Infinite Systems of Differential and Integral Equations: Current State and Some Open Problems

Abstract

This paper presents some topics of the theory of infinite systems of differential and integral equations. Our considerations focus on showing the symmetries that can be encountered in the theory of nonlinear differential and integral equations from the viewpoint of initial conditions, such as the symmetry of the behaviour of solutions of differential equations with respect to initial conditions, the symmetry of the behaviour of solutions in $+\infty$ and $-\infty$ and some other essential properties of solutions of differential and integral equations. First of all, we describe the fundamental facts connected with the theory of infinite systems of both differential and integral equations. Particular attention is paid to the location of infinite systems of the mentioned equations in a suitable Banach space. Indeed, we define the spaces in question and describe the basic properties of those spaces. Next, we discuss conditions imposed on terms of equations of the considered infinite systems that guarantee the existence of solutions of those systems and allow us to obtain some essential information on those solutions. Moreover, after the description of the current state of investigations concerning the theory of infinite systems of differential and integral equations, we formulate a few open problems concerning the mentioned systems of equations.

1. Introduction

It is well known that various real world events generate their described models with help of differential or integral equations or the systems (finite systems) of those equations.

Infinite systems of differential or integral equations create generalizations of finite systems of the equations in question. From a general point of view, an infinite system (IS) of differential equations is a system with the form (cf. Banaś and Mursaleen [1], Deimling [2])

$$\left\{\begin{array}{c}{x}_1^{\prime }\left(t\right)=f_1(t,x_1\left(t\right),x_2\left(t\right),\dots ),\hfill \\ x_2^{\prime }\left(t\right)=f_2(t,x_1\left(t\right),x_2\left(t\right),\dots ),\hfill \\ ..............................\hfill \end{array}\right.$$

(1)

We can represent an IS (1) in the form

$$x_n^{\prime }\left(t\right)=f_n(t,x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right),\dots )$$

(2)

for $n=1,2,\dots$ and $t\in I$ (I is an interval and we assume that I is a bounded interval of the form $I=I_T=[0,T]$ or $[0,T)$ or that I is the unbounded interval of the form $I=I_{\infty }=[0,\infty )$). Obviously, we look for a local solution of IS (1) or (2) that satisfies the initial conditions of the form

$$x_n\left(0\right)=x_n^0$$

for $x_n^0\in \mathbb{R}$ and $n=1,2,\dots$, where $\mathbb{R}$ stands for the set of real numbers.

An analogous situation occurs if we consider an infinite system of (possibly nonlinear) integral equations. Such a system has the form

$$x_n\left(t\right)=a_n\left(t\right)+{\int }_0^t{u}_n(t,s,x_1\left(s\right),x_2\left(s\right),\dots )ds$$

(3)

for $n=1,2,\dots$ and $t\in I_T$ or for $t\in I_{\infty }$ (see Banaś and Mursaleen [1] and Deimling [3] for examples). Obviously, the above IS of nonlinear integral equations is of the Volterra–Urysohn type (on a bounded or unbounded interval) (cf. Jalal and Jan [4], Dehghan and Roshan [5], Das et al. [6], for instance). We can also investigate solutions of the IS of Urysohn integral equations with the form

$$x_n\left(t\right)=a_n\left(t\right)+{\int }_0^T{u}_n(t,s,x_1\left(s\right),x_2\left(s\right),\dots )ds$$

(4)

for $n=1,2,\dots$ and $t\in I_T=[0,T]$.

A more complicated situation takes place when we consider the IS of integral equations on the unbounded interval that has the form

$$x_n\left(t\right)=a_n\left(t\right)+{\int }_0^{\infty }{u}_n(t,s,x_1\left(s\right),x_2\left(s\right),\dots )ds,$$

(5)

where $n=1,2,\dots$ and $t\in {\mathbb{R}}_{+}=I_{\infty }$.

Let us pay attention to the fact that we can also treat more complicated and simultaneously more general forms of ISs of nonlinear integral equations (see Banaś and Mursaleen [1]). Such a system can have the form

$$x_n\left(t\right)=a_n\left(t\right)+f_n(t,x_1\left(t\right),x_2\left(t\right),\dots ){\int }_0^T{u}_n(t,s,x_1\left(s\right),x_2\left(s\right),\dots )ds$$

(6)

or

$$x_n\left(t\right)=a_n\left(t\right)+f_n(t,x_1\left(t\right),x_2\left(t\right),\dots ){\int }_0^{\infty }{u}_n(t,s,x_1\left(s\right),x_2\left(s\right),\dots )ds$$

(7)

for $t\in I_T$ (or $t\in I_{\infty }$) and $n=1,2,\dots$.

