## Compactness Conditions in the Theory of Nonlinear Differential and Integral Equations

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M. Mursaleen, Abdullah Alotaibi, "Infinite System of Differential Equations in Some Spaces", *Abstract and Applied Analysis*, vol. 2012, Article ID 863483, 20 pages, 2012. https://doi.org/10.1155/2012/863483

# Infinite System of Differential Equations in Some Spaces

**Academic Editor:**Beata Rzepka

#### Abstract

The first measure of noncompactness was defined by Kuratowski in 1930 and later the Hausdorff measure of noncompactness was introduced in 1957 by Goldenštein et al. These measures of noncompactness have various applications in several areas of analysis, for example, in operator theory, fixed point theory, and in differential and integral equations. In particular, the Hausdorff measure of noncompactness has been extensively used in the characterizations of compact operators between the infinite-dimensional Banach spaces. In this paper, we present a brief survey on the applications of measures of noncompactness to the theory of infinite system of differential equations in some spaces and .

#### 1. and Spaces

In this section, we give some basic definitions and notations about and spaces for which we refer to [1–3].

We will write for the set of all complex sequences . Let and denote the sets of all finite, bounded, convergent, and null sequences, respectively, and be the set of all convergent series. We write for . By and , we denote the sequences such that for , and and . For any sequence , let be its -section.

Note that and are Banach spaces with the norm , and are Banach spaces with the norm .

A sequence in a linear metric space is called *Schauder basis* if for every , there is a unique sequence of scalars such that . A sequence space with a linear topology is called a if each of the maps defined by is continuous for all . A space is called an *space* if is complete linear metric space; a *space* is a normed space. An space is said to have if every sequence has a unique representation , that is, .

A linear space equipped with a translation invariant metric is called a *linear metric space* if the algebraic operations on are continuous functions with respect to . A complete linear metric space is called a *Fréchet space*. If and are linear metric spaces over the same field, then we write for the class of all continuous linear operators from to . Further, if and are normed spaces then consists of all bounded linear operators , which is a normed space with the operator norm given by for all , where denotes the unit sphere in , that is, . Also, we write for the closed unit ball in a normed space . In particular, if then we write for the set of all continuous linear functionals on with the norm .

The *theory of **spaces* is the most powerful and widely used tool in the characterization of matrix mappings between sequence spaces, and the most important result was that matrix mappings between spaces are continuous.

A sequence space is called an *FK space* if it is a locally convex Fréchet space with continuous coordinates , where denotes the complex field and for all and every . A normed space is called a *space*, that is, a *space* is a Banach sequence space with continuous coordinates.

The famous example of an space which is not a space is the space , where

On the other hand, the classical sequence spaces are spaces with their natural norms. More precisely, the spaces , and are spaces with the sup-norm given by . Also, the space is a space with the usual -norm defined by . Further, the spaces , , and are spaces with the same norm given by , and is a space with .

An space is said to have if every sequence has a unique representation , that is, . This means that is a Schauder basis for any space with such that every sequence, in an space with , coincides with its sequence of coefficients with respect to this basis.

Although the space has no Schauder basis, the spaces , and all have Schauder bases. Moreover, the spaces , , and have , where .

There are following spaces which are closely related to the spaces .

Let denote the space whose elements are finite sets of distinct positive integers. Given any element of , we denote by the sequence such that for , and otherwise. Further that is, is the set of those whose support has cardinality at most , and define

For , the following sequence spaces were introduced by Sargent [4] and further studied in [5–8]: where denotes the set of all sequences that are rearrangements of .

*Remark 1.1. * (i) The spaces and are spaces with their respective norms.

(ii) If for all , then and if for all , then , .

(iii) for all of .

#### 2. Measures of Noncompactness

The first measure of noncompactness was defined and studied by Kuratowski [9] in 1930. The Hausdorff measure of noncompactness was introduced by Goldenštein et al. [10] in 1957, and later studied by Goldenštein and Markus [11] in 1965.

Here, we shall only consider the Hausdorff measure of noncompactness; it is the most suitable one for our purposes. The basic properties of measures of noncompactness can be found in [12–14].

