Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 61, pp. 1–20. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO INFINITE SYSTEMS OF NONLINEAR INTEGRAL EQUATIONS ON THE REAL HALF-AXIS AGNIESZKA CHLEBOWICZ Abstract. In this article we study the solvability of an infinite system of inte- gral equations of Volterra-Hammerstein type in the space of functions defined, continuous and bounded on the real half-axis with values in the sequence space l1. We extend a known existence result for such an infinite system of integral equations and prove a theorem applicable to a wider class of considered infinite systems. We also give an example to show the usefulness of our result. 1. Introduction The theory of integral equations is a significant branch of mathematical analysis and is closely related to the theory of ordinary and partial differential equations. Both these theories have numerous applications to physics, astronomy, chemistry, engineering and other branches of exact sciences (cf. [11, 13, 16, 20, 22], for ex- ample) because most important phenomena in the real world can be described in terms of differential or integral equations. In the study of differential equations we sometimes consider infinite systems of differential equations linked with applications (cf. [8, 11, 12, 19] and references therein). While the theory of infinite systems of differential equations creates rather developed branch of differential equations, the theory of infinite systems of integral equations is quite young field of study. Only a few papers devoted to infinite systems of integral equations have appeared so far, see [3, 4, 6, 7, 9, 18]. The present paper concerns the theory of infinite systems of nonlinear integral equations of Volterra-Hammerstein type. More precisely, we deal with infinite sys- tems of the so-called quadratic integral equations of Volterra-Hammerstein type. We look for solutions of the mentioned infinite systems in the Banach space con- sisting of functions defined, continuous and bounded on the real half-axis R+ with values in the Banach sequence space l1. The main tool used in our study is the technique of measures of noncompactness [1, 2, 5] and the fixed point theorem of Darbo type related to that technique (cf. [5, 10]). The investigations in this paper continue those contained in papers [3, 9]. In particular, in this paper we generalize results from paper [9]. 2010 Mathematics Subject Classification. 47H08, 45G15. Key words and phrases. Infinite system of integral equations; measure of noncompactness; fixed point theorem. c©2021. This work is licensed under a CC BY 4.0 license. Submitted June 7, 2020. Published July 6, 2021. 1 2 A. CHLEBOWICZ EJDE-2021/61 2. Preliminaries At the beginning we establish the notation and recall auxiliary facts used in the paper. Thus, we denote by R the set of real numbers. The symbol N stands for the set of natural numbers (positive integers). Apart from this we put R+ = [0,∞). Next, assume that E is a Banach space with the norm ‖ · ‖E and with the zero vector θ. Denote by B(x, r) the closed ball in E centered at x and with radius r. We write Br to denote the ball B(θ, r). Moreover, if X is a subset of the space E then we denote by X the closure of X and by convX the closed convex hull of the set X. To denote the algebraic operations on subsets X and Y of the space E we use the symbols X + Y and λX (λ ∈ R). If X is a nonempty and bounded subset of E, then the symbol diam X stands for the diameter of X. Next, denote by ME the family of all nonempty and bounded subsets of E and by NE its subfamily consisting of all relatively compact sets. We accept the following definition of the concept of a measure of noncompactness (cf. [5]). Definition 2.1. A function µ : ME → R+ is called a measure of noncompactness in E if it satisfies the following conditions: (i) The family kerµ = {X ∈ME : µ(X) = 0} is nonempty and kerµ ⊂ NE . (ii) X ⊂ Y ⇒ µ(X) ≤ µ(Y ). (iii) µ(X) = µ(X). (iv) µ(convX) = µ(X). (v) µ(λX + (1− λ)Y ) ≤ λµ(X) + (1− λ)µ(Y ) for λ ∈ [0, 1]. (vi) If (Xn) is a sequence of closed sets from ME such that Xn+1 ⊂ Xn for n = 1, 2, . . . and if limn→∞ µ(Xn) = 0 then the set X∞ = ⋂∞ n=1Xn is nonempty. The set kerµ from axiom (i) is called the kernel of the measure of noncompactness µ. Let us notice that the intersection set X∞ in axiom (vi) belongs to the family kerµ [5]. This observation is important in our further investigations. To distinguish some classes of measures of noncompactness we say that the mea- sure µ is sublinear if it satisfies the following extra conditions: (vii) µ(X + Y ) ≤ µ(X) + µ(Y ). (viii) µ(λX) = |λ|µ(X) for λ ∈ R. We say that a measure of noncompactness µ has maximum property if (ix) µ(X ∪ Y ) = max{µ(X), µ(Y )}. We say that a measure of noncompactness µ is full if (x) kerµ = NE . A sublinear measure of noncompactness which is full and has maximum property is called a regular measure of noncompactness [5]. The first measure of noncompactness was defined in 1930 by Kuratowski [17] in the following way α(X) = inf { ε > 0 : X can be covered by a finite family of sets X1, X2, . . . , Xm such that diamXi ≤ ε for i = 1, 2, . . . ,m } . The measure α(X) is called the Kuratowski measure of noncompactness. It is known (see [5]) that the Kuratowski measure of noncompactness is regular. The EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 3 most useful measure of noncompactness is the Hausdorff measure of noncompact- ness which was introduced in [14, 15] by the definition χ(X) = inf{ε > 0 : X has a finite ε− net in E}. It is easily seen that the measure χ(X) is regular and equivalent to the Kuratowski measure α(X) [5]. Additionally, in some Banach spaces such as C([a, b]), lp and c0 we can write the exact formulas for the Hausdorff measure of noncompactness. Each such a formula involves the structure of a considered space. For example, in the sequence space l1 consisting of all sequences (xn) such that ∑∞ n=1 |xn| <∞ we have the formula χ(X) = lim n→∞ { sup { ∞∑ i=n |xi| : x = (xk) ∈ X }} . (2.1) It is worth mentioning that up to now we do not know exact formulas expressing the Kuratowski measure of noncompactness. This fact shows the importance and usefulness of the Hausdorff measure of noncompactness. Later on we will apply the fixed point theorem of Darbo type [5, 10] to prove our main result. Fundamental to this theorem is the concept of measure of non- compactness. 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 ∈ [0, 1) for which µ(QX) ≤ kµ(X) for an arbitrary nonempty subset X of Ω. Then the operator Q has at least one fixed point in the set Ω. Next, we consider the Banach space BC(R+, E). This space consists of all functions x : R+ → E which are continuous and bounded on R+. The space BC(R+, E) is equipped with the classical supremum norm defined in the following way ‖x‖∞ = sup{‖x(t)‖E : t ∈ R+} for x ∈ BC(R+, E). Additionally, we will assume that χ is the Hausdorff measure of noncompactness in the Banach space E. Now let us recall the definition of a measure of noncompactness in the space BC(R+, E) introduced in paper [3]. To this end assume that X is an arbitrary nonempty and bounded subset of the space BC(R+, E). Fix numbers ε > 0 and T > 0. For x ∈ X we denote by ωT (x, ε) the modulus of continuity of the function x on the interval [0, T ] defined by the formula ωT (x, ε) = sup{‖x(t)− x(s)‖E : t, s ∈ [0, T ], |t− s| ≤ ε}. Next, let us put ωT (X, ε) = sup{ωT (x, ε) : x ∈ X}, ωT0 (X) = lim ε→0 ωT (X, ε), ω0(X) = lim T→∞ ωT0 (X). (2.2) For a fixed t ∈ R+ we denote by X(t) = {x(t) : x ∈ X} the so-called cross-section of the set X at the point t. Obviously we have that X(t) ∈ME . For T > 0 we can define