# Infinite System of Differential Equations in SomeBK Spaces

Authors: M. Mursaleen; Abdullah Alotaibi
Journal: Abstract and Applied Analysis
 (2012)

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## 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 someBK spaces c0,c,ℓp(1≤p≤∞) and n(ϕ).


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## Body

## 1.FKand BK Spaces

In this section, we give some basic definitions and notations aboutFK and BK spaces for which we refer to [1–3].We will writew for the set of all complex sequences x=(xk)k=0∞. Let φ,ℓ∞,c and c0 denote the sets of all finite, bounded, convergent, and null sequences, respectively, and cs be the set of all convergent series. We write ℓp:={x∈w:∑k=0∞‍|xk|p<∞} for 1≤p<∞. By e and e(n)(n∈ℕ), we denote the sequences such that ek=1 for k=0,1,…, and en(n)=1 and ek(n)=0(k≠n). For any sequence x=(xk)k=0∞, let x[n]=∑k=0n‍xke(k) be its n-section.Note thatℓ∞,c and c0 are Banach spaces with the norm ∥x∥∞=supk≥0|xk|, and ℓp(1≤p<∞) are Banach spaces with the norm ∥x∥p=(∑k=0∞‍|xk|p)1/p.A sequence(b(n))n=0∞ in a linear metric space X is called Schauder basis if for every x∈X, there is a unique sequence (λn)n=0∞ of scalars such that x=∑n=0∞‍λnb(n). A sequence space X with a linear topology is called a Kspace if each of the maps pi:X→ℂ defined by pi(x)=xi is continuous for all i∈ℕ. A K space is called an FKspace if X is complete linear metric space; a BKspace is a normed FK space. An FKspace X⊃φ is said to have AK if every sequence x=(xk)k=0∞∈X  has a unique representation x=∑k=0∞‍xke(k), that is, x=limn→∞x[n].A linear spaceX equipped with a translation invariant metric d is called a linear metric space if the algebraic operations on X are continuous functions with respect to d. A complete linear metric space is called a Fréchet space. If X and Y are linear metric spaces over the same field, then we write ℬ(X,Y) for the class of all continuous linear operators from X to Y. Further, if X and Y are normed spaces then ℬ(X,Y) consists of all bounded linear operators L:X→Y, which is a normed space with the operator norm given by ∥L∥=supx∈SX∥L(x)∥Yfor all L∈ℬ(X,Y), where SX denotes the unit sphere in X, that is, SX:={x∈X:∥x∥=1}. Also, we write B-X:={x∈X:∥x∥≤1} for the closed unit ball in a normed space X. In particular, if Y=ℂ then we write X* for the set of all continuous linear functionals on X with the norm ∥f∥=supx∈SX|f(x)|.Thetheory ofFKspaces  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 FK spaces are continuous.A sequence spaceX is called an FK space if it is a locally convex Fréchet space with continuous coordinates pn:X→ℂ(n∈ℕ), where ℂ denotes the complex field and pn(x)=xn for all x=(xk)∈X and every n∈ℕ. A normed FK space is called a BKspace, that is, a BKspace is a Banach sequence space with continuous coordinates.The famous example of anFK space which is not a BK space is the space (w,dw), where
(1.1)dw(x,y)=∑k=0∞‍12k(|xk-yk|1+|xk-yk|);(x,y∈w).On the other hand, the classical sequence spaces areBK spaces with their natural norms. More precisely, the spaces ℓ∞,c, and c0 are BK spaces with the sup-norm given by ∥x∥ℓ∞=supk|xk|. Also, the space ℓp(1≤p<∞) is a BK space with the usual ℓp-norm defined by ∥x∥ℓp=(∑k‍|xk|p)1/p. Further, the spaces bs, cs, and cs0 are BK spaces with the same norm given by ∥x∥bs=supn∑k=0n‍|xk|, and bv is a BK space with ∥x∥bv=∑k‍|xk-xk-1|.AnFK space X⊃ϕ is said to have AK if every sequence x=(xk)∈X has a unique representation x=∑k=0∞‍xke(k), that is, limn→∞(∑k=0n‍xke(k))=x. This means that (e(k))k=0∞ is a Schauder basis for any FK space with AK such that every sequence, in an FK space with AK, coincides with its sequence of coefficients with respect to this basis.Although the spaceℓ∞ has no Schauder basis, the spaces w,c0,c, and ℓp all have Schauder bases. Moreover, the spaces w, c0, and ℓp have AK, where 1≤p<∞.There are followingBK spaces which are closely related to the spaces ℓp(1≤p≤∞).Let𝒞 denote the space whose elements are finite sets of distinct positive integers. Given any element σ of 𝒞, we denote by c(σ) the sequence {cn(σ)} such that cn(σ)=1 for n∈σ, and cn(σ)=0 otherwise. Further
(1.2)𝒞s={σ∈𝒞:∑n=1∞‍cn(σ)≤s},
that is, 𝒞s is the set of those σ whose support has cardinality at most s, and define
(1.3)Φ={ϕ=(ϕk)∈w:0<ϕ1≤ϕn≤ϕn+1,(n+1)ϕn≥nϕn+1}.Forϕ∈Φ, the following sequence spaces were introduced by Sargent [4] and further studied in [5–8]:
(1.4)m(ϕ)={x=(xk)∈w:∥x∥m(ϕ)=sups≥1supσ∈𝒞s(1ϕs∑k∈σ‍|xk|)<∞},n(ϕ)={x=(xk)∈w:∥x∥n(ϕ)=supu∈S(x)(∑k=1∞‍|uk|Δϕk)<∞},
where S(x) denotes the set of all sequences that are rearrangements of x.Remark 1.1.
(i) The spacesm(ϕ) and n(ϕ) are BK spaces with their respective norms.
(ii) Ifϕn=1 for all n∈ℕ, then m(ϕ)=l1,n(ϕ)=l∞,and if ϕn=n for all n∈ℕ, then m(ϕ)=l∞, n(ϕ)=l1.
(iii)l1⊆m(ϕ)⊆l∞[l∞⊇n(ϕ)⊇l1] for all ϕ of Φ.


