Infinite System of Differential Equations in Some BK Spaces

and Applied Analysis 3 that is, Cs is the set of those σ whose support has cardinality at most s, and define Φ { φ ( φk ) ∈ w : 0 < φ1 ≤ φn ≤ φn 1, n 1 φn ≥ nφn 1 } . 1.3 For φ ∈ Φ, the following sequence spaces were introduced by Sargent 4 and further studied in 5–8 : m ( φ ) { x xk ∈ w : ‖x‖m φ sup s≥1 sup σ∈Cs ( 1 φs ∑


FK and BK Spaces
In this section, we give some basic definitions and notations about FK and BK spaces for which we refer to 1-3 .
We will write w for the set of all complex sequences x x k ∞ k 0 .Let ϕ, ∞ , c and c 0 denote the sets of all finite, bounded, convergent, and null sequences, respectively, and cs be the set of all convergent series.We write p : {x ∈ w : ∞ k 0 |x k | p < ∞} for 1 ≤ p < ∞.By e and e n n ∈ N , we denote the sequences such that e k 1 for k 0, 1, . .., and e n n n 0 in a linear metric space X is called Schauder basis if for every x ∈ X, there is a unique sequence λ n ∞ n 0 of scalars such that x ∞ n 0 λ n b n .A sequence space X with a linear topology is called a Kspace if each of the maps p i : X → C defined by p i x x i 2 Abstract and Applied Analysis is continuous for all i ∈ N. A K space is called an FK space if X is complete linear metric space; a BK space is a normed FK space.An FK space X ⊃ ϕ is said to have AK if every sequence x x k ∞ k 0 ∈ X has a unique representation x ∞ k 0 x k e k , that is, x lim n → ∞ x n .A linear space X 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 sup x∈S X L x Y for all L ∈ B X, Y , where S X denotes the unit sphere in X, that is, S X : {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 The theory of FK 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 FK spaces are continuous.
A sequence space X is called an FK space if it is a locally convex Fréchet space with continuous coordinates p n : X → C n ∈ N , where C denotes the complex field and p n x x n for all x x k ∈ X and every n ∈ N. A normed FK space is called a BK space, that is, a BK space is a Banach sequence space with continuous coordinates.
The famous example of an FK space which is not a BK space is the space w, d w , where On the other hand, the classical sequence spaces are BK spaces with their natural norms.More precisely, the spaces ∞ , c, and c 0 are BK spaces with the sup-norm given by x ∞ sup k |x k |.Also, the space p 1 ≤ p < ∞ is a BK space with the usual p -norm defined by x p k |x k | p 1/p .Further, the spaces bs, cs, and cs 0 are BK spaces with the same norm given by x bs sup n n k 0 |x k |, and bv is a BK space with 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 space ∞ has no Schauder basis, the spaces w, c 0 , c, and p all have Schauder bases.Moreover, the spaces w, c 0 , and p have AK, where 1 ≤ p < ∞.
There are following BK spaces which are closely related to the spaces p 1 ≤ p ≤ ∞ .Let C 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 {c n σ } such that c n σ 1 for n ∈ σ, and c n σ 0 otherwise.Further Abstract and Applied Analysis 3 that is, C s is the set of those σ whose support has cardinality at most s, and define For φ ∈ Φ, the following sequence spaces were introduced by Sargent 4 and further studied in 5-8 : where S x denotes the set of all sequences that are rearrangements of x.
Remark 1.1.i The spaces m φ and n φ are BK spaces with their respective norms.
ii If

Measures of Noncompactness
The first measure of noncompactness was defined and studied by Kuratowski 9  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 S and M be subsets of a metric space X, d and let > 0. Then S is called an -net of M in X if for every x ∈ M there exists s ∈ S such that d x, s < .Further, if the set S is finite, then the -net S of M is called a finite -net of M, and we say that M has a finite -net in X.A subset M of a metric space X is said to be totally bounded if it has a finite -net for every > 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 write M X for the collection of all bounded subsets of a metric space X, d .If Q ∈ M X , 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 > 0 such that Q can be covered by a finite number of balls of radii < and centers in X.This can equivalently be redefined as follows: The function χ : M X → 0, ∞ is called the Hausdorff measure of noncompactness.
If Q, Q 1 , and Q 2 are bounded subsets of a metric space X, then we have

