Solvability of an in nite system of integral equations on the real half-axis

The aim of the paper is to investigate the solvability of an in nite system of nonlinear integral equations on the real half-axis. The considerations will be located in the space of function sequences which are bounded at every point of the half-axis. The main tool used in the investigations is the technique associated with measures of noncompactness in the space of functions de ned, continuous and bounded on the real half-axis with values in the space l∞ consisting of real bounded sequences endowed with the standard supremum norm. The essential role in our considerations is played by the fact that we will use a measure of noncompactness constructed on the basis of ameasure of noncompactness in thementioned sequence space l∞. An example illustrating our result will be included.


Introduction
The paper is dedicated to the study of the existence of solutions of an in nite system of nonlinear integral equations on the real half-axis R+ = [ , ∞). More precisely, we will look for solutions of the mentioned in nite system in the space BC(R+, l∞) consisting of functions de ned, continuous and bounded on the interval R+ with values in the classical sequence space l∞ which is equipped with the standard supremum norm. Thus, any solution of our in nite system of integral equations will be treated as a sequence of functions (xn(t)) de ned on R+ and such that for any xed t ∈ R+ the sequence (xn(t)) is an element of the space l∞. Further details concerning the properties of the solutions in question will be formulated in the sequel of our paper.
The present paper is a continuation of papers [1,2], where we constructed measures of noncompactness needed in our study. Particularly, in paper [2] we constructed measures of noncompactness in the Banach space BC(R+, E) containing functions de ned, continuous and bounded on the interval R+ and taking values in a given Banach space E. The construction of those measures depends strongly on a given measure of noncompactness in the space E. Additionally, in the mentioned paper [2] we applied one of the constructed measures of noncompactness to the study of the solvability of an in nite system of integral equations in the space BC (R+, l∞).
In this paper we are going to study a more general in nite system of nonlinear integral equations in the same Banach space BC(R+, l∞) but with the use of another measure of noncompactness. Such an approach allows us to obtain an existence result concerning the mentioned in nite system, but under weaker assump-tions than the existence result obtained in [2]. Thus, our result creates a generalization of the existence result contained in paper [2]. It is also worthwhile mentioning that in paper [1] it was also studied the solvability of an in nite system of integral equations in the Banach space BC(R+, E) but we assumed that E is a Banach space with a regular measure of noncompactness being equivalent to the so-called Hausdor measure of noncompactness. Let us point out that in some Banach spaces such measures of noncompactness are not known [3].
Finally, let us mention that the theory of in nite systems of integral equations has recently been intensively developed and up to now there have appeared a lot of papers concerning those in nite systems [4][5][6][7][8][9]. That theory is closely related to the theory of in nite systems of di erential equations (cf. [4,7,10] and references therein).
However, in the papers published up to now the authors investigated mainly in nite systems of integral equations in the Banach space C([ , T], E) consisting of functions de ned and continuous on the bounded interval [ , T] with values in a Banach space E. The generalization to the space BC(R+, E) is rather a quite new branch of the theory of in nite systems of integral equations (cf. [1,2]).

