On solutions of semilinear upper diagonal infinite systems of differential equations

The goal of the paper is to investigate the existence of solutions for semilinear upper diagonal infinite systems of differential equations. We will look for solutions of the mentioned infinite systems in a Banach tempered sequence space. In our considerations we utilize the technique associated with the Hausdorff measure of noncompactness and some existence results from the theory of ordinary differential equations in abstract Banach spaces.


1.
Introduction. The principal goal of the paper is to study the solvability of some kind of infinite systems of differential equations. More precisely, we will investigate semilinear upper diagonal infinite systems of differential equations which are perturbed by nonlinear terms. Our investigations are located in the Banach space consisting or real sequences converging to zero if we tempere them appropriately. Such an approach guarantees that we can use the technique of measures of noncompactness being very convenient and fruitful tool in the study of the existence of solutions of various types of operator equations (differential, integral, integrodifferential etc.) (cf. [1,2,3,5,8], for example).
It is worthwhile mentioning that infinite systems of differential equations can be considered as special cases of ordinary differential equations treated in abstract Banach spaces [5,7]. Indeed, an infinite system of ordinary differential equations can be always considered as a special case of an ordinary differential equation in some Banach sequence space.
In our study conducted in the paper we will investigate infinite systems of differential equations in the Banach space c β 0 consisting of real sequences converging to zero if we tempere them by a suitable tempering sequence β = (β n ). Such an approach is very convenient in applications. Indeed, choosing an appropriate tempering sequence β we can always "reduce" any Banach sequence space to the mentioned Banach sequence space c β 0 containing sequences converging to zero after suitable temperation. On the other hand, in the mentioned Banach tempered sequence space c β 0 we can use the so-called Hausdorff measure of noncompactness being the most convenient measure in applications as far as we know (cf. [1,3,6,9]).
It is worthwhile noticing that numerous real world problems encountered in mathematical physics, mechanics, engineering, the theory of branching processes, the theory of neural nets etc. lead to infinite systems of ordinary differential equations in some Banach sequence space (cf. [4,7]). On the other hand some numerical methods of solving of partial differential equations also lead to investigations connected with ordinary differential equations in abstract Banach spaces [7].
The investigations of the present paper create the continuation and extension of the study conducted in paper [4], where we considered semilinear lower diagonal infinite systems of differential equations. Similarly as in the present paper, the considerations of the mentioned paper [4] were also located in Banach tempered sequence spaces. However, despite of the fact that methods applied in our research of this paper are similar to those from [4], the problem of the solvability of semilinear upper diagonal infinite systems of differential equations considered in this paper forced to develop some new methods and tools than those used in [4]. By these regards the results obtained in this paper create a generalization of those from [4].
2. Auxiliary facts. In this section we provide a few auxiliary facts which will be used in our further considerations.
At the beginning we establish some notation. Namely, by R we will denote the set of real numbers and we put R + = [0, ∞). The symbol N stands for the set of natural numbers (positive integers).
Next, if E is a Banach space with the norm || · || and with the zero element θ, we denote by B(x, r) the closed ball in E centered at x and with radius r. We will write B r to denote the ball B(θ, r). If X is a subset of E then by X, ConvX we denote the closure and convex closure of X, respectively. Moreover, the symbols X + Y , λX (λ ∈ R) stand for algebraic operations on sets X and Y .
Further, let M E denote the family of all nonempty and bounded subsets of the space E and N E its subfamily consisting of relatively compact sets.
For an arbitrary set X ∈ M E we define number χ(X) by putting The quantity χ(X) defined in such a way is called the Hausdorff measure of noncompactness of the set X [3]. It can be shown that this quantity has the following useful properties [3,5]: is a sequence of closed sets from M E such that X n+1 ⊂ X n for n = 1, 2, ... and lim n→∞ χ(X n ) = 0 then the set X ∞ = ∞ n=1 X n is nonempty.
For other properties of the Hausdorff measure of noncompactness χ we refer to [1,3]. Now, we recall some facts concerning the Banach sequence spaces. First of all, let us remind that by the symbol c 0 we denote the classical Banach sequence space consisting of real sequences x = (x n ) coverging to zero and normed via the classical norm ||x|| = ||(x n )|| = sup{|x n | : n = 1, 2, ...} = max{|x n | : n = 1, 2, ...}.
Obviously, in nonlinear analysis we consider also other Banach sequence spaces denoted by c, l p , l ∞ . Since we will work in the space c 0 only we will not remind details concerning those spaces.