2.2
Further, if X is a normed space, then the function χ has some additional properties connected with the linear structure, for example, Let X and Y be Banach spaces and χ 1 and χ 2 be the Hausdorff measures of noncompactness on X and Y , respectively.An operator L : 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: let X 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 and we have that L is compact if and only if Furthermore, the function χ 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 b k ∞ k 0 is a Schauder basis for a Banach space X, then every element x ∈ X has a unique representation x ∞ k 0 φ k x b k , where φ k k ∈ N are called the basis functionals.Moreover, the operator P r : X → X, defined for each r ∈ N by P r x r k 0 φ k x b k x ∈ X , is called the projector onto the linear span of {b 0 , b 1 , . . ., b r }.Besides, all operators P r and I − P r are equibounded, where I denotes the identity operator on X. Theorem 2.1.Let X be a Banach space with a Schauder basis b k ∞ k 0 , E ∈ M X and P n : X → X n ∈ N be the projector onto the linear span of {b 0 , b 1 , . . ., b n }.Then, one has where a lim sup n → ∞ I − P n .
In particular, the following result shows how to compute the Hausdorff measure of noncompactness in the spaces c 0 and p 1 ≤ p < ∞ which are BK spaces with AK.Theorem 2.2.Let E be a bounded subset of the normed space X, where It is easy to see that for Also, it is known that e, e 0 , e 1 , . . . is a Schauder basis for the space c and every sequence z z n , where z lim n → ∞ z n .Thus, we define the projector P r : c → c r ∈ N , onto the linear span of {e, e 0 , e 1 , . . ., e r }, by for all z z n ∈ c with z lim n → ∞ z n .In this situation, we have the following.
Theorem 2.3.Let Q ∈ M c and P r : c → c r ∈ N be the projector onto the linear span of {e, e 0 , e 1 , . . ., e r }.Then, one has where I is the identity operator on c.
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 X 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 x n in X, the sequence L x n 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 ∈ M X .
Further, we write C 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.
Let X 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 L χ χ L S X , 2.12 and we have L is compact if and only if L χ 0.

2.13
Since matrix mappings between BK 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 .

Applications to Infinite Systems of Differential Equations
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 x 0 x 0 .

3.2
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 X is a real Banach space with the norm • .Let us take an interval I 0, T , T > 0 and B x 0 , r the closed ball in X centered at x 0 with radius r.
Theorem A see 34 .Assume that f t, x is a function defined on I × X with values in X such that for any x ∈ X, where Q and R are nonnegative constants.Further, let f be uniformly continuous on the set Moreover, assume that for any nonempty set Y ⊂ B x 0 , s and for almost all t ∈ I the following inequality holds: with a sublinear measure of noncompactness μ such that {x 0 } ∈ ker μ.Then problem 3.1 -3.2 has a solution x such that {x t } ∈ ker μ for t ∈ I 1 , where q t is an integrable function on I, and ker μ {E ∈ M X : μ E 0} is the kernel of the measure μ.
Remark 3.1.In the case when μ 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 Sections 3.1 and 3.2 are from 33 , Section 3.3 from 26 , and Section 3.4 from 32 .

Infinite Systems of Differential Equations in the Space c 0
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 c 0 .It is known that in the space c 0 the Hausdorff measure of noncompactness can be expressed by the following formula 33 : where E ∈ M c 0 .
We will be interested in the existence of solutions x t x i t of the infinite systems of differential equations with the initial condition i 0, 1, 2, . . .which are defined on the interval I 0, T and such that x t ∈ c 0 for each t ∈ I.
An existence theorem for problem 3.6 -3.7 in the space c 0 can be formulated by making the following assumptions.
Assume that the functions f i i 1, 2, . . .are defined on the set I × R ∞ and take real values.Moreover, we assume the following hypotheses: . .acts from the set I × c 0 into c 0 and is continuous, iii there exists an increasing sequence k n of natural numbers obviously k n → ∞ as n → ∞ such that for any t ∈ I, x x i ∈ c 0 and n 1, 2, . . . the following inequality holds: where p i t and q i t are real functions defined and continuous on I such that the sequence p i t converges uniformly on I to the function vanishing identically and the sequence q i t is equibounded on I. Now, let us denote q t sup n≥1 q n t , Q sup t∈I q t , P sup p n t : t ∈ I, n 1, 2, . . . .

