Solvability of an in nite system of nonlinear integral equations of Volterra-Hammerstein type

Abstract: The purpose of the paper is to study the solvability of an in nite system of integral equations of Volterra-Hammerstein type on an unbounded interval. We show that such a system of integral equations has at least one solution in the space of functions de ned, continuous and bounded on the real half-axis with values in the space l1 consisting of all real sequences whose series is absolutely convergent. To prove this result we construct a suitable measure of noncompactness in the mentioned function space and we use that measure together with a xed point theorem of Darbo type.


Introduction
Integral equations create a very signi cant part of nonlinear analysis and applied mathematics ( [1][2][3][4]). Many researchers, not only mathematicians, are interested in the study of the solvability of integral equations and the applicability of such equations to di erent problems arising in the description of real world events ( [2,3,[5][6][7][8][9]). The results obtained in the theory of integral equations are useful and widely utilized in many branches of technical sciences, as mechanics or engineering and exact sciences as physics, for example.
In the theory of integral equations the special and exceptional branch is created by in nite systems of integral equations. On the one hand such systems are very interesting subject of the study for researchers specialized in the theory of integral equations but on the other hand systems of integral equations play very crucial role in applications.
In this paper we deal with an in nite system of nonlinear integral equations of Volterra-Hammerstein type. In [10] we showed that such a system has a solution belonging to the space BC(R+, c ) of functions dened, continuous and bounded on the real half-axis and with values in the sequence space c . In the present paper we prove a stronger result. Namely, we show that in nite system of integral equations of Volterra-Hammerstein type has at least one solution in the space BC(R+, l ) consisting of all functions de ned, continuous and bounded on the interval R+ with values in the sequence space l . Of course each such solution belongs to the space BC(R+, c ) considered in [10]. Let us mention that paper [10] was the rst one in which the study of solvability of in nite systems of integral equations de ned on an unbounded interval was carried out. All known up to now results have been obtained in the space of functions de ned on a bounded interval (see [11][12][13][14], for example).

Notation, de nitions and auxiliary facts
In this section we recall some facts which will be utilized in the paper. Let us start with establishing some notation. The symbols R and N stand for sets of real and natural numbers, respectively. Moreover, we put R+ = [ , ∞).
The letter E means a Banach space normed by norm || · || E and with zero vector θ. The symbol B(x, r) denotes the closed ball in E centered at x with radius r. In the special case when x = θ we write Br instead of B(θ, r). Moreover, if X is a subset of E then we denote by X the closure of X and by Conv X the closed convex hull of the set X. The symbols X + Y, λX (λ ∈ R) will stand for the usual algebraic operations on subsets X and Y of E. For a nonempty and bounded set X ⊂ E we denote by diam X the diameter of the set X. The symbol ||X|| E will stand for the norm of the set X ⊂ E i.e., we have The fundamental notion that we use in this paper is the concept of a measure of noncompactness. We recall now the de nition of a measure of noncompactness which was introduced in monograph [15]. To this end let us 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.
De nition 2.1. A function µ : M E → R+ will be called a measure of noncompactness in 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 set ker µ de ned in axiom (i) is called the kernel of the measure of noncompactness µ. Let us observe that the intersection set X∞ appearing in axiom (vi) is a member of the family ker µ [15]. This simple observation plays an essential role in our further considerations. Now we present some properties of measures of noncompactness [15]. We say that µ is sublinear if it satis es the following additional conditions: Moreover, we say that a measure of noncompactness µ has maximum property if If the measure of noncompactness µ is such that (x) ker µ = N E then it is called full. A sublinear measure of noncompactness which is full and has maximum property is said to be regular measure of noncompactness [15].
Let us mention that the rst measure of noncompactness was de ned in 1930 by Kuratowski [16] in the following way α(X) = inf {ε > : X can be covered by a nite family of sets X , X , . . . , Xm such that diam X i ≤ ε for i = , , . . . , m}.