Let and be subsets of a metric space and let . Then is called an -*net* of in if for every there exists such that . Further, if the set is finite, then the -net of is called a *finite **-net* of , and we say that has a finite -net in . A subset of a metric space is said to be *totally bounded* if it has a finite -net for every and is said to be *relatively compact* if its closure is a compact set. Moreover, if the metric space is complete, then is totally bounded if and only if is relatively compact.

Throughout, we shall write for the collection of all bounded subsets of a metric space . If , then the *Hausdorff measure of noncompactness* of the set , denoted by , is defined to be the infimum of the set of all reals such that can be covered by a finite number of balls of radii and centers in . This can equivalently be redefined as follows:
The function is called the *Hausdorff measure of noncompactness*.

If , , and are bounded subsets of a metric space , then we have

Further, if is a normed space, then the function has some additional properties connected with the linear structure, for example,

Let and be Banach spaces and and be the Hausdorff measures of noncompactness on and , respectively. An operator is said to be ,-*bounded* if for all and there exist a constant such that for all . If an operator is ,-bounded then the number for all is called the ,-*measure of noncompactness of *. If , then we write .

The most effective way in the characterization of compact operators between the Banach spaces is by applying the Hausdorff measure of noncompactness. This can be achieved as follows: let and be Banach spaces and . Then, the Hausdorff measure of noncompactness of , denoted by , can be determined by and we have that is compact if and only if

Furthermore, the function is more applicable when is a Banach space. In fact, there are many formulae which are useful to evaluate the Hausdorff measures of noncompactness of bounded sets in some particular Banach spaces. For example, we have the following result of Goldenštein et al. [10, Theorem 1] which gives an estimate for the Hausdorff measure of noncompactness in Banach spaces with Schauder bases. Before that, let us recall that if is a Schauder basis for a Banach space , then every element has a unique representation , where are called the basis functionals. Moreover, the operator , defined for each by , is called the projector onto the linear span of . Besides, all operators and are equibounded, where denotes the identity operator on .

Theorem 2.1. * Let be a Banach space with a Schauder basis , and be the projector onto the linear span of . Then, one has
**
where .*

In particular, the following result shows how to compute the Hausdorff measure of noncompactness in the spaces and which are spaces with .

Theorem 2.2. * Let be a bounded subset of the normed space , where is for or . If is the operator defined by for all , then one has
*

It is easy to see that for

Also, it is known that is a Schauder basis for the space and every sequence has a unique representation , where . Thus, we define the projector , onto the linear span of , by for all with . In this situation, we have the following.

Theorem 2.3. * Let and be the projector onto the linear span of . Then, one has
**
where is the identity operator on .*

Theorem 2.4. * Let be a bounded subset of . Then
*

The idea of *compact operators* between Banach spaces is closely related to the Hausdorff measure of noncompactness, and it can be given as follows.

Let and be complex Banach spaces. Then, a linear operator is said to be *compact* if the domain of is all of , that is, , and for every bounded sequence in , the sequence has a convergent subsequence in . Equivalently, we say that is compact if its domain is all of and is relatively compact in for every .

Further, we write for the class of all compact operators from to . Let us remark that every compact operator in is bounded, that is, . More precisely, the class is a closed subspace of the Banach space with the operator norm.

The most effective way in the characterization of compact operators between the Banach spaces is by applying the Hausdorff measure of noncompactness. This can be achieved as follows.

The most effective way in the characterization of compact operators between the Banach spaces is by applying the Hausdorff measure of noncompactness. This can be achieved as follows.

Let and be Banach spaces and . Then, the Hausdorff measure of noncompactness of , denoted by , can be given by and we have

Since matrix mappings between spaces define bounded linear operators between these spaces which are Banach spaces, it is natural to use the Hausdorff measure of noncompactness to obtain necessary and sufficient conditions for matrix operators between spaces to be compact operators. This technique has recently been used by several authors in many research papers (see, for instance, [5, 6, 15–32]).

#### 3. Applications to Infinite Systems of Differential Equations

This section is mainly based on the work of Banaś and Lecko [33], Mursaleen and Mohiuddine [26], and Mursaleen [32]. In this section, we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in some Banach sequence spaces , and .

