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Computational Methods in Applied Mathematics

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Volume 20, Issue 1


Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability

André EikmeierORCID iD: https://orcid.org/0000-0002-0270-6491 / Etienne EmmrichORCID iD: https://orcid.org/0000-0001-9869-0334 / Hans-Christian Kreusler
Published Online: 2018-11-21 | DOI: https://doi.org/10.1515/cmam-2018-0268


The initial value problem for an evolution equation of type v+Av+BKv=f is studied, where A:VAVA is a monotone, coercive operator and where B:VBVB induces an inner product. The Banach space VA is not required to be embedded in VB or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

Keywords: Nonlinear Evolution Equation; Monotone Operator; Volterra Operator; Exponentially Decaying Memory; Existence; Uniqueness; Stability; Time Discretisation; Convergence

MSC 2010: 47J35; 45K05; 34K30; 35K90; 35R09; 65J08; 65M12

Dedicated to Professor Rolf D. Grigorieff on the occasion of his 80th birthday

1 Introduction

1.1 Problem Statement and Main Result

We are concerned with initial value problems of the type

v+Av+BKv=fin (0,T),(1.1a)v(0)=v0,(1.1b)where(Kv)(t)=u0+0tk(t-s)v(s)ds,k(z)=λe-λz.(1.1c)

Here, T>0 is the time under consideration, λ>0 is a given parameter, and u0, v0, f are given data of the problem.

The operator A:VAVA is a hemicontinuous, monotone, coercive operator satisfying a certain growth condition, where VA denotes a real, reflexive Banach space. The operator B:VBVB is a linear, symmetric, bounded, strongly positive operator on a real Hilbert space VB. We assume that both VA and VB are densely and continuously embedded in a real Hilbert space H, which is identified with its dual. We emphasise that we do not require VA to be embedded in VB (or vice versa) but only assume that V=VAVB is densely embedded in VA as well as in VB. This yields the scale


of Banach and Hilbert spaces with dense and continuous embeddings. Moreover, we assume that V is separable.

Since the kernel k in (1.1c) is assumed to be of exponential type, (1.1a) can easily be derived from the system


with suitable linear operators C, D such that B=CD (and with u0 being replaced by Du0).

Such systems appear, e.g., in the description of viscoelastic fluid flow (see, e.g., [15, 38]) as well as heat flow in materials with memory (see, e.g., [39, 43]).

Other applications arise in, e.g., non-Fickian diffusion models describing diffusion processes of a penetrant through a viscoelastic material (see, e.g., [17, 18, 46]), where, apart from the usual diffusion rate of the penetrant, the change in the internal structure of the viscoelastic material has to be considered. Non-Fickian diffusion also plays a role in, e.g., mathematical biology (see, e.g., [13, 20, 42]).

Note that all the results obtained in this paper are also valid for kernels of the type k(z)=ce-λz (c,λ>0). The proof only differs for the stability with respect to perturbations of the kernel. However, in many applications, kernels of the type given in (1.1c) naturally arise. In such applications, one often deals with a coupled system of the type (1.3), where 1λ can be interpreted as a relaxation or averaged delay time. If λ tends to 0, then the system decouples with u(t)u0. The limit λ results in a first-order equation for v without memory. In contrast to this, in the case where k(z)=ce-λz, the limit λ0 results in an evolution equation for u of second order in time (see [19]), whereas λ results again in a first-order evolution equation for v.

Our goal in this paper is twofold: On the one hand, we wish to prove existence of generalised solutions to (1.1). On the other hand, we wish to prove convergence of a simple time discretisation method, which relies on the implicit Euler or Rothe method combined with a product quadrature. Moreover, we prove stability of the solution against perturbations of the problem data (including perturbations of λ), which also implies uniqueness.

1.2 Literature Overview

Most of the results on evolutionary Volterra integral equations available so far are, indeed, more general with respect to the integral kernel (often dealing with memory of positive type). Our results, however, include the case of different domains of definition of the underlying operators involved without assuming that VA is embedded in VB (or vice versa). Such a situation occurs, e.g., if A is a spatial differential operator of order lower than that of B. This has, to the best knowledge of the authors, not yet been studied. Dealing with memory of exponentially decaying type can thus be seen as a first step within this more general functional analytic framework.

Well-posedness for linear evolutionary integral equations in Banach or Hilbert spaces has been studied in detail by many authors for a long time. We only refer to the standard monographs [30, 44] and the references cited therein. For the semigroup approach, we also refer to [2]. For the finite-dimensional case (including nonlinear problems), one may also consult [9].

Whereas there are many results available for linear problems, there is less known for the nonlinear case. Classical results for nonlinear problems are due to Clément, Crandall, Dafermos, Desch, Gripenberg, Londen, MacCamy, Nohel, and others. One of the equations that is mostly studied is of the type


where a is a suitable kernel function (e.g., completely monotone or of positive type) and A is a suitable nonlinear operator satisfying certain monotonicity assumptions; see, e.g., [14, 33, 27, 28, 29, 50, 51, 52].

For example, MacCamy, and Wong [40] consider a class of nonlinear integro-differential equations with completely monotone kernel in a Hilbert space setting. Gajewski, Gröger, and Zacharias [22] study rather general classes of nonlinear evolution equations with Volterra operators, but are restricted to the Hilbert space case VA=VB. Crandall, Londen, and Nohel [10] consider a doubly nonlinear problem governed by maximal monotone subdifferential operators, where the domain of definition of one of the operators is dense and continuously embedded into the domain of definition of the other operator; see also [11]. An interesting result for a class of doubly nonlinear integro-differential equations governed by an m-dissipative operator can be found in [26]. The nonlocality in time here is of a special form incorporating this m-dissipative operator again. The domain of definition of the principal part of the equations, however, equals the domain of definition of the nonlocality in time. The method of proof relies upon nonlinear semigroup theory. More recent results can be found, e.g., in [1, 25, 6, 23, 24, 48, 53, 54, 3]. The above-cited references do not cover the class of problems we consider here.