Notations, de nitions and auxiliary facts
In this section we establish the notations used in the paper and we provide de nitions creating the basis of our study conducted further on. Apart from this we give some facts concerning the theory of measures of noncompactness being the basic tool utilized in our considerations.
In the paper we will denote by R the set of real numbers while N stands for the set of natural numbers. We will put R+ = [ , ∞). Further assume that E is a Banach space with the norm · E and the zero element θ. We will denote by B(x, r) the closed ball centered at x and with radius r. We write Br to denote the ball B(θ, r). If X, Y are subsets of the Banach space E and λ ∈ R then the standard algebraic operations on sets will be denoted by X + Y and λX. Moreover, the symbol X denotes the closure of the set X while ConvX stands for the closed convex hull of the set X.
Next, denote by M E the family of all nonempty and bounded subsets of E and by N E its subfamily consisting of all relatively compact sets. In our considerations we will accept the following de nition of the concept of a measure of noncompactness (cf. [3,4]).
De nition 2.1. A function µ : M E → R+ will be called a measure of noncompactness in the space E if it satis es the following conditions: (i) The family ker µ = {X ∈ M E : µ(X) = } is nonempty and ker µ ⊂ N E . The family ker µ from axiom (i) will be called the kernel of the measure of noncompactness µ. Let us also observe that the set X∞ de ned in axiom (vi) is an element of the family ker µ. In fact, it follows easily from the inclusion X∞ ⊂ Xn for n = , , . . . and axiom (ii) which implies the inequality µ(X∞) µ(Xn) for any n ∈ N. Hence we have that µ(X∞) = and consequently, X∞ ∈ ker µ. This simple conclusion is very crucial in applications.
In what follows assume that µ is a measure of noncompactness in the space E. The measure µ is called sublinear [3] if it satis es the following additional conditions If the measure of noncompactness µ satis es the condition (ix) µ(X ∪ Y) = max{µ(X), µ(Y)} then we say that it has the maximum property. Moreover, if ker µ = N E we say that µ is full. If µ is a sublinear and full measure of noncompactness with the maximum property, then µ is said to be regular. Let us recall that from the historical point of view the rst measure of noncompactness was de ned in 1930 by K. Kuratowski [11], but the most important and useful measure of noncompactness is the so-called Hausdor measure of noncompactness which was de ned in [12,13] by the formula where X ∈ M E . The importance of the Hausdor measure χ is caused by the fact that in some Banach spaces like C ([a, b]), c and lp we can give formulas expressing the measure χ in connection with the structure of the mentioned Banach spaces. Let us mention that χ is a regular measure of noncompactness [3,14,15].
However, in a lot of Banach spaces we are not in a position to give formulas for the Hausdor measure of noncompactness χ. Even more, we are not able to provide formulas for full measures of noncompactness [3,4]. Therefore, in such a situation we restrict ourselves to measures of noncompactness in the sense of De nition 2.1, which are not full (cf. also [3,4,14]). In the present paper we will also consider a measure of noncompactness of such a type.
Namely, assume that E is a given Banach space with the norm · E and µ is a measure of noncompactness in the space E. Consider the Banach space BC(R+, E) consisting of all functions x : R+ → E which are continuous and bounded on the interval R+. The space BC(R+, E) will be equipped with the standard supremum norm Further, take an arbitrary nonempty and bounded subset X of the space BC(R+, E). Fix x ∈ X and ε > . We de ne the modulus of the uniform continuity of the function x (cf. [2]) by putting Obviously, lim ε→ ω ∞ (x, ε) = if and only if the function x is uniformly continuous on the interval R+.
Further, let us de ne the following quantities Next, let us consider the function µ ∞ de ned on the family M BC(R+,E) in the following way where µ T (X) is de ned by the formula for any xed T > . It can be shown that the function µa is a measure of noncompactness in the space BC(R+, E) (cf. [2]). The kernel ker µa of the measure µa consists of all bounded subsets X of the space BC(R+, E) such that functions from X are uniformly continuous and equicontinuous on R+ (equivalently we can say that functions from X are equiuniformly continuous on R+) and tend to zero at in nity with the same rate. Apart from this, all cross-sections X(t) = {x(t) : x ∈ X} of the set X belong to the kernel ker µ of the measure of noncompactness µ in the Banach space E (cf. [2]). The measure µa is not full and has the maximum property. If the measure µ is sublinear in E then the measure µa de ned by (2.4) is also sublinear [2].
Let us notice that in the similar way as above we may de ne other measures of noncompactness in the space BC(R+, E) (see [2]). We will not provide the de nitions of those measures since we will use only the measure of noncompactness µa further on.
In what follows, taking into account our further purposes, we will consider as the Banach space E, the sequence space l∞ endowed with the standard supremum norm. Thus, let us consider the Banach space BC(R+, l∞) consisting of functions x : R+ → l∞ which are continuous and bounded on R+. Observe, that if x ∈ BC(R+, l∞), then we can write this function in the form for any t ∈ R+, where the sequence (xn(t)) is an element of the space l∞ for any xed t. The norm of the function x = x(t) = (xn(t)) is de ned by the equality In our further considerations the space BC(R+, l∞) will be denoted by BC∞. Now, we provide the formula expressing the measure of noncompactness µa de ned by (2.4) in the space BC∞, provided the measure of noncompactness µ in the sequence space l∞ is de ned by the formula [3] µ (X) = lim n→∞ sup x=(x i )∈X sup |x k | : k n for X ∈ M l∞ . In such a case the component µ ∞ de ned by (2.2) will be denoted by µ ∞ . Thus, our measure of noncompactness µa de ned by (2.4) will be now denoted by µ a and is de ned as a particular case of (2.4) by the formula µ a (X) = ω ∞ (X) + µ ∞ (X) + a∞(X), (2.5) where the components on the right hand side of formula (2.5) are de ned in the following way: In what follows we give an other formula expressing the quantity µ ∞ de ned by (2.2). To this end we prove the following lemma.