Infinite systems of ordinary differential equations describe numerous world real problems which can be encountered in the theory of branching processes, the theory of neural nets, the theory of dissociation of polymers, and so on (cf. [34–38], e.g.). Let us also mention that several problems investigated in mechanics lead to infinite systems of differential equations [39–41]. Moreover, infinite systems of differential equations can be also used in solving some problems for parabolic differential equations investigated via semidiscretization [42, 43]. The theory of infinite systems of ordinary differential equation seems not to be developed satisfactorily up to now. Indeed, the existence results concerning systems of such a kind were formulated mostly by imposing the Lipschitz condition on right-hand sides of those systems (cf. [10, 11, 39, 40, 44–50]). Obviously, the assumptions formulated in terms of the Lipschitz condition are rather restrictive and not very useful in applications. On the other hand, the infinite systems of ordinary differential equations can be considered as a particular case of ordinary differential equations in Banach spaces. Until now several existence results have been obtained concerning the Cauchy problem for ordinary differential equations in Banach spaces [33, 35, 51–53]. A considerable number of those results were formulated in terms of measures of noncompactness. The results of such a type have a concise form and give the possibility to formulate more general assumptions than those requiring the Lipschitz continuity. But in general those results are not immediately applicable in concrete situations, especially in the theory of infinite systems of ordinary differential equations.

In this section, we adopt the technique of measures of noncompactness to the theory of infinite systems of differential equations. Particularly, we are going to present a few existence results for infinite systems of differential equations formulated with the help of convenient and handy conditions.

Consider the ordinary differential equation with the initial condition

Then the following result for the existence of the Cauchy problem (3.1)-(3.2) was given in [34] which is a slight modification of the result proved in [33].

Assume that is a real Banach space with the norm . Let us take an interval and the closed ball in centered at with radius .

Theorem A (see [34]). * Assume that is a function defined on with values in such that
**
for any , where and are nonnegative constants. Further, let be uniformly continuous on the set , where and . Moreover, assume that for any nonempty set and for almost all the following inequality holds:
**
with a sublinear measure of noncompactness such that . Then problem (3.1)-(3.2) has a solution such that for , where is an integrable function on , and is the kernel of the measure .*

*Remark 3.1. * In the case when (the Hausdorff measure of noncompactness), the assumption of the uniform continuity on can be replaced by the weaker one requiring only the continuity of .

Results of Sections 3.1 and 3.2 are from [33], Section 3.3 from [26], and Section 3.4 from [32].

##### 3.1. Infinite Systems of Differential Equations in the Space

From now on, our discussion is exactly same as in Section 3 of [33].

In this section, we study the solvability of the infinite systems of differential equations in the Banach sequence space . It is known that in the space the Hausdorff measure of noncompactness can be expressed by the following formula [33]: where .

We will be interested in the existence of solutions of the infinite systems of differential equations with the initial condition which are defined on the interval and such that for each .

An existence theorem for problem (3.6)-(3.7) in the space can be formulated by making the following assumptions.

Assume that the functions are defined on the set and take real values. Moreover, we assume the following hypotheses:(i) , (ii) the map acts from the set into and is continuous,(iii) there exists an increasing sequence of natural numbers obviously as such that for any and the following inequality holds: where and are real functions defined and continuous on such that the sequence converges uniformly on to the function vanishing identically and the sequence is equibounded on .

Now, let us denote

Then we have the following result.

Theorem 3.2 (see [33]). *Under the assumptions (i)–(iii), initial value problem (3.6)-(3.7) has at least one solution defined on the interval whenever and . Moreover, for any .*

*Proof. *Let be any arbitrary sequence in . Then, by (i)–(iii), for any and for a fixed we obtain

Hence, we get

In what follows, let us take the ball , where is chosen according to Theorem A. Then, for a subset of and for , we obtain
Hence, by assumptions, we get
Now, using our assumptions and inequalities (3.11) and (3.13), in view of Theorem A and Remark 3.1 we deduce that there exists a solution of the Cauchy problem (3.6)-(3.7) such that for any .