The question of numerical approximation, mostly for linear or semilinear problems with memory of positive type, has been dealt with by McLean, Sloan, Thomée, and Wahlbin (see, e.g., [47, 41, 37, 31]). The focus is on Galerkin finite element methods combined with suitable time discretisation methods based on the backward Euler scheme (see, e.g., [8, 7]). Time discretisation methods have also been studied by Calvo, Lubich, and Palencia (see, e.g., [5, 12]) and, in particular, convolution quadrature by Lubich [34, 35, 36] and Ferreira [21]. Indeed, our proof of existence also relies upon the convergence of a numerical scheme that is based on a convolution quadrature.

1.3 Organisation of the Paper

The paper is organised as follows: The general notation is explained in Section 2. In Section 3, we state the main assumptions on the operators A and B and collect some preliminary results on their properties. The main existence result (Theorem 4.2) is provided in Section 4 by showing (weak) convergence of a suitable time discretisation to (1.1). The crucial point here is an integration-by-parts formula given in Lemma 4.3. In Section 5, we show uniqueness and stability with respect to perturbations of the problem data. In particular, we also consider perturbations of the kernel parameter λ.

2 Notation

For a Banach space X, we denote its norm by X, its dual by X, equipped with the standard norm X, and the duality pairing by ,. We recall that X is reflexive and separable if X is. For a Hilbert space X, we denote the inner product (inducing the norm X) by (,)X. The intersection of two Banach spaces X, Y is equipped with the norm XY=X+Y, whereas the sum X+Y is equipped with the norm

gX+Y=inf{max(gXX,gYY):g=gX+gY with gXX,gYY}.

We recall that (XY)=X+Y; see also [22, pp. 12ff.].

For a real, reflexive, separable Banach space X, the Bochner–Lebesgue spaces Lr(0,T;X) (r[1,]) are defined in the usual way and equipped with the standard norm. Denoting by r=rr-1 the conjugate of r(1,) with r= if r=1, we have that (Lr(0,T;X))=Lr(0,T;X) if r[1,); the duality pairing is given by


see, e.g., [16, Theorem 1 on p. 98, Corollary 13 on p. 76, Theorem 1 on p. 79]. Moreover, Lr(0,T;X) is reflexive if r(1,) (see [16, Corollary 2 on p. 100]) and L1(0,T;X) is separable.

By W1,r(0,T;X) (r[1,]) we denote the Banach space of functions uLr(0,T;X) whose distributional time derivative u is again in Lr(0,T;X); the space is equipped with the standard norm. Note that if uW1,1(0,T;X), then u equals almost everywhere a function that is in 𝒜𝒞([0,T];X), i.e., a function that is absolutely continuous on [0,T] as a function taking values in X. Moreover, W1,1(0,T;X) is continuously embedded in the Banach space 𝒞([0,T];X) of functions that are continuous on [0,T] as functions with values in X (see, e.g., [45, Chapter 7] for more details). By 𝒞w([0,T];X) we denote the space of functions that are continuous on [0,T] with respect to the weak topology in X.

The space of infinitely many times differentiable real functions with compact support in (0,T) is denoted by 𝒞c(0,T). The space of on [0,T] continuously differentiable real functions is denoted by 𝒞1([0,T]). By c we denote a generic positive constant. We set j=10xj=0 for xj whatsoever.

3 Main Assumptions and Preliminary Results

Let VA be a real, reflexive Banach space and let VB and H be real Hilbert spaces satisfying (1.2). Moreover, we assume that V=VAVB is separable.

The structural properties we assume for the operators A and B throughout this paper are as follows.

Assumption A.

The operator A:VAVA satisfies for all u,v,wVA,

  • (i)

    θA(u+θv),w𝒞([0,1]) (hemicontinuity),

and there exist p(2,), μA,βA>0, cA0 such that for all v,wVA,

  • (ii)

    Av-Aw,v-w0 (monotonicity),

  • (iii)

    Av,vμAvVAp-cA (p-coercivity),

  • (iv)

    AvVAβA(1+vVAp-1) ((p-1)-growth).

Assumption B.

The linear operator B:VBVB is symmetric and there exist μB,βB>0 such that for all vVB,

  • (i)

    Bv,vμBvVB2 (strong positivity),

  • (ii)

    BvVBβBvVB (boundedness).

By B=B,1/2 we denote the norm induced by B, which is equivalent to VB. Further, by L2(0,T;B) we denote the space L2(0,T;(VB,B)).

We shall remark that the operators A and B can be extended, as usual, to operators acting on functions defined on [0,T] and taking values in VA and VB, respectively. Since the operator A:VAVA is hemicontinuous and monotone, it is also demicontinuous (see, e.g., [55, Proposition 26.4 on p. 555]). Therefore, the operator A maps, in view of the separability of VA and the theorem of Pettis (see, e.g., [16, Theorem 2 on p. 42]), a Bochner measurable function v:[0,T]VA into a Bochner measurable function Av:[0,T]VA with (Av)(t)=Av(t) (t[0,T]). Because of the growth condition, the operator A then maps Lp(0,T;VA) into (Lp(0,T;VA))=Lp(0,T;VA). The linear, symmetric, bounded, strongly positive operator B:VBVB extends, via (Bu)(t)=Bu(t) for a function u:[0,T]VB, to a linear, symmetric, bounded, strongly positive operator mapping L2(0,T;VB) into its dual. Indeed, B can also be seen as a linear, bounded operator mapping Lr(0,T;VB) into Lr(0,T;VB) for any r[1,].

With respect to the operator K defined by (1.1c), we make the following observations.

Lemma 3.1.

Let k be as in (1.1c) and let u0VB. Then K is an affine-linear, bounded mapping of L2(0,T;VB) into itself with


where kL1(0,T)=1-e-λT. Further, K is an affine-linear, bounded mapping of L1(0,T;VB) into AC([0,T];VB) with


The standard proof is omitted here.

Note that in the assertions above VB can be replaced by H. An immediate consequence of the lemma above is that BK:L2(0,T;VB)L2(0,T;VB) as well as BK:L1(0,T;VB)L(0,T;VB) is affine-linear and bounded if Assumption B is satisfied.

We will often make use of the following relation, which indeed is crucial within this work and reflects the exponential type of the memory kernel. For a function vL1(0,T;X), with X being an arbitrary Banach space, we have that for almost all t(0,T),


If X is a Hilbert space and if vL2(0,T;X), then testing this relation by Kv-u0 immediately implies for all t[0,T],


which shows that the memory term is of positive type.

4 Main Result: Existence via Time Discretisation

In this section, we show existence of generalised solutions to (1.1) by proving weak or weak* convergence of a sequence of approximate solutions constructed from a suitable time discretisation. We commence by studying the corresponding numerical scheme and its properties.

4.1 Time Discretisation

For N, let τ=TN and tn=nτ (n=0,1,,N). Let v0v0, u0u0, and {fn}n=1Nf be given approximations of the problem data v0, u0, and f, respectively. We look for approximations vnv(tn) (n=1,2,,N).

The numerical scheme we consider combines the implicit Euler method with a convolution or product quadrature for the integral operator K and reads


To be precise, Ku0τ acts on a grid function {vn}n=1N and, with a slight abuse of notation, the evaluation at n is denoted by (Ku0τv)n. The coefficients γi (i=1,2,,N) are given by


and hence


The idea behind is the approximation

0tk(t-s)v(s)dsj=1ntj-1tjk(tn-s)dsv(tj)for t(tn-1,tn](n=1,2,,N).

As we deal with kernels k of exponential type given by (1.1c), we can explicitly calculate


which immediately leads to the properties

0<γ1=1-e-λττλ=k(0)as τ0,(4.2)



4.2 Existence, Uniqueness, and A Priori Estimates for the Time Discrete Problem

In what follows, we show existence of solutions for the time discrete equation (4.1) and derive suitable a priori estimates.

Theorem 4.1 (Time Discrete Problem).

Let Assumptions A and B be fulfilled and let u0VB, v0H, and {fn}n=1N={f0n}n=1N+{f1n}n=1NVA+H be given. Then there is a unique solution {vn}n=1NV=VAVB to (4.1). Moreover, there holds for n=1,2,,N,



For better readability, we write Kτ instead of Ku0τ during this proof. We commence with proving existence and uniqueness of a solution step by step. In the n-th step, (4.1a) is equivalent to determine vn from {vj}j=1n-1V and the data of the problem by solving


Because of the continuous embeddings (1.2), we know that the right-hand side of the foregoing relation is in V. The operator M:=1τI+A+τγ1B is easily shown to be hemicontinuous, coercive, and strictly monotone as a mapping of V into V. Here, we make use of the fact that γ1>0. In particular, we observe that for all wV,


The famous theorem of Browder and Minty (see, e.g., [55, Theorem 26.A on p. 557]) now provides existence of a solution vnV; uniqueness immediately follows from the strict monotonicity.

For proving the a priori estimate, we test (4.1a) by vnV (n=1,2,,N). With


the coercivity of A, and Young’s inequality, we find for n=1,2,,N,


By summing up, we obtain for n=1,2,,N,




We observe that (4.1b) together with (4.2), (4.3) implies for n=2,3,,N,


which is the discrete analogue of the crucial relation (3.1). Note that (eλτ-1)/τλ as τ0. By setting (Kτv)0:=u0, relation (4.6) remains true for n=1.

Resolving (4.6) for vn, together with (4.5) (recall that B:VBVB induces an inner product on VB), gives




Altogether, we find for n=1,2,,N,


Taking the maximum over all n{1,,N} first on the right-hand side and then on the left-hand side leads to


where C>0 depends on u0, v0, and {fn}. This yields


which, inserted in (4.7), proves the assertion. ∎

4.3 Convergence of Approximate Solutions and Existence of a Generalised Solution

Let {N} be a sequence of positive integers such that N as . We consider the corresponding sequence of time discrete problems (4.1) with step sizes τ=T/N, starting values v0H with v0v0 in H as well as u0VB with u0u0 in VB, and right-hand sides {fn}n=1NVA+H given by


which is well-defined for f=f0+f1Lp(0,T;VA)+L1(0,T;H). As a slight abuse of notation, in general we do not call the dependence of un,vn,fn, and of the time instances tn on .

Let {vn}n=1NV denote the solution to (4.1) with step size τ. We then consider the piecewise constant functions v with v(t)=vn for t(tn-1,tn] (n=1,2,,N) and v(0)=v1. Moreover, let v^ be the piecewise affine-linear interpolation of the points (tn,vn) (n=0,1,,N) and let f be the piecewise constant function with f(t)=fn for t(tn-1,tn] (n=1,2,,N) and f(0)=f1.

Regarding the integral operator K, we define for any integrable function w the piecewise constant function Kw by means of

(Kw)(t)=u0+0tnk(tn-s)w(s)dsif t(tn-1,tn](n=1,,N),

with (Kw)(0):=u0. As an immediate consequence, with (4.1b) and (4.1c) we obtain

(Kv)(t)=(Ku0τv)nif t(tn-1,tn](n=1,,N).(4.8)

We are now able to state the main result.

Theorem 4.2 (Existence via Convergence).

Let Assumptions A and B be fulfilled, and let u0VB, v0H, and fLp(0,T;VA)+L1(0,T;H). Then there exists a solution vLp(0,T;VA)Cw([0,T];H) to problem (1.1) with KvCw([0,T];VB) such that (1.1a) holds in the sense of Lp(0,T;VA)+L1(0,T;H).

By passing to a subsequence if necessary, both the piecewise constant prolongation v and the piecewise affine-linear prolongation v^ of the approximate solutions to (4.1) converge weakly* in L(0,T;H) to v as . Furthermore, again by passing to a subsequence if necessary and as , the piecewise constant prolongation v converges weakly to v in Lp(0,T;VA) and the approximation Kv of the memory term converges weakly* to Kv in L(0,T;VB). The approximation v^ of the time derivative converges to


in the sense that v^,wv,w for all wLp(0,T;VA)L(0,T;H)L1(0,T;VB).1

Note that if {τv0VAp} is bounded, then also v^ converges weakly in Lp(0,T;VA) to v as (passing to a subsequence if necessary).