## 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].LetS and M be subsets of a metric space (X,d) and let ϵ>0. Then S is called an ϵ-net of M in X if for every x∈M there exists s∈S such that d(x,s)<ϵ. Further, if the set S is finite, then the ϵ-net S of M is called a finiteϵ-net of M, and we say that M has a finite ϵ-net in X. A subset M of a metric space X is said to be totally bounded if it has a finite ϵ-net for every ϵ>0 and is said to be relatively compact if its closure M- is a compact set. Moreover, if the metric space X is complete, then M is totally bounded if and only if M is relatively compact.Throughout, we shall writeℳX for the collection of all bounded subsets of a metric space (X,d). If Q∈ℳX, then the Hausdorff measure of noncompactness of the set Q, denoted by χ(Q), is defined to be the infimum of the set of all reals ϵ>0 such that Q can be covered by a finite number of balls of radii <ϵ and centers in X. This can equivalently be redefined as follows:
(2.1)χ(Q)=inf{ϵ>0:Qhasafiniteϵ-net}.
The function χ:ℳX→[0,∞) is called the Hausdorff measure of noncompactness.IfQ, Q1, and Q2 are bounded subsets of a metric space X, then we have
(2.2)χ(Q)=0if and only ifQis totally bounded,Q1⊂Q2impliesχ(Q1)≤χ(Q2).Further, ifX is a normed space, then the function χ has some additional properties connected with the linear structure, for example,
(2.3)χ(Q1+Q2)≤χ(Q1)+χ(Q2),χ(αQ)=|α|χ(Q),∀α∈ℂ.LetX and Y be Banach spaces and χ1 and χ2 be the Hausdorff measures of noncompactness on X and Y, respectively. An operator L:X→Y is said to be (χ1,χ2)-bounded if L(Q)∈ℳY for all Q∈ℳX and there exist a constant C≥0 such that χ2(L(Q))≤Cχ1(Q) for all Q∈ℳX. If an operator L is (χ1,χ2)-bounded then the number ∥L∥(χ1,χ2):=inf{C≥0:χ2(L(Q))≤Cχ1(Q) for all Q∈ℳX} is called the (χ1,χ2)-measure of noncompactness ofL. If χ1=χ2=χ, then we write ∥L∥(χ1,χ2)=∥L∥χ.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: letX and Y be Banach spaces and L∈B(X,Y). Then, the Hausdorff measure of noncompactness of L, denoted by ∥L∥χ, can be determined by
(2.4)∥L∥χ=χ(L(SX)),
and we have that L is compact if and only if
(2.5)∥L∥χ=0.Furthermore, the functionχ is more applicable when X 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 (bk)k=0∞ is a Schauder basis for a Banach space X, then every element x∈X has a unique representation x=∑k=0∞‍ϕk(x)bk, where ϕk(k∈ℕ) are called the basis functionals. Moreover, the operator Pr:X→X, defined for each r∈ℕ by Pr(x)=∑k=0r‍ϕk(x)bk(x∈X), is called the projector onto the linear span of {b0,b1,…,br}. Besides, all operators Pr and I-Pr are equibounded, where I denotes the identity operator on X.Theorem 2.1.
LetX be a Banach space with a Schauder basis (bk)k=0∞, E∈ℳX and Pn:X→X(n∈ℕ) be the projector onto the linear span of {b0,b1,…,bn}. Then, one has
(2.6)1a·limsupn→∞(supx∈Q∥(I-Pn)(x)∥)≤χ(E)≤limsupn→∞(supx∈Q∥(I-Pn)(x)∥),
where a=limsupn→∞∥I-Pn∥.In particular, the following result shows how to compute the Hausdorff measure of noncompactness in the spacesc0 and ℓp(1≤p<∞) which are BK spaces with AK.Theorem 2.2.
LetE be a bounded subset of the normed space X, where X is ℓp for 1≤p<∞ or c0. If Pn:X→X(n∈ℕ) is the operator defined by Pn(x)=x[n]=(x0,x1,…,xn,0,0,…) for all x=(xk)k=0∞∈X, then one has
(2.7)χ(E)=limn→∞(supx∈Q∥(I-Pn)(x)∥).It is easy to see that forE∈ℳℓp(2.8)χ(E)=limn→∞(supx∈Q∑k≥n‍|xk|p).Also, it is known that(e,e(0),e(1),…) is a Schauder basis for the space c and every sequence z=(zn)n=0∞∈c has a unique representation z=z-e+∑n=0∞‍(zn-z-)e(n), where z-=limn→∞zn. Thus, we define the projector Pr:c→c(r∈ℕ), onto the linear span of {e,e(0),e(1),…,e(r)}, by
(2.9)Pr(z)=z-e+∑n=0r‍(zn-z-)e(n);(r∈ℕ),
for all z=(zn)∈c with z-=limn→∞zn. In this situation, we have the following.Theorem 2.3.
LetQ∈ℳc and Pr:c→c(r∈ℕ) be the projector onto the linear span of {e,e(0),e(1),…,e(r)}. Then, one has
(2.10)12·limr→∞(supx∈Q∥(I-Pr)(x)∥ℓ∞)≤χ(Q)≤limr→∞(supx∈Q∥(I-Pr)(x)∥ℓ∞),
where I is the identity operator on c.Theorem 2.4.
LetQ be a bounded subset of n(ϕ). Then
(2.11)χ(Q)=limk→∞supx∈Q(supu∈S(x)(∑n=k∞‍|un|Δϕn)).The idea ofcompact operators between Banach spaces is closely related to the Hausdorff measure of noncompactness, and it can be given as follows.LetX and Y be complex Banach spaces. Then, a linear operator L:X→Y is said to be compact if the domain of L is all of X, that is, D(L)=X, and for every bounded sequence (xn) in X, the sequence (L(xn)) has a convergent subsequence in Y. Equivalently, we say that L is compact if its domain is all of X and L(Q) is relatively compact in Y for every Q∈ℳX.Further, we write𝒞(X,Y) for the class of all compact operators from X to Y. Let us remark that every compact operator in 𝒞(X,Y) is bounded, that is, 𝒞(X,Y)⊂ℬ(X,Y). More precisely, the class 𝒞(X,Y) is a closed subspace of the Banach space ℬ(X,Y) 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.LetX and Y be Banach spaces and L∈ℬ(X;Y). Then, the Hausdorff measure of noncompactness of L, denoted by ∥L∥χ, can be given by
(2.12)∥L∥χ=χ(L(SX)),
and we have
(2.13)Liscompactifandonlyif∥L∥χ=0.Since matrix mappings betweenBK 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 BK 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 c0,c,ℓp(1≤p<∞), and n(ϕ).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(3.1)x′=f(t,x),
with the initial condition
(3.2)x(0)=x0.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 thatX is a real Banach space with the norm ∥·∥. Let us take an interval I=[0,T],T>0 and B(x0,r) the closed ball in X centered at x0 with radius r.Theorem A (see [34]).
Assume thatf(t,x) is a function defined on I×X with values in X such that
(3.3)∥f(t,x)∥≤Q+R∥x∥,
for any x∈X, where Q and R are nonnegative constants. Further, let f be uniformly continuous on the set I1×B(x0,r), where r=(QT1+RT1∥x0∥)/(1-RT1) and I1=[0,T1]⊂I,RT1<1. Moreover, assume that for any nonempty set Y⊂B(x0,s) and for almost all t∈I the following inequality holds:
(3.4)μ(f(t,Y))≤q(t)μ(Y),
with a sublinear measure of noncompactness μ such that {x0}∈kerμ. Then problem (3.1)-(3.2) has a solution x such that {x(t)}∈kerμ for t∈I1, where q(t) is an integrable function on I, and kerμ={E∈ℳX:μ(E)=0} is the kernel of the measure μ.Remark 3.1.
In the case whenμ=χ (the Hausdorff measure of noncompactness), the assumption of the uniform continuity on f can be replaced by the weaker one requiring only the continuity of f.Results of Sections3.1 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 Spacec0

