≥0,0≥𝑓_𝑖(𝑥)≥𝑓(𝑥)for𝑥≤0,𝑖≥0. (1.2) Note that (1.2) implies that 𝑓(0)=0,𝑓_𝑖(0)=0,𝑖≥0. (1.3) The above difference equation can be considered as the discrete counterpart of the Volterra integral equation whose importance in the applications is well known (see, e.g., [3, 4]), and arises also in the application of numerical methods to Volterra integral and integrodifferential equations. The theory of the qualitative behavior of this type of nonlinear difference equation is very important, in particular for the study of numerical stability of such methods (see, e.g., [5–11] and the references therein).

In this paper, we study some sufficient conditions for the boundedness of the solutions (if they exist) of (1.1), subject to (1.2), and their asymptotic behavior as 𝑖→+∞. In particular, in Section 2 we investigate the asymptotic behavior when |𝑓_𝑖(𝑥)| is upper bounded by a linear function. The case of nonnegative coefficients is investigated in Section 3 and, with additional monotonicity assumptions, in Section 4.

2. Case of |𝑓_𝑖(𝑥)|≤|𝑥|,𝑖≥0

Assume that, in (1.1) with (1.2), the following additional hypotheses hold: inf_𝑖≥0𝑎_𝑖,𝑖>−1,𝑓_𝑖(𝑥)≤|𝑥|,forany𝑥∈(−∞,+∞),𝑖≥0. (2.1) Observe that the second part of (2.1) is true if in (1.2) 𝑓(𝑥)=𝑥. The following lemma can be easily proved.

Lemma 2.1. If 𝑎_𝑖𝑖≥0,𝑡ℎ𝑒𝑛|𝑥|≤𝑥+𝑎_𝑖𝑖𝑓₀,𝑎(𝑥)_𝑖𝑖≤0,𝑡ℎ𝑒𝑛1+𝑎_𝑖𝑖|𝑥|≤𝑥+𝑎_𝑖𝑖𝑓₀(𝑥), (2.2) for all 𝑥∈ℝ.

Here and in the sequel we assume a sum with a negative superscript to be zero. By using (2.1) and Lemma 2.1, from (1.1), we have that 𝑥_𝑖≤𝑏_𝑖+^𝑖−1_𝑗=0𝑎_𝑖,𝑗𝑥_𝑗,𝑖≥0, (2.3) and we set 𝑏_𝑖=𝑏_𝑖1+min0,𝑎_𝑖,𝑖,𝑎_𝑖,𝑗=𝑎_𝑖,𝑗1+min0,𝑎_𝑖,𝑖,0≤𝑗≤𝑖−1. (2.4) This inequality will be useful in order to find sufficient conditions for the boundedness of 𝑥_𝑖 and for its convergence to zero as 𝑖 tends to infinity.

Theorem 2.2. Consider (1.1) with (1.2) and (2.1), if there exists a positive constant 𝐴 such that sup_{0≤𝑗≤𝑖−1}𝑎_𝑖,𝑗≤𝐴<+∞,∀𝑖≥1,𝐵=sup_𝑖≥0𝑏_𝑖<+∞,𝐴₀=sup∑_𝑖≥𝑖0^𝑖−1_𝑗=𝑖0𝑎_𝑖,𝑗<1 (2.5) for some positive integer 𝑖₀, then 𝑥_𝑖 is bounded and 𝑥_𝑖≤(1+𝐴)^𝑖0𝐵1−𝐴₀<+∞,𝑖≥𝑖₀. (2.6) Moreover, if lim_𝑖→∞𝑏_𝑖=0,lim_𝑖→∞𝑎_𝑖,𝑗=0∀𝑗≥0, (2.7) then lim_𝑖→∞𝑥_𝑖=0.

Proof. Let us consider (2.3), by using (2.5), we have that 𝑥₀≤𝑏₀𝑥≤𝐵,₁≤𝑏₁+𝑎_1,0𝑥₀𝑥≤𝐵+𝐴𝐵=(1+𝐴)𝐵,₂≤𝑏₂+𝑎_2,0𝑥₀+𝑎_2,1𝑥₁≤𝐵+𝐴𝐵+𝐴(1+𝐴)𝐵=(1+𝐴)²𝑥𝐵,…,_𝑖≤𝑏_𝑖+^𝑖−1_𝑗=0𝑎_𝑖,𝑗𝑥_𝑗≤𝐵+^𝑖−1_𝑗=0𝐴(1+𝐴)^𝑗𝐵=𝐵+(1+𝐴)^𝑖−1𝐵=(1+𝐴)^𝑖𝐵,𝑖≥0. (2.8) In particular, assume that the third part of (2.5) holds, then |𝑥_𝑖|≤𝑏_𝑖+^𝑖0−1_𝑗=0𝑎_𝑖,𝑗𝑥_𝑗+^𝑖−1_𝑗=𝑖0𝑎_𝑖,𝑗𝑥_𝑗,∀𝑖≥𝑖₀, (2.9) and thus, 𝑥_𝑖≤𝑏_𝑖+^𝑖0−1_𝑗=0𝑎_𝑖,𝑗𝑥_𝑗+𝐴₀max_{𝑖0≤𝑗≤𝑖}𝑥_𝑗. (2.10) Hence, the following inequalities hold for each 𝑘≤𝑖: 𝑥_𝑘≤𝑏_𝑖+^𝑖0−1_𝑗=0𝑎_𝑖,𝑗𝑥_𝑗+𝐴₀max_{𝑖0≤𝑗≤𝑘}𝑥_𝑗≤𝑏_𝑖+^𝑖0−1_𝑗=0𝑎_𝑖,𝑗𝑥_𝑗+𝐴₀max_{𝑖0≤𝑗≤𝑖}𝑥_𝑗. (2.11) For this reason, max_{𝑖0≤𝑗≤𝑖}𝑥_𝑗≤𝑏_𝑖+^𝑖0−1_𝑗=0𝑎_𝑖,𝑗𝑥_𝑗+𝐴₀max_{𝑖0≤𝑗≤𝑖}𝑥_𝑗, (2.12) from which we obtain that max_{𝑖0≤𝑗≤𝑖}𝑥_𝑗≤𝐵+∑^𝑖0−1_𝑗=0𝐴(1+𝐴)^𝑗𝐵1−𝐴₀=(1+𝐴)^𝑖0𝐵1−𝐴₀<+∞,𝑖≥𝑖₀. (2.13) Thus, |𝑥_𝑖| is bounded and satisfies (2.6).
Assume that 𝑥=limsup_𝑖→∞|𝑥_𝑖|>0 and put 𝑟=sup_𝑖≥𝑖0∑^𝑖−1_𝑗=𝑖0|𝑎_𝑖,𝑗|,𝛾=1+min(0,inf_𝑖≥𝑖0𝑎_𝑖,𝑖) and 𝑀=sup_𝑖≥0|𝑥_𝑖|. Then, since |𝑥_𝑖| is bounded and the third of (2.5) holds, we have that 𝑀<+∞ and 𝑟<𝛾. Let's take any 𝜖>0 and consider a continuous function 𝐹(𝑥)=(𝑟+𝑥)(𝑥+𝑥)+2𝑥 on [0,𝜖]. Then, by 𝐹(0)=𝑟𝑥<𝛾𝑥, there exists a constant 0<𝜖₀<𝜖 such that 𝐹𝜖₀=𝑟+𝜖₀𝑥+𝜖₀+2𝜖₀<𝛾𝑥. (2.14) For the above 𝜖₀>0, there exists a positive integer 𝑖₁≥𝑖₀ such that |𝑥_𝑖|<𝑥+𝜖₀ and ∑^𝑖−1_𝑗=𝑖0|𝑎_𝑖,𝑗|<𝑟+𝜖₀, for any 𝑖≥𝑖₁. By assumption (2.7), we have that for 𝜖₁=𝜖₀/(1+𝑀)>0, there exists a positive integer 𝑖₂≥𝑖₁ such that 𝑏_𝑖<𝜖₁,^𝑖2−1_𝑗=0𝑎_𝑖,𝑗<𝜖₁forany𝑖≥𝑖₂. (2.15) Then, for 𝑖≥𝑖₂ and 𝑐_𝑖=𝑏_𝑖−∑^𝑖2−1_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗(𝑥_𝑗), we have that 𝑐_𝑖≤𝑏_𝑖+∑^𝑖2−1_𝑗=0𝑎_𝑖,𝑗𝑀<𝜖₁+𝜖₁𝑀=𝜖₀,𝑥_𝑖≤∑^𝑖−1_𝑗=𝑖2𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗+𝑐_𝑖<𝑟+𝜖₀𝑥+𝜖₀+𝜖₀<𝑥−𝜖₀,𝑖≥𝑖₂, (2.16) which is a contradiction with the limsup definition. Hence, 𝑥=0 and we obtain lim_𝑖→∞𝑥_𝑖=0.

