Infinite first order differential systems with nonlocal initial conditions

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable"sub-linear"growth then the system has at least one solution. The approach relies on the application, in a suitable Fr\'echet space, of the classical Schauder-Tychonoff fixed point theorem. We show that, as a special case, our approach covers the case of a system of a finite number of differential equations. An illustrative example of application is also provided.


Introduction
In the recent paper [6] Bolojan-Nica and co-authors developed a technique that can be used to study the solvability of a system of N first order differential equations of the form x ′ 1 (t) = g 1 (t, x 1 (t) , x 2 (t), ..., x N (t)) , ...
x ′ N (t) = g N (t, x 1 (t) , x 2 (t) , ..., x N (t)) , t ∈ [0, 1], subject to the coupled nonlocal conditions where η ij : C[0, 1] → R are continuous linear functionals. These functionals involve a suitable support and can be written in a form involving Stieltjes integrals, namely, (1.3) η ij , v = wheret ∈ [0, 1] is given. The formulation (1.3) covers, as special cases, initial conditions of the type where η ijk ∈ R and 0 ≤ t 1 < t 2 < ... < t m ≤t. Note that, if all the functionals η ij have discrete expressions, (1.2) gives a multi-point condition. The approach in [6] relies on the fixed point theorems of Perov, Schauder and Schaefer and on a vector method for treating systems which uses matrices with spectral radius less than one.
Note that on non-compact intervals the problem of finding solutions of differential systems becomes more complicated, since the associated integral formulation may display a lack of compactness. For systems of a finite number of differential equations on non-compact intervals, subject to linear (or more general) conditions, we refer to the papers by Andres et al. [2], Cecchi and co-authors [7,8], De Pascale et al. [11], Marino and Pietramala [18] and Marino and Volpe [19].
On the other hand, infinite systems of differential equations often occur in applications, for example in stochastic processes and quantum mechanics [1] (also, see ch. XXIII in [13]) and in physical chemistry of macromolecules [20]. For some recent work on infinite systems on compact intervals we refer the reader to the papers by Frigon [12], by Banaś and co-authors [3,4,5] and references therein.
The methodology to treat initial value problems for infinite systems, often relies on using the theory of differential equations in Banach spaces, but it is also known that this approach can reduce the set of solutions and some of their specific properties. A fruitful alternative is to deal with such systems in the framework of locally convex spaces. For a discussion emphasizing these aspects we refer the reader to the paper by Jebelean and Reghiş [14] and references therein.
The case of initial value problems for infinite systems on non-compact intervals has been investigated in [14], where the authors extended a comparison theorem due to Stokes [24], valid for the scalar case. Here we utilize the framework developed in [14], in order to deal with infinite systems of nonlocal initial value problems on the half line. To be more precise, let us denote by J the half line [0, +∞) and by S the space of all real valued sequences, that which is assumed to be endowed with the product topology. We fix t 0 ∈ (0, +∞) and consider the space C[0, t 0 ] of all continuous real valued functions defined on [0, t 0 ] with the usual supremum norm. For each n ∈ N, let f n : J × S → R be a continuous function and α n : C[0, t 0 ] → R be a continuous linear functional.
Here we deal with the solvability of the nonlocal initial value problem where, in a similar way as above, α n , v denotes the value of the functional α n at v ∈ C[0, t 0 ].
By a solution of problem (1.5) we mean a sequence of functions x n : J → R, n ∈ N, such that each x n is derivable on J and satisfies The rest of the paper is organized as follows. In Section 2 we transform the system (1.5) into a fixed point problem within a suitable Fréchet space. In Section 3 we present the main existence result, we show how this approach can be utilized in the special case of a finite number of differential equations and, finally, we provide an example that illustrates our theory. Our results are new in the context of nonlocal problems for infinite systems and complement earlier results in the literature.