The proof of Theorem 4.2 will be prepared by the following integration-by-parts formula.

Lemma 4.3.

Let u0VB and wLp(0,T;VA)L(0,T;H) such that


Then for all t[0,T] there holds



The difficulty in proving (4.9) is that neither


can be assumed but only the sum w+BKw is known to be in Lp(0,T;VA)+L1(0,T;H). Hence it is not possible to split the sum on the left-hand side and to carry out integration by parts separately for both terms.

As w+BKwLp(0,T;VA)+L1(0,T;H) and BKwL(0,T;VB), it is easy to show that


and thus w𝒜𝒞([0,T];VA+VB). In view of [32, Chapter 3, Lemma 8.1], we find w𝒞w([0,T];H). Because of (3.1), we also see that (Kw)L1(0,T;VA+VB), and thus Kw𝒜𝒞([0,T];VA+VB), which implies Kw𝒞w([0,T];VB). Because of (3.1), we also have (Kw)𝒞w([0,T];H). This justifies evaluating w(t)H, Kw(t)B, and (Kw)(t)H for fixed t.

Further, we remark that Kw-u0L(0,T;V) since wLp(0,T;VA)L(0,T;H). Due to (3.1), we thus have (Kw)Lp(0,T;VA)L(0,T;H). Differentiating (3.1) yields


for almost all t(0,T). Therefore, we have


Since 1λ(Kw)′′+(Kw)+BKwLp(0,T;VA)+L1(0,T;H) and Kw-u0L(0,T;V), we are allowed to split the right-hand side into


For the first term on the right-hand side, we apply the result of Strauss [49, Theorem 4.1] providing the integration-by-parts formula


Since (Kw)Lp(0,T;VA)L(0,T;H), BKwL(0,T;VB), and Kw-u0L(0,T;V), the third term on the right-hand side of (4.10) yields

1λ(Kw)′′+(Kw)+BKw,Kw-u0=1λ(Kw)′′,Kw-u0+(Kw),Kw-u0+BKw,Kw-u0=1λ(Kw-u0)(T),(Kw-u0)(T)-1λ(Kw-u0)(0),(Kw-u0)(0)-1λ(Kw-u0)L2(0,T;H)2   +12(Kw)(T)-u0H2-12(Kw)(0)-u0H2+KwL2(0,T;B)2-BKw,u0.

Altogether, we have


Due to (3.1), there holds


as well as


Inserting this into (4.11) yields the desired integration-by-parts formula for t=T. However, everything above remains true for an arbitrary t[0,T], which proves the assertion. ∎

Proof of Theorem 4.2.

For simplicity, we write τ and N instead of τ and N, respectively, during this proof. Since, by assumption, v0v0 in H, u0u0 in VB, and since ff in Lp(0,T;VA)+L1(0,T;H), the right-hand side of the a priori estimate (4.4) stays bounded as . Since


the a priori estimate (4.4) implies that the sequence {v} is bounded in L(0,T;H)Lp(0,T;VA), the sequence {v^} is bounded in L(0,T;H), and – in view of (4.8) – the sequence {Kv} is bounded in L(0,T;VB). Due to the growth condition on A, we also conclude that the sequence {Av} is bounded in Lp(0,T;VA).

Since L(0,T;H) and L(0,T;VB) are the duals of the separable normed spaces L1(0,T;H) and L1(0,T;VB), respectively, and since Lp(0,T;VA) and Lp(0,T;VA) are reflexive Banach spaces, there exists a common subsequence, again denoted by , and elements vL(0,T;H)Lp(0,T;VA), v^L(0,T;H), uL(0,T;VB), and aLp(0,T;VA) such that, as

v*vin L(0,T;H),vvin Lp(0,T;VA),v^*v^in L(0,T;H),Kv*uin L(0,T;VB),Avain Lp(0,T;VA);

see, e.g., [4, Corollaire III.26, Théorème III.27]. Since B:VBVB is linear and bounded, we also find that BKv*Bu in L(0,T;VB).

A short calculation shows that


which tends to 0 as because of the a priori estimate (4.4). Hence v=v^.

Let us show that u=Kv, and thereby KvL(0,T;VB). We already know that Kv*u in L(0,T;VB), and thus Kvu in L2(0,T;H). The integral operator K^ defined by K^w:=Kw-u0 can easily be shown to be a linear, bounded and thus weakly-weakly continuous mapping of L2(0,T;H) into itself (see also Lemma 3.1). Therefore, in view of v*v in L(0,T;H), and thus vv in L2(0,T;H), we find that Kv-Kv=K^v-K^v0 in L2(0,T;H). Because of


it thus remains to prove that Kv-Kv converges at least weakly in L2(0,T;H) to 0 in order to prove u=Kv. For wL2(0,T;H) and t(tn-1,tn] (n=1,2,,N), we have


Since |k(t-s)|=λe-λ(t-s)λeλτ for s(t,tn)(tn-1,tn) as well as

|k(tn-s)-k(t-s)|λ(eλτ-1)cτfor s(0,tn),

the Cauchy–Schwarz inequality yields


Since u0u0 in VB and v is bounded in L2(0,T;H), we immediately get Kv-Kv0 in L2(0,T;H) as (and thus τ0), which is the last step to prove u=KvL(0,T;VB).