Note that the third part of (2.5) is equivalent to 𝑟=sup_𝑖≥𝑖0∑^𝑖−1_𝑗=𝑖0|𝑎_𝑖,𝑗|<𝛾=1+min(0,inf_𝑖≥𝑖0𝑎_𝑖,𝑖).

The theorem above gives some conditions on the coefficients 𝑎_𝑖𝑗 of (1.1) for the boundedness of 𝑥_𝑖 which supplement the results in [12, Theorem 2.1]. Moreover, it worths while to compare our result with the ones in [9, Theorem 3.1] and [5, Theorem 4.1]. In order to do that, we assume 𝑎_𝑖𝑖=0, 𝑏_𝑖=0 and then 𝑥₀ is given. In this case, following the line of the proof of Theorem 2.2, we can still show that 𝑥_𝑖 vanishes as 𝑖→∞ provided that (2.5) and the second part of (2.7) hold. Observe that this represents an additional result with respect to [9, Theorem 3.1] and [5, Theorem 4.1] which, involving the sum of the coefficients 𝑎_𝑖,𝑗 on the second index, enlarges the set of conditions for 𝑥_𝑖 to be bounded and convergent to zero. As an example, for equation 𝑥_𝑖=^𝑖−1_𝑗=01(𝑖+1)2^𝑖−𝑗𝑥_𝑗, (2.17) (3.2) in [9] or the sufficient condition in [5] is not satisfied, however (2.5) is fulfilled. Moreover, it is easy to see that, in the convolution case 𝑎_𝑖,𝑗=𝑎_𝑖−𝑗, the third of (2.5) coincides with the known one [5, 10] ^+∞_𝑙=0𝑎_𝑙<1, (2.18) and the second part of (2.7) is implied by (2.5).

Theorem 2.2 turns out to be quite useful in the linear case when (1.1) represents the linearized equation for the global error of a numerical method applied to a Volterra integral equation. In this case, 𝑏_𝑖 represents the local truncation error of the method at the step 𝑖. Thus, if 𝑏_𝑖 is bounded for all 𝑖 and if (2.5) holds, then the error 𝑥_𝑖 is bounded and the bound is given in (2.6).

The following theorem provides some sufficient conditions on the coefficients of (1.1) for the summability of {𝑥_𝑖}^+∞_𝑖=0, which turn out to be less restrictive of those stated by [13, Theorem 2.8].

Theorem 2.3. For (1.1) with (1.2), assume (2.1). If 𝐵=^∞_𝑖=0𝑏_𝑖<+∞,𝐴=sup^∞_{𝑗≥0𝑖=𝑗+1}𝑎_𝑖,𝑗<1, (2.19) then ∑^∞_𝑖=0|𝑥_𝑖|≤𝐵/(1−𝐴)<+∞, and consequently, lim_𝑖→∞𝑥_𝑖=0.

Proof. By (2.3), ^𝑖_𝑘=0𝑥_𝑘≤^𝑖_𝑘=0𝑏_𝑘+^𝑖_𝑘=0^𝑘−1_𝑗=0𝑎_𝑘,𝑗𝑥_𝑗≤^𝑖_𝑘=0𝑏_𝑘+^𝑖−1_𝑗=0^𝑖_𝑘=𝑗+1𝑎_𝑘,𝑗𝑥_𝑗≤^𝑖_𝑘=0𝑏_𝑘+sup^∞_{𝑗≥0𝑘=𝑗+1}𝑎_𝑘,𝑗^𝑖−1_𝑗=0𝑥_𝑗. (2.20) Therefore, by (2.19), we have that ^𝑖_𝑘=0𝑥_𝑘≤∑^𝑖_𝑘=0𝑏_𝑘1−sup_𝑗≥0∑^∞_𝑘=𝑗+1𝑎_𝑘,𝑗≤𝐵1−𝐴<+∞, (2.21) and then, lim_𝑖→∞𝑥_𝑖=0.

In the case (1.1) is linear, 𝑥_𝑖=𝑏_𝑖−^𝑖_𝑗=0𝑎_𝑖,𝑗𝑥_𝑗,𝑖≥0, (2.22) the following theorem is easily proved.

Theorem 2.4. For the linear equation (2.22), assume inf_𝑖≥0𝑎_𝑖,𝑖>−1, and for 𝛾=1+min0,inf_𝑖≥0𝑎_𝑖,𝑖,𝑐_𝑖,𝑗=𝑎_{𝑖−1,𝑗}−𝑎_𝑖,𝑗𝛾𝑐,𝑖−2≥𝑗≥0,_{𝑖,𝑖−1}=1+𝑎_{𝑖−1,𝑖−1}−𝑎_{𝑖,𝑖−1}𝛾𝑑,𝑖≥1,₀=𝑏₀𝛾,𝑑_𝑖=𝑏_𝑖−𝑏_𝑖−1𝛾,𝑖≥1. (2.23)
(i)Suppose thatsup_{0≤𝑗≤𝑖−1}𝑐_𝑖,𝑗=𝐶<+∞,𝐷=sup_𝑖≥0𝑑_𝑖<+∞. (2.24)Then, |𝑥_𝑖|≤(1+𝐶)^𝑖𝐷,𝑖≥0. In particular, if there exists a positive integer𝑖₀such that𝐶₀=sup_𝑖≥𝑖0^𝑖−1_𝑗=𝑖0𝑐_𝑖,𝑗<1, (2.25)then𝑥_𝑖is bounded and|𝑥_𝑖|≤(1+𝐶)^𝑖0𝐷1−𝐶₀<+∞,𝑖≥𝑖₀. (2.26)Moreover, iflim_𝑖→∞𝑏_𝑖−𝑏_𝑖−1=0,lim_𝑖→∞^𝑖0−1_𝑗=0𝑎_𝑖,𝑗=0, (2.27)thenlim_𝑖→∞𝑥_𝑖=0.(ii)If𝐶=sup^∞_{𝑗≥0𝑖=𝑗+1}𝑐_𝑖,𝑗<1,∑𝐷=^∞𝑖=0𝑏_𝑖+1−𝑏_𝑖+𝑏₀𝛾<+∞, (2.28)then∑^∞_𝑖=0|𝑥_𝑖|≤𝐷/(1−𝐶)<+∞, and consequently, lim_𝑖→∞𝑥_𝑖=0.

Proof. By (2.22), we obtain that 1+𝑎_𝑖,𝑖𝑥_𝑖=𝑏_𝑖−𝑏_𝑖−1+1+𝑎_{𝑖−1,𝑖−1}−𝑎_{𝑖,𝑖−1}𝑥_𝑖−1+^𝑖−2_𝑗=0𝑎_{𝑖−1,𝑗}−𝑎_𝑖,𝑗𝑥_𝑗,𝑖≥1. (2.29) Then, we have that 𝑥_𝑖≤𝑑_𝑖+^𝑖−1_𝑗=0𝑐_𝑖,𝑗𝑥_𝑗,𝑖≥1. (2.30) Thus, analogously to the proofs of Theorems 2.2 and 2.3, we obtain the conclusion of this theorem.

3. Nonnegative Coefficients

In this section, we focus on the solutions of (1.1) with (1.2) and 𝑎_𝑖,𝑗≥0,𝑖≥𝑗≥0. (3.1) Such discrete equations are useful, above all, in the investigations on the behavior of the solution of some numerical methods when used to solve nonlinear heat flow in a material with memory (see [14] and the bibliography therein). Let us start with the following lemma, which describes some aspects of the solution of (1.1)-(1.2) with (3.1) when 𝑥_𝑖 has a sign eventually constant for all 𝑖≥0. The utility of this lemma is not in itself, but as an instrument to prove some of the next theorems (see Theorems 3.4, 4.1 and 4.3).

Lemma 3.1. Let {𝑥_𝑖}^∞_𝑖=0 be the solution of (1.1) and assume that
(i)|𝑏_𝑖|≤𝐵and𝑎_𝑖,𝑗≤𝐴_𝑗for each𝑖,𝑗≥0;(ii)there exists𝑖₀>0such that𝑥_𝑖≥0 (resp., 𝑥_𝑖≤0) for any𝑖≥𝑖₀,
then |𝑐_𝑖|≤𝐶,0≤𝑥_𝑖≤𝑐_𝑖(𝑟𝑒𝑠𝑝.,0≥𝑥_𝑖≥𝑐_𝑖)∀𝑖≥𝑖₀, (3.2) where 𝑐_𝑖=𝑏_𝑖−∑^𝑖0−1_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗(𝑥_𝑗) and 𝐶 is a positive constant.
Moreover, assume that one of the following conditions holds:
(iii₁)lim_𝑖→∞𝑐_𝑖=0,(iii₂)𝑎=liminf_𝑗→∞(liminf_𝑖→∞𝑎_𝑖,𝑗)>0and there exists a strictly increasing function𝑓(𝑥) on (−∞,+∞)such that𝑓(0)=0andinf_𝑗≥0𝑓_𝑗(𝑥)≥𝑓(𝑥),𝑥∈[0,+∞) (resp., inf_𝑗≥0𝑓_𝑗(𝑥)≤𝑓(𝑥),𝑥∈(−∞,0]), then lim_𝑖→∞𝑥_𝑖=0.
Furthermore, if, in addition to (𝑖𝑖𝑖₂), there exists a positive constant 𝛿 such that 𝑓(𝑥)≥𝛿𝑥,𝑥∈[0,+∞) (resp., 𝑓(𝑥)≤𝛿𝑥,𝑥∈(−∞,0]), then ∑^∞_𝑖=𝑖0|𝑥_𝑖|≤(sup_𝑖≥𝑖0|𝑐_𝑖|)/(𝛿𝑎)<+∞.

Proof. Since 𝑎_𝑖,𝑗≤𝐴_𝑗,|𝑏_𝑖|≤𝐵 and 𝑓(𝑥) is a continuous function, then |𝑐_𝑖| is bounded. Assume that there exists a nonnegative integer 𝑖₀ such that 𝑥_𝑖≥0 for any 𝑖≥𝑖₀ (the analysis of the case 𝑥_𝑖≤0 for all 𝑖≥𝑖₀ is analogous). Then, by the fact that, for the main hypothesis (1.2), 𝑓_𝑖−𝑗(𝑥)>0 whenever 𝑥>0, we have 0≤𝑥_𝑖=𝑏_𝑖−^𝑖0−1_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗−^𝑖_𝑗=𝑖0𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗≤𝑐_𝑖,𝑖≥𝑖₀. (3.3) Hence, the first part of the lemma is proved. Consider now the two cases (iii₁) and (iii₂) separately.
Case (iii₁): lim_𝑖→∞𝑐_𝑖=0 of course implies lim_𝑖→∞𝑥_𝑖=0.
Case (iii₂): put limsup_𝑖→∞𝑥_𝑖=𝑥. Assume that 𝑥>0, and let {𝑥_𝑖𝑘}^∞_𝑘=0 be a subsequence of {𝑥_𝑖}^∞_𝑖=𝑖0 such that lim_𝑘→∞𝑥_𝑖𝑘=𝑥. Then, one can prove that limsup_𝑘→∞∑^𝑖𝑘_𝑗=𝑖0𝑓(𝑥_𝑗)=+∞. By (1.1) and assumptions, we have 𝑥_𝑖𝑘+^𝑖𝑘_𝑗=𝑖0𝑎_{𝑖𝑘,𝑗}𝑓𝑥_𝑗≤𝑐_𝑖𝑘. (3.4) Therefore, +∞>limsup_𝑘→∞𝑐_𝑖𝑘≥limsup_𝑘→∞𝑥_𝑖𝑘+^𝑖𝑘_𝑗=𝑖0𝑎_{𝑖𝑘,𝑗}𝑓𝑥_𝑗≥𝑥+liminf_𝑗→∞liminf_𝑖→∞𝑎_𝑖,𝑗limsup^𝑖_𝑘→∞^𝑘_𝑗=𝑖0𝑓𝑥_𝑗 (3.5) which is a contradiction because liminf_𝑗→∞(liminf_𝑖→∞𝑎_𝑖,𝑗)>0 and limsup_𝑘→∞∑^𝑖𝑘_𝑗=𝑖0𝑓(𝑥_𝑗)=+∞. Hence, we have 𝑥=lim_𝑖→∞𝑥_𝑖=0.
In addition, suppose that there exists a positive constant 𝛿 such that 𝑓(𝑥)≥𝛿𝑥,𝑥∈[0,+∞). Then, we have that 𝑥_𝑖+𝛿^𝑖_𝑗=𝑖0𝑎_𝑖,𝑗𝑥_𝑗≤𝑐_𝑖,𝑖≥𝑖₀. (3.6) Thus, from 𝑎=liminf_𝑗→∞(liminf_𝑖→∞𝑎_𝑖,𝑗)>0, we conclude that 0≤∑^∞_𝑖=𝑖0𝑥_𝑖≤(sup_𝑖≥𝑖0𝑐_𝑖)/(𝛿𝑎)<+∞.
The proof is completely analogous when there exists a nonnegative integer 𝑖₀ such that 𝑥_𝑖≤0 for any 𝑖≥𝑖₀.

Remark 3.2. Observe that in the linear case (2.22), the last conditions of Lemma 3.1 are satisfied whenever 𝛿=1 and 𝑓(𝑥)=𝑥.

Hereafter, we investigate on the boundedness of the solution of (1.1)-(1.2) when 𝑓(𝑥)≠𝑥,lim_{𝑥→−∞}𝑓(𝑥)>−∞. (3.7)

Lemma 3.3. Let {𝑥_𝑖}^∞_𝑖=0 be the solution of (1.1) with (1.2) and (3.7), and assume that sup_𝑖≥0𝑏_𝑖<+∞,𝜆=sup^𝑖_{𝑖≥0𝑗=0}𝑎_𝑖,𝑗<+∞, (3.8) then |𝑥_𝑖| is bounded.

Proof. Let 𝑏 be the bound for |𝑏_𝑖|,𝑖≥0 and lim_{𝑥→−∞}𝑓(𝑥)=−𝛽>−∞. Let us write ∑^𝑖_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗(𝑥_𝑗) as the sum of the following two contributions: 𝑥_𝑖=𝑏_𝑖−^𝑖_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗=𝑏_𝑖−^𝑖𝑝_𝑙=1𝑎_{𝑖,𝑗𝑙}𝑓_{𝑖−𝑗𝑙}𝑥_𝑗𝑙−^𝑖𝑛_𝑙=1𝑎_{𝑖,𝑘𝑙}𝑓_{𝑖−𝑘𝑙}𝑥_𝑘𝑙, (3.9) where 𝑓_{𝑖−𝑗𝑙}(𝑥_𝑗𝑙)≥0,𝑙=1,…,𝑖_𝑝,𝑓_{𝑖−𝑘𝑙}(𝑥_𝑘𝑙)<0,𝑙=1,…,𝑖_𝑛, and 𝑖_𝑝+𝑖_𝑛=𝑖+1. Therefore, since (1.2), (3.1), (3.7), and (3.8) hold, we have that 𝑥_𝑖≤𝑏−^𝑖𝑛_𝑙=1𝑎_{𝑖,𝑘𝑙}𝑓_{𝑖−𝑘𝑙}𝑥_𝑘𝑙≤𝑏−^𝑖𝑛_𝑙=1𝑎_{𝑖,𝑘𝑙}𝑓𝑥_𝑘𝑙≤𝑥𝑏+𝜆𝛽;_𝑖≥−𝑏−^𝑖𝑝_𝑙=1𝑎_{𝑖,𝑗𝑙}𝑓_{𝑖−𝑗𝑙}𝑥_𝑗𝑙≥−𝑏−𝑓𝑏+𝛽𝜆^𝑖𝑝_𝑙=1𝑎_{𝑖,𝑗𝑙}≥−𝑏−𝜆𝑓𝑏+𝛽𝜆. (3.10) Thus, 𝑥_𝑖 is bounded and the proof is complete.

As an example we consider the equation 𝑥_𝑖=1−^𝑖_𝑗=01(𝑖+1)2^𝑖−𝑗𝑒^𝑥𝑗−11+𝑥²_𝑗, (3.11) in this case 𝑓(𝑥)=𝑒^𝑥−1,𝑏=sup_𝑖≥0|𝑏_𝑖|=1,𝜆=sup_𝑖≥0∑^𝑖_𝑗=01/((𝑖+1)2^𝑖−𝑗)=1, and 𝛽=lim_{𝑥→−∞}(𝑒^𝑥−1)=−1. Hence, −1𝑒≤𝑥_𝑖≤0. (3.12) Another example is given by the explicit equation 𝑥_𝑖=(−1)^𝑖−110^𝑖−1_𝑗=01(𝑖+1)2^𝑖−𝑗𝑒^𝑥𝑗−11+𝑥²_𝑗. (3.13) Here 𝑏=1 and 𝜆=1/40, hence 1−1−𝑒40^−1/40−1≤𝑥_𝑖≤39.40 (3.14) From Figure 1 it is clear that the bounds established by Lemma 3.3 (represented by dotted lines) may be quite sharp. We are able to prove the following result.

Figure 1 

Plot of (3.13) and its bounds given by (3.14).

Theorem 3.4. Assume that 𝑓(𝑥)≠𝑥 is continuous on (−∞,+∞), 𝑟=limsup_𝑖→∞^𝑖−1_𝑗=0𝑎_𝑖,𝑗<+∞,−𝑟𝑓−𝑟𝑓(𝐿)>𝐿,𝑓𝑜𝑟𝑎𝑛𝑦𝐿<0,lim_𝑖→∞𝑏_𝑖=0,lim_𝑖→∞𝑎_𝑖,𝑗=0, (3.15)(3.16) for all 𝑗≥0. Then lim_𝑖→∞𝑥_𝑖=0.

Proof. Let 𝑥=liminf_𝑖→∞𝑥_𝑖 and assume 𝑥<0. Since we are in the hypotheses of Lemma 3.3, |𝑥_𝑖| is bounded and then 𝑀=sup_𝑖≥0|𝑓(𝑥_𝑖)|<+∞. For any fixed 𝜖>0, consider a continuous function 𝐹(𝑥)=−(𝑟+𝑥)𝑓(−(𝑟+𝑥)𝑓(𝑥−𝑥)+𝑥)−2𝑥 on [0,𝜖]. Then, by 𝐹(0)=−𝑟𝑓(−𝑟𝑓(𝑥))>𝑥, there exists a constant 0<𝜖₀<𝜖 such that 𝐹𝜖₀=−𝑟+𝜖₀𝑓−𝑟+𝜖₀𝑓𝑥−𝜖₀+𝜖₀−2𝜖₀>𝑥. (3.17) For the above 𝜖₀>0, there exists a positive integer 𝑖₀ such that 𝑥_𝑖>𝑥−𝜖₀ and ∑^𝑖−1_𝑗=0𝑎_𝑖,𝑗<𝑟+𝜖₀, for any 𝑖≥𝑖₀. By Assumption (3.16), we have that for 𝜖₁=𝜖₀/(1+𝑀)>0, there exists a positive integer 𝑖₁≥𝑖₀ such that 𝑏_𝑖<𝜖₁,^𝑖1−1_𝑗=0𝑎_𝑖,𝑗<𝜖₁forany𝑖≥𝑖₁. (3.18) Then, for 𝑖≥𝑖₁ and 𝑐_𝑖=𝑏_𝑖−∑^𝑖1−1_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗(𝑥_𝑗), we have that 𝑐_𝑖≤𝑏_𝑖+^𝑖1−1_𝑗=0𝑎_𝑖,𝑗𝑀<𝜖₁+𝜖₁𝑀=𝜖₀. (3.19) Let us rewrite (1.1) in the following form: 𝑥_𝑖=𝑐_𝑖−^𝑖𝑝_𝑙=1𝑎_{𝑖,𝑗𝑙}𝑓_{𝑖−𝑗𝑙}𝑥_𝑗𝑙−^𝑖𝑛_𝑙=1𝑎_{𝑖,𝑘𝑙}𝑓_{𝑖−𝑘𝑙}𝑥_𝑘𝑙, (3.20) where 𝑓_{𝑖−𝑗𝑙}(𝑥_𝑗𝑙)≥0, for 𝑙=1,…,𝑖_𝑝 and 𝑓_{𝑖−𝑘𝑙}(𝑥_𝑘𝑙)<0, for 𝑙=1,…,𝑖_𝑛 and 𝑖_𝑝+𝑖_𝑛=𝑖−𝑖₁+1. Thus, 𝑥_𝑖≤𝑐_𝑖−^𝑖𝑛_𝑙=0𝑎_{𝑖,𝑘𝑙}𝑓𝑥_𝑘𝑙<𝜖₀−𝑟+𝜖₀𝑓𝑥−𝜖₀,∀𝑖≥𝑖₁ (3.21) and, since 𝑓(𝑥) is an increasing function, we have that, for all 𝑗_𝑙≥𝑖₁, 𝑓𝑥_𝑗𝑙<𝑓𝜖₀−𝑟+𝜖₀𝑓𝑥−𝜖₀. (3.22) Since we are in the hypothesis that the coefficients 𝑎_𝑖,𝑗 are nonnegative, it follows that −^𝑖𝑝_𝑙=0𝑎_{𝑖,𝑗𝑙}𝑓𝑥_𝑗𝑙>−^𝑖𝑝_𝑙=0𝑎_{𝑖,𝑗𝑙}𝑓𝜖₀−𝑟+𝜖₀𝑓𝑥−𝜖₀>−𝑟+𝜖₀𝑓𝜖₀−𝑟+𝜖₀𝑓𝑥−𝜖. (3.23) In conclusion, from −^𝑖𝑝_𝑙=0𝑎_{𝑖,𝑗𝑙}𝑓𝑥_𝑗𝑙>−𝑟+𝜖₀)𝑓𝜖₀−𝑟+𝜖₀𝑓𝑥−𝜖₀=𝐹𝜖₀+2𝜖₀ (3.24) and by using (3.20), (3.19), and (3.17), the following inequality holds: 𝑥_𝑖≥𝑐_𝑖−^𝑖𝑝_𝑙=0𝑎_{𝑖,𝑗𝑙}𝑓𝑥_𝑗𝑙>−𝜖₀−𝑟+𝜖₀𝑓𝜖₀−𝑟+𝜖₀𝑓𝑥−𝜖=𝐹𝜖₀+𝜖₀>𝑥+𝜖₀. (3.25) This result contradicts the liminf definition. Hence, 𝑥≥0, so 𝑥_𝑖 are eventually nonnegative. Since it is easy to see that we are in the hypotheses of Lemma 3.1 (case (iii₁)), then lim_𝑖→∞𝑥_𝑖=0.

Remark 3.5. Once again, in the convolution case, the first part of (3.15) implies the second one of (3.16).

For the special case 𝑓(𝑥)=(𝑒^𝛼𝑥−1)/𝛼,𝛼>0, we establish the following sufficient condition from Theorem 3.4.

Theorem 3.6. Suppose that 𝑓(𝑥)=(𝑒^𝛼𝑥−1)/𝛼,𝛼>0 and assume that (3.16),𝑡ℎ𝑒𝑓𝑖𝑟𝑠𝑡𝑝𝑎𝑟𝑡𝑜𝑓(3.15)𝑤𝑖𝑡ℎ𝑟≤1 (3.26) hold, then the solution 𝑥_𝑖 of (1.1) tends to zero as 𝑖 tends to infinity.

Proof. Put 𝜑(𝑥)=−𝑟𝑓(𝑥),−∞<𝑥<+∞. Since 𝜑^′(𝑥)=−𝑟𝑒^𝛼𝑥<0 for 𝑥∈(−∞,+∞), hence 𝜑(𝑥) is a strictly monotone decreasing function in (−∞,+∞). Now, we will prove that 𝜑(𝜑(𝑥))>𝑥, for −∞<𝑥<0.
Let 𝑔(𝑥)=𝜑(𝜑(𝑥))−𝑥 for −∞<𝑥<0. Then we have that 𝑔(𝑥)=𝑟1−exp−𝑟𝑒^𝛼𝑥−1𝛼𝑔−𝑥,_′(𝑥)=𝑟𝑟𝑒^𝛼𝑥exp𝑟−𝑟𝑒^𝛼𝑥−1. (3.27) By recalling that 𝑟<1 and 𝑥<0, we have 0≤𝑟𝑒^𝛼𝑥<1. Since the function 𝑦𝑒^−𝑦 is increasing for 0≤𝑦≤1, there results (𝑟𝑒^𝛼𝑥)𝑒^{−𝑟𝑒𝛼𝑥}≤𝑒⁻¹. Thus, 𝑔^′(𝑥)≤𝑟exp(𝑟−1)−1≤𝑟−1≤0. Hence, we have that 𝜑(𝜑(𝑥))>𝑥, for −∞<𝑥<0. Thus, for 𝑓(𝑥)=(𝑒^𝛼𝑥−1)/𝛼 with 𝛼>0, the second of (3.15) is true and, by Theorem 3.4, we have lim_𝑖→∞𝑥_𝑖=0.

Remark 3.7. From the proof of Theorem 3.6 it is clear that the second part of (3.15) is satisfied by 𝑓(𝑥)=(𝑒^𝛼𝑥−1)/𝛼(𝛼>0). By playing with 𝛼, this allows us to consider a wide variety of functions 𝑓_𝑖 which satisfy (1.2). For instance, in the cases 𝛼=1/2 and 𝛼=1 the stained areas in Figure 2 represent the admissible regions for the functions 𝑓_𝑖,𝑖=0,1,…, respectively (the solid lines show, as an example, the graphs of 𝑓_𝑖(𝑥)=𝑥/(1+𝑥²) and 𝑓_𝑖(𝑥)=(𝑒^𝑥−1)/(1+𝑥²)).

(a)

(b)

Figure 2 

Plots of some admissible regions for 0≤𝑎𝑖,𝑗≤𝑎𝑖−1,𝑗,0≤𝑗≤𝑖−1,|||𝑓𝑖−1(𝑥)|||≥|||𝑓𝑖(𝑥)|||,𝑖≥1.(4.1) according to Theorem 3.6.

4. Monotonic Nonnegative Coefficients

In this section, for (1.1), first, we consider the case that 0≤𝑎_𝑖,𝑗≤𝑎_{𝑖−1,𝑗},0≤𝑗≤𝑖−1,𝑓_𝑖−1(𝑥)≥𝑓_𝑖(𝑥),𝑖≥1. (4.1) We provide the following theorem which generalizes [15, Theorem 2.1] to the nonlinear case.

Theorem 4.1. In addition to condition (4.1), suppose that 𝑏_𝑖≥𝑏_𝑖−1≥0resp.,𝑏_𝑖≤𝑏_𝑖−1≤0,𝑖≥1. (4.2) Then, any solution 𝑥_𝑖 of (1.1) satisfies 0≤𝑥_𝑖≤𝑏_𝑖 (resp., 0≥𝑥_𝑖≥𝑏_𝑖), 𝑖≥0. Moreover, if |𝑏_𝑖|≤𝐵, for all 𝑖≥0,𝑎=liminf_𝑗→∞(lim_𝑖→∞𝑎_𝑖,𝑗)>0 and there exists a strictly increasing function 𝑓(𝑥) on (−∞,+∞) such that 𝑓(0)=0 and inf_𝑗≥0𝑓_𝑗(𝑥)≥𝑓(𝑥),𝑥∈[0,+∞) (resp., inf_𝑗≥0𝑓_𝑗(𝑥)≤𝑓(𝑥),𝑥∈(−∞,0]), then lim_𝑖→∞𝑥_𝑖=0.
In addition, if there exists a positive constant 𝛿 such that 𝑓(𝑥)≥𝛿𝑥,𝑥∈[0,+∞) (resp., 𝑓(𝑥)≤𝛿𝑥,𝑥∈(−∞,0]), then ∑^∞_𝑖=0|𝑥_𝑖|≤(sup_𝑖≥0|𝑏_𝑖|)/(𝛿𝑎)<+∞.