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 spacec0. It is known that in the space c0 the Hausdorff measure of noncompactness can be expressed by the following formula [33]:
(3.5)χ(E)=limn→∞(supx∈E{supk≥n|xk|}),
where E∈ℳc0.We will be interested in the existence of solutionsx(t)=(xi(t)) of the infinite systems of differential equations
(3.6)xi′=fi(t,x0,x1,x2,…),
with the initial condition
(3.7)xi(0)=xi0,(i=0,1,2,…) which are defined on the interval I=[0,T] and such that x(t)∈c0 for each t∈I.An existence theorem for problem (3.6)-(3.7) in the space c0 can be formulated by making the following assumptions.Assume that the functionsfi(i=1,2,…) are defined on the set I×ℝ∞ and take real values. Moreover, we assume the following hypotheses:(i)
x0=(xi0)∈c0,(ii)
the mapf=(f1,f2,…)acts from the set I×c0 into c0 and is continuous,(iii)
there exists an increasing sequence(kn) of natural numbers (obviously kn→∞ as n→∞) such that for any t∈I,x=(xi)∈c0 and n=1,2,… the following inequality holds:
(3.8)|fn(t,x1,x2,…)|≤pn(t)+qn(t)(supi≥kn|xi|),where(pi(t)) and (qi(t)) are real functions defined and continuous on I such that the sequence (pi(t)) converges uniformly on I to the function vanishing identically and the sequence (qi(t)) is equibounded on I.Now, let us denote(3.9)q(t)=supn≥1qn(t),Q=supt∈Iq(t),P=sup{pn(t):t∈I,n=1,2,…}.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 x=x(t)=(xi(t)) defined on the interval I1=[0,T1] whenever T1<T and QT1<1. Moreover, x(t)∈c0 for any t∈I1.Proof.
Letx=(xi) be any arbitrary sequence in c0. Then, by (i)–(iii), for any t∈I and for a fixed n∈ℕ we obtain
(3.10)|fn(t,x)|=|fn(t,x1,x2,…)|≤pn(t)+qn(t)(supi≥kn|xi|)≤P+Qsupi≥kn|xi|≤P+Q∥x∥∞.
Hence, we get(3.11)∥f(t,x)∥≤P+Q∥x∥∞.
In what follows, let us take the ballB(x0,r), where r is chosen according to Theorem A. Then, for a subset Y of B(x0,r) and for t∈I1, we obtain
(3.12)χ(f(t,Y))=limn→∞supx∈Y(supi≥n|fi(t,x)|),χ(f(t,Y))=limn→∞supx∈Y(supi≥n|fi(t,x1,x2,…)|)≤limn→∞supx∈Y(supi≥n{pi(t)+qi(t)supj≥kn|xj|})≤limn→∞supi≥npi(t)+q(t)limn→∞{supx∈Y(supi≥n{supj≥kn|xj|})}.
Hence, by assumptions, we get
(3.13)χ(f(t,Y))≤q(t)χ(Y).
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 x=x(t) of the Cauchy problem (3.6)-(3.7) such that x(t)∈c0 for any t∈I1.
This completes the proof of the theorem.We illustrate the above result by the following examples.Example 3.3 (see [33]).
Let{kn} be an increasing sequence of natural numbers. Consider the infinite system of differential equations of the form
(3.14)xi′=fi(t,x1,x2,…)+∑j=ki+1∞‍aij(t)xj,
with the initial condition
(3.15)xi(0)=xi0,(i=1,2,…;t∈I=[0,T]).
We will investigate problem (3.14)-(3.15) under the following assumptions:(i)
x0=(x0)∈c0,(ii)
the functionsfi:I×ℝki→ℝ(i=1,2,…) are uniformly equicontinuous and there exists a function sequence (pi(t)) such that pi(t) is continuous on I for any i∈ℕ and (pi(t)) converges uniformly on I to the function vanishing identically. Moreover, the following inequality holds:
(3.16)|fi(t,x1,x2,…xki)|≤pi(t),fort∈I,(x1,x2,…xki)∈ℝki and i∈ℕ,(iii)
the functionsaij(t) are defined and continuous on I and the function series ∑j=ki+1∞‍aij(t) converges absolutely and uniformly on I (to a function ai(t)) for any i=1,2,…,(iv)
the sequence(ai(t)) is equibounded on I,(v)
QT<1, where Q=sup{ai(t):i=1,2,…;t∈I}.
It can be easily seen that the assumptions of Theorem3.2 are satisfied under assumptions (i)–(v). This implies that problem (3.14)-(3.15) has a solution x(t)=(xi(t)) on the interval I belonging to the space c0 for any fixed t∈I.
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 ℓ∞ 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 spacec0 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 xn(t)→0 when n→∞, for any fixed t∈[0,T].
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 ki=i for i=1,2,… and aij≡0 on I for all i,j. Then system (3.14) has the form
(3.17)x1′=f1(t,x1),x2′=f2(t,x1,x2),…,xi′=fi(t,x1,x2,…,xi),…,
and is called a row-finite system [35].
Suppose that there are satisfied assumptions from Example3.3, that is, x0=(xi0)∈c0 and the functions fi act from I×ℝi into ℝ(i=1,2,…) and are uniformly equicontinuous on their domains. Moreover, there exist continuous functions pi(t)(t∈I) such that
(3.18)|fi(t,x1,x2,…xki)|≤pi(t),
for t∈I and x1,x2,…,xi∈ℝ(i=1,2,…). We assume also that the sequence pi(t) converges uniformly on I to the function vanishing identically.
Further, let|·|i denote the maximum norm in ℝi(i=1,2,…). Take fi=(f1,f2,…,fi). Then we have
(3.19)|fi(t,x)|i=max{|f1(t,x1)|,|f2(t,x1,x2)|,…,|fi(t,x1,x2,…,xi)|}≤max{p1(t),p2(t),…,pi(t)}.
Taking Pi(t)=max{p1(t),p2(t),…,pi(t)}, then the above estimate can be written in the form
(3.20)|fi(t,x)|i≤Pi(t).Observe that from our assumptions it follows that the initial value problemu′=Pi(t),u(0)=xi0 has a unique solution on the interval I. Hence, applying a result from [33], we infer that Cauchy problem (3.17)-(3.15) has a solution on the interval I. Obviously from the result contained in Theorem 3.2 and Example 3.3, we deduce additionally that the mentioned solution belongs to the space c0.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:
(3.21)|fi(t,x1,x2,…,xi)|≤pi(t)+qi(t)max{|x1|,|x2|,…,|xi|},
where the functions pi(t) and qi(t)(i=1,2,…) 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 Spacec