The measure α(X) is called the Kuratowski measure of noncompactness. It is known (see [15]) that the Kuratowski measure of noncompactness is a regular measure. In the similar way was de ned the Hausdor measure of noncompactness ( [17,18]): It can be shown that the measure χ(X) is also regular and that both de ned above measures α(X) and χ(X) are equivalent. But despite of these similarities it turns out that the Hausdor measure of noncompactness χ is more convenient in applications than the Kuratowski measure. The main reason is that in some classical Banach spaces we can nd explicit formulas describing the Hausdor measure of noncompactness but we do not know such formulas for the Kuratowski measure of noncompactness in any Banach space [15]. And now, taking into account our further investigations, we recall the formula expressing the Hausdor measure of noncompactness in the space l (cf. [15]). So, let us call to mind that the space l consists of all sequences whose series is absolutely convergent, i.e.
It is normed by the following norm Then we have the following formula for the Hausdor measure of noncompactness of any bounded set X ∈ M l : (2.1) To prove our main result we will also need a xed point theorem involving a measure of noncompactness. The basic theorem in this subject is known xed point theorem proved by Darbo [19]. We will use a modi ed version of Darbo theorem formulated below (cf. [15]). Theorem 2.2. Let µ be an arbitrary measure of noncompactness in the Banach space E. Assume that Ω is a nonempty, bounded, closed and convex subset of E and Q : Ω → Ω is a continuous operator such that there exists a constant k ∈ [ , ) for which µ(QX) ≤ kµ(X) for an arbitrary nonempty subset X of Ω. Then the operator Q has at least one xed point in the set Ω.

Measures of noncompactness in the space BC(R + , l )
In [10] we investigated measures of noncompactness in the space BC(R+, E) consisting of all functions dened, continuous and bounded on R+ with values in a xed Banach space E. Let us pay attention to the fact that the space BC(R+, E) is a generalization of the well-known and often used classical Banach space BC(R+, R), therefore measures of noncompactness in the space BC(R+, E) must be more complicated then known measures in BC(R+, R). And now we recall some basic facts about the space BC(R+, E) and measures of noncompactness in this space.
Let us start with assuming that E is a given Banach space with the norm || · || E whereby we will assume that E is an in nite dimensional space. Consider the Banach space BC(R+, E) equipped with the supremum norm ||x||∞ de ned in the standard way ||x||∞ = sup{||x(t)|| E : t ∈ R+}.
In [10] we de ned three formulas for measures of noncompactness in the space BC(R+, E) and each such formula is a sum of three components. The rst and the second component are the same in each formula and we start to present them. So, let us x a set X ∈ M BC(R+,E) and numbers T > and ε > . For any function x ∈ X we de ne the modulus of continuity ω T (x, ε) of the function x on the interval [ , T] by the classical formula Next, let us de ne and nally, we put ω (X) = lim T→∞ ω T (X).
Notice that both above limits exist (for details see [10]). The quantity ω (X) is the rst component of each of mentioned earlier measures of noncompactness in BC(R+, E). Next, to obtain the second component, assume that γ = γ(X) is a measure of noncompactness in the space E. Fix number t ∈ R+ and denote by X(t) the socalled cross-section of the set X at the point t: Further, for a xed T > let us put Notice that the above limit exists (see [10]). The obtained quantity γ ∞ (X) is the second component of all three formulas for measure of noncompactness in the space BC(R+, E). Now we introduce the third component of the measure of noncompactness in BC(R+, E) which describes the behaviour of the set of functions at in nity. We can do it in three ways and we will describe each of them. So, for a xed T > let us de ne Next, notice that there exists the limit a∞(X) = lim T→∞ a T (X) (3.4) and the quantity a∞(X) is considered as the third component of the measure of noncompactness in the space And now let us consider the functions γa, γ b , γc de ned on the family M BC(R+,E) as follows In [10] we proved that under some assumptions on γ the functions γa, γ b , γc are measures of noncompactness in the space BC(R+, E). More precisely, we have the following result.
for an arbitrary set X ∈ M BC(R+,E) , where χ denotes the Hausdor measure of noncompactness in the space BC(R+, E).