This completes the proof of the theorem.

We illustrate the above result by the following examples.

*Example 3.3 (see [33]). *Let be an increasing sequence of natural numbers. Consider the infinite system of differential equations of the form
with the initial condition
.

We will investigate problem (3.14)-(3.15) under the following assumptions:(i),
(ii) the functions are uniformly equicontinuous and there exists a function sequence such that is continuous on for any and converges uniformly on to the function vanishing identically. Moreover, the following inequality holds:
for , and ,(iii) the functions are defined and continuous on and the function series converges absolutely and uniformly on (to a function ) for any ,(iv) the sequence is equibounded on ,(v), where .

It can be easily seen that the assumptions of Theorem 3.2 are satisfied under assumptions (i)–(v). This implies that problem (3.14)-(3.15) has a solution on the interval belonging to the space for any fixed .

As mentioned in [33], problem (3.14)-(3.15) considered above contains as a special case the infinite system of differential equations occurring in the theory of dissociation of polymers [35]. That system was investigated in [37] in the sequence space under very strong assumptions. The existing result proved in [35] requires also rather restrictive assumptions. Thus, the above result is more general than those quoted above.

Moreover, the choice of the space for the study of the problem (3.14)-(3.15) enables us to obtain partial characterization of solutions of this problem since we have that when , for any fixed .

On the other hand, let us observe that in the study of the heat conduction problem via the method of semidiscretization we can obtain the infinite systems of form (3.14) (see [42] for details).

*Example 3.4 (see [33]). *In this example, we will consider some special cases of problem (3.14)-(3.15). Namely, assume that for and on for all . Then system (3.14) has the form
and is called a row-finite system [35].

Suppose that there are satisfied assumptions from Example 3.3, that is, and the functions act from into and are uniformly equicontinuous on their domains. Moreover, there exist continuous functions such that
for and . We assume also that the sequence converges uniformly on to the function vanishing identically.

Further, let denote the maximum norm in . Take . Then we have
Taking ,, then the above estimate can be written in the form

Observe that from our assumptions it follows that the initial value problem has a unique solution on the interval . Hence, applying a result from [33], we infer that Cauchy problem (3.17)-(3.15) has a solution on the interval . Obviously from the result contained in Theorem 3.2 and Example 3.3, we deduce additionally that the mentioned solution belongs to the space .

Finally, it is noticed [33] that the result described above for row-finite systems of the type (3.17) can be obtained under more general assumptions.

In fact, instead of inequality (3.18), we may assume that the following estimate holds to be true: where the functions and satisfy the hypotheses analogous to those assumed in Theorem 3.2.

*Remark 3.5 (see [33]). * Note that in the birth process one can obtain a special case of the infinite system (3.17) which is lower diagonal linear infinite system [35, 43]. Thus, the result proved above generalizes that from [35, 37].

##### 3.2. Infinite Systems of Differential Equations in the Space

Now, we will study the solvability of the following perturbed diagonal system of differential equations with the initial condition , where .

From now, we are going exactly the same as in Section 4 of [33].

We consider the following measures of noncompactness in which is more convenient, regular, and even equivalent to the Hausdorff measure of noncompactness [33].

For

Let us formulate the hypotheses under which the solvability of problem (3.22)-(3.23) will be investigated in the space . Assume that the following conditions are satisfied.

Assume that the functions are defined on the set and take real values. Moreover, we assume the following hypotheses:(i), (ii) the map acts from the set into and is uniformly continuous on ,(iii) there exists sequence such that for any and the following inequality holds: (iv) the functions are continuous on such that the sequence converges uniformly on .

Further, let us denote

Observe that in view of our assumptions, it follows that the function is continuous on . Hence, .

Then we have the following result which is more general than Theorem 3.2.

Theorem 3.6 (see [33]). * Let assumptions (i)–(iv) be satisfied. If , then the initial value problem (3.12)-(3.13) has a solution on the interval such that for each .*

*Proof. *Let and and

Then, for arbitrarily fixed natural numbers we get
By assumptions (iii) and (iv), from the above estimate we deduce that is a real Cauchy sequence. This implies that .