An equivalent fixed point problem
We firstly introduce some notations that are used in the sequel. For x, y ∈ S, the product and notice that the family of seminorms ([ · ] n ) n∈N on the space S generates a product topology. Endowed with this topology, S is a Fréchet space, that is a locally convex space which is complete with respect to a translation invariant metric.
For the sake of simplicity, we rewrite (1.5) in a more compact form. In order to do this, we define the map f : J × S → S by .., f n (t, y), ...) , ∀ (t, y) ∈ J × S and the operator α : Notice that f is continuous and α is linear and continuous. Therefore the problem (1.5) can be re-written as the derivation ′ being understood component-wise. Integrating (2.2) we get with some c ∈ S. The nonlocal initial condition yields . Thus, assuming that we obtain We consider the space C(J) of all continuous real valued functions defined on J with the topology of the uniform convergence on compacta. Given (t k ) k∈N a strictly increasing sequence in (0, +∞) with t k → +∞ as k → ∞, this topology is generated by the (countable) family of seminorms (ii) T is continuous and maps bounded sets into relatively compact sets.
(ii) It is a standard matter to check that, for all k ∈ N, the operator R k : is continuous. We now show that R : C(J, S) → C(J, S) given by We also have that R maps bounded sets into relatively compact sets. To see this, let M ⊂ C(J, S) be bounded. Using the Arzelà-Ascoli theorem, it is routine to show that for any fixed n ∈ N, the set is relatively compact in C[0, t k ] for each k ∈ N. Hence, the set {R(v) n : v ∈ M} is relatively compact in C(J). By virtue of the Tychonoff theorem, we get that R(M) is relatively compact in the product space C(J, S).

Now, the conclusion follows from the equality
and the above properties of the operator R.
We shall make use of the following consequence of the classical Schauder-Tychonoff fixed point theorem, see for example Corollary 2, page 107 of [10].

Main result
We present now the main result of the paper. In order to do this, we fix a strictly increasing sequence (n p ) p∈N ⊂ N and (t p ) p∈N a sequence of real numbers, such that and, for α as in (2.1), we denote holds for every p ∈ N, then the problem (2.2) (hence (1.5)) has at least one solution x ∈ C(J, S).
Proof. Taking into account Proposition 2.1, it suffices to show that T has a fixed point in C(J, S). In order to do this, we shall apply Theorem 2.1. Firstly, note that for all u ∈ C(J, S) and n ∈ N.
For each p ∈ N, let and θ p > 0 be such that note that such a θ p exists from (3.4).
We introduce the (continuous) semi-norms We denote by we take ρ p such that and define Since each Ω p is closed and convex, the (nonempty) set Ω p has the same properties. Next, we prove that T (Ω) ⊂ Ω.
Let x ∈ Ω and p ∈ N. If t ∈ [0, t 0 ], using (3.5), we have The growth condition (3.3) gives which, taking the maximum over [0, t 0 ], yields If t ∈ [t 0 , t p ], then arguing as above we get Multiplying the inequality (3.8) by e −θp(t−t 0 ) (≤ 1), we obtain and, taking the maximum over [t 0 , t p ], we obtain From (3.7), (3.9) and (3.6), we obtain showing that x ∈ Ω p , for all p ∈ N; hence, x ∈ Ω. Finally, note that Ω is a bounded subset of C(J, S). Then, from Proposition 2.1, the continuous operator T maps Ω into a relatively compact set. Therefore, by virtue of Theorem 2.1, T has a fixed point in Ω and the proof is complete.
Remark 3.1. Note that the result above is also valid in the case of a non-compact interval The only modification is to consider We now illustrate how this methodology can be applied in the case of a finite system of differential equations. The assumptions are relatively easy to be checked and we include this result for completeness.
With the notation above we can state the following existence result.
Theorem 3.2. Let (t p ) p∈N be a sequence satisfying (3.1). Assume the condition (3.11) holds and that g := (g 1 , ..., g N ) satisfies with A ∈ L 1 + (0, t 0 ) and B, C p ∈ R + . If the inequality holds, then the nonlocal initial value problem (3.10) has at least one solution.
Proof. Consider the problem (2. 2) with To prove the solvability of (2.2) with the above choices of f and α, we apply Theorem 3.1 with n p = N + p − 1, ∀p ∈ N. In this view, it follows from (3.12) that the growth condition (3.3) is fulfilled with A p = A and B p = B. Also, we have for all p ∈ N. Hence, the condition (3.4) is satisfied from (3.13) and the proof is complete.
We conclude the paper with an example, where notice that the constants that occur in Theorem 3.1 can be effectively computed.