Now, we will study the solvability of the following perturbed diagonal system of differential equations(3.22)xi′=ai(t)xi+gi(t,x1,x2,…),
with the initial condition
(3.23)xi(0)=xi0,(i=1,2,…), where t∈I=[0,T].From now, we are going exactly the same as in Section 4 of [33].We consider the following measuresμ of noncompactness in c which is more convenient, regular, and even equivalent to the Hausdorff measure of noncompactness [33].ForE∈ℳc(3.24)μ(E)=limp→∞{supx∈E{supn,m≥p|xn-xm|}},χ(E)=limn→∞(supx∈Q{supk≥n|xk|}).Let us formulate the hypotheses under which the solvability of problem (3.22)-(3.23) will be investigated in the space c. Assume that the following conditions are satisfied.Assume that the functionsfi(i=1,2,…) are defined on the set I×ℝ∞ and take real values. Moreover, we assume the following hypotheses:(i)
x0=(xi0)∈c,(ii)
the mapg=(g1,g2,…)acts from the set I×c into c and is uniformly continuous on I×c,(iii)
there exists sequence(bi)∈c0 such that for any t∈I,x=(xi)∈c and n=1,2,… the following inequality holds:
(3.25)|gn(t,x1,x2,…)|≤bi,(iv)
the functionsai(t) are continuous on I such that the sequence (ai(t)) converges uniformly on I.Further, let us denote(3.26)a(t)=supi≥1ai(t),Q=supt∈Ia(t).Observe that in view of our assumptions, it follows that the functiona(t) is continuous on I. Hence, Q<∞.Then we have the following result which is more general than Theorem3.2.Theorem 3.6 (see [33]).
Let assumptions (i)–(iv) be satisfied. IfQT<1, then the initial value problem (3.12)-(3.13) has a solution x(t)=(xi(t)) on the interval I such that x(t)∈c for each t∈I.Proof.
Lett∈I and x=(xi)∈c and
(3.27)fi(t,x)=ai(t)xi+gi(t;x),f(t,x)=(f1(t,x),f2(t;x),…)=(fi(t,x)).
Then, for arbitrarily fixed natural numbersn,m we get
(3.28)|fn(t,x)-fm(t,x)|=|an(t)xn+gn(t;x)-am(t)xm-gm(t;x)|≤|an(t)xn+gn(t;x)|-|am(t)xm+gm(t;x)|≤|an(t)xn-an(t)xm|+|an(t)xm-am(t)xm|+bn+bm≤|an(t)||xn-xm|+|an(t)-am(t)||xm|+bn+bm≤|an(t)||xn-xm|+∥x∥∞|an(t)-am(t)|bn+bm.
By assumptions (iii) and (iv), from the above estimate we deduce that (fi(t,x)) is a real Cauchy sequence. This implies that (fi(t,x))∈c.
Also we obtain the following estimate:(3.29)|fi(t,x)|≤|ai(t)||xi|+|gi(t,x)|≤Q|xi|+bi≤Q∥x∥∞+B,
where B=supi≥1bi. Hence,
(3.30)∥f(t,x)∥≤Q∥x∥∞+B.
In what follows, let us consider the mappingf(t,x) on the set I×B(x0,r), where r is taken according to the assumptions of Theorem A, that is, r=(BT+QT∥x0∥∞)/(1-QT). Further, fix arbitrarily t,s∈I and x,y∈B(x0,r). Then, by our assumptions, for a fixed i, we obtain
(3.31)|fi(t,x)-fi(s,y)|=|ai(t)xi+gi(t,x)-ai(s)yi-gi(s,y)|≤|ai(t)xi-ai(s)yi|+|gi(t,x)-gi(s,y)|≤|ai(t)-ai(s)||xi|+|ai(s)||xi-yi|+|gi(t,x)-gi(s,y)|.
Then,(3.32)∥f(t,x)-f(s,y)∥=supi≥1|fi(t,x)-fi(s,y)|≤(r+∥x0∥∞)·supi≥1|ai(t)-ai(s)|+Q∥x-y∥∞+∥g(t,x)-g(s,y)∥.
Hence, taking into account that the sequence (ai(t)) is equicontinuous on the interval I and g is uniformly continuous on I×c, we conclude that the operator f(t,x) is uniformly continuous on the set I×B(x0,r).
In the sequel, let us take a nonempty subsetE of the ball B(x0,r) and fix t∈I,x∈X. Then, for arbitrarily fixed natural numbers n,m we have
(3.33)|fn(t,x)-fm(t,x)|≤|an(t)||xn-xm|+|xm||an(t)-am(t)|+|gn(t,x)|+|gm(t,x)|≤a(t)|xn-xm|+(r+∥x0∥∞)|an(t)-am(t)|+bn+bm.
Hence, we infer the following inequality:(3.34)μ(f(t,E))≤a(t)μ(E).
Finally, combining (3.30), (3.34) and the fact (proved above) that f is uniformly continuous on I×B(x0,r), in view of Theorem A, we infer that problem (3.22)-(3.23) is solvable in the space c.
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 Spaceℓp