With the help of the piecewise constant and piecewise linear interpolation of the corresponding grid functions, we can now rewrite the numerical scheme (4.1a) as


For any wV and ϕ𝒞c(0,T), we thus obtain


Since ff in Lp(0,T;VA)+L1(0,T;H), and employing the (weak and weak*) convergence just shown, we can pass to the limit in the foregoing equation to come up with


This shows that v is differentiable in the weak sense with


Let us summarise the regularity properties of the limits we have proven so far: As


we conclude that v𝒜𝒞([0,T];V). This implies v𝒞w([0,T];H) (see [32, Chapter 3, Lemma 8.1]). Likewise, from


(see (3.1)), we deduce that Kv𝒜𝒞([0,T];VA+VB), and thus Kv𝒞w([0,T];VB). We remark that, in addition, one can easily show that Kv-u0W1,p(0,T;VA).

By construction, we have v^(0)=v0v0 in H and v^(T)=vN. In view of the a priori estimate (4.4), the sequence {v^(T)} is bounded in H, and hence possesses a subsequence, again denoted by , that converges weakly in H to an element vTH. We shall show that vT=v(T).

For any ϕ𝒞1([0,T]), wV, we test (4.13) with the function tϕ(t)w𝒞1([0,T];V), which allows us to integrate by parts. Together with (4.12), we find


The first two terms on the right-hand side tend to zero as . This yields

(v(T),w)ϕ(T)-(v(0),w)ϕ(0)=(vT,w)ϕ(T)-(v0,w)ϕ(0)for all wV.

Choosing ϕ such that ϕ(T)=0 and ϕ(0)=0, respectively, shows that v(0)=v0 and v(T)=vT, respectively.

By definition, we have (Kv)(0)=u0u0 in VB and (Kv)(T)=(Ku0τv)N. Again, the a priori estimate yields the boundedness of {(Kv)(T)} in VB, and thus the existence of a weakly convergent subsequence, again denoted by , and an element uTVB such that (Kv)(T)uT in VB.

We shall show that uT=(Kv)(T). Analogously to (4.14), choosing ϕ(t)=tT, t[0,T], we have for any wV,


Using (3.1), we obtain


For the first integral, (4.6) yields


For better readability, we write Kτ instead of Ku0τ for the rest of this proof. Splitting up the sum and shifting the index in the second sum yields


Inserting this into (4.15) and using that {u0} is bounded in VB, (Kv)(T)uT in VB, Kv*Kv in L(0,T;VB) as well as λτ(eλτ-1)-11 as (and thus τ0), we end up with


which proves uT=(Kv)(T).

It remains to show that a equals Av. We recall that vL(0,T;V). Hence, for arbitrary wLp(0,T;VA) the monotonicity of A implies


The right-hand side converges to


This already implies

lim infAv,vAw,v-w+a,w.

Below we will show

lim supAv,va,v,(4.16)

hence altogether proving


Taking w=v±rz for any zLp(0,T;VA), r>0, passing to the limit r0, and Lebesgue’s theorem on dominated convergence together with the hemicontinuity and the growth condition of A:VAVA yield

Av-a,z=0for all zLp(0,T;VA),

which proves the assertion.

To finish the proof of existence, it remains to show (4.16). We start with


Since ff in Lp(0,T;VA)+L1(0,T;H), the first term on the right-hand side tends to f,v as . Using (4.5), we estimate the second term as


As v0v0=v(0) and vNvT=v(T) in H, the weak lower semicontinuity of the norm provides

lim infv^,v12v(T)H2-12v(0)H2.

Regarding the last term on the right-hand side of equation (4.17), note that only vv in Lp(0,T;VA) and BKvBKv in L2(0,T;VB). Therefore, a finer examination of that term is necessary. To start with, a short calculation (see (4.6)) shows that


We conclude with (4.5) that


Since (Kτv)0=u0u0=(Kv)(0) in VB and (Kτv)N=(Kv)(T)uT=(Kv)(T) in VB as well as KvKv in L2(0,T;VB), the weak lower semicontinuity of the norms involved shows, by passing to the limit and employing γ1λ (see (4.2)),

lim infBKv,v12λ(Kv)(T)B2-12λ(Kv)(0)B2+KvL2(0,T;B)2-0T(BKv)(t),u0dt.

Altogether, we find

lim supAv,vf,v-12v(T)H2+12v(0)H2-12λ(Kv)(T)B2+12λ(Kv)(0)B2-KvL2(0,T;B)2+0T(BKv)(t),u0dt.

We replace f by v+a+BKv and remind that v𝒞w([0,T];H) as well as Kv𝒞w([0,T];VB). Thus we are allowed to apply Lemma 4.3, which yields (4.16). ∎

5 Uniqueness and Continuous Dependence on the Problem Data

Using again the integration-by-parts formula of Lemma 4.3, we are able to prove a stability and uniqueness result. Further, we are able to prove stability with respect to perturbations of the kernel parameter λ. In order to do so, we again have to prove an integration-by-parts formula using a different approach that requires less structure than the one used in the proof of Lemma 4.3.

Theorem 5.1 (Stability).

Suppose Assumptions A and B are fulfilled and let u0,u^0VB, v0,v^0H, and f,f^Lp(0,T;VA)+L1(0,T;H).

  • (i)

    Let f-f^L1(0,T;H) and let v and v^ be solutions to ( 1.1 ) with data (u0,v0,f) and (u^0,v^0,f^) , respectively. Then the stability estimate


    holds for all t[0,T] , where the index in the notations Ku0 and Ku^0 denotes the dependence of the integral operator on u0 and u^0 , respectively.

  • (ii)

    Let f-f^Lp(0,T;VA) . Assume in addition that A is uniformly monotone in the sense that there is μ>0 such that


    for all v,wVA . Let v and v^ be solutions to ( 1.1 ) with data (u0,v0,f) and (u^0,v^0,f^) , respectively. Then the stability estimate


    holds for all t[0,T].


Let v and v^ be solutions for (u0,v0,f) and (u^0,v^0,f^), respectively. The difference of both equations (1.1a) then reads


Lemma 4.3 implies for all t[0,T],

0t(v-v^)(s)+B((Ku0v)(s)-(Ku^0v^)(s)),(v-v^)(s)ds=12v(t)-v^(t)H2-12v0-v^0H2+12λ(Ku0v)(t)-(Ku^0v^)(t)B2   -12λu0-u^0B2-0tB((Ku0v)(s)-(Ku^0v^)(s)),u0-u^0ds+0t(Ku0v)(s)-(Ku^0v^)(s)B2ds.