Further, let us recall a very convenient formula expressing the Hausdorff measure of noncompactness in the space c 0 . Namely, for X ∈ M c0 we have [3] In what follows let us observe that the sequence space c 0 is not very handy in applications since it is very easy to encounter a situation that investigated problems cannot be located in the space c 0 (cf. [4], for instance). It is caused by the fact that this space is rather small.
To overcome the indicated difficulty we can consider the so-called Banach tempered sequence spaces [4].
To this end let us fix a real sequence β = (β n ) such that β n is positive for n = 1, 2, ... and the sequence (β n ) is nonincreasing. In practice we also assume that β n → 0 as n → ∞, but such an assumption is not needed in general. Next, consider the set c β 0 consisting of all sequences x = (x n ) such that β n x n → 0 as n → ∞. It is easily seen that c β 0 forms a Banach space under the norm ||x|| = ||(x n )|| = sup{β n |x n | : n = 1, 2, ...} = max{β n |x n | : n = 1, 2, ...}.
The essential property of the space c β 0 is connected with the fact that it is isometric to the classical space c 0 . Indeed, it is easy to check that the mapping J : c β 0 → c 0 defined in the following way is an isometry of the spaces c β 0 and c 0 (cf. [4]). This observation enables us to define the Hausdorff measure of noncompactness in the space c β 0 [4]. Indeed, for an arbitrary set X ∈ M c β 0 , in view of 1 we have 3. A few results from the theory of differential equations in Banach spaces. This section is devoted to collect a few results from the theory of ordinary differential equations in Banach spaces [3,7]. We restrict ourselves to present those results which will be needed in our further investigations. Thus, let us assume that (E, || · ||) is a given real Banach space. Let x 0 be a fixed element of E and, as earlier, the symbol B(x 0 , r) denotes the ball in E.
We will consider the differential equation with the initial condition Assume also that χ is the Hausdorff measure of noncompactness in the space E. Now, we recall a result concerning initial value problem 3-4 being not very general but sufficient and convenient for our purposes [3].
Moreover, for an arbitrary nonempty set X ⊂ B(x 0 , r) and for almost all t ∈ I the following inequality holds where p(t) is an integrable function on the interval I. Then problem 3-4 has at least one solution x = x(t) on the interval I.
The below given result is a special case of the above theorem and will be used in our further study [4].
for each t ∈ I = [0, T ] and x ∈ E, where P and Q are positive constants. Further, assume that f is uniformly continuous on the set [0, Moreover, we assume that f satisfies condition 5. Then, initial value problem 3-4 has a solution x = x(t) on the interval Remark 1. It is worthwhile mentioning that in the case when the Banach space E is weakly compactly generated the assumption on the uniform continuity of the function f can be replaced by the weaker one requiring the continuity [10]. Particularly, we can show that any separable Banach space is weakly compactly generated [10]. Since the sequence space c β 0 is separable we can apply Theorems 3.1 and 3.2 with the assumption that the function f is continuous on the set 4. Main results. In this section we will consider the semilinear upper diagonal infinite system of differential equations which has the form with the initial value conditions x n (0) = x n 0 , for n = 1, 2, ... .
Thus, in contrast to paper [4], the infinite system of differential equations 7 represents the semilinear upper diagonal system since all terms of the linear part of equations appearing in 7 are located upper the main diagonal of the matrix (a nni ). It is worthwhile mentioning that in [4] we considered also infinite system 7, but we assumed that 1 ≤ n 1 < n 2 < · · · < n kn ≤ n. Obviously in such a case an infinite system 7 represents the lower diagonal system.
In what follows we will also assume that there exists a natural number K such that k n ≤ K for n = 1, 2, .... Such an assumption means that the linear part of each equation appearing in system 7 contains only finite number of nonzero terms and the number of those term does not exceed K. In the sequel of the paper infinite systems 7 satisfying the above constraint will be called infinite systems of differential equations with linear part of constant width (cf. also [4]).
Hence, replacing n by j and j by i, we can rewrite the above inequality in the form Now, let us observe that from estimate 9 we derive the following inequality where the operator g = g(t, x) is defined on the set I × c β 0 in the following way g(t, x) = (g 1 (t, x), g 2 (t, x), ...).

Obviously, P < ∞ in view of assumption (v).