Observe that for a fixed $n\in \mathbb{N}$ ($\mathbb{N}$ denotes the set of natural numbers), the integral equations (6) or (7) represents the so-called quadratic integral equation (of the Urysohn type). It is well-known that quadratic integral equations create rather complicated and hard forms of integral equations with the classical form, say (4) or (5).

It is worth mentioning that the behaviour of ISs of integral equations of the form (6) or (7) is quite distinct from the behaviour of ISs of integral equations of type (3), (4) or (5) (cf. Banaś and Mursaleen [1], Banaś and Chlebowicz [7], Zabrejko et al. [8], Banaś and Madej [9]). Indeed, the operator generated by (3), (4) or (5) can be completely continuous, while the operator generated by (6) or (7) is not completely continuous but may be condensing with respect to a suitable measure of noncompactness (MNC) (cf. Banaś and Mursaleen [1]).

To continue studying the types of ISs of nonlinear integral equations, let us mention the ISs of integral equations of the Stieltjes type (see Banaś and Rzepka [10] and Banaś and Taktak [11] for examples). Such ISs have the form

$$x_n\left(t\right)=a_n\left(t\right)+f_n(t,x_1\left(t\right),x_2\left(t\right),\dots ){\int }_0^t{g}_n(t,s)h_n(s,x_1\left(s\right),x_2\left(s\right),\dots )d_s{K}_n(t,s)$$

(8)

or

$$x_n\left(t\right)=a_n\left(t\right)+f_n(t,x_1\left(t\right),x_2\left(t\right),\dots ){\int }_0^T{g}_n(t,s)h_n(s,x_1\left(s\right),x_2\left(s\right),\dots )d_s{K}_n(t,s),$$

(9)

where $t\in I_T$ (or $t\in {\mathbb{R}}_{+}$) and $n=1,2,\dots$.

Obviously, instead of the term under the sign of the integral being the counterpart of the Hammerstein type, we may consider the term

$$u_n(t,s,x_1\left(s\right),x_2\left(s\right),\dots )$$

(the counterpart of the Urysohn type).

We would like to emphasize strongly that in ISs of type (8) or (9), the integral is taken in the Stieltjes sense, while the integral in (3)–(7) is the ordinary Riemann (or Lebesgue) integral (cf. Natanson [12], Łojasiewicz [13], Convertito and Cruz-Uribe [14], Cruz-Uribe and Glidewell [15], for example). The Riemann integral in (3)–(7) is a particular case of the Stieltjes integral in (8) and (9). In fact, if we take in (8) (or (9)) $K_n(t,s)=s$ for $n=1,2,\dots$; then, we obtain (3), (4), (5), (6) or (7). Thus, ISs of integral equations of the Stieltjes type (8) or (9) create a strong generalization of ISs of integral equations like (3)–(7). Apart from this, let us also pay attention to other papers in the field of ISs of differential or integral equations that appeared quite recently: Alotaibi et al. [16], Malik and Jalal [17,18], Mehravaran and Kayvanloo [19], Mehravaran et al. [20,21], Mursaleen [22], Mursaleen et al. [23], Mursaleen and Mohiuddine [24], Mursaleen and Rizvi [25], Saadati et al. [26], and Seemab and Rehman [27].

2. Basic Results of the Theory of ISs of Differential or Integral Equations: The Current State

This section presents a description of results obtained up to now concerning the subject of ISs of differential and integral equations. We focus on the topics associated with conditions guaranteeing the existence and characterization of solutions of ISs of the equations in question.

For simplicity, we consider an IS of differential equations (2) as well as an IS of integral equations (6) (or (7)). Observe that any solution of that IS is a function sequence of the form $x\left(t\right)=(x_1\left(t\right),x_2\left(t\right),\dots )=\left(x_n\left(t\right)\right)$ for $t\in I_T$ ($t\in I_{\infty }$). Thus, if we arbitrarily fix $t\in I_T$ ($t\in I_{\infty }$), then the sequence $\left(x_n\left(t\right)\right)$ is a sequence of numbers $(x_1\left(t\right),x_2\left(t\right),\dots )$. In such a situation, we can choose a sequence space (say $l_{\infty }$ or c or $c_0$ and so on) to impose suitable assumptions in our considerations.

To ensure generality, suppose that we consider the sequence space $l_{\infty }$ with the classical supremum norm. Thus, we can assume that we then look for solutions of ISs of integral equations that are functions acting from the interval I with values in the sequence space $l_{\infty }$. An analogous situation takes place if we consider an IS of differential equations.

