Convergence Criteria for Fixed Point Problems and Differential Equations

Abstract

We consider a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. We use a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we establish convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. These criteria provide the necessary and sufficient conditions on a sequence $\left\{u_n\right\}$, which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result.

1. Introduction

Convergence results represent an important topic in Functional Analysis, Numerical Analysis, and Differential and Partial Differential Equations Theory. They are important in the study of mathematical models which arise in Mechanics, Physics, and Engineering Sciences as well. Some elementary examples are specified below: (a) the convergence of the solution of a penalty problem to the solution of the original problem as the penalty parameter converges—a reference in the field is [1], in which the penalty method is used in the study of variational inequalities; (b) the convergence of the discrete solution to the solution of the continuous problem as the time step or the discretization parameter converges to zero—a reference in the field is [2], where error estimates and convergence results for discrete schemes are provided; and (c) the convergence of the solution of a viscoelastic problem to the solution of an elastic problem as the viscosity goes to zero—references in the field include [3,4,5], in which various models of contact with elastic and viscoelastic materials are analyzed.

For all these reasons, a considerable effort was made to obtain convergence results in the study of various mathematical problems including nonlinear equations, inequality problems, inclusions, fixed point problems, optimization problems, and some others. Nevertheless, in most of the cases, such results provide only sufficient conditions which guarantee the convergence of a given sequence $\left\{u_n\right\}$ to the solution of the corresponding problem, denoted in what follows by $\mathcal{P}$. They do not describe all the sequences which have this property. Therefore, we naturally turn to consider the following problem, associated with $\mathcal{P}$.

Problem 1

(${\mathcal{Q}}_{\mathcal{P}}$). Given a Problem $\mathcal{P}$ which has a unique solution u in a metric space Y, describe the convergence of a sequence $\left\{u_n\right\}\subset Y$ to the solution u. In other words, provide necessary and sufficient conditions for the convergence $u_n\to u$ in Y, i.e., provide a convergence criterion.

Note that Problem ${\mathcal{Q}}_{\mathcal{P}}$ represents a major issue in the study of convergence results. Its solution depends on the structure of the initial problem $\mathcal{P}$ and cannot be provided in this general framework. Results in solving Problem ${\mathcal{Q}}_{\mathcal{P}}$ were obtained in [6], in the particular case when $\mathcal{P}$ is one of the three problems: a variational inequality, a fixed point problem, or a minimization problem. The novelty of the current paper arises from the fact that, here, we solve Problem ${\mathcal{Q}}_{\mathcal{P}}$ in the case when $\mathcal{P}$ represents both a fixed point problem with history-dependent operators and a Cauchy problem for a nonlinear differential equation in Hilbert spaces. This allows us to formulate convergence criteria, which represent a powerful tool in the study of various nonlinear problems and lead to interesting applications.

More precisely, in this current paper, we continue our research in [6] with the case when $\mathcal{P}$ is a Cauchy problem of the following form:

$$\begin{array}{ccc}& & \dot{u}\left(t\right)=F(t,u\left(t\right)))\forall t\in I,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & u\left(0\right)=u_0\hfill \end{array}$$

(2)

Throughout this paper, X represents either a Banach space endowed with the norm ${\parallel ·\parallel }_X$ or a Hilbert space endowed with the inner product ${(·,·)}_X$ and the associated norm ${\parallel ·\parallel }_X$, $I\subset \mathbb{R}$ is an interval of time, $F:I\times X\to X$, and $u_0\in X$ is a given initial data. Moreover, the dot above represents the derivative concerning the time variable. We consider both the case when I is a finite interval of the form $I=[0,T]$ with $T>0$ and the case when $I={\mathbb{R}}_{+}$ and, when no specification is made, I will represent either of these intervals. Under appropriate assumptions, which guarantee that problem (1) and (2) has a unique solution $u\in C^1(I;X)$, our aim is to indicate the necessary and sufficient conditions which guarantee the convergence of a given sequence $\left\{u_n\right\}\in C^1(I;X)$ to the solution u, in both the $C(I;X)$ and $C^1(I;X)$ spaces. For this reason, our results below extend the existing results in the literature which, to the best of our knowledge, provide only sufficient conditions which guarantee the convergences above.

Note that the study of problem (1) and (2) is related to the study of the fixed point problem

$$u\left(t\right)=\Lambda u\left(t\right)\forall t\in I,$$

(3)

where $\Lambda :C(I;X)\to C(I;X)$ is an operator which will be specified later. For this reason, we start with convergence results concerning this auxiliary fixed point problem. To conclude, in this paper we shall provide an answer to Problem ${\mathcal{Q}}_{\mathcal{P}}$ in the case when $\mathcal{P}$ is both the Cauchy problem (1) and (2) and the fixed point problem (3), while the space Y is the space $C^1(I;X)$ and the space $C(I;X)$, respectively. Our study is motivated by possible applications in Analysis and Solid Mechanics.

The rest of the manuscript is structured as follows. In Section 2, we introduce some preliminary material. Then, in Section 3, we state and prove a convergence criterion in the study of the fixed problem (3). Next, in Section 4, we state and prove two different convergence criteria in the study of the Cauchy problem (1) and (2), in the space of continuous and continuously differentiable functions, respectively. We use these results in Section 5, in which we consider a particular form of the differential Equation (1), governed by two nonlinear operators. Finally, in Section 6, we provide an application of the abstract results in Section 5 in the study of differential equations arising in viscoelasticity. We end this paper with Section 7, in which we present some concluding remarks.

2. Preliminaries

In this section, we introduce two spaces of functions and the class of history-dependent operators. Then, we state two elementary inequalities which will be used repeatedly in the next sections. We precisely point out that everywhere in this manuscript m will denote a given positive integer and the limits are considered as $n\to \infty$, even if we do not mention it explicitly. For a sequence $\left\{{\epsilon }_n\right\}\subset {\mathrm{I}\mathrm{R}}_{+}$ which converges to zero, we use the simple notation $0\le {\epsilon }_n\to 0$. We extend this notation to a sequence $\left\{{\epsilon }_n^m\right\}\subset {\mathrm{I}\mathrm{R}}_{+}$ (with m given) which converges to zero and, therefore, we write $0\le {\epsilon }_n^m\to 0$.

Space of continuously and continuously differentiable functions. We start with some properties of the spaces $C(I;X)$ and $C^1(I;X)$ defined by

$$\begin{array}{ccc}& & C(I;X)=\{v:I\to X\mid v\mathrm{is}\mathrm{continuous}\},\hfill \\ & & C^1(I;X)=\{v:I\to X\mid v\in C(I;X)\mathrm{and}\dot{v}\in C(I;X)\}.\hfill \end{array}$$

On occasion, these spaces will be denoted by $C\left(\right[0,T];X)$ and $C^1([0,T];X)$, respectively, if $I=[0,T]$. The space $C\left(\right[0,T];X)$ will be equipped with the norm of the uniform convergence, that is

$${\parallel v\parallel }_{C\left(\right[0,T];X)}=\underset{t\in [0,T]}{max}{\parallel v\left(t\right)\parallel }_X.$$

(4)

It is well known that, endowed with this norm, this space is a Banach space. Moreover, the space $C^1([0,T];X)$ is a Banach space with the norm

$${\parallel v\parallel }_{C^1([0,T];X)}=\underset{t\in [0,T]}{max}{\parallel v\left(t\right)\parallel }_X+\underset{t\in [0,T]}{max}{\parallel \dot{v}\left(t\right)\parallel }_X.$$

(5)

We now consider the case $I={\mathbb{R}}_{+}$ and we assume that X is a Banach space. Then, as shown in [7,8], the space $C({\mathbb{R}}_{+};X)$ can be organized in a canonical way, as with a Fréchet space, in which the convergence of a sequence $\left\{v_n\right\}\subset C({\mathbb{R}}_{+};X)$ to the element $v\in C({\mathbb{R}}_{+};X)$ is characterized by the following equivalence:

$$\left\{\begin{array}{c}{v}_n\to v\mathrm{i}\mathrm{n} C({\mathbb{R}}_{+};X)⟺\hfill \\ {\displaystyle \underset{t\in [0,m]}{max}\parallel v_n\left(t\right)-v\left(t\right){\parallel }_X\to 0\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l}m\in \mathbb{N}.}\hfill \end{array}\right.$$

(6)

In other words, the sequence $\left\{v_n\right\}$ converges to the element v in the space $C({\mathbb{R}}_{+};X)$ if and only if it converges to v in the space $C\left(\right[0,m];X)$ for any $m\in \mathbb{N}$. Moreover, for $\left\{v_n\right\}\subset C^1({\mathbb{R}}_{+};X)$ and $v\in C^1({\mathbb{R}}_{+};X)$, the following equivalence holds:

$$\left\{\begin{array}{c}{v}_n\to v\mathrm{in}{C}^1({\mathbb{R}}_{+};X)⟺\hfill \\ {\displaystyle \underset{t\in [0,m]}{max}{\parallel v_n\left(t\right)-v\left(t\right)\parallel }_X\to 0\mathrm{and}}\hfill \\ {\displaystyle \underset{t\in [0,m]}{max}{\parallel {\dot{v}}_n\left(r\right)-\dot{v}\left(t\right)\parallel }_X\to 0,\mathrm{for}\mathrm{all}m\in \mathbb{N}.}\hfill \end{array}\right.$$

(7)

The equivalences (6) and (7) will be used repeatedly in the next sections to prove various convergence results when working on the framework of an unbounded interval of time.

