On the Stability of Numerical Methods for Nonlinear Volterra Integral Equations

Abstract

Here we investigate the behavior of the analytical and numerical solution of a nonlinear second kind Volterra integral equation where the linear part of the kernel has a constant sign and we provide conditions for the boundedness or decay of solutions and approximate solutions obtained by Volterra Runge-Kutta and Direct Quadrature methods.

1. Introduction

In this paper we consider Volterra integral equations (VIEs) of Hammerstein type, that is, 𝑦(𝑡)=𝑔(𝑡)+^𝑡₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠, (1.1) where 𝑔,𝑘, and 𝑓 are given real-valued functions. This type of equation arises in problems related to evolutionary processes in ecology, in nuclear reactors and in control theory (see, e.g., [1], [2, Chapter  1], [3, Chapter  2] and references therein). There has been interest in the literature over many years in studying the asymptotic behavior of the solution of (1.1) and, in particular, many results appeared on the existence of solutions that decay to zero as 𝑡→∞ for the convolution version of (1.1) (𝑘(𝑡,𝑠)=𝑘(𝑡−𝑠), see, e.g., [4, 5]). It is well known that in the numerical solution of this type of equations, the discrete approximation should emulate the features of the analytical solution, hence an analysis is due which correlates the behavior of the continuous and the discrete problem. There is a large literature in this sense, however the most of the results refer to the linear version of (1.1) and many others to the convolution case [6–9] which arises commonly in models of phenomena where no ageing or seasoning and therefore no explicit dependence upon time are present. In this paper the kernel 𝑘 in (1.1) is not necessarily of convolution type and we only assume that the function 𝑓 is regular enough to guarantee the existence and the uniqueness of the solution and that 𝑔 and 𝑘 are continuous functions (see, e.g., [10], [11, Chapter  4]). Our aim is to give sufficient conditions under which the analytical solution of (1.1) and the numerical one provided by the “most popular” linear (Volterra Direct Quadrature (VDQ)) and nonlinear (Volterra Runge-Kutta (VRK)) methods have the same behavior. This of course in order to assure that the numerical solution well emulates the analytical one. The starting idea for this kind of analysis comes out from an investigation that the authors carried out in 2008 on a nonlinear Volterra discrete equation [12] which turns out to be a discrete analogue of (1.1). The paper is organized as follows. Section 2 is devoted to establishing bounds on the analytical solution of (1.1), on [0,𝑇] or on [0,∞), under certain conditions on 𝑘(⋅,⋅),𝑓(⋅), and 𝑔(⋅). Section 3 is devoted to obtaining similar bounds for approximate solutions obtained by the two types of methods mentioned above. In Section 3 some numerical experiments are reported.

2. The Behavior of the Analytical Solution

In this section we prove some results on the behavior of the solution 𝑦(𝑡) of (1.1). It is assumed that (1.1) is satisfied for 𝑡∈[0,𝑇], where 𝑇 is a positive constant of for 𝑡∈[0,∞). The following theorem gives the conditions for 𝑦(𝑡) to be bounded on [0,𝑇], which is the extended nonnegative real line in the latter case.

Theorem 2.1. Suppose that the following set of hypotheses hold:(i)∃𝑓≥0∶𝑓(𝑥)≥−𝑓,∀𝑥∈ℝ,(ii)limsup_{𝑥→−∞}𝑓(𝑥)=𝑓<+∞,(iii)𝑘(𝑡,𝑠)≤0,∀𝑠≤𝑡,𝑡∈[0,𝑇], (iv)∃𝐾>0∶sup_{𝑡∈[0,𝑇]}∫^𝑡₀|𝑘(𝑡,𝑠)|𝑑𝑠≤𝐾,∀𝑡∈[0,𝑇],(v)∃𝐺₁,𝐺₂∶𝐺₁≤𝑔(𝑡)≤𝐺₂,∀𝑡∈[0,𝑇],then 𝑦(𝑡) is bounded for 𝑡∈[0,𝑇] and it is 𝐺₁−𝐾𝐹^∗≤𝑦(𝑡)≤𝐺₂+𝑓𝐾, (2.1) where 𝐹^∗ is a given positive constant.

Proof. From (ii) we have that ∃𝛿∶𝑓(𝑥)≤𝑓,∀𝑥∈(−∞,𝛿). (2.2) From (i) and (iii) there results 𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))≤−𝑓𝑘(𝑡,𝑠), hence 𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠≤𝑓^𝑡₀^𝑡₀|𝑘(𝑡,𝑠)|𝑑𝑠≤𝑓𝐾,∀𝑡∈[0,𝑇]. (2.3) From here and (v) there results 𝑦(𝑡)≤𝐺₂+𝑓𝐾. (2.4) Hence, 𝑓(𝑦)≤max{𝑓,max_{𝑥∈[𝛿,𝐺2+}𝑓𝐾^]𝑓(𝑥)}, (2.5) where 𝛿 is defined in (2.2). This assures that the positive constant 𝐹^∗ given by 𝐹^∗=max{0,max{𝑓,max_{𝑥∈[𝛿,𝐺2+}𝑓𝐾^]𝑓(𝑥)}} (2.6) is such that ∀𝑡∈[0,𝑇],𝑓(𝑦)≤𝐹^∗. Therefore, ^𝑡₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠≥𝐹^∗𝑡₀𝑘(𝑡,𝑠)𝑑𝑠≥−𝐾𝐹^∗. (2.7) Taking into account (v) we obtain 𝑦(𝑡)≥𝐺₁−𝐾𝐹^∗, (2.8) which, together with (2.4) prove the result.