In this section, we study the solvability of the infinite systems of differential equations (3.6)-(3.7) in the Banach sequence space ℓp(1≤p<∞) such that x(t)∈ℓp for each t∈I.An existence theorem for problem (3.6)-(3.7) in the space ℓp can be formulated by making the following assumptions:(i)
x0=(xi0)∈ℓp,(ii)
fi:I×ℝ∞→ℝ(i=0,1,2,…) maps continuously the set I×ℓp into ℓp,(iii)
there exist nonnegative functionsqi(t) and ri(t) defined on I such that
(3.35)|fi(t,x)|p=|fi(t,x0,x1,x2,…)|p≤qi(t)+ri(t)|xi|p,fort∈I;x=(xi)∈ℓp and i=0,1,2,…,(iv)
the functionsqi(t) are continuous on I and the function series ∑i=0∞‍qi(t) converges uniformly on I,(v)
the sequence(ri(t)) is equibounded on the interval I and the function r(t)=limsupi→∞ri(t) is integrable on I.Now, we prove the following result.Theorem 3.8 (see [26]).
Under the assumptions (i)–(v), problem (3.6)-(3.7) has a solution x(t)=(xi(t)) defined on the interval I=[0,T] whenever RT<1, where R is defined as the number
(3.36)R=sup{ri(t):t∈I,i=0,1,2,…}.
Moreover, x(t)∈ℓp for any t∈I.Proof.
For anyx(t)∈ℓp and t∈I, under the above assumptions, we have
(3.37)∥f(t,x)∥pp=∑i=0∞‍|fi(t,x)|p≤∑i=0∞‍[qi(t)+ri(t)|xi|p]≤∑i=0∞‍qi(t)+(supi≥0ri(t))(∑i=0∞‍|xi|p)≤Q+R∥x∥pp,
where Q=supt∈I(∑i=0∞‍qi(t)).
Now, choose the numbers=(QT+RT∥x0∥pp)/(1-RT) as defined in Theorem A. Consider the operator f=(fi) on the set I×B(x0;s). Let us take a set Y∈ℳℓp. Then by using (2.8), we get
(3.38)χ(f(t,Y))=limn→∞supx∈Y(∑i≥n‍|fi(t,x1,x2,…)|p)≤limn→∞(∑i≥n‍qi(t)+(supi≥nri(t))(∑i≥n‍|xi|p)).
Hence, by assumptions (iv)-(v), we get
(3.39)χ(f(t,Y))≤r(t)χ(Y),
that is, the operator f satisfies condition (3.4) of Theorem A. Hence, by Theorem A and Remark 3.1 we conclude that there exists a solution x=x(t) of problem (3.6)-(3.7) such that x(t)∈ℓp for any t∈I.
This completes the proof of the theorem.Remark 3.9.
(I) Forp=1, 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(3.40)xi′=ai(t)xi+gi(t,x0,x1,x2,…),with the initial condition(3.41)xi(0)=xi0,(i=0,1,2,…) where t∈I.
An existence theorem for problem (3.6)-(3.7) in the space ℓp can be formulated by making the following assumptions:(i)
x0=(xi0)∈ℓp,(ii)
the sequence(|ai(t)|) is defined and equibounded on the interval I=[0,T]. Moreover, the function a(t)=limsupi→∞sup|ai(t)| is integrable on I,(iii)
the mappingg=(gi) maps continuously the set I×ℓp into ℓp,(iv)
there exist nonnegative functionsbi(t) such that
(3.42)|fi(t,x0,x1,x2,…)|p≤bi(t),fort∈I;x=(xi)∈ℓp and i=0,1,2,…,(v)
the functionsbi(t) are continuous on I and the function series ∑i=0∞‍bi(t) converges uniformly on I.


### 3.4. Infinite Systems of Differential Equations in the Spacen(ϕ)

An existence theorem for problem (3.6)-(3.7) in the space n(ϕ) can be formulated by making the following assumptions:(i)
x0=(xi0)∈n(ϕ),(ii)
fi:I×ℝ∞→ℝ(i=1,2,…) maps continuously the set I×n(ϕ) into n(ϕ),(iii)
there exist nonnegative functionspi(t) and qi(t) defined on I such that
(3.43)|fi(t,u)|=|fi(t,u1,u2,…)|≤pi(t)+qi(t)|ui|,fort∈I;x=(xi)∈n(ϕ) and i=1,2,…, where u=(ui) is a sequence of rearrangement of x=(xi),(iv)
the functionspi(t) are continuous on I and the function series ∑i=1∞‍pi(t)Δϕi converges uniformly on I,(v)
the sequence(qi(t)) is equibounded on the interval I and the function q(t)=limsupi→∞qi(t) is integrable on I.Now, we prove the following result.Theorem 3.10 (see [32]).
Under the assumptions (i)–(v), problem (3.6)-(3.7) has a solution x(t)=(xi(t)) defined on the interval I=[0,T] whenever QT<1, where Q is defined as the number
(3.44)Q=sup{qi(t):t∈I,i=1,2,…}.
Moreover,x(t)∈n(ϕ) for any t∈I.Proof.
For anyx(t)∈n(ϕ) and t∈I, under the above assumptions, we have
(3.45)∥f(t,x)∥n(ϕ)=supu∈S(x)∑i=1∞‍|fi(t,u)|Δϕi≤supu∈S(x)∑i=1∞‍[pi(t)+qi(t)|ui|]Δϕi≤∑i=1∞‍pi(t)Δϕi+(supiqi(t))(supu∈S(x)∑i=1∞‍|ui|Δϕi)≤P+Q∥x∥n(ϕ),
where P=supt∈I∑i=1∞‍pi(t)Δϕi.
Now, choose the numberr defined according to Theorem A, that is, r=(PT+QT∥x0∥n(ϕ))/(1-QT). Consider the operator f=(fi) on the set I×B(x0;r). Let us take a set X∈ℳn(ϕ). Then by using Theorem 2.4, we get
(3.46)χ(f(t,X))=limk→∞supx∈X(supu∈S(x)(∑n=k∞‍|fn(t,u1,u2,…)|Δϕn))≤limk→∞(∑n=k∞‍pn(t)Δϕn+(supn≥kqn(t))(supu∈S(x)∑n=k∞‍|un|Δϕn)).
Hence, by assumptions (iv)-(v), we get
(3.47)χ(f(t,X))≤q(t)χ(X),
that is, the operator f satisfies condition (3.4) of Theorem A. Hence, the problem (3.6)-(3.7) has a solution x(t)=(xi(t)).
This completes the proof of the theorem.Remark 3.11.
Similarly, we can establish such type of result for the spacem(ϕ).