Using the properties of the integral, it is easy to see that, if $f\in C(I;X)$, then the function $g:I\to X$ given by

$$g\left(t\right)={\int }_0^{t}f\left(s\right)ds\mathrm{for} \mathrm{all}t\in I$$

belongs to $C^1(I;X)$ and, moreover, $\dot{g}=f$. In addition, we recall that, for a function $v\in C^1(I;X)$, the following equality holds

$$v\left(t\right)={\int }_0^t\dot{v}\left(s\right)ds+v\left(0\right)\mathrm{for} \mathrm{all}t\in I.$$

(8)

Finally, we mention that, when no confusion arises, we shall use the notation $0_X$ for the zero element in both spaces X, $C(I;X)$, $C^1(I;X)$, $C\left(\right[0,m];X)$ and $C^1([0,m];X)$, for any $m\in \mathbb{N}$.

History-dependent operators. We now introduce a class of operators defined on the space of continuous functions $C(I;X)$.

Definition 1.

An operator $\Lambda :C(I;X)\to C(I;X)$ is called history-dependent if

(a) $I=[0,T]$ and there exists $L>0$ such that

$$\begin{array}{ccc}& & {\displaystyle \parallel \Lambda u_1\left(t\right)-\Lambda u_2{\left(t\right)\parallel }_X\le L{\int }_0^t{\parallel u_1\left(s\right)-u_2\left(s\right)\parallel }_{X}ds}\hfill \\ & & \mathrm{for}\mathrm{all}{u}_1,u_2\in C([0,T];X),t\in [0,T].\hfill \end{array}$$

(9)

(b) $I={\mathbb{R}}_{+}$ and for any $m\in \mathbb{N}$ there exists $L_m>0$ such that

$$\begin{array}{ccc}& & {\displaystyle \parallel \Lambda u_1\left(t\right)-\Lambda u_2{\left(t\right)\parallel }_X\le L_m{\int }_0^t{\parallel u_1\left(s\right)-u_2\left(s\right)\parallel }_{X}ds}\hfill \\ & & \mathrm{for}\mathrm{all}{u}_1,u_2\in C({\mathbb{R}}_{+};X),t\in [0,m].\hfill \end{array}$$

(10)

Note that here and below, when no confusion arises, we use the shorthand notation $\Lambda u\left(t\right)$ to represent the value of the function $\Lambda u$ at the point t, i.e., $\Lambda u\left(t\right)=(\Lambda u)\left(t\right)$, for all $t\in I$. Also, we recall that the term “history-dependent operator” was introduced in [9]; since then, it has been used in many papers (see [10] and the references therein). Examples of history-dependent operators will be provided in the next sections of this manuscript.

Finally, using Definition 1 and the convergences (4) and (6), it is easy to see that any history operator $\Lambda :C(I;X)\to C(I;X)$ is continuous, that is

$$u_n\to u\mathrm{in}C(I;X)⟹\Lambda u_n\to \Lambda u\mathrm{in}C(I;X).$$

(11)

An important property of history-dependent operators is the following fixed point property, proved in [10].

Theorem 1.

Let X be a Banach space and $\Lambda :C(I;X)\to C(I;X)$ be a history-dependent operator. Then, Λ has a unique fixed point, i.e., there exists a unique element $u\in C(I;X)$ such that $\Lambda u=u$.

Theorem 1 can be used to prove the unique solvability of various nonlinear problems. An example is provided by the following result, which will be used in Section 6 in this paper.

Theorem 2.

Let X be a Hilbert space, $A:X\to X$ a strongly monotone Lipschitz continuous operator and $\Lambda :C(I;X)\to C(I;X)$ a history-dependent operator. Then, for any $f\in C(I;X)$, there exists a unique function $u\in C(I;X)$ such that

$$Au\left(t\right)+\Lambda u\left(t\right)=f\left(t\right)\forall t\in I.$$

(12)

The proof of Theorem 2 can be found in [10], based on the fixed point result provided by Theorem 1.

Two elementary inequalities. We now recall two elementary inequalities which will be used in many places below. To this end, for simplicity, we use the notation $C\left(I\right)=C(I;\mathbb{R})$. The first inequality we recall is the well-known Gronwall inequality and is stated as follows.

Lemma 1.

Let f, $g\in C\left(I\right)$ and assume that there exists $c>0$ such that

$$f\left(t\right)\le g\left(t\right)+c{\int }_0^{t}f\left(s\right)ds\mathrm{for}\mathrm{all}t\in I.$$

(13)

Then,

$$f\left(t\right)\le g\left(t\right)+c{\int }_0^{t}g\left(s\right)e^{c(t-s)}ds\mathrm{for}\mathrm{all}t\in I.$$

(14)

Moreover, if g is nondecreasing, then

$$f\left(t\right)\le g\left(t\right)e^{ct}\mathrm{for}\mathrm{all}t\in I.$$

A proof of Lemma 1 can be found in ([10] p. 60), and, therefore, we have skipped it.

The second inequality we need is the following.

Lemma 2.

Let X be a Hilbert space, $x\in X$ and $\epsilon \ge 0$. Then, the following equivalence holds:

$${\parallel x\parallel }_X\le \epsilon ⟺{(x,v)}_X+\epsilon {\parallel v\parallel }_X\ge 0\forall v\in X.$$

(15)

Proof. 

Assume that ${\parallel x\parallel }_X\le \epsilon$ and $v\in X$. Then, it is easy to see that

$${(x,v)}_X+{\epsilon \parallel v\parallel }_X\ge -{\parallel x\parallel }_X{\parallel v\parallel }_X+{\epsilon \parallel v\parallel }_X={(\epsilon -\parallel x\parallel }_X{)\parallel v\parallel }_X$$

and, therefore, ${(x,v)}_X+\epsilon {\parallel v\parallel }_X\ge 0$. Conversely, assume that ${(x,v)}_X+\epsilon {\parallel v\parallel }_X\ge 0$ for any $v\in X$. We take $v=-x$ in this inequality to find that $-{(x,x)}_X+\epsilon {\parallel x\parallel }_X\ge 0$, which implies that ${\parallel x\parallel }_X^2\le \epsilon {\parallel x\parallel }_X$. We deduce from this that ${\parallel x\parallel }_X\le \epsilon$, which concludes the proof. □

3. The Fixed Point Problem

In this section, we deal with the fixed point problem (3). To this end, we assume that X is a Hibert space and, under the assumption of Theorem 1, we denote by $u\in C(I;X)$ the fixed point of operator $\Lambda$. Moreover, given an arbitrary sequence $\left\{u_n\right\}\subset C(I;X)$, we consider the following statements:

$$\begin{array}{ccc}& & u_n\to u\mathrm{in}C(I;X).\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & u_n-\Lambda u_n\to 0_X\mathrm{in}C(I;X).\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I=[0;T]\mathrm{and} \mathrm{there} \mathrm{exists}0\le {\epsilon }_n\to 0\mathrm{such} \mathrm{that}\hfill \\ {(u_n\left(t\right),v)}_X+{\epsilon }_n{\parallel v\parallel }_X\ge {(\Lambda u_n\left(t\right),v)}_X\forall v\in X,n\in \mathbb{N},t\in I.\hfill \end{array}\right.\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I={\mathbb{R}}_{+}\mathrm{and} \mathrm{for} \mathrm{any} m\in \mathbb{N}\mathrm{there} \mathrm{exists}0\le {\epsilon }_n^m\to 0\mathrm{such} \mathrm{that}\hfill \\ {(u_n\left(t\right),v)}_X+{\epsilon }_n^m{\parallel v\parallel }_X\ge {(\Lambda u_n\left(t\right),v)}_X\forall v\in X,n\in \mathbb{N},t\in [0,m].\hfill \end{array}\right.\hfill \end{array}$$

(19)

We now state and prove our main result in this section.

Theorem 3.

Let X be a Hilbert space, $T>0$ and $\Lambda :C(I;X)$ a history-dependent operator.

(a) If $I=[0;T]$, then the statements (16)–(18) are equivalent.

(b) If $I={\mathbb{R}}_{+}$, then the statements (16), (17), and (19) are equivalent.

Proof. 

(a) We start with the case $I=[0,T]$. Assume that (16) holds. Then, from (11), it is easy to see that $u_n-\Lambda u_n\to u-\Lambda u$ in $C(I;X)$ and, since u is the solution of the fixed point problem (3), we deduce that (17) holds.

Next, assume that (17) holds, which shows that

$$\underset{s\in [0,T]}{max}{\parallel u_n\left(s\right)-\Lambda u_n\left(s\right)\parallel }_X\to 0\mathrm{as}n\to \infty .$$

(20)

For each $n\in \mathbb{N}$ we denote

$${\epsilon }_n=\underset{s\in [0,T]}{max}{\parallel u_n\left(s\right)-\Lambda u_n\left(s\right)\parallel }_X.$$

(21)

Then, (20) shows that $0\le {\epsilon }_n\to 0$, and definition (21) implies that for any $t\in I$ we have

$$\parallel u_n\left(t\right)-\Lambda u_n{\left(t\right)\parallel }_X\le {\epsilon }_n.$$

(22)

We now use inequality (22) and Lemma 2 to see that condition (18) holds.