We test in equation (5.1) with v-v^. To prove (i), the monotonicity of A and Young’s inequality imply


An argument analogous to the one used in the proof of Theorem 4.1 yields the first statement. The second one follows analogously from the uniform monotonicity of A and


which proves the theorem. ∎

As usually, the stability estimates directly provide a uniqueness result, which we formulate in the following corollary.

Corollary 5.2 (Uniqueness).

Under the assumptions of Theorem 4.2, the solution to problem (1.1) is unique. Furthermore, the whole sequences {v} and {v^} of piecewise constant and of piecewise affine-linear prolongations of solutions to the discrete problem (4.1) converge to the solution in the sense stated in Theorem 4.2.


We consider u0=u^0VB, v0=v^0H and f=f^Lp(0,T;VA)+L1(0,T;H) with corresponding solutions v and v^. As f-f^=0L1(0,T;H), the first assumption of Theorem 5.1 is fulfilled, which shows


for all t[0,T]. This proves the uniqueness. Convergence of the whole sequences then follows as usual by contradiction. ∎

Finally, we aim to derive Lipschitz-type dependence on the kernel parameter 1λ, that is, Lipschitz dependence on the average relaxation-time.

Theorem 5.3 (Perturbation of λ).

Let the assumptions of Theorem 4.2 be fulfilled and let v0V. For λ>0, let Kλ denote the operator related to the kernel kλ(t)=λe-λt and let vλLp(0,T;VA)L(0,T;H) denote the corresponding solution to problem (1.1). Moreover, assume that vλL2(0,T;VB). Then for all μ>0 and corresponding solutions vμLp(0,T;VA)L(0,T;H) there holds


for almost all t.


Taking the difference of the equations (1.1a) for vλ and vμ, respectively, and testing with vλ-vμ leads to


To deal with the integral on the left-hand side, we need to prove an integration-by-parts formula similar to (4.9). Unfortunately, due to the fact that we have two different parameters λ and μ instead of one, we are not able to apply the same method as in the proof of Lemma 4.3. Therefore, we use another method of proof using the centered Steklov average.2

In order to prove the integration-by-parts formula on (0,t), we first have to extend the functions considered to the interval (-η,0) for a fixed η>0. We then prove the formula on an arbitrary interval (α,β) with


At the end, instead of a fixed α, we consider a sequence {αk} with αk<0 and αk0 as k. If we considered a sequence of positive αk, we would not be able to identify the limit properly.

We fix h0>0. The centered Steklov average for a function zLp(-η,T;X), with X being an arbitrary Banach space, is defined by


where 0<h<h0 and t[-η+h0,T-h0]. It is well known (see [16, Theorem 9, p. 49]) that (Shz)(t)z(t) in X for almost all t(-η+h0,T-h0). In addition, it is easy to show that ShzLp(-η+h0,T-h0;X) as well as, by using Fubini’s theorem,


and thus Shzz in Lp(-η+h0,T-h0;X).

First, note that both vλ and vμ are in 𝒞w([0,T];H), and take the same value v0 at t=0. Hence, the difference vλ-vμ fulfills the initial condition (vλ-vμ)(0)=0. We assume both vλ and vμ to be extended by v0V for t<0, and thus the difference to be extended by 0 for t<0. We conclude that vλ-vμLp(-η,T;VA)𝒞w([-η,T];H) and similarly Kλvλ-Kμvμ𝒞w([-η,T];VB). Relation (3.1) still holds for both vλ and vμ for almost all t(-η,0).



it is easy to show that Sh(vλ-vμ) is bounded in Lp(-η+h0,T-h0;VA)L(-η+h0,T-h0;H). Therefore, there exists a subsequence, again denoted by h, such that Sh(vλ-vμ)*vλ-vμ in


In view of (3.1), we have


such that Kλvλ,Kμvμ𝒞w([-η,T];VB) provide Sh(vλ-vμ)𝒞w([-η+h0,T-h0];VB). Thus, in contrast to the difference vλ-vμ itself, the Steklov average Sh(vλ-vμ) is an element of


which provides the regularity needed.

For all α,β with -η+h0<α<β<T-h0 we find

αβ(vλ-vμ)(t)+B((Kλvλ)(t)-(Kμvμ)(t)),(vλ-vμ)(t)dt=αβ(vλ-vμ)(t)+B((Kλvλ)(t)-(Kμvμ)(t)),(vλ-vμ)(t)-(Sh(vλ-vμ))(t)dt   +αβ(vλ-vμ)(t)+B((Kλvλ)(t)-(Kμvμ)(t))   -(Sh((vλ-vμ)+B(Kλvλ-Kμvμ)))(t),(Sh(vλ-vμ))(t)dt   +αβ(Sh(vλ-vμ))(t)+B(Sh(Kλvλ-Kμvμ))(t),(Sh(vλ-vμ))(t)dt.(5.3)

The first term on the right-hand side tends to zero as h0 since




in Lp(-η+h0,T-h0;VA)L(-η+h0,T-h0;H) as h0, the second one since {Sh(vλ-vμ)} is bounded in Lp(-η+h0,T-h0;VA)L(-η+h0,T-h0;H), and


in Lp(-η+h0,T-h0;VA)+L1(-η+h0,T-h0;H) as h0. The latter follows from the linearity of Sh.