χT (X) = sup{χ(X(t)) : t ∈ [0, T ]} 4 A. CHLEBOWICZ EJDE-2021/61 and χ∞(X) = lim T→∞ χT (X). (2.3) For a fixed T > 0 we define aT (X) = sup x∈X { sup{‖x(t)‖E : t ≥ T} } and next we put a∞(X) = lim T→∞ aT (X). (2.4) Finally, gathering (2.2), (2.3) and (2.4) we can define χa(X) = ω0(X) + χ∞(X) + a∞(X). (2.5) It can be shown [3] that the function χa defined on the family MBC(R+,E) by formula (2.5) is a measure of noncompactness in the space BC(R+, E). The kernel kerχa of this measure consists of nonempty and bounded subsets X of the space BC(R+, E) such that functions from X are locally equicontinuous on R+ and for any t ∈ R+ the cross-section X(t) is relatively compact in the space E. Moreover, all functions belonging to X tend to zero at infinity with the same rate i.e, the following condition ∀ε>0 ∃T>0∀x∈X∀t≥T ‖x(t)‖E ≤ ε is satisfied. In what follows we will work in the Banach space BC(R+, E) for E = l1, where l1 is a classical sequence space. Thus, our considerations will be carried out in the Banach space BC(R+, l1). This space is denoted by BC1. The norm in the space BC1 is defined in the following way ‖x‖BC1 = sup{‖x(t)‖l1 : t ∈ R+} = sup { ∞∑ n=1 |xn(t)| : t ∈ R+ } , where x(t) = ( xn(t) ) ∈ BC1. In this case, based on formula (2.1), we can express the measure of noncompactness χa in the space BC1 by formula (2.5), where the components ω0(X), χ∞(X) and a∞(X) are represented in the following way (cf. [3]): ω0(X) = lim T→∞ { lim ε→0 { sup x=(xn)∈X { sup { ∞∑ n=1 |xn(t)− xn(s)| : t, s ∈ [0, T ], |t− s| ≤ ε }}}} , χ∞(X) = lim T→∞ { sup t∈[0,T ] { lim n→∞ { sup x=(xk)∈X { ∞∑ k=n |xk(t)| }}}} , (2.6) a∞(X) = lim T→∞ { sup x=(xn)∈X { sup t≥T { ∞∑ n=1 |xn(t)| }}} . (2.7) 3. Solutions of an infinite system of integral equations on the real half-axis In this section we study the existence of solutions to the infinite system of the nonlinear quadratic integral equations of Volterra-Hammerstein type of the form xn(t) = an(t) + fn(t, x1(t), x2(t), . . .) ∫ t 0 kn(t, s)gn(s, x1(s), x2(s), . . .) ds (3.1) EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 5 for n = 1, 2, . . . and for t ∈ R+. It is worth mentioning that in paper [9] we considered a particular case of equation (3.1) in the space BC1 and in paper [3] we considered also the same particular case of equation (3.1) but in the space BC0 = BC(R+, c0). Therefore our further result is a generalization of results obtained in [3, 9]. In this section we work in the Banach space BC1 = BC(R+, l1) described in Section 2. In what follows we formulate assumptions under which infinite system of integral equations (3.1) will be investigated. (i) The function sequence (an(t)) is an element of the space BC1 satisfying limt→∞ ∑∞ n=1 an(t) = 0. (ii) The functions kn(t, s) = kn : R2 + → R are continuous on the set R2 + (n = 1, 2, . . .). Moreover, the functions t→ kn(t, s) are locally equicontinuous on the set R+ uniformly with respect to s ∈ R+ i.e., the following condition is satisfied ∀T>0∀ε>0 ∃δ>0∀n∈N∀t1,t2∈[0,T ]∀s∈R+ [|t2 − t1| ≤ δ ⇒ |kn(t2, s)− kn(t1, s)| ≤ ε]. (iii) There exists a constant K1 > 0 such that ∞∑ n=1 ∫ t 0 |kn(t, s)|ds ≤ K1 for any t ∈ R+. (iv) The sequence (kn(t, s)) is equibounded on R2 + i.e., there exists a constant K2 > 0 such that |kn(t, s)| ≤ K2 for t, s ∈ R+ and n = 1, 2, . . .. (v) The function ∑∞ n=1 fn is defined on the set R+ × R∞ and takes real values. Moreover, the function t → ∑∞ n=1 fn(t, x1, x2, . . .) is locally uniformly continuous on R+ uniformly with respect to x = (xn) ∈ l1 i.e., the following condition is satisfied ∀ε>0∀T>0∃δ>0∀(xi)∈l1∀t,s∈[0,T ] [|t− s| ≤ δ ⇒ ∞∑ n=1 |fn(t, x1, x2, . . .)− fn(s, x1, x2, . . .)| ≤ ε]. (vi) There exists a function l : R+ → R+ which is nondecreasing on R+, con- tinuous at 0 and there exist a natural number p and a nonnegative integer q such that for any r > 0 and for x = (xi), y = (yi) ∈ l1 with ‖x‖l1 ≤ r, ‖y‖l1 ≤ r and for t ∈ R+, n ∈ N, n ≥ p+ 1 the following inequality |fn(t, x1, x2, . . .)