Finally, assume that (18) holds. Let $n\in \mathbb{N}$ and $t\in [0,T]$. We take $v=u\left(t\right)-u_n\left(t\right)$ in this inequality to see that

$${(u_n\left(t\right),u\left(t\right)-u_n\left(t\right))}_X+{\epsilon }_n{\parallel u\left(t\right)-u_n\left(t\right)\parallel }_X\ge {(\Lambda u_n\left(t\right),u\left(t\right)-u_n\left(t\right))}_X$$

and, using equality $u\left(t\right)=\Lambda u\left(t\right)$, we find that

$${(u_n\left(t\right)-u\left(t\right),u\left(t\right)-u_n\left(t\right))}_X+{\epsilon }_n{\parallel u\left(t\right)-u_n\left(t\right)\parallel }_X\ge {(\Lambda u_n\left(t\right)-\Lambda u\left(t\right),u\left(t\right)-u_n\left(t\right))}_X.$$

Thus,

$$\parallel u_n{\left(t\right)-u\left(t\right)\parallel }_X^2\le {\epsilon }_n\parallel u_n{\left(t\right)-u\left(t\right)\parallel }_X+\parallel \Lambda u_n{\left(t\right)-\Lambda u\left(t\right)\parallel }_X{\parallel u_n\left(t\right)-u\left(t\right)\parallel }_X$$

and, therefore,

$$\parallel u_n{\left(t\right)-u\left(t\right)\parallel }_X\le {\epsilon }_n+{\parallel \Lambda u_n\left(t\right)-\Lambda u\left(t\right)\parallel }_X.$$

We now use inequality (9) to see that

$$\parallel u_n{\left(t\right)-u\left(t\right)\parallel }_X\le {\epsilon }_n+L{\int }_0^t{\parallel u_n\left(s\right)-u\left(s\right)\parallel }_{X}ds$$

and, employing the Gronwall argument provided by Lemma 1, we find that

$$\parallel u_n{\left(t\right)-u\left(t\right)\parallel }_X\le {\epsilon }_n{e}^{Lt}.$$

We finally use the convergence ${\epsilon }_n\to 0$ and inequality $t\le T$ to see that

$$\underset{t\in [0,T]}{max}{\parallel u_n\left(t\right)-u\left(t\right)\parallel }_X\to 0\mathrm{as}n\to \infty ,$$

which implies that (16) holds.

To conclude, we proved the implications (16) $⟹$ (17) $⟹$ (18) $⟹$ (16), which shows the equivalence of the statements (16)–(18).

(b) We continue with the case $I={\mathbb{R}}_{+}$. To this end, we fix $m\in \mathbb{N}$ and we use the first part of the theorem with $T=m$, combined with the remark that the quantity ${\epsilon }_n$ defined in (21) depends on T and, therefore, since $T=m$, we denote it by ${\epsilon }_n^m$. We deduce from here the equivalences of the following statements:

$$\begin{array}{ccc}& & u_n\to u\mathrm{in}C([0,m];X).\hfill \end{array}$$

(23)

$$\begin{array}{ccc}& & u_n-\Lambda u_n\to 0_X\mathrm{in}C([0,m];X).\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}\mathrm{there}\mathrm{exists}0\le {\epsilon }_n^m\to 0\mathrm{such} \mathrm{that}\hfill \\ {(u_n\left(t\right),v)}_X+{\epsilon }_n^m{\parallel v\parallel }_X\ge {(\Lambda u_n\left(t\right),v)}_X\forall v\in X,n\in \mathbb{N},t\in [0,m].\hfill \end{array}\right.\hfill \end{array}$$

(25)

Recall that the equivalence of these statements is valid for any $m\in \mathbb{N}$. We now use (6) to see that the convergences (23) and (24) can be replaced by the convergences (16) and (17), respectively, which concludes the proof. □

We remark that Theorem 3 provides an answer to Problem ${\mathcal{Q}}_{\mathcal{P}}$ in the particular case when Problem $\mathcal{P}$ is the fixed point problem (3). Indeed, it provides a convergence criterion for the solution of this problem, in the case of both $I=[0,T]$ and $I={\mathbb{R}}_{+}$. For this reason, Theorem 3 has a specific advantage with respect to the existing literature which, to the best of our knowledge, gives only sufficient conditions which guarantee the convergence above.

We end this section with the remark that, in the case of $I={\mathbb{R}}_{+}$, we cannot skip the dependence on m for the constants ${\epsilon }_n^m$ which appear in (25). More precisely, we claim that, in the case of $I={\mathbb{R}}_{+}$, condition

$$\left\{\begin{array}{c}\mathrm{there} \mathrm{exists}0\le {\epsilon }_n\to 0\mathrm{such} \mathrm{that}\hfill \\ {(u_n\left(t\right),v)}_X+{\epsilon }_n{\parallel v\parallel }_X\ge {(\Lambda u_n\left(t\right),v)}_X\forall v\in X,n\in \mathbb{N},t\in I\hfill \end{array}\right.$$

(26)

is not equivalent to the convergence (16). The proof of this claim follows from the following example.

Example 1.

Let $X=\mathbb{R}$, $I={\mathbb{R}}_{+}$ and let $\Lambda :C\left(I\right)\to C\left(I\right)$ be the operator defined by

$$\Lambda u\left(t\right)={\int }_0^{t}u\left(s\right)ds$$

(27)

for all $u\in C\left(I\right)$, $t\in I$, and $n\in \mathbb{N}$. Then, it is easy to see that Λ is a history-dependent operator and its unique fixed point is the function $u\left(t\right)=0$ for all $t\in I$. Consider, now, the function

$$u_n\left(t\right)=\frac{1}{n}{e}^{\frac{n+1}{n}t}\forall n\in \mathbb{N},t\in {\mathbb{R}}_{+}.$$

(28)

Then, it is easy to see that

$$\underset{t\in [0,m]}{max}\left|u_n\left(t\right)\right|=\frac{1}{n}{e}^{\frac{n+1}{n}m}\le \frac{1}{n}{e}^{2m}\to 0\mathrm{as}n\to \infty ,\forall m\in \mathbb{N}$$

and, therefore, (6) shows that $u_n\to 0$ in the space $C\left(I\right)$. Nevertheless, we shall prove that condition (26) does not hold. Indeed, arguing the contrary, assume that the sequence $\left\{u_n\right\}$ satisfies this condition. Then, there exists a sequence $0\le {\epsilon }_n\to 0$ such that the inequality in (26) holds and, using Lemma 2, we deduce that

$$|\Lambda u_n\left(t\right)-u_n\left(t\right)|\le {\epsilon }_n\forall n\in \mathbb{N},t\in {\mathbb{R}}_{+}.$$

(29)

Using, now, (27)–(29), we deduce that

$$\frac{1}{n(n+1)}{e}^{\frac{n+1}{n}t}+\frac{1}{n+1}\le {\epsilon }_n\forall n\in \mathbb{N},t\in {\mathbb{R}}_{+}.$$

We now take $t=n^2$ in the previous inequality; then, we pass to the limit as $n\to \infty$ and arrive at a contradiction. We conclude from here that condition (26) does not hold.

4. The Cauchy Problem

We now proceed with the study of the Cauchy problem (1) and (2) and, to this end, we consider the following assumptions.

$$\left\{\begin{array}{c}\left(\mathrm{a}\right)F:I\times X\to X.\hfill \\ \left(\mathrm{b}\right)\mathrm{The}\mathrm{mapping}t\mapsto F(t,u):I\to X\mathrm{is}\mathrm{continuous}\hfill \\ \mathrm{for}\mathrm{all}u\in X.\hfill \\ \left(\mathrm{c}\right)IfI=[0,T]\mathrm{then} \mathrm{there} \mathrm{exists}{L}_F>0\mathrm{such}\mathrm{that}\hfill \\ \parallel F(t,u_1)-F(t,u_2){\parallel }_X\le L_F{\parallel u_1-u_2\parallel }_X\hfill \\ \mathrm{for}\mathrm{all}{u}_1,u_2\in X,t\in [0,T].\hfill \\ \left(\mathrm{d}\right)IfI={\mathbb{R}}_{+}\mathrm{then} \mathrm{for} \mathrm{any}m\in \mathbb{N}\mathrm{there} \mathrm{exists}{L}_F^m>0\mathrm{such}\mathrm{that}\hfill \\ \parallel F(t,u_1)-F(t,u_2){\parallel }_X\le L_F^m{\parallel u_1-u_2\parallel }_X\hfill \\ \mathrm{for}\mathrm{all}{u}_1,u_2\in X,t\in [0,m].\hfill \end{array}\right.$$

(30)

$$u_0\in X.$$

(31)

Our first result in this section is the following.

Theorem 4.

Let X be a Banach space, $T>0$ and assume (30) (a), (b), and (31). Then, problems (1) and (2) have a unique solution $u\in C^1(I;X)$ in the following two cases:

(a) $I=[0;T]$ and F satisfy condition (30) (c);

(b) $I={\mathbb{R}}_{+}$ and F satisfy condition (30) (d).

Proof. 

Let $u_0\in X$ and let $\Lambda :C(I;X)\to C(I;X)$ be the operator defined by

$$\Lambda u\left(t\right)={\int }_0^{t}F(s,u\left(s\right))ds+u_0\mathrm{for} \mathrm{all}u\in C(I;X),t\in I.$$

(32)

Note that assumptions (30) (a) and (b) imply that, for any function $u\in C(I;X)$, the function $t\mapsto F(t,u(t\left)\right)$ is continuous on I and, therefore, the operator $\Lambda$ is well defined. In addition, using condition (30), (c) it is easy to see that, in the case when $I=[0,T]$, this operator satisfies inequality (9) and, therefore, Definition 1 (a) guarantees that it is a history-dependent operator. Moreover, if $I={\mathbb{R}}_{+}$, using condition (30), (d) it follows that the operator $\Lambda$ satisfies inequality (10) and, therefore, Definition 1 (b) guarantees that it is a history-dependent operator, too. Therefore, using Theorem 1, we deduce that there exists a unique function $u\in C(I;X)$ such that

$$u\left(t\right)=\Lambda \left(t\right)\mathrm{for} \mathrm{all}t\in I.$$

(33)

Hence, using (32) and (33), we deduce the existence of a unique function $u\in C(I;X)$ such that

$$u\left(t\right)={\int }_0^{t}F(s,u\left(s\right))ds+u_0\mathrm{for} \mathrm{all}t\in I.$$

(34)

On the other hand, it is easy to see that a function $u\in C^1(I;X)$ is a solution to the Cauchy problem (1) and (2) if and only if $u\in C(I;X)$ and (34) hold. We combine this equivalence with the unique solvability of the integral Equation (34) to end the proof. □

The proof of Theorem 4 establishes a link between the Cauchy problem (1) and (2) and the fixed point problem (3) with $\Lambda$ given by (32). Based on this link, in the case when X is a Hilbert space, we can easily deduce a convergence criterion for the solution of the Cauchy problem (1) and (2). More precisely, we write the statements (16)–(19) in the particular case of the operator (32):

$$\begin{array}{ccc}& & u_n\to u\mathrm{in}C(I;X).\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & u_n-{\int }_0^{t}F(s,u_n\left(s\right))ds-u_0\to 0\mathrm{in}C(I;X).\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I=[0;T]\mathrm{and} \mathrm{there} \mathrm{exists}0\le {\epsilon }_n\to 0\mathrm{such} \mathrm{that}\hfill \\ {\displaystyle {(u_n\left(t\right),v)}_X+{\epsilon }_n{\parallel v\parallel }_X\ge {({\int }_0^{t}F(s,u_n\left(s\right))ds+u_0,v)}_X}\hfill \\ \forall v\in V,n\in \mathbb{N},t\in I.\hfill \end{array}\right.\hfill \end{array}$$

(37)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I={\mathbb{R}}_{+}\mathrm{and} \mathrm{for} \mathrm{any} m\in \mathbb{N}\mathrm{there} \mathrm{exists}0\le {\epsilon }_n^m\to 0\mathrm{such} \mathrm{that}\hfill \\ {\displaystyle {(u_n\left(t\right),v)}_X+{\epsilon }_n^m{\parallel v\parallel }_X\ge {({\int }_0^{t}F(s,u\left(s\right))ds+u_0,v)}_X}\hfill \\ \forall v\in V,n\in \mathbb{N},t\in [0,m].\hfill \end{array}\right.\hfill \end{array}$$

(38)

Then, using the convergence criterion provided by Theorem 3, we deduce the following result.

Corollary 1.

Let X be a Hilbert space, $T>0$ and assume (30) (a) and (b) and (31).

(a) If $I=[0;T]$ and (30) (c) holds, then the statements (35)–(37) are equivalent.

(b) If $I={\mathbb{R}}_{+}$ and (30) (d) holds, then the statements (35), (36) and (38) are equivalent.

Note that Corollary 1 provides a convergence criterion for the solution of the Cauchy problem (1) and (2), in the space $C(I;X)$. Nevertheless, recall that the solution u of the problem belongs to the space $C^1(I;X)$. The example below shows that this criterion is not valid in the space $C^1(I;X)$.

Example 2.

Let X be a Hilbert space, $I=[0,T]$, $f\in X$, $f\ne 0_X$ and consider the Cauchy problem of finding a function $u:I\to X$ such that

$$\dot{u}\left(t\right)+u\left(t\right)=f\forall t\in [0,T],u\left(0\right)=f.$$

(39)

Then, it is easy to see that this problem is of the form (1) and (2) with $F(t,u)=f-u$ for each $t\in I$, $u\in X$ and $u_0=f$. It is easy to see that the assumptions of Corollary 1 a) are satisfied and, moreover, the solution of this problem is given by

$$u\left(t\right)=f\forall t\in I.$$

Consider, now, the sequence $\left\{u_n\right\}\subset C^1(I;X)$ defined by

$$u_n\left(t\right)=\left(1+\frac{1}{n}\sin nt\right)f\forall t\in I.$$

Then, it is easy to see that conditions (35) and (36) are satisfied. Nevertheless, the convergence $u_n\to u$ in $C^1([0,T];X)$ does not hold since, for instance, the sequence of derivatives $\left\{{\dot{u}}_n\right\}$ does not converge to zero in the space $C\left(\right[0,T];X)$.

To provide a convergence criterion for the solution the Cauchy problem (1) and (2) in the space $C^1(I;X)$, we consider the following statements.

$$\begin{array}{ccc}& & u_n\to u\mathrm{in}{C}^1(I;X).\hfill \end{array}$$

(40)

$$\begin{array}{ccc}& & {\dot{u}}_n-F(·,u_n)\to 0_X\mathrm{in}C(I;X)\mathrm{and}{u}_n\left(0\right)\to u_0\mathrm{in}X.\hfill \end{array}$$

(41)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I=[0;T]\mathrm{and} \mathrm{there} \mathrm{exists}0\le {\epsilon }_n\to 0\mathrm{such} \mathrm{that}\hfill \\ {({\dot{u}}_n\left(t\right),v)}_X+{\epsilon }_n{\parallel v\parallel }_X\ge {(F(t,u_n\left(t\right)),v)}_X\forall v\in X,n\in \mathbb{N},t\in I,\hfill \\ \parallel u_n\left(0\right)-u_0{\parallel }_X\le {\epsilon }_n\forall n\in \mathbb{N}.\hfill \end{array}\right.\hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I={\mathbb{R}}_{+}\mathrm{and} \mathrm{for} \mathrm{any} m\in \mathbb{N}\mathrm{there} \mathrm{exists}0\le {\epsilon }_n^m\to 0\mathrm{such} \mathrm{that}\hfill \\ {({\dot{u}}_n\left(t\right),v)}_X+{\epsilon }_n^m{\parallel v\parallel }_X\ge {(F(t,u_n\left(t\right)),v)}_X\forall v\in X,n\in \mathbb{N},t\in [0,m],\hfill \\ \parallel u_n\left(0\right)-u_0{\parallel }_X\le {\epsilon }_n^m\forall n\in \mathbb{N}.\hfill \end{array}\right.\hfill \end{array}$$

(43)

Our next result in this section is the following.

Theorem 5.

Let X be a Hilbert space, $T>0$ and assume (30) (a), (b), and (31).

(a) If $I=[0;T]$ and (30) (c) holds, then the statements (40)–(42) are equivalent.

(b) If $I={\mathbb{R}}_{+}$ and (30) (d) holds, then the statements (40), (41), and (43) are equivalent.

Proof. 

(a) We start with the case $I=[0,T]$. Assume that (40) holds. Then, using (30) (a)–(c), it is easy to see that ${\dot{u}}_n-F(·,u_n)\to \dot{u}-F(·,u)$ in $C(I;X)$ and, since u is the solution of the Cauchy problem problem (3), we deduce that

$${\dot{u}}_n-F(·,u_n)\to 0\mathrm{in}C(I;X).$$

In addition, $u_n\left(0\right)\to u\left(0\right)$ in X and, since $u\left(0\right)=u_0$, we find that $u_n\left(0\right)\to u_0$ in X. It follows from here that (41) holds.