Remark 2.2. The essence of the hypotheses of Theorem 2.1 are the following: the integrand 𝑘(𝑡,𝑠)𝑓(𝑦(𝑠)) has a negative linear part (𝑘(𝑡,𝑠)), hence, when 𝑓(𝑦) is allowed to become positive and large, it is multiplied by a negative quantity so that the positive growth of 𝑦 is avoided. On the other hand, when 𝑓(𝑦) is negative, it is also bounded from below by hypothesis (i).

In the examples below we see some applications of the results proved in Theorem 2.1. Of course, the bound (2.1) is not necessarily sharp (it depends on the shape of the function 𝑔) and this is clear in Example 2.3, nevertheless the purpose of the previous result is to give some information about the qualitative behavior of the solution also on an unbounded interval.

Example 2.3. Consider (1.1) with 𝑡∈[0,1],𝑘(𝑡,𝑠)=−𝑒^−𝑠/(1+𝑡²),𝑓(𝑦)=(𝑒^3𝑦−1)/3 and 𝑔(𝑡) such that 𝑦(𝑡)=𝑡−0.5. By Theorem 2.1 we obtain −1.12≤𝑦(𝑡)≤1.85,∀𝑡∈[0,1].

Finer bounds are obtained in the following examples.

Example 2.4. Consider (1.1) with 𝑇∈[0,1],𝑘(𝑡,𝑠)=−𝑒^−𝑠/(1+𝑡²),𝑓(𝑦)=−sin²𝑦, and 𝑔(𝑡) such that 𝑦(𝑡)=𝜋/2. By (2.1) we obtain 1.23≤𝑦(𝑡)≤1.98,∀𝑡∈[0,1].

Example 2.5. Consider (1.1) with 𝑇∈[0,1],𝑘(𝑡,𝑠)=−𝑒^−𝑠/(1+𝑡²),𝑓(𝑦)=𝑒^𝑦, and 𝑔(𝑡) such that 𝑦(𝑡)=𝑡²−0.5. By (2.1) we obtain −1.22≤𝑦(𝑡)≤0.75,∀𝑡∈[0,1].

In order to prove that the solution 𝑦(𝑡) of (1.1) tends to zero as 𝑡→∞, we need the following lemma.

Lemma 2.6. Assume that(i)∃𝑓≥0∶𝑓(𝑥)≥−𝑓,∀𝑥∈ℝ,(ii)𝑥𝑓(𝑥)≥0,∀𝑥∈ℝ,(iii)𝑘(𝑡,𝑠)≤0,∀𝑠≤𝑡,𝑡≥0,(iv)lim_𝑡→∞𝑘(𝑡,𝑠)=0,(v)lim_𝑡→∞𝑔(𝑡)=0,(vi)𝑦(𝑡) ultimately nonnegative for 𝑡≥0,then lim_𝑡→∞𝑦(𝑡)=0.

Proof. From hypothesis (vi) there exists 𝑡 such that 𝑦(𝑡)≥0, for 𝑡>𝑡. Assume 𝑡>𝑡 and write 𝑦(𝑡)=𝑔(𝑡)+^𝑡₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠+^𝑡_𝑡𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠, (2.9) in view of (ii) and (iii), we have 0≤𝑦(𝑡)≤𝑔(𝑡)+^𝑡₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠, (2.10) which, taking into account (i), gives 0≤𝑦(𝑡)≤𝑔(𝑡)−𝑓^𝑡₀𝑘(𝑡,𝑠)𝑑𝑠. (2.11) Finally, hypotheses (iv) and (v) and the Squeeze theorem prove the result stated in the theorem.

Now the following theorem, which is the continuous analogue of Theorem 3.4 in [12] can be proved.

Theorem 2.7. Assume that (i)∃𝑓≥0∶𝑓(𝑥)≥−𝑓,∀𝑥∈ℝ,(ii)𝑓 is nondecreasing, 𝑥𝑓(𝑥)≥0,∀𝑥∈ℝ,(iii)𝑘(𝑡,𝑠)≤0 for 𝑠≤𝑡,𝑡≥0,(iv)∃𝐾>0∶sup_𝑡≥0∫^𝑡₀|𝑘(𝑡,𝑠)|𝑑𝑠≤𝐾,∀𝑡≥0,(v)lim_𝑡→∞𝑔(𝑡)=0,(vi)lim_𝑡→∞𝑘(𝑡,𝑠)=0,(vii)−𝐾𝑓(−𝐾𝑓(−𝑥))>−𝑥,∀𝑥>0,then lim_𝑡→∞𝑦(𝑡)=0.