For other properties of the above introduced measures of noncompactness we refer to [10]. We recall only that Theorem 3.1 remains valid if in the construction of the component γ ∞ we replace the Hausdor measure of noncompactness χ by an arbitrary regular measure of noncompactness µ equivalent to the Hausdor measure χ [10]. Now, we are going to present formulas (3.7), (3.8) and (3.9) in the special case, for E = l . The space l is one of the Banach sequence spaces and we deal with this space (and, in general, with sequence spaces) because it is strictly connected with the form of solutions of in nite systems of integral equations (see Theorem 4.2). Therefore, we will work in the Banach space x is continuous and bounded on R+}.
The fundamental fact in our investigations is that every function x ∈ BC(R+, l ) can be regarded as a function sequence for t ∈ R+, where for any xed t the sequence (xn(t)) is a real sequence being an element of the space l .
Obviously, it means that According to the formula for the norm in the space BC(R+, E) given earlier we have Now, we are going to present explicitly the consecutive components ω (X), χ ∞ (X) and a∞(X) of the measure of noncompactness χa(X) for any set X ∈ M BC(R+,l ) . So, let us start with ω (X). Fix arbitrarily numbers T > and ε > . For any x = x(t) = (xn(t)) ∈ X we have Hence, we get Finally, we obtain Next, we are going to de ne the second component occuring in the formula for the measure χa(X). To this end let us assume that X ∈ M BC(R+,l ) and t ∈ R+ is arbitrarily xed. Using (2.1) we have Next, for a xed T > utilizing (3.1) we get Finally, in view of (3.2) we obtain the following expression: And now let us write the third component of the measure χa(X). Thus, x an arbitrary number T > . Then, on the basis of (3.3), we have Next, by (3.4) we obtain Finally, based on Theorem 3.1 we get that the function is a measure of noncompactness in the space BC(R+, l ), where ω (X), χ ∞ (X) and a∞(X) are given by formulas (3.10), (3.11) and (3.12), respectively. Observe, that keeping in mind formulas (3.5) and (3.6) together with Theorem 3.1 we obtain that the functions and are also measures of noncompactness in the space BC(R+, l ), where

Theorem on the existence of solutions of in nite systems of integral equations on the real half-axis
Let us consider the following in nite system of nonlinear quadratic integral equations of the Volterra-Hammerstein type for t ∈ R+ and for n = , , . . .. In [10] we proved that system (4.1) has at least one solution in the space BC(R+, c ) := {x : R+ → c : x is continuous and bounded on R+}. In this paper we prove the other result, namely we show that the system (4.1) has at least one solution in the space BC(R+, l ). For the convenience, from now on the space BC(R+, l ) will be denoted by BC . At the beginning we provide a lemma which we will utilize in the proof of our main result. Proof. The proof is an immediate consequence of the Cauchy condition for real sequences.
In what follows we will investigate the solvability of system (4.1) in the space BC under the below listed assumptions.
(i) The function sequence (an(t)) is an element of the space BC such that lim (iii) There exists a constant K > such that (vi) There exists a function l : R+ → R+ such that l is nondecreasing on R+, continuous at and the following condition is satis ed transforms the space R+×l into l and is such that the family of functions {(gx)(t)} t∈R+ is equicontinuous at every point of the space l i.e., for each arbitrarily xed x ∈ l and for a given ε > there exists δ > such that ||(gy)(t) − (gx)(t)|| l ≤ ε for every t ∈ R+ and for any y ∈ l such that ||y − x|| l ≤ δ. (ix) The operator g de ned in assumption (viii) is bounded on the space R+ × l . More precisely, there exists a positive constant G such that ||(gx)(t)|| l ≤ G for any x ∈ l and for each t ∈ R+. (x) There exists a positive solution r of the inequality A + F G K + G K r l(r) ≤ r such that G K l(r ) < , where the constants F, G, K were de ned above and the constant A is de ned in the following way Now we can present our main result concerning the solvability of in nite system of integral equations (4.1).
Our proof will be conducted in several steps. At the begining we show that operator Q transforms the space BC into itself. Thus, let us take x = x(t) = (xn(t)) ∈ BC . Obviously this means that ∞ n= |xn(t)| < ∞.
Then, for arbitrary t ∈ R+, using assumption (vi), we get Thus we have that (Fx)(t) ∈ l . Moreover, from (4.2) we infer that the function Fx is bounded on the set R+. Further, we are going to show that the function Fx is continuous on the interval R+. To this end, let us take arbitrary t ∈ R+ and ε > . It follows from the continuity of the function x that the below given condition is satis ed Thus, let us choose δ > according to (4.4). Then, for t ∈ R+ such that |t − t | ≤ δ we obtain Next, let us choose a number δ > according to assumption (v). Then for Thus, taking δ = min{δ , δ } we can deduce that the following estimate is satis ed for any t ∈ R+. Therefore, we can write for any t ∈ R+. It means that Fx is continuous on R+.
Linking above established facts we obtain that the operator F transforms the space BC into itself. Next, we are going to show that the operator V maps the space BC into BC . So, let us take an arbitrary function x = x(t) = (xn(t)) ∈ BC . We are going to prove that a function Vx ∈ BC . We start with showing the boundness of the function Vx on R+. To this end observe that for an arbitrary number t ∈ R+, using assumptions (iii) and (ix), we get Therefore we have that for any t ∈ R+ the following inequality holds This means the function Vx is bounded on R+.
To prove the continuity of the function Vx on the interval R+ let us x ε > , T > and t ∈ [ , T). In virtue of (4.4) we can choose a number δ > according to the continuity of the function x = x(t) at the point t . Next, take t ∈ [ , T) satisfying |t − t | ≤ δ (without loss of generality we can assume that t > t ). Then, keeping in mind assumptions (iv), (viii) and (ix) and using the Lebesgue monotone convergence theorem [20], we arrive at the following estimates: where ω T k (ε) denotes a common modulus of continuity of the function sequence t → kn(t, s) on the set [ , T] (according to assumption (ii)). Obviously ω T k (ε) → as ε → . Hence, on the basis of (4.7) we derive the following inequality ||(Vx)(t) − (Vx)(t )|| l ≤ ω T k (ε) G T + K G ε, which means that the function Vx is continuous on [ , T). The number T was choosen arbitrarily therefore we deduce that the function Vx is continuous on the real half-axis R+.
Futher on, we are going to prove that the function Q maps the space BC into itself. In order to prove this fact, notice that we can treat the space BC as a Banach algebra with respect to the coordinatewise multiplication of sequences. Therefore, take any function x ∈ BC and consider a function Qx. Keeping in mind the de nition of the operator Q and established facts that the function Fx and the function Vx are continuous on R+ we obtain that the function Qx is also continuous on R+. Similarly, taking into account the boundness of functions Fx and Vx on the set R+ we infer that Qx is also bounded on R+. In order to show that Qx : R+ → l let us notice that using asumption (i) and (4. so for any xed t ∈ R+ we obtain that (Qx)(t) ∈ l and hence Qx : Finally, linking all the above established properties of the function Qx we derive that the operator Q transforms the space BC into itself.
In what follows we show the existence of a number r > such that Q transforms the ball Br in the space BC into itself. For arbitrary t ∈ R+, utilizing estimates (4.2) and (4.6) as well as assumptions (x) and (vi) we obtain This yields to estimate Taking into account the last inequality and assumption (x) we deduce that there exists a number r > such that the operator Q transforms the ball Br into itself.
In what follows we show that the operator Q is continuous on the ball Br . In order to prove this fact x arbitrarily x ∈ Br , ε > and take a function y ∈ Br such that ||x − y|| BC ≤ ε. Fix t ∈ R+. Then, in view of assumption (vi) we have This means that F is continuous on the ball Br . Further, let us consider the function δ = δ(ε) de ned for ε > in the following way δ(ε) = sup {|gn(t, x) − gn(t, y)| : x, y ∈ l , ||x − y|| l ≤ ε, t ∈ R+, n ∈ N}.
This proves the continuity of the operator V on the ball Br . Now, linking the continuity of the operators F and V on the ball Br and keeping in mind the representation of the operator Q written at the beginning of the proof we infer that the operator Q is continuous on Br . And now we have the last step of our proof in which we show that the inequality from Theorem 2.2 is satis ed for any set X ⊂ Br and for measure of noncompactness χa de ned by formula (3.7) for γ = χ l .
To this end take a nonempty subset X of the ball Br and x numbers ε > and T > . Choose t, s ∈ [ , T] such that |t − s| ≤ ε and consider a function x = x(t) = (xn(t)) ∈ X. Then, proceeding similarly as in (4.5) we obtain the following estimate: where Obviously, in view of assumption (v) we have that ω (f , ε) → as ε → .
Taking supremum with respect to t, s ∈ [ , T], |t − s| ≤ ε on the left side of (4.8), we get the following estimate Next, let us take t, s as above. Assuming additionaly that t > s and following the estimate (4.7) we get where the function ω T k (ε) was introduced earlier. Consequently this implies the following inequality: Now, take any function x ∈ X and t, s ∈ R+. Keeping in mind the representation of the operator Q we derive the following estimate: where we denoted a(t) = (an(t)).
Finally, taking limit as T → ∞, we get ω (QX) ≤ G K l(r ) ω (X). (4.11) Notice that we obtained the estimate for the rst component ω (X) of the measure of noncompactness χa(X) expressed by formula (3.7).
In what follows we obtain two consecutive estimations for the second and the third component of the measure of noncompactness χa(X). To this end, similarly as before, x a set X ⊂ Br and a function x ∈ X. Take an arbitrary number T > and x t ∈ [ , T]. Then, for any n ∈ N, utilizing estimates (4.2) and (4.6) (for series from i = n to ∞), we get Further, taking supremum over all x = (x i ) ∈ X, we derive the following evaluation Passing with n → ∞ and utilizing assumptions (i), (vii) and Lemma 4.1, we get Finally, taking supremum over t ∈ [ , T] on both sides of the above inequality and letting with T → ∞, in view of formula (3.11) we deduce the following estimate Now, we are going to estimate the third component a∞(X) of the measure of noncompactness χa(X). Assume, as earlier, that X ⊂ Br , x ∈ X and T > . Moreover, take t ≥ T. Then, keeping in mind inequalities (4.2) and (4.6), we have Taking supremum over t ≥ T and x = (xn) ∈ X, we get Letting with T → ∞ and utilizing assumptions (i) and (vii), we derive the following inequality a∞(QX) ≤ l(r ) G K a∞(X). (4.13) Finally, combining (4.11), (4.12), (4.13) and formula (3.7) for γ = χ, we get the following inequality Hence, in view of the previously established fact that the operator Q is a continuous self-mapping of the ball Br , utilizing Theorem 2.2 we conclude that the in nite system of Volterra-Hammerstein integral equations (4.1) has at least one solution x(t) = (xn(t)) in the space BC = BC(R+, l ). Obviously x ∈ Br . The proof is complete.
Hence we infer that the functions fn (n = , , . . .) satisfy assumption (vi) with the function l(r) de ned by the equality l(r) = γ max{ , r}.
Further on, we are going to verify assumption (viii). To this end, let us rst notice that for a xed natural number n the function gn = gn(t, x , x , . . .) de ned by (5.2) on the set R+×R ∞ takes real values (n = , , . . .).
Thus we have proved that there is satis ed assumption (viii). Finally, gathering all the above obtained constants A, F, G, K and taking into account the function l(r) = γ max{ , r} indicated in the above calculations, we conclude that the inequality from assumption (x) has the form β + γ π r max{ , r} ≤ r. (5.4) Further, let us assume that we are looking for a solution r of inequality (5.4) such that r ≤ . In such a case inequality (5.4) has the form β + γ π r ≤ r. (5.5) Assuming that γ < /π we infer that, for example, the number r = β/( −γπ ) is a solution of inequality (5.4) provided β < ( − γπ )/ . It easily seen that in this case we have that GK l(r ) < which proves that assumption (x) is thoroughly satis ed. Now, applying Theorem 4.2 we deduce that in nite system of integral equations (5.1) has at least one solution x = x(t) = (xn(t)) in the space BC = BC(R+, l ).