Lemma 2.2. The following equality is satis ed
where µ ∞ is de ned by formula (2.2).
Proof. Obviously, for any xed T > we have To prove the converse inequality, let us denote Further, x an arbitrary number ε > . Then we can nd a number t ∈ R+ such that Hence, for T t we obtain Combining (2.10) and (2.11), we have Consequently, in view of the arbitrariness of the number ε, we derive the following inequality Finally, linking (2.9) and (2.12) we obtain the desired equality.
Now, let us notice that taking into account Lemma 2.2 and formula (2.7) expressing the quantity µ ∞ in the case of the space BC∞, we obtain the following corollary. At the end of this section we recall a useful xed point theorem of Darbo type [3,16].
To this end let us assume that E is a Banach space and µ is a measure of noncompactness (in the sense of De nition 2.1) in the space E.

Solvability of an in nite system of quadratic integral equations on the real half-axis
The aim of this section is to investigate the in nite system of the quadratic integral equations of Volterra-Hammerstein type having the form where t ∈ R+ and n = , , . . . . Our goal is to prove the existence of solutions of the in nite system of integral equations (3.1). Considerations conducted further on will be situated in the Banach space BC∞ = BC(R+, l∞) described in details previously. Moreover, in our study we are going to use the measure of noncompactness µ a (X) de ned by formula (2.5). Now, we formulate the assumptions under which the in nite system (3.1) will be considered. (i) The sequence (an(t)) is an element of the space BC∞ such that lim t→∞ an(t) = uniformly with respect to n ∈ N i.e., the following condition is satis ed Moreover, lim n→∞ an(t) = for any t ∈ R+.
(ii) The functions kn(t, s) = kn : R + → R are continuous on the set R + (n = , , . . . ). Moreover, the functions t → kn(t, s) are equicontinuous on the set R+ uniformly with respect to s ∈ R+ i.e., the following condition is satis ed (iii) There exists a constant K > such that t |kn(t, s)|ds K for any t ∈ R+ and n = , , . . . . (iv) The sequence (kn(t, s)) is equibounded on R + i.e., there exists a constant K > such that |kn(t, s)| K for t, s ∈ R+ and n = , , . . . . (v) The function fn is de ned on the set R+ × R ∞ and takes real values for n = , , . . . . Moreover, the function t → fn(t, x , x , . . . ) is uniformly continuous on R+ uniformly with respect to x = (xn) ∈ l∞ and uniformly with respect to n ∈ N i.e., the following condition is satis ed (vi) There exists a function l : R+ → R+ such that l is nondecreasing on R+, continuous at and there exists a sequence of functions (f n ) being an element of the space BC∞, taking nonnegative values and such that lim t→∞ f n (t) = uniformly with respect to n ∈ N (cf. assumption (i)) and lim n→∞ f n (t) = for any t ∈ R+.
Moreover, for any r > the following inequality is satis ed Before formulating our main result we indicate some consequences of assumption (i).

Lemma 3.1. Let the function x(t) = (xn(t)) be an element of the space BC∞. Then the sequence (xn) is equibounded and locally equicontinuous on R+.
The proof can be conducted analogously as the proof of Lemma 4.1 in paper [1] and is therefore omitted.

Lemma 3.2. Let the sequence (an(t)) be an element of the space BC∞ such that lim
t→∞ an(t) = uniformly with respect to n ∈ N (cf. assumption (i)). Then the sequence (an) is equibounded and equicontinuous on R+.
Proof. The equiboundedness of the sequence (an) on R+ follows from Lemma 3.1. To prove the equicontinuity of (an) on R+ let us x ε > . Keeping in mind the remaining part of the assumption in our lemma we can nd a number T > such that |an(t)| ε/ for t T and n = , , . . . . On the other hand, in view of Lemma 3.1 we deduce that the sequence (an) is locally equicontinuous on R+. Thus, we can nd a number δ > such that |an(t ) − an(t )| ε/ for t , t ∈ [ , T] such that |t − t | δ and for any n = , , . . . . Now, let us take arbitrary t , t ∈ R+ such that |t − t | δ. Without loss of generality we can assume that t < t . If t , t ∈ [ , T], then in view of the above established fact we have that |an(t ) − an(t )| ε/ for n = , , . . . . If t , t T, then we obtain |an(t ) − an(t )| |an(t )| + |an(t )| ε for any n = , , . . . . Assume now that t < T t . Then, xing arbitrarily n ∈ N and taking into account the above derived facts we get |an(t ) − an(t )| |an(t ) − an(T)| + |an(T) − an(t )| ε + ε = ε.

This shows that the sequence (an) is equicontinuous on R+.
In what follows let us notice that as an immediate consequence of Lemma 3.1 we obtain the following corollary.

Corollary 3.3. The constant A de ned by equality (3.2) is nite.
Now, we are in a position to formulate an existence theorem concerning the in nite system of integral equations (3.1).
To this end x the function x = x(t) = (xn(t)) ∈ BC∞. Then, in view of assumption (vi) we have the following estimate for t ∈ R+ and for n = , , . . . , where the functions f n and l = l(r) were speci ed in assumption (vi). Hence, on the basis of (3.3) we deduce that for any x ∈ BC∞. This shows that the function Fx is bounded on R+.
In order to prove the continuity of the function Fx on the interval R+, let us x ε > . Then, from assumption (v) we obtain that there exists δ > such that for t, s ∈ R+ and |t − s| δ the following inequality is satis ed provided t, s ∈ R+ are such that |t − s| δ. But this means that the function Fx is continuous (even uniformly continuous) on R+. Finally we conclude that the operator F transforms the space BC∞ into itself. Now, we are going to show that the operator V acts from the space BC∞ into itself. Thus, similarly as above, x a function x = x(t) = (xn(t)) ∈ BC∞. Then, for arbitrarily xed numbers t ∈ R+ and n ∈ N, in view of assumptions (iii) and (ix), we get Particularly, the above estimate yields that the function Vx is bounded on the interval R+.
Next, x ε > and choose a number δ > according to assumption (ii). Then, for arbitrarily xed numbers t , t ∈ R+ such that |t − t | δ, on the basis of assumptions (ii) and (ix) (assuming additionally that t < t ), we obtain where K is a constant appearing in assuption (iv) and ω k (δ) denotes a common modulus of continuity of the sequence of functions t → kn(t, s) on the interval R+ (according to assumption (ii)). Obviously we have that ω k (δ) → as δ → . Further on, taking into account estimate (3.6) and assumption (ix), we obtain Hence we deduce that the function Vx is continuous on the interval R+. Linking boundedness of the function Vx with its continuity on R+ we conclude that the operator V transforms the space BC∞ into itself. Now, keeping in mind the fact that the space BC∞ = BC(R+, l∞) forms a Banach algebra with respect to the coordinatewise multiplication of function sequences and taking into account the de nition of the operator Q as well as assumption (i), we infer that for an arbitrarily xed function x = x(t) ∈ BC∞ the function (Qx)(t) = ((Qn x)(t)) = an(t) + (Fn x)(t)(Vn x)(t) acts from the interval R+ into the space l∞. Indeed, in view of the fact that ((Fn x)(t)) ∈ l∞ for any t ∈ R+ and in the light of estimate (3.5) we get Hence, applying (3.3) we deduce that (Qx)(t) = ((Qn x)(t)) ∈ l∞ for each t ∈ R+.
Next, let us notice that the continuity of the function Qx on R+ is a simple consequence of the fact that both the function Fx and the function Vx are continuous on R+. Similarly we can derive that the function Qx is bounded on the interval R+. Indeed, it is only su cient to make use assumption (i) and Lemma 3.1.
Finally, let us observe that combining all the above established properties of the function Qx we conclude that the operator Q transforms the space BC∞ into itself.
Further, let us note that based on estimates (3.4) and (3.5), for arbitrarily xed n ∈ N and t ∈ R+ we get From the above estimate and assumption (x) we infer that there exists a number r > such that the operator Q transforms the ball Br into itself.
In what follows we are going to show that the operator Q is continuous on the ball Br . Keeping in mind the representation of the operator Q given at the beginning of our proof we see it is su cient to prove the continuity of the operators F and V, separately.
To this end let us x ε > and x ∈ Br . Next, choose an arbitrary point y ∈ Br such that x − y BC∞ ε. Then, for each xed t ∈ R+, in virtue of assumption (vii) we get (Fy)(t) − (Fx)(t) l∞ m(r ) x − y l∞ εm(r ).
Particularly this shows that the operator F is continuous at every point of the ball Br .
To prove the continuity of the operator V on the ball Br let us de ne the function δ = δ(ε) by putting Obviously, in view of assumption (viii) we have that δ(ε) → as ε → . Next, taking x, y ∈ Br such that y − x BC∞ ε and t ∈ R+, for arbitrary n ∈ N we obtain
Thus we see that the operator V is continuous on the ball Br .