Proof. We prove the theorem in the case 𝑏_𝑖≥𝑏_𝑖−1≥0,𝑖≥1, the proof for 𝑏_𝑖≤𝑏_𝑖−1≤0,𝑖≥1 is perfectly symmetric. Then by (1.1), we have that 𝑥₀+𝑎_0,0𝑓₀𝑥₀=𝑏₀≥0, (4.3) hence, for the properties of 𝑓_𝑖(𝑥) described in (1.2), it has to be 𝑥₀≥0.
Proceeding by induction, suppose that 𝑥_𝑗≥0,0≤𝑗≤𝑖−1,𝑖≥1. By (1.1), 𝑥_𝑖+𝑎_𝑖,𝑖𝑓₀𝑥_𝑖=𝑏_𝑖−∑^𝑖−1_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗,0=𝑥_𝑖−1−𝑏_𝑖−1+∑^𝑖−1_𝑗=0𝑎_{𝑖−1,𝑗}𝑓_{𝑖−𝑗−1}𝑥_𝑗, (4.4) and hence, by adding the two relations and taking into account that, for the second part of (4.1), 𝑓_{𝑖−𝑗−1}(𝑥)≥𝑓_𝑖−𝑗(𝑥), we have that 𝑥_𝑖+𝑎_𝑖,𝑖𝑓₀𝑥_𝑖≥𝑏_𝑖−𝑏_𝑖−1+𝑥_𝑖−1+^𝑖−2_𝑗=0𝑎_{𝑖−1,𝑗}−𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗≥𝑏_𝑖−𝑏_𝑖−1≥0,𝑖≥1. (4.5) So we have that 𝑥_𝑖≥0,𝑖≥0 and, from (1.1), 𝑥_𝑖≤𝑏_𝑖,𝑖≥0.
Thus, we are in the hypotheses of Lemma 3.1 part (iii₂) and, hence, we get the thesis.

Observe that when (1.1) is linear, the last condition of Theorem 4.1 is satisfied by choosing 𝛿=1 and 𝑓(𝑥)=𝑥. In this case, the hypotheses of Theorem 4.1 include, as particular cases, those of [15, Theorem 2.1]. In particular we note that, as Theorem 4.1 prove the summability of the solution 𝑥_𝑖, it is interesting when applied to the equation satisfied by the fundamental matrix of a Volterra difference equation (see, e.g., [15, equation (1.4)]). Namely, in [15] it is underlined that such a result can be employed in the study of the stability of some numerical methods.

A simple application of Theorem 4.1 in the linear case is given by the following example.

Example 4.2. Let us consider the difference equation 𝑥_𝑖=𝑏−^𝑖_𝑗=0𝑐𝑥_𝑗,𝑖≥0, (4.6) whose solution is given by 𝑥_𝑖=𝑏(1+𝑐)^𝑖+1,𝑖≥0. (4.7) Then, for 𝑎>0 and 𝑏≥0, all the conditions in Theorem 4.1 are satisfied with 𝛿=1, which implies lim_𝑖→∞𝑥_𝑖=0 and ∑^+∞_𝑖=0𝑥_𝑖≤(𝑏/𝑎)<+∞. Observe that in this case the bound coincides with the exact value of the sum of the series.

Next, we provide the following theorem whose proof is a direct extension of the proof of Crisci et al. [6, Theorem 2.1], which gives a priori bound for the solution 𝑥_𝑖 of (1.1) depending on the forcing terms 𝑏_𝑖.

Theorem 4.3. In addition to the conditions (4.1), assume that sup_𝑖≥1^𝑖−1_𝑗=0𝑏_𝑗+1−𝑏_𝑗<+∞. (4.8) Then, any solution 𝑥_𝑖 of (1.1) is bounded and satisfies 𝑥_𝑖≤^𝑖−1_𝑗=0𝑏_𝑗+1−𝑏_𝑗+𝑏₀,𝑖≥0. (4.9) Moreover, suppose that 𝑎=liminf_𝑗→∞(lim_𝑖→∞𝑎_𝑖,𝑗)>0 and there exists a strictly increasing function 𝑓(𝑥) on (−∞,+∞) such that 𝑓(0)=0 and inf_𝑗≥0𝑓_𝑗(𝑥)≥𝑓(𝑥),𝑥∈[0,+∞) (resp., inf_𝑗≥0𝑓_𝑗(𝑥)≤𝑓(𝑥),𝑥∈(−∞,0]), then lim_𝑖→∞𝑥_𝑖=0.
In addition, if there exists a positive constant 𝛿 such that 𝑓(𝑥)≥𝛿𝑥,𝑥∈[0,+∞) (resp., 𝑓(𝑥)≤𝛿𝑥,𝑥∈(−∞,0]), then ∑^∞_𝑖=0|𝑥_𝑖|≤(2/𝛿𝑎)(∑^∞_𝑗=0|𝑏_𝑗+1−𝑏_𝑗|+|𝑏₀|)<+∞.