In what follows we show that the operator g acts continuously from the set I × c β 0 into the space c β 0 . To this end we represent the operator g in the form where the operators L and f are defined as follows: At first we show that f is continuous on the set I × c β 0 . To realize this goal fix arbitrarily a number ε > 0 and x ∈ c β 0 , t ∈ I. Then, in view of assumption (v) we can find a natural number n 0 such that β n p n < ε 2 (11) for n ≥ n 0 . Further, in virtue of assumption (iv) we can find a number δ i (i = 1, 2, ..., n 0 ) such that for any y ∈ c β 0 such that ||x − y|| ≤ δ i and for s ∈ I such that |t − s| ≤ δ i we have Next, take δ = min{δ 1 , δ 2 , ..., δ n }. Then, for arbitrary y ∈ c β 0 such that ||x − y|| ≤ δ and for s ∈ I such that |t − s| ≤ δ, we get Linking 11 and 12, for y ∈ c β 0 with ||x − y|| ≤ δ and for s ∈ I with |t − s| ≤ δ, we obtain This shows that the operator f is continuous on the set I × c β 0 . Now, we show that the operator L is continuous on the set I × c β 0 . Similarly as previously, fix arbitrary x ∈ c β 0 , t ∈ I and a number ε > 0. Then, for y ∈ c β 0 with ||x − y|| ≤ ε, for s ∈ I with |t − s| ≤ ε and for an arbitrarily fixed natural number n, in virtue of our assumptions we get where the symbol ω = ω(ε) denotes the common modulus of continuity of the functions a nni (t) on the interval I. Such a modulus exists in view of assumption (i). Further, keeping in mind assumption (vi), we obtain The above obtained estimate allows us to infer that the operator L is continuous on the set I × c β 0 . Joining this fact with the continuity of the operator f established before, we deduce that the operator g is continuous on the set I × c β 0 . Next, let us take a positive number T 1 such that T 1 ≤ T and AKM T 1 < 1. Denote I 1 = [0, T 1 ]. Keeping in mind the above established facts and Theorem 3.2 let us take the number Now, consider the ball B(x 0 , r) and choose an arbitrary nonempty subset X of B(x 0 , r). Then, for a fixed element x ∈ X and for an arbitrary number t ∈ I 1 , in view of estimates 9 and 10, for arbitrarily fixed natural number n, we obtain: Consequently, we arrive at the following estimate: Now, passing with n → ∞ and bearing in mind that j 1 → ∞ as j → ∞, in view of formula 2 expressing the Hausdorff measure of noncompactness in the space c β 0 , we derive the following inequality χ(g(t, X)) ≤ AKM χ(X).
Finally, gathering all the above stated facts, in view of Theorem 3.2 we complete the proof.
Observe that 13 is a semilinear upper diagonal infinite system of differential equations with linear parts of constant width K = 3. Apart from this it is easily seen that system 13 is a particular case of system 7 if we take a nn (t) = 1 for n ∈ N and for t ∈ I 1 = [0, T 1 ], where T 1 ≤ T is a number chosen according to assumptions of Theorem 3.2. Moreover, a nni (t) = t ni+t for i = 2 and n 2 = n + 1, while for i = 3 we have that n 3 = 2n. Obviously |a nni (t)| ≤ 1 for t ∈ I 1 and n = 1, 2, ..., i = 1, 2, 3. This implies that the functions a nni (t) satisfy assumption (ii) of Theorem 4.1.
On the other hand it is easy to check that the functions a nni (t) satisfy the Lipschitz condition with the constant 1 for n = 1, 2, ... and i = 1, 2, 3. Hence we see that there is satisfied assumption (i) of our theorem.
The above established facts allows us to deduce that functions f n satisfy assumptions (iv) and (v) with p n = 1 2 n for n = 1, 2, .... On the other hand we have that β n β n kn = β n β 2n = 4.
Thus we see that assumption (vi) is satisfied with M = 4. Finally, in view of Theorem 4.1 we deduce that there exists at least one solution x(t) = (x n (t)) of initial value problem 13-14 defined on some interval I 1 = [0, T 1 ] such that for any t ∈ I 1 the sequence (x n (t)) belongs to the space c β 0 with β = 1 n 2 . Obviously, we can easily calculate that T 1 < min{T, 1/12}.
In what follows we provide an analogon of the result formulated in paper [4] as Theorem 6.3, for infinite semilinear upper diagonal system of differential equations.
Namely, we will consider problem 7-8 for infinite upper diagonal system of differential equations and we dispense with the assumption requiring that system 7 has linear parts with constant width.
In the sequel we are going to show that the operator g is continuous on the set I 1 × c β 0 . To this end let us observe that the proof of the continuity of the operator f runs exactly in the same way as in a suitable part of the proof of Theorem 4.1. Therefore, it is sufficient to prove the continuity of the operator L on the set I 1 ×c β 0 . In order to conduct this proof let us fix arbitrarily x ∈ c β 0 , t ∈ I 1 and a number ε > 0. Next, take an arbitrary element y ∈ c β 0 with ||x − y|| ≤ ε and a number s ∈ I 1 with |t − s| ≤ ε. Without loss of generality we may assume that s < t. Then, for an arbitrary natural number n we obtain: