The superposition operator in the space of functions continuous and converging at in nity on the real half-axis

Wewill consider the so-called superposition operator in the space CC(R+) of real functions de ned, continuous on the real half-axis R+ and converging to nite limits at in nity. We will assume that the function f = f (t, x) generating the mentioned superposition operator is locally uniformly continuous with respect to the variable x uniformly for t ∈ R+. Moreover, we require that the function t → f (t, x) satis es the Cauchy condition at in nity uniformly with respect to the variable x. Under the above indicated assumptions a few properties of the superposition operator in question are derived. Examples illustrating our considerations will be included.


Introduction
The superposition operator plays an important role in numerous mathematical investigations connected with operator theory, theory of integral equations and with considerations conducted in nonlinear functional analysis (cf. [1][2][3][4]). Especially, a lot of properties related to the superposition operator are utilized in the theory of Hammerstein integral equations [4]. Our aim in this paper is to establish some properties of the superposition operator acting in the space CC(R+) consisting of real functions de ned, continuous on the real half-axis R+ and converging to nite limits at innity. Firstly, we are going to establish conditions guaranteeing that the superposition operator transforms the space CC(R+) into itself and is continuous. In order to nd those conditions we will impose appropriate assumptions on the function f (t, x) generating the superposition operator in question. It is worthwhile mentioning that in several research works devoted to the superposition operator one can nd a lot of papers stating properties in the mentioned spirit which are dedicated to investigations of properties of the superposition operator in several function spaces [2]. However, the research in the case of the space CC(R+) seems to be not thoroughly conducted up to now. Secondly, we will look for conditions (imposed on the generating function f (t, x) which ensure that the superposition operator generated by f (t, x) transforms some kind of relatively compact sets in the space CC(R+) into itself. Such investigations are, in general, not easy what is mainly caused by the fact that we do not know necessary and su cient conditions for relative compactness in the space CC(R+) which are related to the structure of this space (such, for example, as Arzelà-Ascoli criterion in the classical space C([a, b]) or Kolmogorov or Riesz criteria in the space L p (a, b)).
Results which will be obtained in the realization of the above formulated goals will be applied in the formulation of additional properties of the superposition operator connected with the property of transforming continuously some class of subsets of the space CC(R+) being relatively compact into that class.
The results of the paper create an extension of those ones obtained up to now in the theory of superposition operators (cf. [1,2,5,6], for example).

Notation, de nitions and auxiliary facts
This section is devoted to establish some notation and to present de nitions of basic concepts utilized in the paper. Moreover, we are going to recall a few results connected with investigations conducted in the paper.
At the beginning let us present some notation. Denote by R the set of real numbers and put R+ = [ , ∞). The symbol N will stand for the set of natural numbers (positive integers). In this paper we will mainly deal with the so-called superposition operator. To formulate the de nition of that operator let us take a function f : R+ × R → R. Next, consider the set X consisting of functions x : R+ → R. Then the operator F : X → X de ned on the set X in the following way is called the superposition operator generated by the function f = f (t, x).
Let us pay attention to the fact that we can de ne the superposition operator in a more general setting [1,2], but for our further purposes the above given de nition will be entirely su cient.
Considerations of this paper will be located in the space CC(R+) consisting of all functions x(t) = x : R+ → R which are continuous on R+ and converging at in nity to nite limits. The space CC(R+) forms a Banach space with the standard supremum norm Observe that the space CC(R+) can be de ned equivalently as the space consisting of real functions x = x(t) de ned, continuous and bounded on R+ and converging at in nity. We can also consider the space BC(R+) of all functions which are de ned, continuous and bounded on R+. Obviously, BC(R+) forms a Banach space with the norm · ∞ de ned above. It is also worthwhile mentioning that CC(R+) forms a closed subspace of the space BC(R+). We omit the standard proof of this assertion. Keeping in mind our further considerations we now recall a convenient necessary and su cient condition for the function x : R+ → R to be convergent at in nity to a nite limit. It is easy to recognize this condition as the classical Cauchy condition.
Finally, we recall a su cient condition for a bounded subset of the space CC(R+) to be relatively compact in CC(R+). That condition can be easily deduced from a suitable condition in the space BC(R+) given in [7]. In order to formulate a su cient condition mentioned above we quote de nitions of two concepts which will be needed in the presentation of the announced condition.

De nition 2.2.
Let X be a subset of the space BC(R+). We say that functions of the set X are locally equicontinuous on the interval R+ if the following condition is satis ed

De nition 2.3.
Let X be a subset of the space BC(R+). We will say that functions of the set X satisfy uniformly the Cauchy condition at in nity if Observe that the above given de nitions remain the same if we formulate them in the space CC(R+). Moreover, it is worthwhile mentioning that De nition 2.3 is closely related to Lemma 2.1. Now, we formulate a theorem providing the above announced su cient condition for a bounded subset of the space CC(R+) to be relatively compact in CC(R+).

Theorem 2.4. Let X be a bounded set in the space CC(R+). If functions of the set X are locally equicontinuous on the interval R+ and satisfy uniformly the Cauchy condition at in nity then the set X is relatively compact in the space CC(R+).
In what follows we will call the condition contained in the above theorem as the condition (A) for relative compactness.

Results concerning the superposition operator
In this section we are going to present the main results of the paper which are concerning some properties of the superposition operator in the Banach function space CC(R+) described in the previous section. Throughout this section we will assume that f : R+ × R → R is a given function. The superposition operator generated by the function f (t, x) will be denoted by F.
In what follows we will assume that the function f = f (t, x) satis es the below presented assumptions. (i) The function f is continuous on the set R+ × R.
(ii) For any r > the function x → f (t, x) is uniformly continuous on the interval [−r, r] uniformly with respect to the variable t ∈ R+ i.e., the following condition is satis ed ε .
(iii) The function t → f (t, x) satis es the Cauchy condition at in nity uniformly with respect to the variable x ∈ R what means that the following condition is satis ed Now, let us scrutinize assumptions imposed on the function f (t, x) which are formulated above. First of all let us observe that from assumption (iii) and Lemma 2.1 we infer that for each xed x ∈ R there exists a nite limit lim t→∞ f (t, x). Let us denote this limit by fx i.e., let us put Then we have the following theorem.
Proof. Fix arbitrarily numbers ε > and r > . Then, in view of the de nition of the limit of a function at in nity and (3.1), we can choose to the number ε a number T > such that for t T . Similarly, we can nd a number T > such that for t T .
Next, let us take T = max {T , T }. Then, in virtue of (3.2) and (3.3), for t T we infer that the following inequalities are satis ed Further, using assumption (ii), we can choose δ > to the numbers ε and r (r > ) such that for x, y ∈ [−r, r], |x − y| δ and for an arbitrary number t ∈ R+ the following inequality holds Now, keeping in mind (3.4), (3.5) and taking arbitrary numbers x, y ∈ [−r, r] such that |x − y| δ, for an arbitrary number t T we get Thus the proof of our theorem is complete.
In the sequel of this section we will consider an assumption being stronger than assumption (ii). Namely, let us formulate the announced assumption. (ii') The function x → f (t, x) is uniformly continuous on R uniformly with respect to the variable t ∈ R+ i.e., the following condition is satis ed Then, remaining assumptions (i), (iii) and replacing assumption (ii) by the above formulated assumption (ii'), we obtain the following counterpart of Theorem 3.1. Proof. The proof runs in the same way as the proof of Theorem 3.1 to the moment when we obtain inequalities (3.4). Next, taking into account assumption (ii'), for a number ε we can choose δ > such that if x, y ∈ R and |x − y| δ, then for an arbitrary number t ∈ R+ the following inequality is satis ed Further, based on (3.4) and (3.6), for arbitrary numbers x, y ∈ R such that |x − y| δ and for t T we obtain Since the choice of the number T has not an e ect on the validity of the inequality |fx − fy| ε which is true for all x, y ∈ R such that |x − y| δ. The proof is complete.

Example 3.3. Let us consider the function f (t, x)
where the functions a : R+ → R, b : R → R satisfy the following assumptions: (α) a ∈ CC(R+).
(β) The function b is continuous and bounded on R.
We show that the function f satis es assumptions (i), (ii) and (iii). Obviously, the function f satis es assumption (i). In order to show that the function f satis es assumption (ii), let us x arbitrarily a number r > . Next, choose arbitrary numbers x, y ∈ [−r, r] and t ∈ R+. Then we obtain Since a ∈ CC(R+) then the function a is bounded on R+ i.e., there exists a constant A > such that |a(t)| A for t ∈ R+.
Next, x arbitrarily ε > and choose a number δ > in such a way that there is satis ed assumption (β). Further, taking into account that the function b is uniformly continuous on the interval [−r, r], in view of (3.7) we obtain This shows that assumption (ii) is met.
To show that f satis es assumption (iii) observe that based on assumption (β) we can indicate a constant B > such that |b(x)| B for x ∈ R. Further, x arbitrarily ε > . Then, in virtue of assumption (α) we can nd T > such that for t, s T we have |a(t) − a(s)| ε B . Then, for arbitrary numbers x ∈ R and t, s T we obtain This proves that assumption (iii) is met.
Now, let us observe that a slight modi cation of assumption (β) causes that the function f (t, x) satis es assumptions (i), (ii') and (iii). It is shown by the next example. In what follows we are going to investigate further properties of the superposition operator F generated by the function f = f (t, x) satisfying assumptions (i)-(iii). We have the following theorem.

Theorem 3.5. Let the function f satisfy assumptions (i)-(iii). Then, the superposition operator F generated by the function f maps the space CC(R+) into itself.
Proof. Fix arbitrarily a function x ∈ CC(R+). In view of assumption (i) we deduce that the function (Fx)(t) = f (t, x(t)) is continuous on R+ as the composition of continuous functions. Thus, in order to prove our theorem it is su cient to show that the function f (t, x(t)) is convergent to a nite limit at in nity. Applying Lemma 2.1 we see that to realize this goal it is su cient to show that To this end let us x arbitrarily ε > . Since the function t → f (t, x) satis es condition (iii) we infer that there exists T > such that for arbitrary numbers t , t T and x ∈ R we have Obviously the function x is bounded on R+ which implies that there exists a number M > such that In view of assumption (ii) we conclude that for any r > the function x → f (t, x) is uniformly continuous on the interval [−r, r] uniformly with respect to the variable t ∈ R+. Hence it follows that taking r = M, on the basis of (ii) we have that there exists a number δ > such that for arbitrary numbers x, y ∈ [−M, M] and t ∈ R+ we obtain Since the function x = x(t) is convergent to a proper limit at in nity we infer that for the number δ from condition (3.9) there exists a number T > such that for arbitrary t , t T we have Now, taking T = max {T , T } and applying (3.8) and (3.9), for arbitrary numbers t , t T we obtain Thus the proof is complete.
Our next result is devoted to the continuity of the superposition operator F. Proof. Fix a function x ∈ CC(R+) and take a sequence (xn) ⊂ CC(R+) converging to x in the space CC(R+). This implies that for an arbitrary t ∈ R+ the real sequence xn(t) is convergent to the limit x (t). Thus, for an arbitrarily xed number ε > there exists n ∈ N such that xn(t) − x (t) ε for n n , n ∈ N. Taking, for example, ε = we can nd n ∈ N such that for n n . Since x ∈ CC(R+) we infer that there exists a nite limit a = lim t→∞ x (t). Hence it follows that we can nd a number T > such that for t T we have Utilizing the above inequality and (3.10), for n n and t T we obtain This implies that xn(t) ∈ [a − , a + ] for n n and t T.
Further, let us notice that in view of the fact that the function x is an element of the space CC(R+) we can nd a constant M > such that x (t) M for t ∈ R+. Hence, in virtue of (3.10), we get for n n and for t ∈ R+.
Next, let us observe that since the functions x , x , . . . , x n − are bounded, thus we can nd a number P > which creates the common bound of the functions x i for i = , , . . . , n − . More precisely, let us put p for i = , , . . . , n − .
In what follows x an arbitrary number ε > and choose δ > according to assumption (ii) with the above de ned number r > . Since the sequence (xn) is convergent to x in the space CC(R+), there exists a number n ∈ N such that for n n , n ∈ N, we have Obviously, this yields that for t ∈ R+ and n n . Thus, applying assumption (ii) we obtain for n n and for t ∈ R+. On the other hand we have Finally, joining (3.11) and (3.12) we conclude that Fxn − Fx ∞ ε for n n and the proof is complete.
Further on, let us notice that both Theorem 3.5 and Theorem 3.6 remain true if we replace assumption (ii) by the stronger assumption (ii'). Obviously, assumptions (i) and (iii) remain the same.
In the sequel of this section we prove that the image of a subset of the space CC(R+), which is locally equicontinuous on the interval R+ (cf. De nition 2.2), by the superposition operator F is also locally equicontinuous on R+.

Theorem 3.7.
Assume that the function f (t, x) = f : R+ × R → R satis es assumptions (i)-(iii). Let X be a subset of the space CC(R+) which is bounded and locally equicontinuous on R+. Then the image F(X) of the set X by the superposition operator F generated by the function f (t, x) is locally equicontinuous on R+.
Proof. Since the set X is bounded in the space CC(R+), there exists a constant M > such that |x(t)| M for each x ∈ X and for any t ∈ R+. Keeping in mind the fact that X is locally equicontinuous on R+, on the basis of De nition 2.2 we infer that the following condition holds In what follows let us x numbers ε > and T > . Next, choose a number δ > according to condition (3.13). Then, for arbitrary t, s ∈ [ , T] such that |t − s| δ and for arbitrary x ∈ X we get (Fx)(t) − (Fx)(s) = f (t, x(t)) − f (s, x(s)) f (t, x(t)) − f (t, x(s)) + f (t, x(s)) − f (s, x(s)) . (3.14) Further, let us observe that in view of our assumptions, for any t ∈ R+ we have that −M This shows that functions of the set F(X) satisfy uniformly the Cauchy condition at in nity. The proof is complete.
In what follows linking the results contained in Theorems 3.7 and 3.8 and taking into account Theorem 2.4 we obtain the main result of this section.