Proof. Consider the two possible subcases: (a) 𝑥₀≥0, and (b) 𝑥₀<0.
(a) Assume 𝑥₀≥0.
If 𝑥_𝑗≥0,0≤𝑗≤𝑖, then by (1.1), we get 𝑏_𝑖≥0 and 0≤𝑥_𝑖≤𝑏_𝑖. Hence, (4.9) holds if {𝑥_𝑗}^𝑖_𝑗=0 is oscillatory about 0. Let 𝑚₀=−1 and denote by 𝑙₁ the time moment of the first passage of the solution 𝑥_𝑖 through the zero, that is, 𝑥_𝑗≥0for𝑚₀+1=0≤𝑗≤𝑙₁,𝑥_𝑙1+1<0. (4.10) The time moment of the following passage through the zero of the solution after 𝑙₁ is denoted by 𝑚₁, that is, 𝑥_𝑗≤0for𝑙₁+1≤𝑗≤𝑚₁,𝑥_𝑚1+1>0. (4.11) In a similar way, we introduce the indexes 𝑙_𝑝,𝑚_𝑝,𝑝≥1 as follows: 𝑥_𝑗≥0for𝑚_𝑝−1+1≤𝑗≤𝑙_𝑝,𝑥_𝑙𝑝+1𝑥<0,_𝑗≤0for𝑙_𝑝+1≤𝑗≤𝑚_𝑝,𝑥_𝑚𝑝+1>0. (4.12)
(1) Consider that 𝑚_𝑘+1≤𝑖≤𝑙_𝑘+1, hence 𝑥_𝑖≥0. Then, from (1.1), we have that 𝑥_𝑖=𝑏_𝑖−^𝑖_{𝑗=𝑚𝑘+1}𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗−^𝑚𝑘_𝑗=0𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗,0=𝑥_𝑚𝑘−𝑏_𝑚𝑘+^𝑚𝑘_{𝑗=𝑙𝑘+1}𝑎_{𝑚𝑘,𝑗}𝑓_{𝑚𝑘−𝑗}𝑥_𝑗+^𝑙𝑘_𝑗=0𝑎_{𝑚𝑘,𝑗}𝑓_{𝑚𝑘−𝑗}𝑥_𝑗,0=−𝑥_𝑙𝑘+𝑏_𝑙𝑘−^𝑙𝑘_{𝑗=𝑚𝑘−1+1}𝑎_{𝑙𝑘,𝑗}𝑓_{𝑙𝑘−𝑗}𝑥_𝑗−^𝑚𝑘−1_𝑗=0𝑎_{𝑙𝑘,𝑗}𝑓_{𝑙𝑘−𝑗}𝑥_𝑗),0=𝑥_𝑚𝑘−1−𝑏_𝑚𝑘−1+^𝑚𝑘−1_{𝑗=𝑙𝑘−1+1}𝑎_{𝑚𝑘−1,𝑗}𝑓_{𝑚𝑘−1−𝑗}𝑥_𝑗+^𝑙𝑘−1_𝑗=0𝑎_{𝑚𝑘−1,𝑗}𝑓_{𝑚𝑘−1−𝑗}𝑥_𝑗,⋯0=𝑥_𝑚1−𝑏_𝑚1+^𝑚1_𝑗=𝑙1+1𝑎_𝑚1,𝑗𝑓_𝑚1−𝑗𝑥_𝑗+^𝑙1_𝑗=0𝑎_𝑚1,𝑗𝑓_𝑚1−𝑗𝑥_𝑗,0=−𝑥_𝑙1+𝑏_𝑙1−^𝑙1_𝑗=0𝑎_𝑙1,𝑗𝑓_𝑙1−𝑗𝑥_𝑗, (4.13) where every summation of the type ∑^𝑙𝑟_{𝑗=𝑚𝑟−1+1} involves only positive 𝑥_𝑗, while ∑^𝑚𝑟_{𝑗=𝑙𝑟+1} the negative ones. Now observe that, for 𝑙_𝑘+1≤𝑗≤𝑚_𝑘, by using (4.1) and the fact that 𝑚_𝑘<𝑖, we have |𝑓_{𝑚𝑘−𝑗}(𝑥_𝑗)|≥|𝑓_𝑖−𝑗(𝑥_𝑗)|, furthermore 𝑥_𝑗≤0, because of (4.12), hence both 𝑓_𝑖−𝑗(𝑥_𝑗) and 𝑓_{𝑚𝑘−𝑗}(𝑥_𝑗) are less than or equal to zero, thus 𝑓_{𝑚𝑘−𝑗}(𝑥_𝑗)≤𝑓_𝑖−𝑗(𝑥_𝑗). By using these considerations it is easy to see that the following inequality holds: ^𝑚𝑘_{𝑗=𝑙𝑘+1}𝑎_{𝑚𝑘,𝑗}𝑓_{𝑚𝑘−𝑗}𝑥_𝑗−𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗≤^𝑚𝑘_{𝑗=𝑙𝑘+1}𝑎_{𝑚𝑘,𝑗}−𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗. (4.14) With analogues considerations we get −^𝑙𝑘_{𝑗=𝑚𝑘−1+1}𝑎_{𝑙𝑘,𝑗}𝑓_{𝑙𝑘−𝑗}𝑥_𝑗−𝑎_{𝑚𝑘,𝑗}𝑓_{𝑚𝑘−𝑗}𝑥_𝑗+𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗≤−^𝑙𝑘_{𝑗=𝑚𝑘−1+1}𝑎_{𝑙𝑘,𝑗}−𝑎_{𝑚𝑘,𝑗}+𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗,⋯−^𝑙1_𝑗=0𝑎_𝑙1,𝑗𝑓_𝑙1−𝑗𝑥_𝑗−𝑎_𝑚1,𝑗𝑓_𝑚1−𝑗𝑥_𝑗+⋯+𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗≤−^𝑙1_𝑗=0𝑎_𝑙1,𝑗−𝑎_𝑚1,𝑗+⋯+𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗. (4.15) By adding each side of (4.13) and taking into account (4.14), (4.15), it comes out that 0≤𝑥_𝑖≤𝑏_𝑖−𝑏_𝑚𝑘+𝑏_𝑙𝑘−𝑏_𝑚𝑘−1+⋯−𝑏_𝑚1+𝑏_𝑙1+𝑥_𝑚𝑘−𝑥_𝑙𝑘+𝑥_𝑚𝑘−1−⋯+𝑥_𝑚1−𝑥_𝑙1−^𝑖_{𝑗=𝑚𝑘+1}𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗+^𝑚𝑘_{𝑗=𝑙𝑘+1}𝑎_{𝑚𝑘,𝑗}−𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗−^𝑙𝑘_{𝑗=𝑚𝑘−1+1}𝑎_{𝑙𝑘,𝑗}−𝑎_{𝑚𝑘,𝑗}+𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗+^𝑚𝑘−1_{𝑗=𝑙𝑘−1+1}𝑎_{𝑚𝑘−1,𝑗}−𝑎_{𝑙𝑘,𝑗}+𝑎_{𝑚𝑘,𝑗}−𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗⋯+^𝑚1_𝑗=𝑙1+1𝑎_𝑚1,𝑗−𝑎_𝑙2,𝑗+𝑎_𝑚2,𝑗−𝑎_𝑙3,𝑗+⋯+𝑎_{𝑚𝑘,𝑗}−𝑎_𝑖,𝑗𝑓_𝑖−𝑗𝑥_𝑗−^𝑙1_𝑗=0𝑎_𝑙1,𝑗−𝑎_𝑚1,𝑗+𝑎_𝑙2,𝑗−𝑎_𝑚2,𝑗+⋯−𝑎_{𝑚𝑘,𝑗}+𝑎_𝑖,𝑗𝑓_𝑖−𝑗(𝑥_𝑗. (4.16) By using the monotonicity of 𝑎_𝑖,𝑗 stated by (4.1) and the main hypothesis (1.2), taking into account (4.12), we have that 0≤𝑥_𝑖≤𝑏_𝑖−𝑏_𝑚𝑘+𝑏_𝑙𝑘−𝑏_𝑚𝑘−1+⋯−𝑏_𝑚1+𝑏_𝑙1≤^𝑖−1_𝑗=0𝑏_𝑗−𝑏_𝑗−1+𝑏₀. (4.17)