It turns out that we can assume that solutions of our ISs are functions belonging to the Banach space $BC(I,l_{\infty })$ consisting of functions $x:I\to l_{\infty }$, which are continuous and bounded on I, i.e., $x=x\left(t\right)=(x_1\left(t\right),x_2\left(t\right),\dots )$, and x is continuous and bounded on I. Moreover, the space $BC(I,l_{\infty })$ is equipped with the classical norm

$$\begin{array}{cc}\hfill \parallel x\parallel & =\parallel \left(x_n\right)\parallel =sup\left\{{\parallel x\left(t\right)\parallel }_{l_{\infty }}:t\in I\right\}\hfill \\ \hfill & =sup\left\{sup\left\{\left|x_n\left(t\right)\right|:n=1,2,\dots \right\}:t\in I\right\}.\hfill \end{array}$$

(10)

More generally, we can consider the Banach space $(E,{\parallel ·\parallel }_E)$, and by $BC(I,E)$, we denote the space of functions $x:I\to E$, which are continuous and bounded on the interval I, furnished with the norm $\parallel ·\parallel$, defined by the formula

$$\parallel x\parallel =sup\left\{{\parallel x\left(t\right)\parallel }_E:t\in I\right\}.$$

(11)

Particularly, with $E=l_{\infty }$, we obtain the Banach space $BC(I,l_{\infty })$ with the norm given by (10). We can also consider the spaces $BC(I,c)$ or $BC(I,c_0)$, where c and $c_0$ are the classical Banach sequence spaces.

In order to prove the existence of solutions of ISs of differential or integral equations, we can apply the classical, well-known, fixed-point theorems (FPTs) of Banach or Schauder. However, the use of a Banach FPT requires a very strong assumption that is not recommended nowadays (the tool is a bit trivial). On the other hand, the application of the Schauder FPT requires a compact convex subset of the Banach space $BC(I,E)$, which is transformed continuously into itself by an operator associated with IS of differential or integral equations. In general, it is rather difficult to find such a subset. Therefore, we are forced usually to apply the technique connected with measures of noncompactness (MNCs).

We do not develop and present here the theory of MNCs since it is a rather extensive part of nonlinear analysis. Let us assert that if E is a given Banach space, then we denote by ${\mathfrak{M}}_E$ the family of all nonempty and bounded subsets of E and by ${\mathfrak{N}}_E$ its subfamily consisting of relatively compact sets. Then, we approve the following definition (see Banaś and Mursaleen [1]):

Definition 1.

A function $\mu :{\mathfrak{M}}_E\to {\mathbb{R}}_{+}$ is said to be a measure of noncompactness in the space E if it satisfies the following conditions:

(i) 

The family $ker\mu =\left\{X\in {\mathfrak{M}}_E:\mu \left(X\right)=0\right\}$ is nonempty and $ker\mu \subset {\mathfrak{N}}_E$.

(ii) 

$X\subset Y\Rightarrow \mu \left(X\right)\le \mu \left(Y\right).$

(iii) 

$\mu \left(\overline{X}\right)=\mu \left(X\right).$

(iv) 

$\mu \left(\mathit{Conv}X\right)=\mu \left(X\right).$

(v) 

$\mu \left(\lambda X+\left(1-\lambda \right)Y\right)\le \lambda \mu \left(X\right)+\left(1-\lambda \right)\mu \left(Y\right)$ for $\lambda \in [0,1].$

(vi) 

If ${\left(X_n\right)}_{n⩾1}$ is a sequence of closed sets from ${\mathfrak{M}}_E$ such that $X_{n+1}\subset X_n$ for $n=1,2,\dots$ and $\underset{n\to \infty }{lim}\mu \left(X_n\right)=0,$ then the set $X_{\infty }={\displaystyle \bigcap _{n=1}^{\infty }}{X}_n$ is nonempty.

The main tool associated with the theory of MNCs is the FPT of the Darbo type, being a counterpart of the Banach contraction theorem but expressed in terms of MNCs (cf. Banaś and Mursaleen [1]).

Theorem 1.

Let Ω be a nonempty, bounded, closed and convex subset of a Banach space E. Assume that $Q:\Omega \to \Omega$ is a continuous operator and there exists a constant $k\in [0,1)$ such that $\mu \left(QX\right)\le k\mu \left(X\right)$ for any nonempty subset X of Ω, where μ is a measure of noncompactness in the space E. Then, Q has at least one fixed point in the set Ω.

The success in the use of the technique of MNCs depends on the choice of a suitable MNC in a given Banach space E and on imposing appropriate conditions on components (terms) of a considered operator equation. Let us recall how the MNCs look in the spaces $BC(I_T,l_{\infty })$ and $BC(I_{\infty },l_{\infty })$.

Lemma 1.

In the space $BC(I_T,l_{\infty })$ (or in other one), the MNC has the form

$${\mu }_{\infty }\left(X\right)={\omega }_0^{\infty }\left(X\right)+{\overline{\mu }}_{\infty }\left(X\right),$$

(12)

where ${\omega }_0^{\infty }\left(X\right)$ represents the modulus of continuity of functions belonging to the set X and ${\overline{\mu }}_{\infty }\left(X\right)$ is defined by the formula

$${\overline{\mu }}_{\infty }\left(X\right)=sup\left\{\mu \left(X\left(t\right)\right):t\in I_T\right\}$$

(where μ is a MNC given in the space $l_{\infty }$ and $X\left(t\right)$ indicates the set $X\left(t\right)=\left\{x\left(t\right):x\in X\right\}$, the cross-section of the set X at the point t).

In the case when $I=I_{\infty }$, the MNC given by (12) additionally has a third component connected with the behaviour of functions of the set X at infinity.

When we consider the IS of differential equations (2) (with suitable initial conditions of the Cauchy type), we usually transfer that problem to an equivalent IS of integral equations and meet the case discussed above. However, if we consider appropriate tools of the theory of differential equations in the Banach space (see Deimling [2]), we impose other conditions that are expressed in a form depending on the used tools.

Let us describe briefly the IS of differential equations (2) when we apply the so-called Hausdorff MNC in the space $c_0^{\beta }$ consisting of sequences $x=\left(x_n\right)$ such that ${\beta }_n{x}_n\to 0$ as $n\to \infty$. Here, a real sequence $\beta =\left({\beta }_n\right)$ is such that ${\beta }_n>0$ for $n=1,2,\dots$, and it is nonincreasing. We assume also that ${\beta }_n\to 0$ as $n\to \infty$ (cf. Banaś and Krajewska [28,29]). It is easily seen that $c_0^{\beta }$ forms a Banach space under the norm

$$\parallel x\parallel =\parallel \left(x_n\right)\parallel =sup\{{\beta }_n\left|x_n\right|:n=1,2,\dots \}=max\{{\beta }_n\left|x_n\right|:n=1,2,\dots \}.$$

For further purposes, let us recall that the mentioned Hausdorff MNC $\chi$ is the function defined on the family ${\mathfrak{M}}_E$ by the following formula:

$$\chi \left(X\right)=inf\{\epsilon >0:X\mathrm{has}\mathrm{a}\mathrm{finite}\epsilon \text{-}\mathrm{net}\mathrm{in}E\}.$$

The Hausdorff MNC has several useful properties (see Akhmerov et al. [30], Ayerbe Toledano et al. [31]). Moreover, in some Banach spaces such as $C\left(\right[a,b\left]\right)$, $c_0$, $l_p$ ($1<p<\infty$), there are known convenient formulas expressing the MNC $\chi$ with help of the structure of those spaces. In this regard, the Hausdorff MNC $\chi$ is the most important MNC that is used in nonlinear analysis (cf. Malkowsky and Rakočević [32], Banaś and Mursaleen [1]).

Now, let us recall the formulas expressing the MNCs $\chi$ in the space $c_0$ consisting of real sequences converging to zero with the supremum norm and in the abovementioned space $c_0^{\beta }$.

Lemma 2.

The Hausdorff MNC χ in the space $c_0$ has the form

$$\chi \left(X\right)=\underset{n\to \infty }{lim}\left\{\underset{\left(x_k\right)\in X}{sup}\left\{sup\left\{\right|x_i|:i\ge n\}\right\}\right\},$$

where $X\in {\mathfrak{M}}_{c_0}$.

Similarly, in the space $c_0^{\beta }$, the Hausdorff MNC χ has the following form (see Banaś and Krajewska [28,29]):

$$\chi \left(X\right)=\underset{n\to \infty }{lim}\left\{\underset{\left(x_k\right)\in X}{sup}\left\{sup\left\{{\beta }_i\right|x_i|:i\ge n\}\right\}\right\},$$

where $X\in {\mathfrak{M}}_{c_0^{\beta }}$.

Now, let us point out that if we take into account the IS of differential equations investigated in the paper by Banaś and Krajewska [28],

$$x_n^{\prime }=\sum _{i=1}^{k_n}{a}_{n{n}_i}\left(t\right)x_{n_i}+f_n(t,x_1,x_2,\dots )$$

(13)

with the initial conditions

$$x_n\left(0\right)=x_0^n$$

(14)

for $n=1,2,\dots$, then we assume that the functions $a_{n{n}_i}=a_{n{n}_i}\left(t\right)$ are equicontinuous on the interval $I=I_T$ ($n=1,2,\dots$ and $i=1,2,\dots ,k_n$). Apart from this, we assume that for any natural number n, the sequence $(n_1,n_2,\dots ,n_{k_n})$ satisfies the inequalities $n\le n_1<n_2<\dots <n_{k_n}$. We assume also that the functions $a_{n{n}_i}\left(t\right)$ are uniformly bounded on the interval I and that the sequence $\left(x_0^n\right)$ is an element of the space $c_0^{\beta }$. Additionally, we require that there exists a sequence $\left(p_n\right)$ of nonnegative terms such that ${\beta }_n{p}_n\to 0$ as $n\to \infty$ and $\left|f_n(t,x_1,x_2,\dots )\right|\le p_n$ for $t\in I$ and $\left(x_n\right)\in c_0^{\beta }$ ($n=1,2,\dots$).

We omit here a detailed discussion of other assumptions imposed in connection to problem (13)–(14) (cf. Banaś and Krajewska [28,29]).

Let us mention that the IS of differential equations (13) represents the semilinear upper diagonal system since all terms of the linear part of equations appearing in (13) are located up the main diagonal of the matrix $\left(a_{n{n}_i}\right)$. It is worth pointing out that in the paper by Banaś and Krajewska [29], IS (13) is considered, but assuming that $1\le n_1<n_2<\dots <n_{k_n}\le n$.

Observe that in such a case, the IS (13) represents the lower diagonal IS of differential equations.

Now, we proceed to a discussion of ISs of integral equations with the form (6)–(9). We focus on the IS of integral equations (7).

Let us pay attention to assumptions imposed in considerations associated with such ISs of integral equations. First of all, let us point out that IS (7) is considered in the space $B{C}_0=BC({\mathbb{R}}_{+},c_0)$ (see Banaś and Madej [33]) and in the space $B{C}_{\infty }=BC({\mathbb{R}}_{+},l_{\infty })$ in the paper by Banaś and Madej [9]. In the first case, we assume that the sequence $\left(a_n\left(t\right)\right)$ is an element of the space $B{C}_0$, while, in the second case, that sequence is an element of the space $B{C}_{\infty }$. Moreover, we assume (in the first case) that the function $t\to f_n(t,x_1,x_2,\dots )$ is continuous on ${\mathbb{R}}_{+}$ uniformly with respect to $\left(x_n\right)\in c_0$, while, in the second case, those functions are equicontinuous on ${\mathbb{R}}_{+}$ uniformly with respect to $\left(x_n\right)\in l_{\infty }$.

Let us also assert that in the paper by Banaś and Madej [33], it is assumed that for any fixed natural number n, the function $u_n$ acts from the set ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times c_0$ into $\mathbb{R}$ and the family of functions $\left\{u_n(t,s,x_1,x_2,\dots )\right\}$ is locally equicontinuous for $t\in {\mathbb{R}}_{+}$ uniformly with respect to $s\in {\mathbb{R}}_{+}$ and $\left(x_n\right)\in c_0$. In the second paper mentioned above (Banaś and Madej [9]), IS of integral equations of the Hammerstein type is considered, being an important special case of IS (7). We do not provide here details concerning that IS of integral equations.

Let us also mention that considerations concerning ISs of integral equations of the Stieltjes type (8) or (9) have a more complicated form and we do not present here details of them. Readers interested in those details are directed to the paper by Banaś and Taktak [11]. Moreover, we return to the subject connected with ISs of integral equations (8) or (9) in the next section, where some open problems concerning those ISs are formulated.

In general, the investigation of ISs of differential or integral equations is rather a hard task in view of the fact that we consider ISs of those equations. Therefore, we are forced to formulate a lot of assumptions that describe our requirements imposed on all components of ISs of the considered integral (or differential) equations and regulate the behaviour of those components on the interval $I_{\infty }$. It is also important that we verify the continuity of the operator generated by an IS of integral equations on a suitable subset of the space $BC({\mathbb{R}}_{+},E)$. This point is often lost in several manuscripts sent for publication in journals.

Up to now, in the last two decades, there have appeared some papers concerning the discussed subject. Nevertheless, there are a lot of problems waiting to be solved. As we mentioned above, some open problems concerning the discussed subject are presented in the next section.

3. Open Problems

As we announced previously, the goal of this section is to raise a few open problems in the field of the theory of integral equations and ISs both of differential and integral equations. The difficulty level of these open problems is rather high. Moreover, other mathematicians working in this field formulate similar open problems but, unfortunately, we do not know any numerical simulation results for these problems.

Problem 1.

As we indicated in the previous section, there are some existence theorems concerning the IS of differential equations of the form (13), i.e., the IS of the so-called semilinear upper diagonal or lower diagonal systems of differential equations

$$x_n^{\prime }=\sum _{i=1}^{k_n}{a}_{n{n}_i}\left(t\right)x_{n_i}+f_n(t,x_1,x_2,\dots )$$

with the initial conditions (14)

$$x_n\left(0\right)=x_0^n$$

for $n=1,2,\dots$. In the above system of differential equations, $a_{n{n}_i}:[0,T]\to \mathbb{R}$ for $n=1,2,\dots$ and $i=1,2,\dots ,k_n$, $f_n:[0,T]\times {\mathbb{R}}^{\infty }\to \mathbb{R}$ for $n=1,2,\dots$ are given functions and $x_n:[0,T]\to \mathbb{R}$ ($n=1,2,\dots$) are unknown functions.

As we mentioned before, in the papers by Banaś and Krajewska [28,29], both upper diagonal systems and lower diagonal systems of the form (13) are studied.

However, up to now, there are no papers considering the case of around diagonal IS of differential equations of the form (13). This means that the sequence $(n_1,n_2,\dots ,n_{k_n})$ appearing in any equation of IS (13) is such that

$$n_1<n_2<\dots <n_l\le n<n_{l+1}<\dots <n_{k_n}$$

or

$$n_1<n_2<\dots <n_l<n\le n_{l+1}<\dots <n_{k_n}.$$

In other words, we can say that some terms of the linear part of equations in (13) can be located under the main diagonal of the matrix $\left(a_{n{n}_i}\right)$ and other terms of the linear part of those equations can be located up the main diagonal of the mentioned matrix $\left(a_{n{n}_i}\right)$.

Problem 2.

In the paper by Banaś et al. [34], the IS of differential equations associated with the so-called birth-and-death stochastic process was considered. That IS of differential equations was obtained during the process of modelling the mentioned stochastic process. We do not repeat here the details of creating of the mentioned IS of differential equations. Readers interested in those details are directed to the paper quoted above.

Let us recall only that the mentioned IS of differential equations has the form

$$\left\{\begin{array}{c}{c}_1^{\prime }\left(t\right)=-\lambda c_1\left(t\right)+\mu c_2\left(t\right)+f_1(t,c_1\left(t\right),c_2\left(t\right),\dots ),\hfill \\ c_2^{\prime }\left(t\right)=\lambda c_1\left(t\right)-(\lambda +2\mu )c_2\left(t\right)+3\mu c_3\left(t\right)+f_2(t,c_1\left(t\right),c_2\left(t\right),\dots ),\hfill \\ c_3^{\prime }\left(t\right)=\lambda c_2\left(t\right)-(\lambda +3\mu )c_3\left(t\right)+4\mu c_4\left(t\right)+f_3(t,c_1\left(t\right),c_2\left(t\right),\dots ),\hfill \\ .....................................................................,\hfill \\ c_n^{\prime }\left(t\right)=\lambda c_{n-1}\left(t\right)-(\lambda +n\mu )c_n\left(t\right)+(n+1)\mu c_{n+1}\left(t\right)+f_n(t,c_1\left(t\right),c_2\left(t\right),\dots ),\hfill \\ .......................................................................,\hfill \end{array}\right.$$

(15)

where $\mu >0$, $\lambda >0$ are some constants and $t\in I_T=[0,T]$. Moreover, the function sequence $c\left(t\right)=\left(c_n\left(t\right)\right)=(c_1\left(t\right),c_2\left(t\right),\dots )$ represents an unknown function that is looked for as a solution of the IS of differential equations (15). We can assume, for example, that for any fixed $t\in I_T$, the sequence $\left(c_n\left(t\right)\right)$ is an element of the sequence space $l_{\infty }$ or even some tempered sequence space. We also assume that $f_n=f_n(t,x_1,x_2,\dots )$ is a given function defined on the Cartesian product $I_T\times l_{\infty }$. Let us notice that the IS of differential equations (15) represents a semilinear around diagonal IS of differential equations (cf. Problem 1 formulated above).

It turns out that if we consider the IS of differential equations (15), we obtain the operator generated by the right hand side of (15) which is not bounded on the spaces $BC(I,l_{\infty })$, $BC(I,c_0)$, even on some spaces that contain sequences from ${\mathbb{R}}^{\infty }$ that are appropriately tempered (cf. Banaś et al. [34] for details).

Therefore, our problem is to find conditions and, simultaneously, a suitable Banach space that allow us to obtain an existence result concerning the IS of differential equations (15).

Problem 3.

In Section 2, we quoted a lot of results on the existence of solutions of ISs of integral equations considered mainly in the spaces $BC(I,l_{\infty })$ or $BC({\mathbb{R}}_{+},l_{\infty })$. Indeed, we recall results of this type concerning the IS of integral equations (6) in the space $BC({\mathbb{R}}_{+},l_{\infty })$ obtained by Banaś and Madej [9,33] as well results from the paper by Banaś et al. [35]. The mentioned results contain theorems on the existence of solutions of the IS of integral equations (7), where we showed that under suitable assumptions, the IS of integral equations (7) has solutions in the Banach spaces $B{C}_{\infty }=BC({\mathbb{R}}_{+},l_{\infty })$ or $B{C}_0=BC({\mathbb{R}}_{+},c_0)$.

We raise the problem to formulate results on the existence of solutions of the IS of integral equations (7) that converge at infinity to proper limits. In such a case, we are forced to consider the mentioned IS in the Banach space $BC({\mathbb{R}}_{+},c)$, where c is the classical space containing real sequences converging to proper limits, with the classical supremum norm.

It seems that such a problem is not easy since we have to use a suitable MNC in the space $BC({\mathbb{R}}_{+},c)$ (cf. the paper by Banaś and Chlebowicz [7], where such an MNC was constructed).

As we mentioned above, we raise also the problem of a formulation of suitable results concerning the existence of solutions of IS of integral equations with the form

$$x_n\left(t\right)=a_n\left(t\right)+f_n(t,x_1\left(t\right),x_2\left(t\right),\dots ){\int }_0^t{u}_n(t,s,x_1\left(s\right),x_2\left(s\right),\dots )ds$$

(16)

for $n=1,2,\dots$ and $t\in {\mathbb{R}}_{+}$.

In the above quoted paper by Banaś et al. [35], existence results for ISs of integral equations being the special case of IS (16) were obtained. Namely, in those papers, the existence of solutions of IS of Volterra–Hammerstein integral equations of the form

$$x_n\left(t\right)=a_n\left(t\right)+f_n(t,x_1\left(t\right),x_2\left(t\right),\dots ){\int }_0^t{g}_n(t,s)h_n(s,x_1\left(s\right),x_2\left(s\right),\dots )ds$$

for $n=1,2,\dots$ and $t\in {\mathbb{R}}_{+}$ was discussed.

Problem 4.

In almost all papers and monographs dedicated to the theory of functions of bounded variation, the concept of the bounded variation is only considered on a bounded interval. This means that there the variation of functions on the bounded and closed interval $[a,b]$ is considered (see Appell, Banaś and Merentes [36]; Natanson [12]; Łojasiewicz [13]; Convertito and Cruz-Uribe [14]; Cruz-Uribe and Glidewell [15]; and Dunford and Schwartz [37], for example). The authors do not consider the concept of the variation of a function on an unbounded interval of the type $(-\infty ,b]$ or $[a,\infty )$.

In connection with the above assertion, we raise the following problem: Consider a function $f:[a,\infty )\to \mathbb{R}$ (the case when $f:(-\infty ,b]\to \mathbb{R}$ can be treated similarly). We assume that the function f is bounded on the interval $[a,\infty )$.

The problem is to define the variation of the function f for a reasonable, defined partition of the interval $[a,\infty )$. Next, we can define the variation $\mathit{Var}(f,[a,\infty \left)\right)$ of the function f on the interval $[a,\infty )$ as the supremum of variations of f with respect to partitions of the interval $[a,\infty )$.

After defining this, we can consider the class $BV\left(\right[a,\infty \left)\right)$ of functions of bounded variation on the interval $[a,\infty )$.

We raise also the problem of establishing nontrivial properties of the variation on the interval $[a,\infty )$ and the properties of the class $BV\left(\right[a,\infty \left)\right)$. Moreover, we look for some relations between the class $BV\left(\right[a,b\left]\right)$ and the class $BV\left(\right[a,\infty \left)\right)$, where $b\to \infty$.

Problem 5.

When Problem 4 is satisfactorily solved, we can proceed to the next problem concerning the definition of the Riemann–Stieltjes (or even Lebesgue–Stieltjes) integral on an unbounded interval. To present this problem, let us assume that the function $\alpha :[a,\infty )\to \mathbb{R}$ is of bounded variation on the interval $[a,\infty )$ in the sense of a solution of Problem 4.

Next, let $f:[a,\infty )\to \mathbb{R}$ be a given function bounded on the interval $[a,\infty )$. We raise the problem of defining the Riemann–Stieltjes integral of the function f with respect to a function α of bounded variation on the interval $[a,\infty )$, i.e., we expect the definition of the Riemann–Stieltjes integral to be of the form

$${\int }_a^{\infty }f\left(t\right)d\alpha \left(t\right).$$

(17)

If we define the Riemann–Stieltjes integral in the style suggested above, we expect that in the case when f will be continuous and bounded on the interval $[a,\infty )$ and the function α will be of bounded variation on $[a,\infty )$, then the Riemann–Stieltjes integral (17) is well defined. Obviously, taking into account the applications of the Riemann–Stieltjes integral on an unbounded interval like that of (17), we can also extend our definition for the Riemann–Stieltjes integral of the type

$${\int }_a^{\infty }f(t,s)d_s\alpha (t,s)$$

for $t\in [a,\infty )$.

Problem 6.

Now, we proceed to the applications of the concepts of functions of bounded variation on the unbounded interval $[a,\infty )$ and the Riemann–Stieltjes integral of a function on the interval $[a,\infty )$ in the theory of integral equations.

We start with the problem of a formulation of reasonable assumptions guaranteeing the existence of solutions of the quadratic Urysohn integral equation of the Stieltjes type with the form

$$x\left(t\right)=a\left(t\right)+f(t,x_1\left(t\right),x_2\left(t\right),\dots ){\int }_a^{\infty }u(t,s,x_1\left(s\right),x_2\left(s\right),\dots )d_{s}K(t,s)$$

(18)

for $t\in [a,\infty )$.

Here, we assume that the function $K:[a,\infty )\times [a,\infty )\to \mathbb{R}$ is such that the function $s\to K(t,s)$ is of bounded variation on the interval $[a,\infty )$ for any $t\in {\mathbb{R}}_{+}$.

Of course, we leave to the reader the formulation of other assumptions concerning the terms involved in integral equations (18) that guarantee that the considered equation has a solution in the Banach space $BC([a,\infty ),l_{\infty })$ or $BC([a,\infty ),c_0)$ or even $BC\left(\right[a,\infty ),c)$.

It seems that the above formulated problem is not easy to solve. Let us only suggest, as a possible starting point, the paper by Banaś and Taktak [11], where the IS of integral equations of the Volterra–Hammerstein type (8) was studied. Obviously, for a fixed natural number n, any integral equation of IS (8) is a particular case of integral equations (18).

Problem 7.

Our last open problem is devoted to the IS of integral equations (18). More precisely, we consider the IS of quadratic Urysohn integral equations of the Stieltjes type that has the form

$$x_n\left(t\right)=a_n\left(t\right)+f_n(t,x_1\left(t\right),x_2\left(t\right),\dots ){\int }_a^{\infty }{u}_n(t,s,x_1\left(s\right),x_2\left(s\right),\dots )d_s{K}_n(t,s),$$

(19)

where $t\in [a,\infty )$ and $n=1,2,\dots$.

It is clear that our considerations concerning IS (19) will depend on solutions of Problems 4 and 5. Thus, we assume that for any fixed natural number n, the function $K_n=K_n(t,s)$ is of bounded variation with respect to $s\in [a,\infty )$ for any fixed $t\in [a,\infty )$. Moreover, it seems that we have to make an assumption requiring that the variations of the functions $s\to K_n(t,s)$ are equibounded on the interval $[a,\infty )$ with respect to the variable $t\in [a,\infty )$.

Summing up, we raise the problem of the formulation of assumptions imposed on components of the IS of integral equations (19) that ensure that IS (19) has solutions in the space $BC([a,\infty ),l_{\infty })$, $BC([a,\infty ),c_0)$ or $BC\left(\right[a,\infty ),c)$.

4. Discussion

The aim of this paper was to present the current state of the theory of ISs of differential and integral equations. While the theory of ISs of differential equations has been developed over seventy years, the theory of ISs of integral equations is generally very young.

Our discussion focused on those ISs of differential or integral equations that can be investigated with help of the tools of nonlinear functional analysis.

In this paper, we concentrated on the following aspects.

• We discussed the results that ensure the existence of solutions of ISs of both differential and integral equations.
• We mainly considered the Banach spaces containing function sequences defined on the interval (bounded or not) with values in classical Banach sequence spaces such as $l_{\infty }$, $c_0$ or c.
• The basic tool used in our approach was the technique of MNCs developed in the abovementioned Banach spaces.
• The important and essential part of our presentation was to raise some open problems that appeared naturally in our study and were closely connected with the current state of the theory of ISs of differential or integral equations.
• In our opinion, the solutions of raised open problems will give impetus to the fruitful development of the discussed theory of ISs of differential and integral equations.

5. Conclusions

This paper raised open problems that are closely related with the basic results of the theory of ISs of differential or integral equations that have been obtained up to now. Some of the mentioned open problems create a natural extension of the results that appeared quite recently. Those problems are connected with expecting theorems on the existence of solutions of ISs of integral equations in the Banach spaces $BC(I,c)$ or $BC({\mathbb{R}}_{+},c)$ (those spaces are described in detail in this paper).

However, for a few open problems, we challenge researchers to obtain theorems on the existence of solutions of a Urysohn integral equation of the Stieltjes type on the unbounded interval $[a,\infty )$. We also raise the problem connected with the existence of solutions of ISs of Urysohn integral equations of the Stieltjes type on the unbounded interval $[a,\infty )$.

It is worth mentioning that the last indicated open problems are strongly connected with the need to define the variation of functions on the unbounded interval $[a,\infty )$, with the development of the theory of functions of bounded variation on the interval $[a,\infty )$.

We hope that the open problems formulated in this paper will contribute to the development of the theory of integral equations and the ISs of those equations.