## 3.1. Infinite Systems of Differential Equations in the Spacec0

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 spacec0. It is known that in the space c0 the Hausdorff measure of noncompactness can be expressed by the following formula [33]:
(3.5)χ(E)=limn→∞(supx∈E{supk≥n|xk|}),
where E∈ℳc0.We will be interested in the existence of solutionsx(t)=(xi(t)) of the infinite systems of differential equations
(3.6)xi′=fi(t,x0,x1,x2,…),
with the initial condition
(3.7)xi(0)=xi0,(i=0,1,2,…) which are defined on the interval I=[0,T] and such that x(t)∈c0 for each t∈I.An existence theorem for problem (3.6)-(3.7) in the space c0 can be formulated by making the following assumptions.Assume that the functionsfi(i=1,2,…) are defined on the set I×ℝ∞ and take real values. Moreover, we assume the following hypotheses:(i)
x0=(xi0)∈c0,(ii)
the mapf=(f1,f2,…)acts from the set I×c0 into c0 and is continuous,(iii)
there exists an increasing sequence(kn) of natural numbers (obviously kn→∞ as n→∞) such that for any t∈I,x=(xi)∈c0 and n=1,2,… the following inequality holds:
(3.8)|fn(t,x1,x2,…)|≤pn(t)+qn(t)(supi≥kn|xi|),where(pi(t)) and (qi(t)) are real functions defined and continuous on I such that the sequence (pi(t)) converges uniformly on I to the function vanishing identically and the sequence (qi(t)) is equibounded on I.Now, let us denote(3.9)q(t)=supn≥1qn(t),Q=supt∈Iq(t),P=sup{pn(t):t∈I,n=1,2,…}.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 x=x(t)=(xi(t)) defined on the interval I1=[0,T1] whenever T1<T and QT1<1. Moreover, x(t)∈c0 for any t∈I1.Proof.
Letx=(xi) be any arbitrary sequence in c0. Then, by (i)–(iii), for any t∈I and for a fixed n∈ℕ we obtain
(3.10)|fn(t,x)|=|fn(t,x1,x2,…)|≤pn(t)+qn(t)(supi≥kn|xi|)≤P+Qsupi≥kn|xi|≤P+Q∥x∥∞.
Hence, we get(3.11)∥f(t,x)∥≤P+Q∥x∥∞.
In what follows, let us take the ballB(x0,r), where r is chosen according to Theorem A. Then, for a subset Y of B(x0,r) and for t∈I1, we obtain
(3.12)χ(f(t,Y))=limn→∞supx∈Y(supi≥n|fi(t,x)|),χ(f(t,Y))=limn→∞supx∈Y(supi≥n|fi(t,x1,x2,…)|)≤limn→∞supx∈Y(supi≥n{pi(t)+qi(t)supj≥kn|xj|})≤limn→∞supi≥npi(t)+q(t)limn→∞{supx∈Y(supi≥n{supj≥kn|xj|})}.
Hence, by assumptions, we get
(3.13)χ(f(t,Y))≤q(t)χ(Y).
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 x=x(t) of the Cauchy problem (3.6)-(3.7) such that x(t)∈c0 for any t∈I1.
This completes the proof of the theorem.We illustrate the above result by the following examples.Example 3.3 (see [33]).
Let{kn} be an increasing sequence of natural numbers. Consider the infinite system of differential equations of the form
(3.14)xi′=fi(t,x1,x2,…)+∑j=ki+1∞‍aij(t)xj,
with the initial condition
(3.15)xi(0)=xi0,(i=1,2,…;t∈I=[0,T]).
We will investigate problem (3.14)-(3.15) under the following assumptions:(i)
x0=(x0)∈c0,(ii)
the functionsfi:I×ℝki→ℝ(i=1,2,…) are uniformly equicontinuous and there exists a function sequence (pi(t)) such that pi(t) is continuous on I for any i∈ℕ and (pi(t)) converges uniformly on I to the function vanishing identically. Moreover, the following inequality holds:
(3.16)|fi(t,x1,x2,…xki)|≤pi(t),fort∈I,(x1,x2,…xki)∈ℝki and i∈ℕ,(iii)
the functionsaij(t) are defined and continuous on I and the function series ∑j=ki+1∞‍aij(t) converges absolutely and uniformly on I (to a function ai(t)) for any i=1,2,…,(iv)
the sequence(ai(t)) is equibounded on I,(v)
QT<1, where Q=sup{ai(t):i=1,2,…;t∈I}.
It can be easily seen that the assumptions of Theorem3.2 are satisfied under assumptions (i)–(v). This implies that problem (3.14)-(3.15) has a solution x(t)=(xi(t)) on the interval I belonging to the space c0 for any fixed t∈I.
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 ℓ∞ 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 spacec0 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 xn(t)→0 when n→∞, for any fixed t∈[0,T].
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 ki=i for i=1,2,… and aij≡0 on I for all i,j. Then system (3.14) has the form
(3.17)x1′=f1(t,x1),x2′=f2(t,x1,x2),…,xi′=fi(t,x1,x2,…,xi),…,
and is called a row-finite system [35].
Suppose that there are satisfied assumptions from Example3.3, that is, x0=(xi0)∈c0 and the functions fi act from I×ℝi into ℝ(i=1,2,…) and are uniformly equicontinuous on their domains. Moreover, there exist continuous functions pi(t)(t∈I) such that
(3.18)|fi(t,x1,x2,…xki)|≤pi(t),
for t∈I and x1,x2,…,xi∈ℝ(i=1,2,…). We assume also that the sequence pi(t) converges uniformly on I to the function vanishing identically.
Further, let|·|i denote the maximum norm in ℝi(i=1,2,…). Take fi=(f1,f2,…,fi). Then we have
(3.19)|fi(t,x)|i=max{|f1(t,x1)|,|f2(t,x1,x2)|,…,|fi(t,x1,x2,…,xi)|}≤max{p1(t),p2(t),…,pi(t)}.
Taking Pi(t)=max{p1(t),p2(t),…,pi(t)}, then the above estimate can be written in the form
(3.20)|fi(t,x)|i≤Pi(t).Observe that from our assumptions it follows that the initial value problemu′=Pi(t),u(0)=xi0 has a unique solution on the interval I. Hence, applying a result from [33], we infer that Cauchy problem (3.17)-(3.15) has a solution on the interval I. Obviously from the result contained in Theorem 3.2 and Example 3.3, we deduce additionally that the mentioned solution belongs to the space c0.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:
(3.21)|fi(t,x1,x2,…,xi)|≤pi(t)+qi(t)max{|x1|,|x2|,…,|xi|},
where the functions pi(t) and qi(t)(i=1,2,…) 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 Spacec

Now, we will study the solvability of the following perturbed diagonal system of differential equations(3.22)xi′=ai(t)xi+gi(t,x1,x2,…),
with the initial condition
(3.23)xi(0)=xi0,(i=1,2,…), where t∈I=[0,T].From now, we are going exactly the same as in Section 4 of [33].We consider the following measuresμ of noncompactness in c which is more convenient, regular, and even equivalent to the Hausdorff measure of noncompactness [33].ForE∈ℳc(3.24)μ(E)=limp→∞{supx∈E{supn,m≥p|xn-xm|}},χ(E)=limn→∞(supx∈Q{supk≥n|xk|}).Let us formulate the hypotheses under which the solvability of problem (3.22)-(3.23) will be investigated in the space c. Assume that the following conditions are satisfied.Assume that the functionsfi(i=1,2,…) are defined on the set I×ℝ∞ and take real values. Moreover, we assume the following hypotheses:(i)
x0=(xi0)∈c,(ii)
the mapg=(g1,g2,…)acts from the set I×c into c and is uniformly continuous on I×c,(iii)
there exists sequence(bi)∈c0 such that for any t∈I,x=(xi)∈c and n=1,2,… the following inequality holds:
(3.25)|gn(t,x1,x2,…)|≤bi,(iv)
the functionsai(t) are continuous on I such that the sequence (ai(t)) converges uniformly on I.Further, let us denote(3.26)a(t)=supi≥1ai(t),Q=supt∈Ia(t).Observe that in view of our assumptions, it follows that the functiona(t) is continuous on I. Hence, Q<∞.Then we have the following result which is more general than Theorem3.2.Theorem 3.6 (see [33]).
Let assumptions (i)–(iv) be satisfied. IfQT<1, then the initial value problem (3.12)-(3.13) has a solution x(t)=(xi(t)) on the interval I such that x(t)∈c for each t∈I.Proof.
Lett∈I and x=(xi)∈c and
(3.27)fi(t,x)=ai(t)xi+gi(t;x),f(t,x)=(f1(t,x),f2(t;x),…)=(fi(t,x)).
Then, for arbitrarily fixed natural numbersn,m we get
(3.28)|fn(t,x)-fm(t,x)|=|an(t)xn+gn(t;x)-am(t)xm-gm(t;x)|≤|an(t)xn+gn(t;x)|-|am(t)xm+gm(t;x)|≤|an(t)xn-an(t)xm|+|an(t)xm-am(t)xm|+bn+bm≤|an(t)||xn-xm|+|an(t)-am(t)||xm|+bn+bm≤|an(t)||xn-xm|+∥x∥∞|an(t)-am(t)|bn+bm.
By assumptions (iii) and (iv), from the above estimate we deduce that (fi(t,x)) is a real Cauchy sequence. This implies that (fi(t,x))∈c.
Also we obtain the following estimate:(3.29)|fi(t,x)|≤|ai(t)||xi|+|gi(t,x)|≤Q|xi|+bi≤Q∥x∥∞+B,
where B=supi≥1bi. Hence,
(3.30)∥f(t,x)∥≤Q∥x∥∞+B.
In what follows, let us consider the mappingf(t,x) on the set I×B(x0,r), where r is taken according to the assumptions of Theorem A, that is, r=(BT+QT∥x0∥∞)/(1-QT). Further, fix arbitrarily t,s∈I and x,y∈B(x0,r). Then, by our assumptions, for a fixed i, we obtain
(3.31)|fi(t,x)-fi(s,y)|=|ai(t)xi+gi(t,x)-ai(s)yi-gi(s,y)|≤|ai(t)xi-ai(s)yi|+|gi(t,x)-gi(s,y)|≤|ai(t)-ai(s)||xi|+|ai(s)||xi-yi|+|gi(t,x)-gi(s,y)|.
Then,(3.32)∥f(t,x)-f(s,y)∥=supi≥1|fi(t,x)-fi(s,y)|≤(r+∥x0∥∞)·supi≥1|ai(t)-ai(s)|+Q∥x-y∥∞+∥g(t,x)-g(s,y)∥.
Hence, taking into account that the sequence (ai(t)) is equicontinuous on the interval I and g is uniformly continuous on I×c, we conclude that the operator f(t,x) is uniformly continuous on the set I×B(x0,r).
In the sequel, let us take a nonempty subsetE of the ball B(x0,r) and fix t∈I,x∈X. Then, for arbitrarily fixed natural numbers n,m we have
(3.33)|fn(t,x)-fm(t,x)|≤|an(t)||xn-xm|+|xm||an(t)-am(t)|+|gn(t,x)|+|gm(t,x)|≤a(t)|xn-xm|+(r+∥x0∥∞)|an(t)-am(t)|+bn+bm.
Hence, we infer the following inequality:(3.34)μ(f(t,E))≤a(t)μ(E).
Finally, combining (3.30), (3.34) and the fact (proved above) that f is uniformly continuous on I×B(x0,r), in view of Theorem A, we infer that problem (3.22)-(3.23) is solvable in the space c.
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 Spaceℓp