In the sequel let us x an arbitrary number ε > . Next, choose a number δ > according to assumption (v). Further x a nonempty subset X of the ball Br and take an arbitrary function x ∈ X and n ∈ N. Then, for arbitrary t, s ∈ R+ such that |t − s| δ, in virtue of assumption (v) we get Now, on the basis of the above estimate we obtain ω ∞ (Fx, ε) ε + m(r )ω ∞ (x, δ). (3.8) Further, let us x a number ε > and choose t , t ∈ R+ such that |t − t | ε. Without loss of generality we may assume that t < t . Then, in view of estimate (3.7) we derive the following inequality This yields the estimate ω ∞ (Vx, ε) Gω k (ε) + GK ε. (3.9) Now, taking into account the representation of the operator Q, for an arbitrary function x ∈ X and for arbitrary numbers t, s ∈ R+, we obtain where a(t) = (an(t)).
Further, x ε > and assume that |t − s| ε. Then, from the above inequality and estimates (3.8), (3.9), (3.4), and (3.5), we get Now, in view of Lemma 3.2 we infer that ω ∞ (a, ε) → as ε → . Next, keeping in mind that ω k (ε) → as ε → , from the above obtained estimate we derive the following inequality ω ∞ (QX) GK m(r )ω ∞ (X). (3.10) In what follows we will consider the second term of the measure of noncompactness µ a (cf. formula (2.5)) which is denoted by µ ∞ and is de ned by formula (2.7). To this end x a nonempty subset X of the ball Br and choose an arbitrary function x = x(t) ∈ X. Further, take a natural number n and T > . Then, for any xed t ∈ [ , T], in view of the representation of the operator Q and estimates (3.3) and (3.5), we obtain Now, taking supremum over all x ∈ X, from the above estimate we get Hence, taking into account assumption (i) and (vi), we derive the following inequality Finally, taking supremum over t ∈ [ , T] on both sides of the above inequality and next, passing with T → ∞, in virtue of formula (2.7) we get µ ∞ (QX) GK l(r )µ ∞ (X). (3.11) In order to estimate the last term a∞ of the measure of noncompactness µ a (cf. formula (2.5)) expressed by formula (2.8), let us take a nonempty subset X (X ⊂ Br ) and choose a function x ∈ X. Further, x arbitrarily T > . Then, taking t T and keeping in mind the previously obtained inequalities Next, let us observe that for each n ∈ N and for arbitrary t, s ∈ R+ we have the following estimate Hence it follows that the sequence (kn(t, s)) is equibounded on R+ with the constant K = . This shows that there is satis ed assumption (iv).
On the other hand we obtain t |kn(t, s)|ds = t s + n(s + t ) ds = ln + nt + nt n ln ln .
This proves that the function sequence (kn(t, s)) satis es assumption (iii) with the constant K = ln . Now, let us take into account the function t → fn(t, x , x , . . . ) de ned by formula (4.3) for n = , , . . . . Fix arbitrary t , t ∈ R+ and x = (xn) ∈ l∞. Then, we get for any n = , , . . . . This shows that the functions fn (n = , , . . . ) satisfy assumption (v). In order to verify assumption (vi) let us x a number r > and choose x = (x i ) ∈ l∞ such that x l∞ r. Then, for arbitrarily xed n ∈ N and t ∈ R+, we obtain This shows that the inequality from assumption (vi) is satis ed with the following functions f n (t) = β n + t , l(r) = γ for n = , , . . . . Since f n (t) β/( + t ) we infer that lim t→∞ f n (t) = uniformly with respect to n ∈ N. Apart from this we have that lim n→∞ f n (t) = for any t ∈ R+.
Thus we see that assumption (vii) is satis ed with the function m(r) = γ( + r).
In the next step of our proof we are going to verify assumption (viii). To this end x arbitrarily n ∈ N and consider the function gn(t, x) = gn(t, x , x , . . . ) de ned by formula (4.5) i.e., gn(t, x , x , . . . ) = arctan x + xn n + t .
From the above estimate we infer that the operator g satis es assumption (viii). Moreover, it is easily seen that for an arbitrary x ∈ l∞ and t ∈ R+ we get (gx)(t) l∞ = sup |gn(t, x)| : n ∈ N π .
This means that the operator g satis es assumption (ix) with the constant G = π/ . Finally, let us consider the rst inequality from assumption (x). Obviously, in our case that inequality has the form α √ + π ln (β + γr) < r. On the other hand, taking the second inequality required in assumption (x), we get γ π ln ( + r ) < . (4.7) It is easy to check that choosing γ < π ln and taking r > α √ + β γ , we can easily verify that both inequalities (4.6) and (4.7) are satis ed. Thus, in the light of Theorem 3.4 we infer that in nite system of nonlinear integral equations (4.1) has at least one solution belonging to the ball Br in the space BC(R+, l∞).