Proof. From Lemma 2.6 it is obvious that if 𝑦(𝑡) is ultimately nonnegative the desired result is true. So, let us proceed by contradiction and assume that liminf_𝑡→+∞𝑦(𝑡)=𝑦<0, (2.12) which leads to ∀𝜖>0,∃𝛿∶𝑡>𝛿⇒𝑦(𝑡)>𝑦−𝜖. (2.13) Let us denote by Λ(𝑥) the continuous function Λ(𝑥)=−𝐾𝑓(𝑥−𝐾𝑓(𝑦−𝑥))−2𝑥, (2.14) for which, by virtue of (vii), Λ(0)=−𝐾𝑓(−𝐾𝑓(𝑦))>𝑦. Since Λ is a continuous function, there exists 𝜖₀>0 such that Λ(𝜖₀)>𝑦 or equivalently ∃𝜖₀∶−𝜖₀−𝐾𝑓(𝜖₀−𝐾𝑓(𝑦−𝜖₀))>𝑦+𝜖₀. (2.15) Moreover, by the definition (2.13) of the inferior limit, for the same 𝜖₀ there exists 𝛿₀ such that 𝑡>𝛿₀⇒𝑦(𝑡)>𝑦−𝜖₀. Since 𝑓 is nondecreasing (ii)), then 𝑓(𝑦(𝑡))≥𝑓(𝑦−𝜖₀),𝑡>𝛿₀, (2.16) with 𝑓(𝑦−𝜖₀)<0 in view of (ii) (recall that 𝑦<0).
Now, assume that 𝑡>𝛿₀ and rewrite (1.1) in the following form: 𝑦(𝑡)=𝑔(𝑡)+^𝛿0₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠+^𝑡_𝛿0𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠. (2.17) Then, from (2.16) and (iii) we obtain 𝑦(𝑡)≤𝑔(𝑡)+^𝛿0₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠−𝐾𝑓(𝑦−𝜖₀),𝑡>𝛿₀. (2.18) Since the hypotheses (v) and (vi) assure that, for any fixed 𝑇lim_𝑡→∞(𝑔(𝑡)+^𝑇₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠)=0, (2.19) we have that ∃𝛿₁∶∀𝑡>𝛿₁,−𝜖₀<𝑔(𝑡)+∫^𝛿0₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠<𝜖₀ and hence, from (2.18) ∀𝑡>𝛿₂=max{𝛿₁,𝛿₀} there results 𝑦(𝑡)<𝜖₀−𝐾𝑓(𝑦−𝜖₀). Since 𝑓 is nondecreasing, we have 𝑓(𝑦(𝑡))<𝑓(𝜖₀−𝐾𝑓(𝑦−𝜖₀)),∀𝑡>𝛿₂, (2.20) and hence ^𝑡_𝛿2𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠≥−𝑓(𝜖₀−𝐾𝑓(𝑦−𝜖₀))𝐾. (2.21) From (2.21) it is 𝑦(𝑡)≥𝑔(𝑡)+^𝛿2₀𝑘(𝑡,𝑠)𝑓(𝑦(𝑠))𝑑𝑠−𝑓(𝜖₀−𝐾𝑓(𝑦−𝜖₀))𝐾,𝑡>𝛿₂, (2.22) and, once again, from (2.19) we get that there exists 𝜖₀ and 𝛿₃ such that for all 𝑡>𝛿₃𝑦(𝑡)≥−𝜖₀−𝑓(𝜖₀−𝐾𝑓(𝑦−𝜖₀))𝐾, (2.23) and therefore, for (2.15), 𝑦(𝑡)>𝑦+𝜖₀. This contradicts the assumption (2.12), therefore 𝑦(𝑡) is ultimately nonnegative and the desired result comes out.

Of course hypothesis (vii) may appear quite restrictive, however, the following remark is useful to find examples of functions satisfying it.

Remark 2.8. It can be easily seen that Theorem 2.7 remains valid if instead of (i), (ii), and (vii) we assume that there exists a function 𝜙, satisfying (i), (ii), and (vii) such that 0≤𝑓(𝑦)≤𝜙(𝑦),𝑦≥0,0≥𝑓(𝑦)≥𝜙(𝑦),𝑦<0. (2.24) This, of course, enlarge the class of problems to which Theorem 2.7 can be applied. Namely, the function 𝜙(𝑦)=(𝑒^𝛼𝑦−1)/𝛼 satisfies (i), (ii), and (vii) for any 𝛼>0 and 𝐾<1. For different 𝛼 we can construct a variety of functions 𝑓 such that (2.24) holds (see [12, Remark  3.7] for some plots of admissible functions 𝑓). Moreover, we note that functions of the type 𝜙, which satisfies (2.24), directly arise in many applications concerning the spread of an epidemic in a population (see, e.g., [2, Example  2.3]).

3. The Behavior of the Numerical Solution

In this section we want to prove that under some additional mild hypotheses on the kernel 𝑘 the most popular numerical methods applied to (1.1) inherit the asymptotic behavior of the analytical solution. We consider here the partitionΠ_𝑁={𝑡_𝑛∶0=𝑡₀<𝑡₁<⋯𝑡_𝑁=𝑇} (3.1) of the time interval [0,𝑇] and we assume that the stepsize ℎ=𝑡_𝑛+1−𝑡_𝑛,𝑛=0,…,𝑁−1 is constant. In agreement with the continuous case we prove the following theorems in [0,𝑇] and the results obtained can be easily generalized to the case 𝑡∈[0,+∞). So, 𝑁 can be the natural infinity if 𝑇 is the real point at plus infinity.

3.1. Volterra Runge-Kutta Methods

Let us consider the classical 𝑚-stage Volterra Runge-Kutta methods of extended type [13, Chapter  4]:𝑦_𝑛+1=𝐹_𝑛(𝑡_𝑛+1)+ℎ^𝑚_𝑗=1𝑏_𝑗𝑘(𝑡_𝑛+1+(𝛾_𝑗−1)ℎ,𝑡_𝑛+𝑐_𝑗ℎ)𝑓(𝑌_𝑛𝑗𝑌),𝑛=0,…,𝑁−1,_𝑛𝑖=𝐹_𝑛(𝑡_𝑛+𝜃_𝑖ℎ)+ℎ^𝑚_𝑗=1𝑎_𝑖𝑗𝑘(𝑡_𝑛+𝑑_𝑖𝑗ℎ,𝑡_𝑛+𝑐_𝑗ℎ)𝑓(𝑌_𝑛𝑗),𝑖=1,…,𝑚, (3.2)(3.3) with𝐹_𝑛(𝑡)=𝑔(𝑡)+ℎ^𝑛−1𝑚_𝑙=0𝑗=1𝑏_𝑗𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗ℎ)𝑓(𝑌_𝑙𝑗),𝑛=0,1,…,𝑁−1, (3.4) where 𝑦_𝑛≈𝑦(𝑡_𝑛) and 𝑏=(𝑏₁,…,𝑏_𝑚)^𝑇,𝛾=(𝛾₁,…,𝛾_𝑚)^𝑇,𝜃=(𝜃₁,…,𝜃_𝑚)^𝑇, 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗=1,…,𝑚}, and 𝐷=(𝑑_𝑖𝑗)_{𝑖,𝑗=1,…,𝑚} are the given coefficients of the method [13, page 170].

The following lemmas are useful to prove the main result of this section.

Lemma 3.1. Assume that(i)∃𝐾>0∶sup_𝑡≥0∫^𝑡₀|𝑘(𝑡,𝑠)|𝑑𝑠≤𝐾,𝑡∈[0,𝑇],(ii)|𝑘(𝑡,𝑠)| is ultimately nonincreasing with respect to 𝑠, ∀𝑡∈[0,𝑇].Then ∀𝑠≤𝑡,∃𝐾^∗(𝑠) such that |𝑘(𝑡,𝑠)|≤𝐾^∗(𝑠),∀𝑡∈[0,𝑇].

Proof. The proof is obvious and therefore is omitted.

Lemma 3.2. Assume that the following hypotheses on the problem (1.1) and on the method coefficients in (3.1), (3.4) hold: (i)∃𝑓>0∶𝑓(𝑥)≥−𝑓,∀𝑥∈ℝ,(ii)limsup_{𝑥→−∞}𝑓(𝑥)=𝑓<+∞,(iii)𝑘(𝑡,𝑠)≤0,∀𝑠≤𝑡,𝑡∈[0,𝑇],(iv)∃𝐾>0∶sup_𝑡≥0∫^𝑡₀|𝑘(𝑡,𝑠)|𝑑𝑠≤𝐾,∀𝑡∈[0,𝑇],(v)∃𝐺₁,𝐺₂∶𝐺₁≤𝑔(𝑡)≤𝐺₂,𝑡∈[0,𝑇],(vi)|𝑘(𝑡,𝑠)| ultimately nonincreasing with respect to 𝑠,∀𝑡∈[0,𝑇],(vii)𝑏_𝑗≥0,𝑗=1,…,𝑚. Then, ∀𝑡∈[0,𝑇] and 𝑛=0,1,…,𝑁 the following upper bound for the lag term 𝐹_𝑛(𝑡) holds: 𝐹_𝑛(𝑡)≤𝐺₂+𝑓𝐾+ℎ𝐶₃, (3.5) where 𝐶₃ is a positive constant.

Proof. For (i) it is 𝑓(𝑌_𝑙𝑗)>−𝑓, then for (iii) and (vii) 𝐹_𝑛(𝑡)≤𝑔(𝑡)+ℎ𝑓^𝑛−1𝑚_𝑙=0𝑗=1𝑏_𝑗|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗ℎ)|. (3.6) According to (vi), there exists 𝑛 such that ∀𝑠≥𝑡_𝑛,|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑠)| is nonincreasing with respect to 𝑠. Hence, for 𝑛=𝑛+1,…,𝑁 let us rewrite (3.4) 𝐹_𝑛(𝑡)≤𝑔(𝑡)+ℎ𝑓^𝑛−1𝑚_𝑙=0𝑗=1𝑏_𝑗|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗𝑓ℎ)|+ℎ^𝑚_𝑗=1𝑏_𝑗^𝑛−1_𝑙=𝑛|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗ℎ)|. (3.7) and recall that for any nonincreasing function 𝜂(𝑡)≥0 in [𝑡₁,𝑡_𝑛−1], we have ℎ∑^𝑛_𝑗=1𝜂(𝑡_𝑗)≤∫^𝑡𝑛_𝑡0𝜂(𝑡)𝑑𝑡. Then, for 𝑛=𝑛+1,…,𝑁𝐹_𝑛(𝑡)≤𝑔(𝑡)+ℎ𝑓^𝑛−1𝑚_𝑙=0𝑗=1𝑏_𝑗|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗𝑓ℎ)|+ℎ^𝑚_𝑗=1𝑏_𝑗|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑛+𝑐_𝑗ℎ)|+𝑓^𝑚_𝑗=1𝑏_𝑗^𝑡𝑛_𝑡𝑛|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑠)|𝑑𝑠. (3.8)
Now, recalling that ∑^𝑚_𝑗=1𝑏_𝑗=1, according to Lemma 3.1, and using hypothesis (iv) we have that there exists a constant 𝐶>0 such that 𝐹_𝑛(𝑡)≤𝑔(𝑡)+ℎ𝑓^𝑛−1𝑚_𝑙=0𝑗=1𝑏_𝑗|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗ℎ)|+ℎ𝐶+𝑓𝐾,𝑛=𝑛+1,…,𝑁. (3.9) Since the summation in (3.9) is finite, by hypothesis (v) and, once again, using the result in Lemma 3.1, we get 𝐹_𝑛(𝑡)≤𝐺₂+𝑓𝐾+ℎ𝐶₁,𝑛=𝑛+1,…,𝑁. (3.10) For 𝑛≤𝑛 the whole summation appearing in (3.4) is finite and so there exists a constant 𝐶₂ such that 𝐹_𝑛(𝑡)≤𝐺₂+ℎ𝐶₂,𝑛≤𝑛. (3.11) The result stated in the theorem follows with 𝐶₃=min{𝐶₁,𝐶₂}.

Lemma 3.3. Assume that hypotheses (i)–(vii) of Lemma 3.2 hold, moreover let (viii)𝑎_𝑖𝑗≥0,for𝑖≥𝑗.Then there exists a positive constant 𝐶 such that 𝑌_𝑛𝑗≤𝐺₂+𝑓𝐾+ℎ𝐶,𝑛=0,…,𝑁,𝑗=1,…,𝑚. (3.12)

Proof. Let us consider the last stage 𝑌_𝑛,𝑚=𝐹_𝑛(𝑡_𝑛+𝜃_𝑚ℎ)+ℎ^𝑚_𝑗=1𝑎_𝑚𝑗𝑘(𝑡_𝑛+𝑑_𝑚𝑗ℎ,𝑡_𝑛+𝑐_𝑗ℎ)𝑓(𝑌_𝑛,𝑗), (3.13) it can be easily seen that, because of (i), (iii), and (viii) it is 𝑌_𝑛,𝑚≤𝐹_𝑛(𝑡_𝑛+𝜃_𝑚ℎ)+ℎ𝑓^𝑚_𝑗=1𝑎_𝑚𝑗|𝑘(𝑡_𝑛+𝑑_𝑚𝑗ℎ,𝑡_𝑛+𝑐_𝑗ℎ)|. (3.14) Moreover, because of (vi) and Lemma 3.1 there exists 𝐾^∗ and 𝛼_𝑚 such that |𝑘(𝑡_𝑛+𝑑_𝑚𝑗ℎ,𝑡_𝑛+𝑐_𝑗ℎ)|≤𝐾^∗𝑌,∀𝑛,_𝑛,𝑚≤𝐺₂+𝑓𝐾+ℎ𝛼_𝑚. (3.15)(3.16) Proceeding as in the proof of Theorem 2.1, it follows that 𝑓(𝑌_𝑛,𝑚)≤𝐹_𝑚(ℎ),𝑛=0…,𝑁, (3.17) where 𝐹_𝑚(ℎ)=max{0,max{𝑓,max_{𝑥∈[𝛿,𝐺+}𝑓𝐾_{+ℎ𝛼𝑚]}𝑓(𝑥)}} and 𝛿 is defined in the proof of Theorem 2.1. Now, rewrite 𝑌_{𝑛,𝑚−1} in the following way: 𝑌_{𝑛,𝑚−1}=𝐹_𝑛(𝑡_𝑛+𝜃_𝑚−1ℎ)+ℎ^𝑚−1_𝑗=1𝑎_{𝑚−1,𝑗}𝑘(𝑡_𝑛+𝑑_{𝑚−1,𝑗}ℎ,𝑡_𝑛+𝑐_𝑗ℎ)𝑓(𝑌_𝑛,𝑗)+ℎ𝑎_{𝑚−1,𝑚}𝑘(𝑡_𝑛+𝑑_{𝑚−1,𝑚}ℎ,𝑡_𝑛+𝑐_𝑚ℎ)𝑓(𝑌_𝑛,𝑚), (3.18) and observe the two possible cases that may occur.
(a) 𝑎_{𝑚−1,𝑚}<0. Then 𝑎_{𝑚−1,𝑚}𝑘(𝑡_𝑛+𝑑_{𝑚−1,𝑚}ℎ,𝑡_𝑛+𝑐_𝑚ℎ)≥0 and hence, for (3.17) and (3.15), ℎ𝑎_{𝑚−1,𝑚}𝑘(𝑡_𝑛+𝑑_{𝑚−1,𝑚}ℎ,𝑡_𝑛+𝑐_𝑚ℎ)𝑓(𝑌_𝑛,𝑚)≤ℎ𝐹_𝑚(ℎ)|𝑎_{𝑚−1,𝑚}|⋅|𝑘(𝑡_𝑛+𝑑_{𝑚−1,𝑚}ℎ,𝑡_𝑛+𝑐_𝑚ℎ)|≤ℎ𝐹_𝑚(ℎ)|𝑎_{𝑚−1,𝑚}|𝐾^∗. (3.19) By recalling that 𝑎_{𝑚−1,𝑗}≥0,𝑗=1,…,𝑚−1, the first summation in (3.18) can be bounded by proceeding as for the previous stage, therefore there exists 𝛼_𝑚−1 such that 𝑌_{𝑛,𝑚−1}≤𝐺+𝑓𝐾+ℎ𝛼_𝑚−1.
(b) 𝑎_{𝑚−1,𝑚}≥0. Then, the same procedure as the one used for the 𝑌_𝑛,𝑚 stage can be applied.
Applying the same procedure to 𝑌_{𝑛,𝑚−2},𝑌_{𝑛,𝑚−3},…,𝑌_𝑛,1, the desired result follows with 𝐶=max{𝛼₁,…,𝛼_𝑚}.

Lemma 3.4. Assume that all the hypotheses of Lemma 3.3 hold. Then there exist two positive constants 𝐶₄ and 𝐶₅ such that 𝐹_𝑛(𝑡)≥𝐺₁−𝐹(ℎ)𝐾+ℎ𝐶₄𝑌,𝑛=0,1,…,𝑁,𝑡∈[0,𝑇],_𝑛,𝑗≥𝐺₁−𝐹(ℎ)𝐾+ℎ𝐶₅,𝑛=0,1,…,𝑁,𝑗=0,…,𝑚. (3.20)

Proof. In view of Lemma 3.3 and by taking into account hypothesis (ii) we can define 𝐹(ℎ)=max{0,max{𝑓,max_{𝑥∈[𝛿,𝐺2+}𝑓𝐾^+ℎ𝐶]𝑓(𝑥)}}, (3.21) with 𝛿 given in (2.2), such that 𝑓(𝑌_𝑙,𝑗)≤𝐹(ℎ),𝑙≥0,1≤𝑗≤𝑚. Therefore, ℎ^𝑛−1𝑚_𝑙=0𝑗=1𝑏_𝑗𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗ℎ)𝑓(𝑌_𝑙,𝑗)≥𝐹(ℎ)^𝑛−1𝑚_𝑙=0𝑗=1𝑏_𝑗𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗ℎ)=−𝐹(ℎ)^𝑛−1𝑚_𝑙=0𝑗=1𝑏_𝑗|𝑘(𝑡+(𝛾_𝑗−1)ℎ,𝑡_𝑙+𝑐_𝑗ℎ)|. (3.22) Now, proceeding as in the proof of Lemma 3.2 and because of (v), we have 𝐹_𝑛(𝑡)≥𝐺₁−𝐹(ℎ)[𝐾+ℎ𝐶₁], (3.23) which is is equivalent to the first of (3.20). What is more, from (i) and (3.21), we have −𝑓<𝑓(𝑌_𝑛,𝑗)≤𝐹(ℎ),∀𝑛≤𝑗≤𝑚, (3.24) and, taking into account (3.15), it can be easily proved that there exists a constant 𝛽 such that ℎ^𝑚_𝑗=1𝑎_𝑖,𝑗𝑘(𝑡_𝑛+𝑑_𝑖,𝑗ℎ,𝑡_𝑛+𝑐_𝑗ℎ)𝑓(𝑌_𝑛,𝑗)≥ℎ𝛽,∀𝑖=1,…,𝑚. (3.25) From here and (3.6) for 𝑌_𝑛,𝑖 given in (3.3) holds: 𝑌_𝑛,𝑖≥𝐺₁−𝐹(ℎ)𝐾+ℎ𝐶₅, (3.26) which corresponds to the second of (3.20).

Now, we are ready to prove our result on the boundedness of the numerical solution 𝑦_𝑛 of (3.1)–(3.4).

Theorem 3.5. Assume that all the hypotheses of Lemma 3.3 hold. Then there exist two positive constants 𝐴 and 𝐵 such that 𝐺₁−𝐹(ℎ)𝐾+𝐴ℎ≤𝑦_𝑛≤𝐺₂+𝑓𝐾+𝐵ℎ,𝑛=0,1,…,𝑁. (3.27)

Proof. From Lemma 3.2 it is 𝑦_𝑛+1≤𝐺₂+𝑓𝐾+ℎ𝐶₁+ℎ^𝑚_𝑗=1𝑏_𝑗𝑘(𝑡_𝑛+1+(𝛾_𝑗−1)ℎ,𝑡_𝑛,𝑗)𝑓(𝑌_𝑛,𝑗). (3.28) Since (i), (iii), (vii), and Lemma 3.1 hold, there exists a positive constant 𝐵 such that 𝑦_𝑛+1≤𝐺₂+𝑓𝐾+𝐵ℎ. (3.29) Moreover, from (3.20) and (3.21), we immediately have 𝑦_𝑛+1≥𝐺₁+𝐹(ℎ)𝐾+ℎ𝐶₄+ℎ𝐶𝐹(ℎ), (3.30) that, together with (3.29), gives the result stated in the theorem.

Observe that, in agreement with the continuous case, 𝑁 is allowed to be ∞.

As we could expected because of the convergence of the method the following result can be observed.

Corollary 3.6. The numerical and analytical solutions are bounded by the same constants as the step-size ℎ goes to zero.

Proof. The desired result comes immediately from (3.27) and (2.1) and observing that for 𝐹(ℎ) defined in (3.21), we have lim_ℎ→0𝐹(ℎ)=𝐹^∗, (3.31) with 𝐹^∗ given in (2.6).

Observe that the hypotheses (vii) and (viii) on the coefficients of the methods are not restrictive at all because they are satisfied by any VRK method.

A similar result can be easily proved on VRK methods of mixed type. Compared with (3.1)–(3.4), such methods present a different expression of the lag-term formula, that is,𝐹_𝑛(𝑡)=𝑔(𝑡)+ℎ^𝑛_𝑙=0𝑤_𝑛,𝑙𝑘(𝑡,𝑡_𝑙)𝑓(𝑦_𝑙), (3.32) where 𝑤_𝑛,𝑙 are given weights of a quadrature formula.

The proof of the following theorem has the same plot as the one of Theorem 3.5 and therefore, it is omitted.

Theorem 3.7. Assume that (i)∃𝑓≥0∶𝑓(𝑥)≥−𝑓,∀𝑥∈ℝ,(ii)limsup_{𝑥→−∞}𝑓(𝑥)=𝑓<+∞,(iii)𝑘(𝑡,𝑠)≤0,∀𝑠≤𝑡,𝑡∈[0,𝑇],(iv)∃𝐾>0∶sup_𝑡≥0∫^𝑡₀|𝑘(𝑡,𝑠)|𝑑𝑠≤𝐾,∀𝑡∈[0,𝑇],(v)∃𝐺₁,𝐺₂∶𝐺₁≤𝑔(𝑡)≤𝐺₂,∀𝑡∈[0,𝑇],(vi)|𝑘(𝑡,𝑠)| nonincreasing with respect to 𝑠,𝑡∈[0,𝑇],(vii)𝑤_𝑛,𝑗≥0,𝑛=0,…,𝑁,𝑗=0,…,𝑛,(viii)𝑎_𝑖,𝑗≥0,𝑖≥𝑗,then, for 𝑦_𝑛 solution of (3.1)–(3.3) with (3.32), we have the following bounds for 𝑛=0,…,𝑁: 𝐺₁−𝐹(ℎ)𝐾+𝐴₁ℎ≤𝑦_𝑛≤𝐺₂+𝐾𝑓+𝐵₁ℎ, (3.33) where 𝐴₁ and 𝐵₁ are two positive constants.

Remark 3.8. If the hypotheses of Theorem 2.1, which assure the boundedness of 𝑦(𝑡), are compared with those of Theorems 3.5 and 3.7 for the boundedness of 𝑦_𝑛, we note the following. Besides the natural assumption on the coefficients of the numerical methods, only the hypothesis (vi) on the monotonicity of |𝑘(𝑡,𝑠)| has been added. In this sense we can claim that under mild additional conditions on the kernel 𝑘, the behavior of the analytical solution is inherited by the numerical one.

3.2. Volterra Direct Quadrature Methods

A general VDQ method for solving VIEs of the type (1.1), defined on the mesh (3.1), reads𝑦_𝑛=𝑔(𝑡_𝑛)+ℎ^𝑛_𝑙=0𝑤_𝑛,𝑙𝑘(𝑡_𝑛,𝑡_𝑙)𝑓(𝑦_𝑙),𝑛=0,1,…,𝑁, (3.34) where 𝑤_𝑛,𝑙 are given weights that satisfy0<𝑤_𝑛,𝑙<1,𝑙=0,1,…,𝑛,𝑛=0,1,…,𝑁,. (3.35) The following theorems represent the discrete analogues of Theorems 2.1 and 2.7, where, due to the linearity of the method, the Volterra discrete equation (3.34) can be interpreted as the discrete counterpart of (1.1).

Theorem 3.9. Assume that(i)∃𝑓≥0∶𝑓(𝑥)≥−𝑓,∀𝑥∈ℝ,(ii)limsup_{𝑥→−∞}𝑓(𝑥)=𝑓<+∞,(iii)𝑘(𝑡,𝑠)≤0,∀𝑠≤𝑡,𝑡∈[0,𝑇],(iv)∃𝐾>0∶sup_𝑡≥0∫^𝑡₀|𝑘(𝑡,𝑠)|𝑑𝑠≤𝐾,∀𝑡∈[0,𝑇],(v)∃𝐺₁,𝐺₂∶𝐺₁≤𝑔(𝑡)≤𝐺₂,∀𝑡∈[0,𝑇],(vi)|𝑘(𝑡,𝑠)| nonincreasing with respect to 𝑠,𝑡∈[0,𝑇],(vii)𝑤_𝑛,𝑙≥0,𝑛=0,1,…,𝑁,𝑙=0,…,𝑛, then 𝑦_𝑛 is bounded and it is, for 𝑛=0,…,𝑁𝐺₁−𝐹(ℎ)𝐾+𝐴₂ℎ≤𝑦_𝑛≤𝐺₂+𝑓𝐾+𝐵₂ℎ, (3.36) where 𝐴₂ and 𝐵₂ are positive constants and lim_ℎ→0𝐹(ℎ)=𝐹^∗.

Proof. From (i) and (iii) there results 𝑘(𝑡_𝑛,𝑡_𝑙)𝑓(𝑦_𝑙)<−𝑓𝑘(𝑡_𝑛,𝑡_𝑙) hence, by taking into account (vi), we get ℎ^𝑛_𝑙=0𝑤_𝑛𝑙|𝑘(𝑡_𝑛,𝑡_𝑙)|𝑓(𝑦_𝑙)≤𝑓[ℎ|𝑘(𝑡_𝑛,0)|+^𝑡𝑛₀|𝑘(𝑡_𝑛,𝑠)|𝑑𝑠]. (3.37) Thus, taking into account Lemma 3.1 and the hypothesis (v) of the theorem, we have 𝑦_𝑛≤𝐺₂+𝑓𝐾+𝐶₆ℎ. (3.38) Once again 𝑓(𝑦_𝑛)≤max{𝑓,max_{𝑦∈[𝛿,𝐺2+}𝑓𝐾^+𝐶₆^ℎ]𝑓(𝑦)}, (3.39) with 𝛿 given in (2.2). If we define 𝐹(ℎ)=max{0,max{𝑓,max_{𝑦∈[𝛿,𝐺2+}𝑓𝐾^+𝐶₆^ℎ]𝑓(𝑦)}}, (3.40) then 𝑓(𝑦_𝑛)≤𝐹(ℎ),𝑛=0…,𝑁,lim_ℎ→0𝐹(ℎ)=𝐹^∗, (3.41) where 𝐹^∗ is given in (2.6). Proceeding as in the proof of Theorem 2.1, we get 𝑦_𝑛≥𝐺₁+^𝑛_𝑙=0𝑘(𝑡_𝑛,𝑡_𝑙)𝑓(𝑦_𝑙)≥𝐺₁−𝐹(ℎ)^𝑛_𝑙=0|𝑘(𝑡_𝑛,𝑡_𝑙)|≥𝐺₁−𝐹(ℎ)(𝐾+𝐶₁ℎ). (3.42) This last inequality, together with (3.38) and the second of (3.41) gives the desired result.

In the case of VDQ methods we are also able to prove that, under the hypothesis of Theorem 2.7, also the numerical solution vanishes. This, of course, assures that the asymptotic behavior of the exact and the approximate solutions coincide. First of all, we enunciate the following Lemma which represents the discrete analogue of Lemma 2.6 and whose proof is then omitted.

Lemma 3.10. Assume that (i)∃𝑓≥0∶𝑓(𝑥)≥−𝑓,∀𝑥∈ℝ,(ii)𝑥𝑓(𝑥)≥0,∀𝑥∈ℝ,(iii)𝑘(𝑡,𝑠)≤0,∀𝑠≤𝑡,𝑡≥0,(iv)lim_𝑡→∞𝑘(𝑡,𝑠)=0,(v)lim_𝑡→∞𝑔(𝑡)=0,(vi)𝑦_𝑛 ultimately nonnegative, then lim_𝑛→∞𝑦_𝑛=0,𝑛=0,…,𝑁.

The following result is the discrete counterpart of Theorem 2.7 and could be derived from the proof of Theorem 3.4 in [12]. We report here the proof for the particular form of the Volterra discrete equation (3.34).

Theorem 3.11. Assume that(i)∃𝑓≥0∶𝑓(𝑥)≥−𝑓,∀𝑥∈ℝ,(ii)𝑓 is nondecreasing, 𝑥𝑓(𝑥)≥0,∀𝑥∈ℝ,(iii)𝑘(𝑡,𝑠)≤0 for 𝑠≤𝑡 and 𝑡≥0,(iv)∃𝐾>0∶sup_𝑡≥0∫^𝑡₀|𝑘(𝑡,𝑠)|𝑑𝑠≤𝐾,𝑡≥0,(v)lim_𝑡→∞𝑔(𝑡)=0,(vi)lim_𝑡→∞𝑘(𝑡,𝑠)=0,(vii)−𝐾𝑓(−𝐾𝑓(−𝑥))>−𝑥,∀𝑥>0,(viii)|𝑘(𝑡,𝑠)| ultimately nonincreasing with respect to 𝑠, for 𝑡≥0,(ix)𝑤_𝑛,𝑙≥0,𝑛=0,…,𝑁,𝑙=0,…,𝑛, then lim_𝑛→∞𝑦_𝑛=0.

Proof. By absurd, assume that the sequence 𝑦_𝑛 given by (3.34) is not ultimately nonnegative, that is, liminf_𝑛→+∞𝑦_𝑛=𝛾<0, (3.43) which leads to ∀𝜖>0,∃𝜌>0∶𝑛>𝜌⇒𝑦_𝑛>𝛾−𝜖. (3.44) Denote by Λ(𝑥) the function Λ(𝑥)=−𝐾𝑓(𝑥−𝐾𝑓(𝛾−𝑥))−2𝑥. (3.45) Now, proceeding as in Theorem 2.7, we can prove that ∃𝜖₀>0, and ∃𝑛₃>0 such that ∀𝑛>𝑛₃𝑦_𝑛≥−𝜖₀−𝑓(𝜖₀−𝐾𝑓(𝛾−𝜖₀))𝐾, (3.46) and hence, 𝑦_𝑛≥𝛾+𝜖₀. This contradicts (3.43), which is therefore absurd. Hence, 𝑦_𝑛 is ultimately nonnegative and Lemma 3.10 holds.

3.3. Numerical Experiments

In this section, we report some numerical experiments that show the application of our results in the study of the behavior of the numerical solutions provided by VRK and VDQ methods for some nonlinear problems. First we consider VRK methods and numerically solve, by a Radau IIa type formula of order 3 (see [13]) the problems described in Examples 2.3 and 2.5 of Section 2. In Figures 1 and 2 we observe that the behavior of the numerical solution 𝑦_𝑛 with respect to the analytical bounds given in Theorem 2.1 is consistent with the theory developed in Section 3.

Figure 1 

Numerical solution for the problem (1.1) described in Example 2.3.

Figure 2 

Numerical solution for the problem (1.1) described in Example 2.5.

In Figure 3 we report the result of an experiment on (1.1) with𝑘(𝑡,𝑠)=−𝑒^−𝑠(1+𝑡²),𝑓(𝑦)=−sin²1𝑦,𝑔(𝑡)=(1+𝑡²), (3.47) by a DQ method based on the trapezoidal rule of order 2. We do not know the solution, but thanks to Theorems 3.11 and 2.7, we can predict that the numerical solution 𝑦_𝑛 tends to zero as 𝑛→+∞ just like the analytical one, and this is exactly the behavior we observe in the plot.

Figure 3 

Numerical solution for the problem (1.1) described in (3.47).