3.9
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 x i t defined on the interval I 1 0, T 1 whenever T 1 < T and QT 1 < 1.Moreover, x t ∈ c 0 for any t ∈ I 1 .

Proof. Let x
x i be any arbitrary sequence in c 0 .Then, by i -iii , for any t ∈ I and for a fixed n ∈ N we obtain

3.10
Hence, we get

3.11
In what follows, let us take the ball B x 0 , r , where r is chosen according to Theorem A. Then, for a subset Y of B x 0 , r and for t ∈ I 1 , we obtain x j .

3.12
Hence, by assumptions, we get χ f t, Y ≤ q t χ Y .

3.13
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 ∈ c 0 for any t ∈ I 1 .This completes the proof of the theorem.
We illustrate the above result by the following examples.
Example 3.3 see 33 .Let {k n } be an increasing sequence of natural numbers.Consider the infinite system of differential equations of the form a ij t x j , 3.14 with the initial condition i 1, 2, . . .; t ∈ I 0, T .We will investigate problem 3.14 -3.15 under the following assumptions: ii the functions f i : . .are uniformly equicontinuous and there exists a function sequence p i t such that p i t is continuous on I for any i ∈ N and p i t converges uniformly on I to the function vanishing identically.Moreover, the following inequality holds: iii the functions a ij t are defined and continuous on I and the function series ∞ j k i 1 a ij t converges absolutely and uniformly on I to a function a i t for any i 1, 2, . .., iv the sequence a i t is equibounded on I, 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 x t x i t on the interval I belonging to the space c 0 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 ∞ 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 c 0 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 x n 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 k i i for i 1, 2, . . .and a ij ≡ 0 on I for all i, j.Then system 3.14 has the form x i f i t, x 1 , x 2 , . . ., x i , . . .,

3.17
and is called a row-finite system 35 .
Suppose that there are satisfied assumptions from Example 3.3, that is, x 0 x 0 i ∈ c 0 and the functions f i act from I × R i into R i 1, 2, . . .and are uniformly equicontinuous on their domains.Moreover, there exist continuous functions p i t t ∈ I such that for t ∈ I and x 1 , x 2 , . . ., x i ∈ R i 1, 2, . . . .We assume also that the sequence p i t converges uniformly on I to the function vanishing identically.

3.19
Taking P i t max{p 1 t ,p 2 t , . . ., p i t }, then the above estimate can be written in the form

3.20
Observe that from our assumptions it follows that the initial value problem u P i t , u 0 x 0 i 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 c 0 .
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 p i t and q i 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 .

Infinite Systems of Differential Equations in the Space c
Now, we will study the solvability of the following perturbed diagonal system of differential equations x i a i t x i g i t, x 1 , x 2 , . . ., 3.22 with the initial condition . ., where t ∈ I 0, T .From now, we are going exactly the same as in Section 4 of 33 .We consider the following measures μ of noncompactness in c which is more convenient, regular, and even equivalent to the Hausdorff measure of noncompactness 33 . For

3.24
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 functions f i i 1, 2, . . .are defined on the set I × R ∞ and take real values.Moreover, we assume the following hypotheses: . .acts from the set I × c into c and is uniformly continuous on I × c, iii there exists sequence b i ∈ c 0 such that for any t ∈ I, x x i ∈ c and n 1, 2, . . . the following inequality holds: iv the functions a i t are continuous on I such that the sequence a i t converges uniformly on I.
Further, let us denote a t sup i≥1 a i t , Q sup t∈I a t .

3.26
Observe that in view of our assumptions, it follows that the function a t is continuous on I. Hence, Q < ∞.
Then we have the following result which is more general than Theorem 3.2.

3.28
By assumptions iii and iv , from the above estimate we deduce that f i t, x is a real Cauchy sequence.This implies that f i t, x ∈ c.Also we obtain the following estimate: where B sup i≥1 b i .Hence,

3.30
In what follows, let us consider the mapping f t, x on the set I × B x 0 , r , where r is taken according to the assumptions of Theorem A, that is, r BT QT x 0 ∞ / 1 − QT .
Further, fix arbitrarily t, s ∈ I and x, y ∈ B x 0 , r .Then, by our assumptions, for a fixed i, we obtain f i t, x − f i s, y a i t x i g i t, x − a i s y i − g i s, y ≤ a i t x i − a i s y i g i t, x − g i s, y x − g i s, y .