Due to the regularity properties of Sh(vλ-vμ), we are allowed to split the last integrand in (5.3). We use


as well as Sh(z)=(Shz) for any integrable function z on (α,β)(-η+h0,T-h0). We rewrite the third term as

αβ(Sh(vλ-vμ))(t)+B(Sh(Kλvλ-Kμvμ))(t),(Sh(vλ-vμ))(t)dt=αβ(Sh(vλ-vμ))(t),(Sh(vλ-vμ))(t)dt+αβB(Sh(Kλvλ-Kμvμ))(t),(Sh(Kλvλ-Kμvμ))(t)dt   +αβB(Sh(Kλvλ-Kμvμ))(t),1λ(Sh(Kλvλ))(t)-1μ(Sh(Kμvμ))(t)dt=12(Sh(vλ-vμ))(β)H2-12(Sh(vλ-vμ))(α)H2+αβ(Sh(Kλvλ-Kμvμ))(t)B2dt   +1μαβB(Sh(Kλvλ-Kμvμ))(t),(Sh(Kλvλ-Kμvμ))(t)dt   +(1λ-1μ)αβB(Sh(Kλvλ-Kμvμ))(t),(Sh(Kλvλ))(t)dt12(Sh(vλ-vμ))(β)H2-12(Sh(vλ-vμ))(α)H2+αβ(Sh(Kλvλ-Kμvμ))(t)B2dt   +12μ(Sh(Kλvλ-Kμvμ))(β)B2-12μ(Sh(Kλvλ-Kμvμ))(α)B2   -|1λ-1μ|αβ(Sh(Kλvλ-Kμvμ))(t)B(Sh(Kλvλ))(t)Bdt.

Taking the limit h and employing (5.2) results in

12(vλ-vμ)(β)H2+αβ(Kλvλ-Kμvμ)(t)B2dt+12μ(Kλvλ-Kμvμ)(β)B212(vλ-vμ)(α)H2+12μ(Kλvλ-Kμvμ)(α)B2   +|1λ-1μ|αβ(Kλvλ-Kμvμ)(t)B(Kλvλ)(t)Bdt(5.4)

for almost all α,β, namely for those that are Lebesgue points of vλ-vμ and Kλvλ-Kμvμ. Finally, we consider a sequence {αk} with αk<0 and αk0 as k. We remark that vλ-vμ𝒞w([-η,T];H) and Kλvλ-Kμvμ𝒞w([-η,T];VB). Due to the choice of the extension for t<0, equation (5.4) shows for β=t that


for almost all t(0,T), and thus


It is easy to show that Kλvλ-u0L2(0,T;B)2λ2T2vλL2(0,T;B)2, which proves the assertion. ∎


  • [1]

    V. Barbu, P. Colli, G. Gilardi and M. Grasselli, Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation, Differential Integral Equations 13 (2000), no. 10–12, 1233–1262.  Google Scholar

  • [2]

    A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Res. Notes Math. 10, A K Peters, Wellesley, 2005.  Google Scholar

  • [3]

    S. Bonaccorsi and G. Desch, Volterra equations in Banach spaces with completely monotone kernels, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 3, 557–594.  CrossrefGoogle Scholar

  • [4]

    H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Dunod, Paris, 1999.  Google Scholar

  • [5]

    M. P. Calvo, E. Cuesta and C. Palencia, Runge–Kutta convolution quadrature methods for well-posed equations with memory, Numer. Math. 107 (2007), no. 4, 589–614.  CrossrefGoogle Scholar

  • [6]

    P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms, NoDEA Nonlinear Differential Equations Appl. 10 (2003), no. 4, 399–430.  CrossrefGoogle Scholar

  • [7]

    C. Chen and T. Shih, Finite Element Methods for Integrodifferential Equations, Ser. Appl. Math. 9, World Scientific, River Edge, 1998.  Google Scholar

  • [8]

    C. Chen, V. Thomée and L. B. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comp. 58 (1992), no. 198, 587–602.  CrossrefGoogle Scholar

  • [9]

    C. Corduneanu, Functional Equations with Causal Operators, Stab. Control Theory Methods Appl. 16, Taylor & Francis, London, 2002.  Google Scholar

  • [10]

    M. G. Crandall, S.-O. Londen and J. A. Nohel, An abstract nonlinear Volterra integrodifferential equation, J. Math. Anal. Appl. 64 (1978), no. 3, 701–735.  CrossrefGoogle Scholar

  • [11]

    M. G. Crandall and J. A. Nohel, An abstract functional differential equation and a related nonlinear Volterra equation, Israel J. Math. 29 (1978), no. 4, 313–328.  CrossrefGoogle Scholar

  • [12]

    E. Cuesta and C. Palencia, A numerical method for an integro-differential equation with memory in Banach spaces: Qualitative properties, SIAM J. Numer. Anal. 41 (2003), no. 4, 1232–1241.  CrossrefGoogle Scholar

  • [13]

    J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer, Berlin, 1977.  Google Scholar

  • [14]

    C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations 4 (1979), no. 3, 219–278.  CrossrefGoogle Scholar

  • [15]

    W. Desch, R. Grimmer and W. Schappacher, Well-posedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations 74 (1988), no. 2, 391–411.  CrossrefGoogle Scholar

  • [16]

    J. Diestel and J. J. Uhl, Jr., Vector Measures, American Mathematical Society, Providence, 1977.  Google Scholar

  • [17]

    D. A. Edwards, Non-Fickian diffusion in thin polymer films, J. Polym. Sci. Part B Polym. Phys. 34 (1996), 981–997.  CrossrefGoogle Scholar

  • [18]

    D. A. Edwards and D. S. Cohen, A mathematical model for a dissolving polymer, AIChE Journal 41 (1995), no. 11, 2345–2355.  CrossrefGoogle Scholar

  • [19]

    E. Emmrich and M. Thalhammer, Doubly nonlinear evolution equations of second order: existence and fully discrete approximation, J. Differential Equations 251 (2011), no. 1, 82–118.  CrossrefGoogle Scholar

  • [20]

    S. Fedotov and A. Iomin, Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion, Phys. Rev. E (3) 77 (2008), no. 3, Article ID 031911.  Google Scholar

  • [21]

    J. A. Ferreira, E. Gudiño and P. de Oliveira, A second order approximation for quasilinear non-Fickian diffusion models, Comput. Methods Appl. Math. 13 (2013), no. 4, 471–493.  Google Scholar

  • [22]

    H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie, Berlin, 1974.  Google Scholar

  • [23]