Next, assume that (41) holds, which shows that

$$\underset{s\in [0,T]}{max}{\parallel {\dot{u}}_n\left(s\right)-F(s,u_n\left(s\right))\parallel }_X\to 0\mathrm{as}n\to \infty .$$

(44)

For $n\in \mathbb{N}$, we denote

$${\theta }_n=\underset{s\in [0,T]}{max}{\parallel {\dot{u}}_n\left(s\right)-F(s,u_n\left(s\right))\parallel }_X.$$

(45)

Then, (44) shows that $0\le {\theta }_n\to 0$, and definition (45) implies that, for any $t\in I$, we have

$$\parallel {\dot{u}}_n\left(t\right)-F(t,u_n\left(t\right)){\parallel }_X\le {\theta }_n.$$

(46)

We now use inequality (46) and Lemma 2 to see that

$${({\dot{u}}_n\left(t\right),v)}_X+{\theta }_n{\parallel v\parallel }_X\ge {(F(t,u_n\left(t\right)),v)}_X\forall v\in X,n\in \mathbb{N},t\in I$$

(47)

Then, it is easy to see that condition (42) holds with

$${\epsilon }_n=max\{{\theta }_n,\parallel u_n\left(0\right)-u\left(0\right){\parallel }_X\}.$$

(48)

Finally, assume that (42) holds. Let $n\in \mathbb{N}$ and $t\in [0,T]$. We take $v=\dot{u}\left(t\right)-{\dot{u}}_n\left(t\right)$ in this inequality to see that

$${({\dot{u}}_n\left(t\right),\dot{u}\left(t\right)-{\dot{u}}_n\left(t\right))}_X+{\epsilon }_n{\parallel \dot{u}\left(t\right)-{\dot{u}}_n\left(t\right)\parallel }_X\ge {(F(t,u_n\left(t\right)),\dot{u}\left(t\right)-{\dot{u}}_n\left(t\right))}_X$$

and, using equality $\dot{u}\left(t\right)=F(t,u\left(t\right))$, we find that

$$\begin{array}{ccc}& & {({\dot{u}}_n\left(t\right)-\dot{u}\left(t\right),\dot{u}\left(t\right)-{\dot{u}}_n\left(t\right))}_X+{\epsilon }_n{\parallel \dot{u}\left(t\right)-{\dot{u}}_n\left(t\right)\parallel }_X\hfill \\ & & \ge {(F(t,u_n\left(t\right))-F(t,u\left(t\right)),\dot{u}\left(t\right)-{\dot{u}}_n\left(t\right))}_X.\hfill \end{array}$$

Thus,

$$\parallel {\dot{u}}_n\left(t\right)-\dot{u}{\left(t\right)\parallel }_X^2\le {\epsilon }_n\parallel {\dot{u}}_n\left(t\right)-\dot{u}{\left(t\right)\parallel }_X+\parallel F(t,u_n\left(t\right))-F(t,u\left(t\right)){\parallel }_X{\parallel {\dot{u}}_n\left(t\right)-\dot{u}\left(t\right)\parallel }_X$$

and, therefore,

$$\parallel {\dot{u}}_n\left(t\right)-\dot{u}{\left(t\right)\parallel }_X\le {\epsilon }_n+\parallel F(t,u_n{\left(t\right)-F(t,u\left(t\right))\parallel }_X.$$

We now use assumption (30) (c) to see that

$$\parallel {\dot{u}}_n\left(t\right)-\dot{u}{\left(t\right)\parallel }_X\le {\epsilon }_n+L_F{\parallel u_n\left(t\right)-u\left(t\right)\parallel }_X$$

and, keeping in mind (8), after some algebra, we find that

$$\parallel {\dot{u}}_n\left(t\right)-\dot{u}{\left(t\right)\parallel }_X\le {\epsilon }_n+L_F{\int }_0^t\parallel {\dot{u}}_n\left(s\right)-\dot{u}{\left(s\right)\parallel }_{X}ds+L_F{\parallel u_n\left(0\right)-u_0\parallel }_X.$$

Next, we use the Gronwall lemma and inequality $\parallel u_n\left(0\right)-u_0{\parallel }_X\le {\epsilon }_n$ in (42) to find that

$$\parallel {\dot{u}}_n\left(t\right)-\dot{u}{\left(t\right)\parallel }_X\le (1+L_F)e^{L_{F}t}{\epsilon }_n.$$

We now use the convergences ${\epsilon }_n\to 0$ and inequality $t\le T$ to see that

$$\underset{t\in [0,T]}{max}{\parallel {\dot{u}}_n\left(t\right)-\dot{u}\left(t\right)\parallel }_X\to 0\mathrm{as}n\to \infty .$$

(49)

On the other hand, using the identity

$$u_n\left(t\right)-u\left(t\right)={\int }_0^t({\dot{u}}_n\left(s\right)-\dot{u}\left(s\right))ds+u_n\left(0\right)-u_0$$

we find that

$$\parallel u_n{\left(t\right)-u\left(t\right)\parallel }_X\le {\int }_0^t\parallel {\dot{u}}_n\left(s\right)-\dot{u}{\left(s\right))\parallel }_{X}ds+{\parallel u_n\left(0\right)-u_0\parallel }_X.$$

Therefore, (49) and (42) imply that

$$\underset{t\in [0,T]}{max}{\parallel u_n\left(t\right)-u\left(t\right)\parallel }_X\to 0\mathrm{as}n\to \infty .$$

(50)

The convergences (49) and (50) show that $u_n\to u$ in $C^1(I;X)$ and, therefore, (40) holds.

To conclude, we proved the implications (40) $⟹$ (41) $⟹$ (42) $⟹$ (40), which shows the equivalence of the statements (40)–(42).

(b) We proceed with the case $I={\mathbb{R}}_{+}$. To this end, we fix $m\in \mathbb{N}$ and we use the first part of the theorem with $T=m$, combined with the remark that the quantity ${\epsilon }_n$ defined by (48), (45) depends on T and, since $T=m$, we denote it in what follows by ${\epsilon }_n^m$. We deduce from here the equivalences of the following statements:

$$\begin{array}{ccc}& & u_n\to u\mathrm{in}{C}^1([0,m];X).\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& & {\dot{u}}_n-F(·,u_n)\to 0_X\mathrm{in}C([0,m];X),u_n\left(0\right)\to u\left(0\right)\mathrm{in}X.\hfill \end{array}$$

(52)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}\mathrm{there} \mathrm{exists}0\le {\epsilon }_n^m\to 0\mathrm{such} \mathrm{that}\hfill \\ {({\dot{u}}_n\left(t\right),v)}_X+{\epsilon }_n^m{\parallel v\parallel }_X\ge {(F(t,u_n\left(t\right)),v)}_X\forall n\in \mathbb{N},t\in [0,m],\hfill \\ \parallel u_n\left(0\right)-u_0{\parallel }_X\le {\epsilon }_n^m\forall n\in \mathbb{N}.\hfill \end{array}\right.\hfill \end{array}$$

(53)

Recall that the equivalence of these statements is valid for any $m\in \mathbb{N}$. We now use (5) and the equivalence (6) to see that the convergences (51) and (52) can be replaced by the convergences (40) and (41), respectively, which concludes the proof. □

Note that Theorem 5 provides a convergence criterion to the solution of the Cauchy problem (1) and (2), in the space $C^1(I;X)$. Therefore, it gives an answer to Problem ${\mathcal{Q}}_{\mathcal{Q}}$ in the case when $\mathcal{P}$ represents the above mentioned Cauchy problem. For this reason, it provides a specific advantage with respect to the existing literature which, to the best of our knowledge, point out only sufficient conditions which guarantee the convergence above.

5. A Particular Case

Everywhere in this section, we assume that X is a Hilbert space. We use the results in Section 4 in the study of the Cauchy problem

$$\begin{array}{ccc}& & A\dot{u}\left(t\right)+Bu\left(t\right)=f\left(t\right)\forall t\in I,\hfill \end{array}$$

(54)

$$\begin{array}{ccc}& & u\left(0\right)=u_0,\hfill \end{array}$$

(55)

in which $A:X\to X$ and $B:X\to X$ are given nonlinear operators, and $f:I\to X$ and $u_0$ are the initial data. In the study of this problem, we assume that A is a strongly monotone Lipschitz continuous operator, that is, there exists two constants $m_A>0$ and $L_A>0$ such that

$$\begin{array}{ccc}\hfill & & {(Au-Av,u-v)}_X\ge m_A{\parallel u-v\parallel }_X^2\forall u,v\in X,\hfill \end{array}$$

(56)

$$\begin{array}{ccc}& & {\parallel Au-Av\parallel }_X\le L_A{\parallel u-v\parallel }_X\forall u,v\in X.\hfill \end{array}$$

(57)

We also assume that B is a Lipschitz continuous operator with constant $L_B>0$, i.e.,

$${\parallel Bu-Bv\parallel }_X\le L_B{\parallel u-v\parallel }_X\forall u,v\in X$$

(58)

and, finally, we assume that the function f and the initial data have the following regularity:

$$\begin{array}{ccc}& & f\in C(I;X),\hfill \end{array}$$

(59)

$$\begin{array}{ccc}& & u_0\in X.\hfill \end{array}$$

(60)

It is well known that conditions (56) and (57) imply that the operator is invertible and, moreover, its inverse $A^{-1}:X\to X$ is a strongly monotone Lipschitz continuous operator, with constants $\frac{m_A^2}{L_A}$ and $\frac{1}{m_A}$, respectively. A proof of this result can be found in ([10] p. 23), for instance. Therefore,

$$\begin{array}{ccc}& & {(A^{-1}u-A^{-1}v,u-v)}_X\ge \frac{m_A}{L_A^2}{\parallel u-v\parallel }_X^2\forall u,v\in X,\hfill \end{array}$$

(61)

$$\begin{array}{ccc}& & \parallel A^{-1}u-A^{-1}{v\parallel }_X\le \frac{1}{m_A}{\parallel u-v\parallel }_X\forall u,v\in X.\hfill \end{array}$$

(62)

We have the following result.

Theorem 6.

Let X be a Hilbert space and assume (56)–(60). Then, problem (54) and (55) has a unique solution $u\in C^1(I;X)$.

Proof. 