(2) Consider that 𝑙_𝑘+1≤𝑖≤𝑚_𝑘, hence 𝑥_𝑖≤0. Proceeding as above, we have 0≥𝑥_𝑖≥−^𝑖−1_𝑗=0𝑏_𝑗−𝑏_𝑗−1+𝑏₀. (4.18) Hence, from (1) and (2), we obtain (4.9). Part (b) of the proof is essentially mirror-like of part (a) and leads once again to (4.9). Thus, any solution 𝑥_𝑖 of (1.1) is bounded and satisfies (4.9).
Moreover, suppose that 𝑎=liminf_𝑗→∞(lim_𝑖→∞𝑎_𝑖,𝑗)>0 and there exists a strictly increasing function 𝑓(𝑥) on (−∞,+∞) such that 𝑓(0)=0 and inf_𝑗≥0𝑓_𝑗(𝑥)≥𝑓(𝑥),𝑥∈[0,+∞) (resp., inf_𝑗≥0𝑓_𝑗(𝑥)≤𝑓(𝑥),𝑥∈(−∞,0]).
If there exists a nonnegative integer 𝑖₀ such that 𝑥_𝑖≥0 (resp., 𝑥_𝑖≤0) for any 𝑖≥𝑖₀, since |𝑏_𝑖|≤∑^𝑖−1_𝑗=0|𝑏_𝑗−𝑏_𝑗−1|+|𝑏₀| and 0≤𝑎_𝑖,𝑗≤𝑎_0,𝑗, for all 𝑗, we are in the hypotheses of Lemma 3.1 part (iii₂) and we obtain lim_𝑖→∞𝑥_𝑖=0. On the contrary, if such an index does not exist, let 𝑥₀≥0 and consider the extract subsequence {𝑥_𝑖𝑝}^∞_𝑝=0 of all the positive values in {𝑥_𝑖}^∞_𝑖=0. Assume that 𝑥=limsup_𝑝→+∞𝑥_𝑖𝑝>0, then limsup^𝑝_{𝑝→+∞𝑗=0}𝑓𝑥_𝑖𝑗=+∞. (4.19) Taking into account (4.12), there exists an index 𝑘≥0 such that 𝑚_𝑘+1≤𝑖_𝑝≤𝑙_𝑘+1, then 𝑥_𝑖𝑝 plays the role of 𝑥_𝑖 in (4.13) and, analogously to (4.16), we have 𝑥_𝑖𝑝+^𝑖𝑝_{𝑗=𝑚𝑘+1}𝑎_{𝑖𝑝,𝑗}𝑓_{𝑖𝑝−𝑗}𝑥_𝑗+^𝑙𝑘_{𝑗=𝑚𝑘−1+1}𝑎_{𝑙𝑘,𝑗}−𝑎_{𝑚𝑘,𝑗}+𝑎_{𝑖𝑝,𝑗}𝑓_{𝑖𝑝−𝑗}𝑥_𝑗++⋯_𝑙1𝑗=0𝑎_𝑙1,𝑗−𝑎_𝑚1,𝑗+𝑎_𝑙2,𝑗−𝑎_𝑚2,𝑗+⋯−𝑎_{𝑚𝑘,𝑗}+𝑎_{𝑖𝑝,𝑗}𝑓_{𝑖𝑝−𝑗}𝑥_𝑗≤𝑏_𝑖𝑝−𝑏_𝑚𝑘+𝑏_𝑙𝑘−𝑏_𝑚𝑘−1+⋯−𝑏_𝑚1+𝑏_𝑙1+𝑥_𝑚𝑘−𝑥_𝑙𝑘+𝑥_𝑚𝑘−1−⋯+𝑥_𝑚1−𝑥_𝑙1+^𝑚𝑘_{𝑗=𝑙𝑘+1}𝑎_{𝑚𝑘,𝑗}−𝑎_{𝑖𝑝,𝑗}𝑓_{𝑖𝑝−𝑗}𝑥_𝑗+^𝑚𝑘−1_{𝑗=𝑙𝑘−1+1}𝑎_{𝑚𝑘−1,𝑗}−𝑎_{𝑙𝑘,𝑗}+𝑎_{𝑚𝑘,𝑗}−𝑎_{𝑖𝑝,𝑗}𝑓_{𝑖𝑝−𝑗}𝑥_𝑗⋯+^𝑚1_𝑗=𝑙1+1𝑎_𝑚1,𝑗−𝑎_𝑙2,𝑗+𝑎_𝑚2,𝑗−𝑎_𝑙3,𝑗+⋯+𝑎_{𝑚𝑘,𝑗}−𝑎_{𝑖𝑝,𝑗}𝑓_{𝑖𝑝−𝑗}𝑥_𝑗≤𝑏_𝑖𝑝−𝑏_𝑚𝑘+𝑏_𝑙𝑘−𝑏_𝑚𝑘−1+⋯−𝑏_𝑚1+𝑏_𝑙1≤^𝑖𝑝−1_𝑗=0𝑏_𝑗+1−𝑏_𝑗+𝑏₀<+∞. (4.20) Hence, since 𝑓(𝑥)≤𝑓_𝑗(𝑥), for all 𝑥≥0, we have that 𝑥_𝑖𝑝+^𝑖𝑝_{𝑗=𝑚𝑘+1}𝑎_{𝑖𝑝,𝑗}𝑓𝑥_𝑗+^𝑙𝑘_{𝑗=𝑚𝑘−1+1}𝑎_{𝑙𝑘,𝑗}−𝑎_{𝑚𝑘,𝑗}+𝑎_{𝑖𝑝,𝑗}𝑓𝑥_𝑗++⋯_𝑙1𝑗=0𝑎_𝑙1,𝑗−𝑎_𝑚1,𝑗+𝑎_𝑙2,𝑗−𝑎_𝑚2,𝑗+⋯−𝑎_{𝑚𝑘,𝑗}+𝑎_{𝑖𝑝,𝑗}𝑓𝑥_𝑗≤^𝑖𝑝−1_𝑗=0𝑏_𝑗+1−𝑏_𝑗+𝑏₀<+∞, (4.21) and so, since only positive quantities are involved, we get 𝑎^𝑖𝑝_{𝑗=𝑚𝑘+1}𝑓𝑥_𝑗+^𝑙𝑘_{𝑗=𝑚𝑘−1+1}𝑓𝑥_𝑗+⋯+^𝑙1_𝑗=0𝑓𝑥_𝑗≤^𝑖𝑝−1_𝑗=0𝑏_𝑗+1−𝑏_𝑗+𝑏₀<+∞. (4.22)Passing to the limsup as 𝑝→+∞, we have that𝑎limsup_𝑝→+∞^𝑖𝑝_{𝑗=𝑚𝑘+1}𝑓𝑥_𝑗+^𝑙𝑘_{𝑗=𝑚𝑘−1+1}𝑓𝑥_𝑗+⋯+^𝑙1_𝑗=0𝑓𝑥_𝑗<+∞. (4.23) Taking into account that all the 𝑥_𝑗 involved in the summations above form the extract {𝑥_𝑖𝑝}^+∞_𝑝=0 of the positive values in {𝑥_𝑖}^+∞_𝑖=0, we get limsup_𝑝→+∞∑^𝑝_𝑗=0𝑓(𝑥_𝑖𝑗)<+∞, which is a contradiction with (4.19), so 𝑥=0. An analogous proof on the extract subsequence of all negative values of {𝑥_𝑖}^+∞_𝑖=0 leads to liminf_𝑛→+∞𝑥_𝑖𝑛=0. The same happens when 𝑥₀<0. Hence, in conclusion, we obtain that lim_𝑖→∞𝑥_𝑖=0.
In addition, suppose that there exists a positive constant 𝛿 such that 𝑓(𝑥)≥𝛿𝑥,𝑥∈[0,+∞). Then, by (4.22) and the fact that 𝑎 is strictly positive, we conclude that 0≤∑^∞_𝑖=0max(0,𝑥_𝑖)≤1/𝛿𝑎(∑^∞_𝑗=0|𝑏_𝑗+1−𝑏_𝑗|+|𝑏₀|). Similarly, we obtain that 0≥∑^∞_𝑖=0min(0,𝑥_𝑖)≥−1/𝛿𝑎(∑^∞_𝑗=0|𝑏_𝑗+1−𝑏_𝑗|+|𝑏₀|). Hence, ∑^∞_𝑖=0|𝑥_𝑖|≤2/𝛿𝑎(∑^∞_𝑗=0|𝑏_𝑗+1−𝑏_𝑗|+|𝑏₀|)<+∞.

Acknowledgment

This research was partially supported by Waseda University grant for special research Projects 2006B–167 and Scientific Research (c), no. 19540229 of Japan Society for the Promotion of Science.