In this section, we study the solvability of the infinite systems of differential equations (3.6)-(3.7) in the Banach sequence space ℓp(1≤p<∞) such that x(t)∈ℓp for each t∈I.An existence theorem for problem (3.6)-(3.7) in the space ℓp can be formulated by making the following assumptions:(i)
x0=(xi0)∈ℓp,(ii)
fi:I×ℝ∞→ℝ(i=0,1,2,…) maps continuously the set I×ℓp into ℓp,(iii)
there exist nonnegative functionsqi(t) and ri(t) defined on I such that
(3.35)|fi(t,x)|p=|fi(t,x0,x1,x2,…)|p≤qi(t)+ri(t)|xi|p,fort∈I;x=(xi)∈ℓp and i=0,1,2,…,(iv)
the functionsqi(t) are continuous on I and the function series ∑i=0∞‍qi(t) converges uniformly on I,(v)
the sequence(ri(t)) is equibounded on the interval I and the function r(t)=limsupi→∞ri(t) is integrable on I.Now, we prove the following result.Theorem 3.8 (see [26]).
Under the assumptions (i)–(v), problem (3.6)-(3.7) has a solution x(t)=(xi(t)) defined on the interval I=[0,T] whenever RT<1, where R is defined as the number
(3.36)R=sup{ri(t):t∈I,i=0,1,2,…}.
Moreover, x(t)∈ℓp for any t∈I.Proof.
For anyx(t)∈ℓp and t∈I, under the above assumptions, we have
(3.37)∥f(t,x)∥pp=∑i=0∞‍|fi(t,x)|p≤∑i=0∞‍[qi(t)+ri(t)|xi|p]≤∑i=0∞‍qi(t)+(supi≥0ri(t))(∑i=0∞‍|xi|p)≤Q+R∥x∥pp,
where Q=supt∈I(∑i=0∞‍qi(t)).
Now, choose the numbers=(QT+RT∥x0∥pp)/(1-RT) as defined in Theorem A. Consider the operator f=(fi) on the set I×B(x0;s). Let us take a set Y∈ℳℓp. Then by using (2.8), we get
(3.38)χ(f(t,Y))=limn→∞supx∈Y(∑i≥n‍|fi(t,x1,x2,…)|p)≤limn→∞(∑i≥n‍qi(t)+(supi≥nri(t))(∑i≥n‍|xi|p)).
Hence, by assumptions (iv)-(v), we get
(3.39)χ(f(t,Y))≤r(t)χ(Y),
that is, the operator f satisfies condition (3.4) of Theorem A. Hence, by Theorem A and Remark 3.1 we conclude that there exists a solution x=x(t) of problem (3.6)-(3.7) such that x(t)∈ℓp for any t∈I.
This completes the proof of the theorem.Remark 3.9.
(I) Forp=1, 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(3.40)xi′=ai(t)xi+gi(t,x0,x1,x2,…),with the initial condition(3.41)xi(0)=xi0,(i=0,1,2,…) where t∈I.
An existence theorem for problem (3.6)-(3.7) in the space ℓp can be formulated by making the following assumptions:(i)
x0=(xi0)∈ℓp,(ii)
the sequence(|ai(t)|) is defined and equibounded on the interval I=[0,T]. Moreover, the function a(t)=limsupi→∞sup|ai(t)| is integrable on I,(iii)
the mappingg=(gi) maps continuously the set I×ℓp into ℓp,(iv)
there exist nonnegative functionsbi(t) such that
(3.42)|fi(t,x0,x1,x2,…)|p≤bi(t),fort∈I;x=(xi)∈ℓp and i=0,1,2,…,(v)
the functionsbi(t) are continuous on I and the function series ∑i=0∞‍bi(t) converges uniformly on I.


## 3.4. Infinite Systems of Differential Equations in the Spacen(ϕ)

An existence theorem for problem (3.6)-(3.7) in the space n(ϕ) can be formulated by making the following assumptions:(i)
x0=(xi0)∈n(ϕ),(ii)
fi:I×ℝ∞→ℝ(i=1,2,…) maps continuously the set I×n(ϕ) into n(ϕ),(iii)
there exist nonnegative functionspi(t) and qi(t) defined on I such that
(3.43)|fi(t,u)|=|fi(t,u1,u2,…)|≤pi(t)+qi(t)|ui|,fort∈I;x=(xi)∈n(ϕ) and i=1,2,…, where u=(ui) is a sequence of rearrangement of x=(xi),(iv)
the functionspi(t) are continuous on I and the function series ∑i=1∞‍pi(t)Δϕi converges uniformly on I,(v)
the sequence(qi(t)) is equibounded on the interval I and the function q(t)=limsupi→∞qi(t) is integrable on I.Now, we prove the following result.Theorem 3.10 (see [32]).
Under the assumptions (i)–(v), problem (3.6)-(3.7) has a solution x(t)=(xi(t)) defined on the interval I=[0,T] whenever QT<1, where Q is defined as the number
(3.44)Q=sup{qi(t):t∈I,i=1,2,…}.
Moreover,x(t)∈n(ϕ) for any t∈I.Proof.
For anyx(t)∈n(ϕ) and t∈I, under the above assumptions, we have
(3.45)∥f(t,x)∥n(ϕ)=supu∈S(x)∑i=1∞‍|fi(t,u)|Δϕi≤supu∈S(x)∑i=1∞‍[pi(t)+qi(t)|ui|]Δϕi≤∑i=1∞‍pi(t)Δϕi+(supiqi(t))(supu∈S(x)∑i=1∞‍|ui|Δϕi)≤P+Q∥x∥n(ϕ),
where P=supt∈I∑i=1∞‍pi(t)Δϕi.
Now, choose the numberr defined according to Theorem A, that is, r=(PT+QT∥x0∥n(ϕ))/(1-QT). Consider the operator f=(fi) on the set I×B(x0;r). Let us take a set X∈ℳn(ϕ). Then by using Theorem 2.4, we get
(3.46)χ(f(t,X))=limk→∞supx∈X(supu∈S(x)(∑n=k∞‍|fn(t,u1,u2,…)|Δϕn))≤limk→∞(∑n=k∞‍pn(t)Δϕn+(supn≥kqn(t))(supu∈S(x)∑n=k∞‍|un|Δϕn)).
Hence, by assumptions (iv)-(v), we get
(3.47)χ(f(t,X))≤q(t)χ(X),
that is, the operator f satisfies condition (3.4) of Theorem A. Hence, the problem (3.6)-(3.7) has a solution x(t)=(xi(t)).
This completes the proof of the theorem.Remark 3.11.
Similarly, we can establish such type of result for the spacem(ϕ).



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*Source: 863483-2012-11-19.xml*