We use the inverse of the operator A to see that problem (54) and (55) is equivalent to the problem of finding a function $u\in C^1(I;X)$ such that

$$\begin{array}{ccc}& & \dot{u}\left(t\right)=A^{-1}(f\left(t\right)-Bu\left(t\right))\mathrm{for} \mathrm{all}t\in I,\hfill \end{array}$$

(63)

$$\begin{array}{ccc}& & u\left(0\right)=u_0.\hfill \end{array}$$

(64)

Denote by $F:I\times X\to X$ the function given by

$$F(t,u)=A^{-1}(f\left(t\right)-Bu)\forall t\in I,u\in X.$$

(65)

Then, using the properties (62), (58) of the operators $A^{-1}$ and B, respectively, as well as the regularity (59) of the function f, it is easy to see that the previously defined function F satisfies conditions (30). Therefore, Theorem 6 is a direct consequence of Theorem 4, which guarantees the unique solvability of the Cauchy problem (63) and (64). □

We provide a convergence criterion for the solution of the Cauchy problem (63) and (64) and, to this end, we consider the following statements.

$$\begin{array}{ccc}& & u_n\to u\mathrm{in}{C}^1(I;X).\hfill \end{array}$$

(66)

$$\begin{array}{ccc}& & A{\dot{u}}_n+B{u}_n\to f\mathrm{in}C(I;X)\mathrm{and}{u}_n\left(0\right)\to u_0\mathrm{in}X.\hfill \end{array}$$

(67)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I=[0;T]\mathrm{and} \mathrm{there} \mathrm{exists}0\le {\epsilon }_n\to 0\mathrm{such} \mathrm{that}\hfill \\ {(A{\dot{u}}_n\left(t\right),v)}_X+{(B{u}_n\left(t\right),v)}_X+{\epsilon }_n{\parallel v\parallel }_X\ge {(f\left(t\right),v)}_X\hfill \\ \forall v\in X,n\in \mathbb{N},t\in I,\hfill \\ \parallel u_n\left(0\right)-u_0{\parallel }_X\le {\epsilon }_n\forall n\in \mathbb{N}.\hfill \end{array}\right.\hfill \end{array}$$

(68)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I={\mathbb{R}}_{+}\mathrm{and} \mathrm{for} \mathrm{any} m\in \mathbb{N}\mathrm{there} \mathrm{exists}0\le {\epsilon }_n^m\to 0\mathrm{such} \mathrm{that}\hfill \\ {(A{\dot{u}}_n\left(t\right),v)}_X+{(B{u}_n\left(t\right),v)}_X+{\epsilon }_n^m{\parallel v\parallel }_X\ge {(f\left(t\right),v)}_X\hfill \\ \forall v\in X,n\in \mathbb{N},t\in [0,m],\hfill \\ \parallel u_n\left(0\right)-u_0{\parallel }_X\le {\epsilon }_n^m\forall n\in \mathbb{N}.\hfill \end{array}\right.\hfill \end{array}$$

(69)

We have the following result.

Theorem 7.

Let X be a Hilbert space, $T>0$ and assume (56)–(60).

(a) If $I=[0;T]$, then the statements (66), (67), and (68) are equivalent.

(b) If $I={\mathbb{R}}_{+}$, then the statements (66), (67), and (69) are equivalent.

Proof. 

(a) We assume that $I=[0,T]$. We use Theorem 5 with F given by (65) to see that the the statements below are equivalent.

$$\begin{array}{ccc}& & u_n\to u\mathrm{in}{C}^1(I;X).\hfill \end{array}$$

(70)

$$\begin{array}{ccc}& & {\dot{u}}_n-A^{-1}(f-B{u}_n)\to 0_X\mathrm{in}C(I;X)\mathrm{and}{u}_n\to u_0\mathrm{in}X.\hfill \end{array}$$

(71)

$$\begin{array}{ccc}& & \left\{\begin{array}{c}I=[0;T]\mathrm{and} \mathrm{there} \mathrm{exists}0\le {\theta }_n\to 0\mathrm{such} \mathrm{that}\hfill \\ {({\dot{u}}_n\left(t\right),v)}_X+{\theta }_n{\parallel v\parallel }_X\ge {(A^{-1}(f\left(t\right)-B{u}_n\left(t\right)),v)}_X\hfill \\ \forall v\in X,n\in \mathbb{N},t\in I,\hfill \\ \parallel u_n\left(0\right)-u_0{\parallel }_X\le {\theta }_n\forall n\in \mathbb{N}.\hfill \end{array}\right.\hfill \end{array}$$

(72)

Let $n\in \mathbb{N}$ and $t\in I$. We write

$${\dot{u}}_n\left(t\right)-A^{-1}(f\left(t\right)-B{u}_n\left(t\right))=A^{-1}\left(A{\dot{u}}_n\left(t\right)\right)-A^{-1}(f\left(t\right)-B{u}_n\left(t\right))$$

Then, we use the property (62) of the operator $A^{-1}$ to deduce that

$$\parallel {\dot{u}}_n\left(t\right)-A^{-1}(f\left(t\right)-B{u}_n\left(t\right)){\parallel }_X\le \frac{1}{m_A}{\parallel A{\dot{u}}_n\left(t\right)+B{u}_n\left(t\right)-f\left(t\right)\parallel }_X.$$

(73)

A similar argument, based on the identity

$$A{\dot{u}}_n\left(t\right)+B{u}_n\left(t\right)-f\left(t\right)=A{\dot{u}}_n\left(t\right)-A\left(A^{-1}(f\left(t\right)-B{u}_n\left(t\right))\right)$$

and the property (57) of the operator A, yields

$$\parallel A{\dot{u}}_n\left(t\right)+B{u}_n{\left(t\right)-f\left(t\right)\parallel }_X\le L_A{\parallel {\dot{u}}_n\left(t\right)-A^{-1}(f\left(t\right)-B{u}_n\left(t\right))\parallel }_X.$$

(74)

Therefore, inequalities (73) and (74) show that the convergence (67) holds if and only the convergence (71) holds.

Assume, now, that (72) holds. Then, Lemma 2 guarantees that

$$\parallel {\dot{u}}_n\left(t\right)-A^{-1}(f\left(t\right)-B{u}_n\left(t\right)){\parallel }_X\le {\theta }_n\forall n\in \mathbb{N},t\in I$$

and, using (74), we deduce that

$$\parallel A{\dot{u}}_n\left(t\right)+B{u}_n{\left(t\right)-f\left(t\right)\parallel }_X\le L_A{\theta }_n\forall n\in \mathbb{N},t\in I.$$

Then, using again Lemme 2, we deduce that

$${(A{\dot{u}}_n\left(t\right),v)}_X+{(B{u}_n\left(t\right),v)}_X+L_A{\theta }_n{\parallel v\parallel }_X\ge {(f\left(t\right),v)}_X\forall v\in X,n\in \mathbb{N},t\in I.$$

It follows from here that the statement (68) holds with ${\epsilon }_n=max\{L_A{\theta }_n,{\theta }_n\}$, for all $n\in \mathbb{N}$. This shows that the statement (72) implies the statement (68). A similar argument, based on inequality (73), shows that the converse of this implication holds, too. We conclude from here that the statement (68) holds if and only the statement (72) holds.

The equivalence of the statements (66), (67), and (68) is now a direct consequence of the equivalence of the statements (70), (71), and (72), guaranteed by Theorem 5, combined with the equivalences of (67) and (71), on one hand, and (68) and (72), on the other hand.

(b) Assume, now, that $I={\mathbb{R}}_{+}$. Then, the equivalences of the statements (66), (67), and (69) follow arguments similar to those used in the first part of the theorem. Since the modifications are straight, we have skipped the details. □

Consider, now, two sequences $\left\{f_n\right\}$ and $\left\{u_{0n}\right\}$ such that, for each $n\in \mathbb{N}$, the following condition holds.

$$\begin{array}{ccc}& & f_n\in C(I;X),\hfill \end{array}$$

(75)

$$\begin{array}{ccc}& & u_{0n}\in X.\hfill \end{array}$$

(76)

Then, it follows from Theorem 6 that, for each $n\in \mathbb{N}$, there exists a unique function $u_n\in C^1(I;X)$ such that

$$\begin{array}{ccc}& & A{\dot{u}}_n\left(t\right)+B{u}_n\left(t\right)=f_n\left(t\right)\forall t\in I,\hfill \end{array}$$

(77)

$$\begin{array}{ccc}& & u_n\left(0\right)=u_{0n},\hfill \end{array}$$

(78)

We have the following result.

Corollary 2.

Let X be a Hilbert space, and assume (56)–(60), (75), and (76). Then, the solution $u_n$ of Problem (77) and (78) converges in $C^1(I;X)$ to the solution u of Problem (54) and (55) if and only if

$$f_n\to f\mathrm{in}{C}^1(I;X)\mathrm{and}{u}_{0n}\to u\mathrm{in}X.$$

Proof. 

Corollary 2 is a direct consequence of the equivalence of statements (66) and (67) in Theorem 7. □

Note that Corollary 2 provides, in particular, a continuous dependence result for the solution of the Cauchy problem (54) and (55) with respect to the date f and $u_0$. Similar results can be obtained by considering the perturbation of the operators A or B as well as various perturbations of the left hand side of the differential Equation (54). Such an example will be presented in the next section, in the study of a viscoelastic problem.

6. An Application in Solid Mechanics

Our results in the previous sections are useful in the study of various boundary value problems in Solid Mechanics. References in the field are [4,5,10], for instance. Here, to keep the paper a reasonable length, we provide only one simplified example and, to this end, we need to introduce some additional notations.

Let $\mathsf{\Omega }\subset {\mathbb{R}}^d$ ($d=1,2,3$) and denote by ${\mathbb{S}}^d$ the space of second-order symmetric tensors on ${\mathbb{R}}^d$. The canonical inner product and the corresponding norm on ${\mathbb{S}}^d$ are given by

$$\begin{array}{ccc}& & \mathit{\sigma }·\mathit{\tau }={\sigma }_{ij}{\tau }_{ij},\parallel \mathit{\tau }\parallel ={(\mathit{\tau }·\mathit{\tau })}^{1/2}\forall \mathit{\sigma }=\left({\sigma }_{ij}\right),\mathit{\tau }=\left({\tau }_{ij}\right)\in {\mathbb{S}}^d.\hfill \end{array}$$

Here, and below in this section, the indices i, j, k, l run between 1 and 3, and, unless stated otherwise, the summation convention over repeated indices is used. We consider the space

$$\begin{array}{ccc}& & Q=L^2{\left(\mathsf{\Omega }\right)}_s^{d\times d}=\left\{\mathit{\tau }=\left({\tau }_{ij}\right)\mid {\tau }_{ij}={\tau }_{ji}\in L^2\left(\mathsf{\Omega }\right),1\le i,j\le d\right\}\hfill \end{array}$$

which, as we will recall, is a Hilbert space with the canonical inner product

$${(\mathit{\sigma },\mathit{\tau })}_Q={\int }_{\mathsf{\Omega }}{\sigma }_{ij}{\tau }_{ij}dx={\int }_{\mathsf{\Omega }}\mathit{\sigma }·\mathit{\tau }dx$$

and the associated norm, denoted by ${\parallel ·\parallel }_Q$. Moreover, we need the space of symmetric fourth-order tensors ${\mathbf{Q}}_{\infty }$ given by

$${\mathbf{Q}}_{\infty }=\{\mathcal{C}=\left(c_{ijkl}\right)\mid c_{ijkl}=c_{jikl}=e_{klij}\in L^{\infty }\left(\mathsf{\Omega }\right),1\le i,j,k,l\le d\}.$$

It is easy to see that ${\mathbf{Q}}_{\infty }$ is a real Banach space with the norm

$${\displaystyle {\parallel \mathcal{C}\parallel }_{{\mathbf{Q}}_{\infty }}=\underset{0\le i,j,k,l\le d}{max}{\parallel c_{ijkl}\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}}$$

and, in addition,

$${\parallel \mathcal{C}\mathit{\tau }\parallel }_Q\le {d\parallel \mathcal{C}\parallel }_{{\mathbf{Q}}_{\infty }}{\parallel \mathit{\tau }\parallel }_Q\forall \mathcal{C}\in {\mathbf{Q}}_{\infty },\mathit{\tau }\in Q.$$

(79)

Below, we denote by ${\mathbf{0}}_{\infty }$ the zero element of the spaces $C(I;{\mathbf{Q}}_{\infty })$ and $C([0,m];{\mathbf{Q}}_{\infty })$ with $m\in \mathbb{N}$. Moreover, ${\mathbf{0}}_Q$ will represent the zero element of the space Q. Finally, let I be a time interval of interest which can be either bounded (i.e., of the form $I=[0,T]$ with $T>0$), or unbounded (i.e., $I={\mathbb{R}}_{+}$) and recall that, as usual, we use a dot above to denote the derivative with respect to the time variable.

We now turn to the considered viscoelastic problem, which is governed by two given operators, $A:Q\to Q$ and $B:Q\to Q$. It can be formulated as follows.

Problem 2

($\mathcal{P}$). Given a function $\mathit{\sigma }\in C(I;Q)$ and an element ${\mathit{\epsilon }}_0\in Q$, find a function $\mathit{\epsilon }\in C(I;Q)$ such that

$$\begin{array}{ccc}& & \mathit{\sigma }\left(t\right)=A\dot{\mathit{\epsilon }}\left(t\right)+B\mathit{\epsilon }\left(t\right)\forall t\in I,\hfill \end{array}$$

(80)

$$\begin{array}{ccc}& & \mathit{\epsilon }\left(0\right)={\mathit{\epsilon }}_0.\hfill \end{array}$$

(81)

This problem describes the behaviour of a viscoelastic body in the time interval I. Here, $\mathsf{\Omega }$ represents the reference configuration of the body, $\mathit{\sigma }$ is the stress tensor, $\mathit{\epsilon }$ represents the linearized strain tensor, and Equation (80) is related to the constitutive law of the material, assumed to be viscoelastic with a short memory. Operator A represents the viscosity operator and B is the elasticity operator. Finally, the function ${\mathit{\epsilon }}_0$ is the initial deformation. More details on the constitutive laws which describe the bahaviour of viscoelastic materials can be found in [10,11,12,13], for instance.

We now consider a sequence $\left\{{\mathcal{C}}_n\right\}$ of functions defined on I with values in the space ${\mathbf{Q}}_{\infty }$ and, for each $n\in \mathbb{N}$, we consider the following problem.

Problem 3

(${\mathcal{P}}_n$). Given a function $\mathit{\sigma }\in C(I;Q)$ and an element ${\mathit{\epsilon }}_0\in Q$, find a function ${\mathit{\epsilon }}_n\in C(I;Q)$ such that

$$\begin{array}{ccc}& & \mathit{\sigma }\left(t\right)=A{\dot{\mathit{\epsilon }}}_n\left(t\right)+B{\mathit{\epsilon }}_n\left(t\right)+{\int }_0^t{\mathcal{C}}_n(t-s){\dot{\mathit{\epsilon }}}_n\left(s\right)ds\forall t\in I,\hfill \end{array}$$

(82)

$$\begin{array}{ccc}& & \mathit{\epsilon }\left(0\right)={\mathit{\epsilon }}_0.\hfill \end{array}$$

(83)

Note that the mechanical significance of Problem ${\mathcal{P}}_n$ is similar to that of Problem $\mathcal{P}$. The difference arises in the fact that the viscoelastic constitutive law with short memory (80) was replaced by the viscoelastic constitutive law with long memory (82), in which ${\mathcal{C}}_n$ represents a relaxation tensor. Such constitutive laws have been used in the literature to model the behavior of real materials like rubbers, rocks, metals, pastes, and polymers. References in the field are [14,15,16], for instance.

In the study of Problems $\mathcal{P}$ and ${\mathcal{P}}_n$, we consider the following assumptions:

$$\begin{array}{ccc}& & A:Q\to Q \mathrm{is} \mathrm{a} \mathrm{strongly} \mathrm{monotone} \mathrm{Lipschitz} \mathrm{continuous} \mathrm{operator}.\hfill \end{array}$$

(84)

$$\begin{array}{ccc}& & B:Q\to Q \mathrm{is} \mathrm{a} \mathrm{Lipschitz} \mathrm{continuous} \mathrm{operator}.\hfill \end{array}$$

(85)

$$\begin{array}{ccc}& & \mathit{\sigma }\in C(I;Q).\hfill \end{array}$$

(86)

$$\begin{array}{ccc}& & {\mathit{\epsilon }}_0\in Q.\hfill \end{array}$$

(87)

$$\begin{array}{ccc}& & {\mathcal{C}}_n\in C(I;{\mathbf{Q}}_{\infty })\forall n\in \mathbb{N}.\hfill \end{array}$$

(88)

$$\begin{array}{ccc}& & {\mathcal{C}}_n\to 0_{\infty }\mathrm{in}C(I;{\mathbf{Q}}_{\infty }).\hfill \end{array}$$

(89)

Our main result in this section is the following.

Theorem 8.

Assume (84)–(88). Then,

(a) Problem $\mathcal{P}$ has a unique solution $\mathit{\epsilon }\in C^1(I;Q)$ and, for each $n\in \mathbb{N}$, Problem ${\mathcal{P}}_n$ has a unique solution ${\mathit{\epsilon }}_n\in C^1(I;Q)$.

(b) If, moreover, (89) holds, then

$${\mathit{\epsilon }}_n\to \mathit{\epsilon }\mathrm{in}{C}^1(I;Q).$$

(90)

Proof. 

(a) The unique solvability of Problem $\mathcal{P}$ is a direct consequence of Theorem 6. Let $n\in \mathbb{N}$. To prove the unique solvability of Problem ${\mathcal{P}}_n$, we consider the operator ${\Lambda }_n:C(I;Q)\to C(I;Q)$ defined by

$${\Lambda }_n\mathit{\eta }\left(t\right)=B\left({\int }_0^t\mathit{\eta }\left(s\right)ds+{\mathit{\epsilon }}_0\right)+{\int }_0^t{\mathcal{C}}_n(t-s)\mathit{\eta }\left(s\right)ds\forall t\in I,\mathit{\eta }\in C(I:Q).$$

(91)

Then, using assumptions (85), (88), and inequality (79) it is easy to see that ${\Lambda }_n$ is a history-dependent operator. We now use Theorem 2 to deduce that there exists a unique function ${\mathit{\eta }}_n\in C(I;Q)$ such that

$$A{\mathit{\eta }}_n\left(t\right)+{\Lambda }_n{\mathit{\eta }}_n\left(t\right)=\mathit{\sigma }\left(t\right)\forall t\in I$$

or, equivalently,

$$A{\mathit{\eta }}_n\left(t\right)+B\left({\int }_0^t{\mathit{\eta }}_n\left(s\right)ds+{\mathit{\epsilon }}_0\right)+{\int }_0^t{\mathcal{C}}_n(t-s){\mathit{\eta }}_n\left(s\right)ds=\mathit{\sigma }\left(t\right)\forall t\in I.$$

(92)

Denote by ${\mathit{\epsilon }}_n$ the function given by

$${\mathit{\epsilon }}_n\left(t\right)={\int }_0^t{\mathit{\eta }}_n\left(s\right)ds+{\mathit{\epsilon }}_0\forall t\in I.$$

(93)

It follows from (92) and (93) that ${\mathit{\epsilon }}_n$ is a solution to Problem ${\mathcal{P}}_n$ with regularity ${\mathit{\epsilon }}_n\in C^1(I;Q)$. This proves the existence of the solution of Problem ${\mathcal{P}}_n$. The uniqueness follows from the uniqueness of the solution of Equation (92), guaranteed by Theorem 6.