3.32
Hence, taking into account that the sequence a i 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 x 0 , r .In the sequel, let us take a nonempty subset E of the ball B x 0 , r and fix t ∈ I, x ∈ X.Then, for arbitrarily fixed natural numbers n, m we have

3.33
Hence, we infer the following inequality: Finally, combining 3.30 , 3.34 and the fact proved above that f is uniformly continuous on I × B x 0 , 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.

Infinite Systems of Differential Equations in the Space p
In this section, we study the solvability of the infinite systems of differential equations 3.6 -3.7 in the Banach sequence space p 1 ≤ p < ∞ such that x t ∈ p for each t ∈ I.
An existence theorem for problem 3.6 -3.7 in the space p can be formulated by making the following assumptions: iii there exist nonnegative functions q i t and r i t defined on I such that

Infinite Systems of Differential Equations in the Space n φ
An existence theorem for problem 3.6 -3.7 in the space n φ can be formulated by making the following assumptions: k e k be its n-section.Note that ∞ , c and c 0 are Banach spaces with the norm x ∞ sup k≥0 |x k |, and p 1 ≤ p < ∞ are Banach spaces with the norm x p ∞ k 0 |x k | p 1/p .A sequence b n ∞ 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.
Theorem 3.6 see 33 .Let assumptions (i)-(iv) be satisfied.If QT < 1, then the initial value problem 3.12 -3.13 has a solution x t x i t on the interval I such that x t ∈ c for each t ∈ I.
r i t |x i | p , 3.35 for t ∈ I; x x i ∈ p and i 0, 1, 2, . .., iv the functions q i t are continuous on I and the function series ∞ i 0 q i t converges uniformly on I, v the sequence r i t is equibounded on the interval I and the function r t lim sup i → ∞ r i t is integrable on I.
Theorem 3.8 see 26 .Under the assumptions (i)-(v), problem 3.6 -3.7 has a solution x t x i t defined on the interval I 0, T whenever RT < 1, where R is defined as the number R sup{r i t : t ∈ I, i 0, 1, 2, . ..}. 3.36Moreover, x t ∈ p for any t ∈ I. Proof.For any x t ∈ p and t ∈ I, under the above assumptions, we have Remark 3.9.I For p 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.IIIWe observe that the above theorem can be applied to the perturbed diagonal infinite system of differential equations of the formx i a i t x i g i t, x 0 , x 1 , x 2 , ...,An existence theorem for problem 3.6 -3.7 in the space p can be formulated by making the following assumptions:i x 0 x 0 i ∈ p , iithe sequence |a i t | is defined and equibounded on the interval I 0, T .Moreover, the function a t lim sup i → ∞ sup |a i t | is integrable on I, iii the mapping g g i maps continuously the set I × p into p , iv there exist nonnegative functions b i t such that f i t, x 0 , x 1 , x 2 , . . .p ≤ b i t , 3.42 for t ∈ I; x x i ∈ p and i 0, 1, 2, . .., v the functions b i t are continuous on I and the function series ∞ i 0 b i t converges uniformly on I.
2, ...maps continuously the set I × n φ into n φ , iii there exist nonnegative functions p i t and q i t defined on I such thatf i t, u f i t,u 1 , u 2 , . ..≤ p i t q i t |u i |, 3.43 for t ∈ I; x x i ∈ n φ and i 1, 2, ..., where u u i is a sequence of rearrangement of x x i , iv the functions p i t are continuous on I and the function series ∞ i 1 p i t Δφ i converges uniformly on I,v the sequence q i t is equibounded on the interval I and the function q t lim sup i → ∞ q i t is integrable on I. Now, we prove the following result.Δφ i .Now, choose the number r defined according to Theorem A, that is, r PT QT x 0 n φ / 1 − QT .Consider the operator f f i on the set I × B x 0 ; r .Let us take a set X ∈ M n φ .Then by using Theorem 2.4, we get Theorem 3.10 see 32 .Under the assumptions (i)-(v), problem 3.6 -3.7 has a solution x t x i t defined on the interval I 0, T whenever QT < 1, where Q is defined as the number Q sup q i t : t ∈ I, i 1, 2, ....3.44Moreover,x t ∈ n φ for any t ∈ I.Proof.For any x t ∈ n φ and t ∈ I, under the above assumptions, we have