Also we obtain the following estimate:
where . Hence,

In what follows, let us consider the mapping on the set , where is taken according to the assumptions of Theorem A, that is, . Further, fix arbitrarily and . Then, by our assumptions, for a fixed , we obtain

Then,
Hence, taking into account that the sequence is equicontinuous on the interval and is uniformly continuous on , we conclude that the operator is uniformly continuous on the set .

In the sequel, let us take a nonempty subset of the ball and fix . Then, for arbitrarily fixed natural numbers we have

Hence, we infer the following inequality:

Finally, combining (3.30), (3.34) and the fact (proved above) that is uniformly continuous on , in view of Theorem A, we infer that problem (3.22)-(3.23) is solvable in the space .

This completes the proof of the theorem.

*Remark 3.7 (see [33, Remark 4]). * The infinite systems of differential equations (3.22)-(3.23) considered above contain as special cases the systems studied in the theory of neural sets (cf. [35, pages 86-87] and [37], e.g.). It is easy to notice that the existence results proved in [35, 37] are obtained under stronger and more restrictive assumptions than our one.

##### 3.3. Infinite Systems of Differential Equations in the Space

In this section, we study the solvability of the infinite systems of differential equations (3.6)-(3.7) in the Banach sequence space such that for each .

An existence theorem for problem (3.6)-(3.7) in the space can be formulated by making the following assumptions:(i), (ii) maps continuously the set into ,(iii) there exist nonnegative functions and defined on such that for and ,(iv) the functions are continuous on and the function series converges uniformly on ,(v) the sequence is equibounded on the interval and the function is integrable on .

Now, we prove the following result.

Theorem 3.8 (see [26]). * Under the assumptions (i)–(v), problem (3.6)-(3.7) has a solution defined on the interval whenever , where is defined as the number
**
Moreover, for any .*

*Proof. * For any and , under the above assumptions, we have
where .

Now, choose the number as defined in Theorem A. Consider the operator on the set . Let us take a set . Then by using (2.8), we get
Hence, by assumptions (iv)-(v), we get
that is, the operator satisfies condition (3.4) of Theorem A. Hence, by Theorem A and Remark 3.1 we conclude that there exists a solution of problem (3.6)-(3.7) such that for any .

This completes the proof of the theorem.

*Remark 3.9. *(I) For , we get Theorem 5 of [33].(II) It is easy to notice that the existence results proved in [44] are obtained under stronger and more restrictive assumptions than our one.(III) We observe that the above theorem can be applied to the perturbed diagonal infinite system of differential equations of the form
with the initial condition
where .

An existence theorem for problem (3.6)-(3.7) in the space can be formulated by making the following assumptions:(i),
(ii) the sequence is defined and equibounded on the interval . Moreover, the function is integrable on ,(iii) the mapping maps continuously the set into ,(iv) there exist nonnegative functions such that
for and ,(v) the functions are continuous on and the function series converges uniformly on .

##### 3.4. Infinite Systems of Differential Equations in the Space

An existence theorem for problem (3.6)-(3.7) in the space can be formulated by making the following assumptions:(i), (ii) maps continuously the set into ,(iii) there exist nonnegative functions and defined on such that for and , where is a sequence of rearrangement of ,(iv) the functions are continuous on and the function series converges uniformly on ,(v) the sequence is equibounded on the interval and the function is integrable on .

Now, we prove the following result.

Theorem 3.10 (see [32]). * Under the assumptions (i)–(v), problem (3.6)-(3.7) has a solution defined on the interval whenever , where is defined as the number
**Moreover, for any .*

*Proof. *For any and , under the above assumptions, we have
where .

Now, choose the number defined according to Theorem A, that is, . Consider the operator on the set . Let us take a set . Then by using Theorem 2.4, we get
Hence, by assumptions (iv)-(v), we get
that is, the operator satisfies condition (3.4) of Theorem A. Hence, the problem (3.6)-(3.7) has a solution .

This completes the proof of the theorem.

*Remark 3.11. * Similarly, we can establish such type of result for the space .

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