(b) Assume, now, that (89) holds. We start with the case when $I=[0,T]$ with $T>0$. First, we prove that the sequence $\left\{{\dot{\mathit{\epsilon }}}_n\right\}$ is bounded in the space $C(I;Q)$ (see the inequality (96) below). To this end, we fix $n\in \mathbb{N}$ and $t\in [0,T]$. Then, using (82), we obtain that

$$\begin{array}{ccc}& & {(A{\dot{\mathit{\epsilon }}}_n\left(t\right),{\dot{\mathit{\epsilon }}}_n\left(t\right))}_Q+{(B{\mathit{\epsilon }}_n\left(t\right),{\dot{\mathit{\epsilon }}}_n\left(t\right))}_Q+{({\int }_0^t{\mathcal{C}}_n(t-s){\dot{\mathit{\epsilon }}}_n\left(s\right)ds,{\dot{\mathit{\epsilon }}}_n\left(t\right))}_Q={(\mathit{\sigma }\left(t\right),{\dot{\mathit{\epsilon }}}_n\left(s\right))}_Q\hfill \end{array}$$

and, therefore,

$$\begin{array}{ccc}& & {(A{\dot{\mathit{\epsilon }}}_n\left(t\right)-A{0}_Q,{\dot{\mathit{\epsilon }}}_n\left(t\right))}_Q=(\mathit{\sigma }\left(t\right),{\dot{\mathit{\epsilon }}}_n\left(s\right))-{(A{0}_Q,{\dot{\mathit{\epsilon }}}_n\left(t\right))}_Q-{(B{\mathit{\epsilon }}_n\left(t\right),{\dot{\mathit{\epsilon }}}_n\left(t\right))}_Q\hfill \\ & & -{({\int }_0^t{\mathcal{C}}_n(t-s){\dot{\mathit{\epsilon }}}_n\left(s\right)ds,{\dot{\mathit{\epsilon }}}_n\left(t\right))}_Q.\hfill \end{array}$$

Next, we use assumption (84) to deduce that

$$m_A\parallel {\dot{\mathit{\epsilon }}}_n{\left(t\right)\parallel }_Q^2\le ({\parallel \mathit{\sigma }\left(t\right)\parallel }_Q+\parallel A{0}_Q{\parallel }_Q+\parallel B{\mathit{\epsilon }}_n{\left(t\right)\parallel }_Q+{\int }_0^t{\parallel {\mathcal{C}}_n(t-s){\dot{\mathit{\epsilon }}}_n\left(s\right)ds\parallel }_Q){\parallel {\dot{\mathit{\epsilon }}}_n\left(t\right)\parallel }_Q,$$

which implies that

$$m_A\parallel {\dot{\mathit{\epsilon }}}_n{\left(t\right)\parallel }_Q\le {\parallel \mathit{\sigma }\left(t\right)\parallel }_Q+\parallel A{0}_Q{\parallel }_Q+\parallel B{\mathit{\epsilon }}_n{\left(t\right)\parallel }_Q+{\int }_0^t{\parallel {\mathcal{C}}_n(t-s){\dot{\mathit{\epsilon }}}_n\left(s\right)ds\parallel }_Q.$$

(94)

We now use assumption (85) and inequality (79) to find that

$$\parallel {\dot{\mathit{\epsilon }}}_n{\left(t\right)\parallel }_Q\le D+\parallel {\mathit{\epsilon }}_n{\left(t\right)\parallel }_Q+d{\int }_0^t\parallel {\mathcal{C}}_n{(t-s)\parallel }_{{\mathbf{Q}}_{\infty }}{\parallel {\dot{\mathit{\epsilon }}}_n\left(s\right)\parallel }_{Q}ds.$$

(95)

In (95) and below, D represents various positive constants that do not depend on n. On the other hand, inequality (95), combined with assumption (89) and identity (8), yields

$$\parallel {\dot{\mathit{\epsilon }}}_n{\left(t\right)\parallel }_X\le D+D{\int }_0^t{\parallel {\dot{\mathit{\epsilon }}}_n\left(s\right)\parallel }_{Q}ds.$$

We now use the Gronwall argument to see that

$$\parallel {\dot{\mathit{\epsilon }}}_n{\left(t\right)\parallel }_Q\le D.$$

(96)

Next, we use Equation (82), again, inequality (79) and inequality (96), valid for any $t\in [0,T]$, to see that

$$\begin{array}{ccc}& & \parallel A{\dot{\mathit{\epsilon }}}_n\left(t\right)+B{\mathit{\epsilon }}_n{\left(t\right)-\mathit{\sigma }\left(t\right)\parallel }_Q={\parallel {\int }_0^t{\mathcal{C}}_n(t-s){\dot{\mathit{\epsilon }}}_n\left(s\right)ds\parallel }_Q\hfill \\ & & \le {\int }_0^t\parallel {\mathcal{C}}_n(t-s){\dot{\mathit{\epsilon }}}_n{\left(s\right)\parallel }_{Q}ds\le d{\int }_0^t\parallel {\mathcal{C}}_n{(t-s)\parallel }_{{\mathbf{Q}}_{\infty }}{\parallel {\mathit{\epsilon }}_n\left(s\right)\parallel }_{Q}ds\hfill \\ & & \le d\underset{r\in [0,T]}{max}\parallel {\mathcal{C}}_n{\left(r\right)\parallel }_{{\mathbf{Q}}_{\infty }}{\int }_0^t\parallel {\dot{\mathit{\epsilon }}}_n{\left(s\right)\parallel }_{Q}ds\le D\underset{r\in [0,T]}{max}{\parallel {\mathcal{C}}_n\left(r\right)\parallel }_{{\mathbf{Q}}_{\infty }}.\hfill \end{array}$$

It follows, now, from assumption (89) that

$$\underset{t\in [0,T]}{max}{\parallel A{\dot{\mathit{\epsilon }}}_n\left(t\right)+B{\mathit{\epsilon }}_n\left(t\right)-\mathit{\sigma }\left(t\right)\parallel }_Q\to 0$$

and, therefore, $A{\dot{\mathit{\epsilon }}}_n+B{\mathit{\epsilon }}_n\to \mathit{\sigma }$ in $C(I;X)$. We now use Theorem 7 (a) to deduce that the convergence (90) holds.

Assume, now, that $I={\mathbb{R}}_{+}$. Then, assumption (89) guarantees that ${\mathcal{C}}_n\to {\mathbf{0}}_{\infty }$ in $C([0,m];{\mathbf{Q}}_{\infty })$, for any $m\in \mathbb{N}$. Therefore, using part a) of the theorem, we deduce that ${\mathit{\epsilon }}_n\to \mathit{\epsilon }$ in $C^1([0,m];Q)$ for any $m\in \mathbb{N}$. This implies that (90) holds, which concludes the proof. □

The convergence result (89) is important from a mechanical point of view since it shows that the viscoelastic constitutive law with short memory (80) can be approached by the viscoelastic constitutive law with long memory (82) for a small relaxation tensor.

7. Conclusions

In this paper, we studied a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. The unique solvability of the problem follows from a fixed point argument for history-dependent operators. Then, we established convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. Our main result is Theorem 5, which provides necessary and sufficient conditions on a sequence, and which guarantees its convergence to the solution of the Cauchy problem in the space of continuously differentiable functions. We used this theorem to deduce continuous dependence results for the solution with respect to the data and provided an application in viscoelasticity.

Our results in this work deserve to be extended in several directions. A first direction would be to extend these results in the framework of reflexive Banach spaces, by using a version of Lemma 2 in which the inner product and the identity operator are replaced by the duality pairing and the duality mapping, respectively. A second direction would be to relax the assumption (30) on the function F and to work in the framework of $L^p(I;X)$ and $W^{1,p}(I;X)$ spaces. A third direction of research could be to study the same problem but with a fractional derivative. Results in whichever of these directions would open up methods in various applications in Solid and Contact Mechanics, involving more general constitutive laws and materials. The use of Theorem 5 in the study of the convergence of the solution of a discrete version of the Cauchy problem (1) and (2), as the discretization parameter converges, would also represent a problem which deserves to be studied in the future. Computer simulations of the theoretical convergence results presented in this paper would be welcome, too. Finally, we mention that solving Problem ${\mathcal{Q}}_{\mathcal{P}}$ for various initial problems $\mathcal{P}$ represents a challenging research direction which, clearly, deserves to be investigated in the future.