    G. Gilardi and U. Stefanelli, Time-discretization and global solution for a doubly nonlinear Volterra equation, J. Differential Equations 228 (2006), no. 2, 707–736.  CrossrefGoogle Scholar

  • [24]

    G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation, J. Math. Anal. Appl. 333 (2007), no. 2, 839–862.  CrossrefGoogle Scholar

  • [25]

    M. Grasselli and A. Lorenzi, Abstract nonlinear Volterra integrodifferential equations with nonsmooth kernels, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 2 (1991), no. 1, 43–53.  Google Scholar

  • [26]

    R. Grimmer and M. Zeman, Nonlinear Volterra integro-differential equations in a Banach space, Israel J. Math. 42 (1982), no. 1–2, 162–176.  CrossrefGoogle Scholar

  • [27]

    G. Gripenberg, On a nonlinear Volterra integral equation in a Banach space, J. Math. Anal. Appl. 66 (1978), no. 1, 207–219.  CrossrefGoogle Scholar

  • [28]

    G. Gripenberg, An abstract nonlinear Volterra equation, Israel J. Math. 34 (1979), no. 3, 198–212.  CrossrefGoogle Scholar

  • [29]

    G. Gripenberg, Nonlinear Volterra equations of parabolic type due to singular kernels, J. Differential Equations 112 (1994), no. 1, 154–169.  CrossrefGoogle Scholar

  • [30]

    G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl. 34, Cambridge University Press, Cambridge, 1990.  Google Scholar

  • [31]

    S. Larsson, V. Thomée and L. B. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp. 67 (1998), no. 221, 45–71.  CrossrefGoogle Scholar

  • [32]

    J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer, New York, 1972.  Google Scholar

  • [33]

    S.-O. Londen, On an integrodifferential Volterra equation with a maximal monotone mapping, J. Differential Equations 27 (1978), no. 3, 405–420.  CrossrefGoogle Scholar

  • [34]

    C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129–145.  CrossrefGoogle Scholar

  • [35]

    C. Lubich, Convolution quadrature and discretized operational calculus. II, Numer. Math. 52 (1988), no. 4, 413–425.  CrossrefGoogle Scholar

  • [36]

    C. Lubich, Convolution quadrature revisited, BIT 44 (2004), no. 3, 503–514.  CrossrefGoogle Scholar

  • [37]

    C. Lubich, I. H. Sloan and V. Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comp. 65 (1996), no. 213, 1–17.  CrossrefGoogle Scholar

  • [38]

    R. C. MacCamy, A model for one-dimensional, nonlinear viscoelasticity, Quart. Appl. Math. 35 (1977/78), no. 1, 21–33.  CrossrefGoogle Scholar

  • [39]

    R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math. 35 (1977/78), no. 1, 1–19.  CrossrefGoogle Scholar

  • [40]

    R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc. 164 (1972), 1–37.  CrossrefGoogle Scholar

  • [41]

    W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term, J. Aust. Math. Soc. Ser. B 35 (1993), no. 1, 23–70.  CrossrefGoogle Scholar

  • [42]

    A. Mehrabian and Y. Abousleiman, General solutions to poroviscoelastic model of hydrocephalic human brain tissue, J. Theoret. Biol. 291 (2011), 105–118.  CrossrefGoogle Scholar

  • [43]

    R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), no. 2, 313–332.  CrossrefGoogle Scholar

  • [44]

    J. Prüß, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993.  Google Scholar

  • [45]

    T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2005.  Google Scholar

  • [46]

    S. Shaw and J. R. Whiteman, Some partial differential Volterra equation problems arising in viscoelasticity, Proceedings of Equadiff 9, Masaryk University, Brno (1998), 183–200.  Google Scholar

  • [47]

    I. H. Sloan and V. Thomée, Time discretization of an integro-differential equation of parabolic type, SIAM J. Numer. Anal. 23 (1986), no. 5, 1052–1061.  CrossrefGoogle Scholar

  • [48]

    U. Stefanelli, On some nonlocal evolution equations in Banach spaces, J. Evol. Equ. 4 (2004), no. 1, 1–26.  CrossrefGoogle Scholar

  • [49]

    W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966), 543–551.  CrossrefGoogle Scholar

  • [50]

    G. F. Webb, Functional differential equations and nonlinear semigroups in Lp-spaces, J. Differential Equations 20 (1976), no. 1, 71–89.  Google Scholar

  • [51]

    G. F. Webb, Volterra integral equations and nonlinear semigroups, Nonlinear Anal. 1 (1976/77), no. 4, 415–427.  Google Scholar

  • [52]

    G. F. Webb, An abstract semilinear Volterra integrodifferential equation, Proc. Amer. Math. Soc. 69 (1978), no. 2, 255–260.  CrossrefGoogle Scholar

  • [53]

    R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl. 348 (2008), no. 1, 137–149.  CrossrefGoogle Scholar

  • [54]

    R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcial. Ekvac. 52 (2009), no. 1, 1–18.  CrossrefGoogle Scholar

  • [55]

    E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B, Springer, New York, 1990.  Google Scholar


  • 1

    Here, we slightly abuse the notation of the duality pairing because Lp(0,T;VA)+L1(0,T;H)+L(0,T;VB) is not the dual space of Lp(0,T;VA)L(0,T;H)L1(0,T;VB). However, we can consider the sum of the duality pairings between Lp(0,T;VA)+L1(0,T;H) and L1(0,T;VB) and their respective dual spaces. 

  • 2

    We could have also used this method to prove Lemma 4.3, but it only provides the integration-by-parts formula to hold on (α,β) for almost all α,β(0,T). 

About the article

Received: 2018-06-20

Accepted: 2018-10-20

Published Online: 2018-11-21

Published in Print: 2020-01-01

This work has been supported by the Deutsche Forschungsgemeinschaft through Collaborative Research Center 910 “Control of self-organizing nonlinear systems: Theoretical methods and concepts of application”.

Citation Information: Computational Methods in Applied Mathematics, Volume 20, Issue 1, Pages 89–108, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0268.

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