− fn(t, y1, y2, . . .)| ≤ l(r) n+q∑ i=n−p |xi − yi| is satisfied. Moreover, the following inequality |fn(t, x1, x2, . . .)− fn(t, y1, y2, . . .)| ≤ l(r) n+q∑ i=n |xi − yi| holds for x = (xi), y = (yi) ∈ l1 with ‖x‖l1 ≤ r, ‖y‖l1 ≤ r and for t ∈ R+, 1 ≤ n ≤ p. (vii) There exists a function m : R+ → R+ nondecreasing on R+, continuous at 0 and there exists a sequence (fn) of nonnegative functions, belonging to the space 6 A. CHLEBOWICZ EJDE-2021/61 BC1 with the property limt→∞ ∑∞ n=1 fn(t) = 0 and such that for any r > 0 and for each x = (xi) ∈ l1 with ‖x‖l1 ≤ r the following inequality |fn(t, x1, x2, . . .)| ≤ fn(t) +m(r) n+q∑ i=n |xi| holds for n = 1, 2, . . . and for t ∈ R+, where q is the number from assumption (vi). On the basis of assumption (vii) we infer that there exists the finite constant F = sup{ ∑∞ n=1 fn(t) : t ∈ R+}. In view of assumption (i) we may define the finite constant A = sup{ ∑∞ n=1 |an(t)| : t ∈ R+}. Now we can formulate our other assumptions. (viii) The function gn = gn(t, x1, x1, . . .) is defined on the set R+×R∞ and takes real values for n = 1, 2, . . .. Moreover, the operator g defined on the set R+ × l1 by the equality (gx)(t) = (gn(t, x)) = (g1(t, x), g2(t, x), . . .) transforms the set R+ × l1 into l1 and is bounded on the set R+ × l1 by a positive G i.e., for any x ∈ l1 and for each t ∈ R+ we have that ‖(gx)(t)‖l1 ≤ G. Apart from this the family of functions {(gx)(t)}t∈R+ is equicontinuous at every point of the space l1. More precisely, for any arbitrarily fixed x ∈ l1 and for a given ε > 0 there exists δ > 0 such that ‖(gy)(t)− (gx)(t)‖l1 ≤ ε for every t ∈ R+ and for any y ∈ l1 such that ‖y − x‖l1 ≤ δ. (ix) There exists a positive solution r0 of the inequality A+ FGK1 + (p+ 1)GK1rm(r) ≤ r such that GK1(p+ q + 1) max{l(r0),m(r0)} < 1. In the sequel we will use in our considerations the following simple lemma (cf. [9]). Lemma 3.1. If the sequence (an) belongs to the space l1 then limn→∞ ∑∞ i=n |ai| = 0. The proof of the above lemma is an immediate consequence of the Cauchy condi- tion for real sequences. Now we formulate the main result of the paper concerning the solvability of infinite system of integral equations (3.1). Theorem 3.2. Under assumptions (i)–(ix) the infinite system of integral equations (3.1) has at least one solution x(t) = (xn(t)) in the space BC1 = BC(R+, l1). Proof. We define three operators F , V , Q on the space BC1 as follows: (Fx)(t) = ((Fnx)(t)) = (fn(t, x(t))) = (fn(t, x1(t), x2(t), . . .)), (V x)(t) = ((Vnx)(t)) = ( ∫ t 0 kn(t, s)gn(s, x1(s), x2(s), . . .) ds), (Qx)(t) = ((Qnx)(t)) = (an(t) + (Fnx)(t) (Vnx)(t)) for an arbitrary function x = (xn) ∈ BC1 and for t ∈ R+. EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 7 At the beginning we show that operator Q transforms the space BC1 into itself. Thus, let us fix a function x = x(t) = (xn(t)) ∈ BC1. Then, based on assumption (vii) for an arbitrary t ∈ R+ we obtain ∞∑ n=1 |(Fnx)(t)| = ∞∑ n=1 |fn(t, x1(t), x2(t), . . .)| ≤ ∞∑ n=1 [ fn(t) +m(‖x(t)‖l1) n+p∑ i=n |xi(t)| ] ≤ ∞∑ n=1 fn(t) +m(‖x‖BC1) ∞∑ n=1 ( n+p∑ i=n |xi(t)|) ≤ ∞∑ n=1 fn(t) + (p+ 1)m(‖x‖BC1 ) ∞∑ n=1 |xn(t)|. Hence we derive the estimate ‖Fx‖BC1 ≤ F + (p+ 1)m(‖x‖BC1 )‖x‖BC1 . (3.2) This shows that the function (Fx)(t) is bounded on the set R+. Next we show that the function Fx is continuous on the interval R+. To this end let us fix T0 and ε > 0. Next, choose a number δ > 0 according to assumption (v). Since the function x = x(t) belongs to the space BC1 so we can choose δ > 0 in such a way that the following condition ∀t,s∈[0,T ] [|t− s| ≤ δ ⇒ ‖x(t)− x(s)‖l1 ≤ ε] (3.3) is satisfied. Thus, taking t, s ∈ [0, T ] such that |t − s| ≤ δ and using assumptions (v) and (vi) we obtain ‖(Fx)(t)− (Fx)(s)‖l1 = ∞∑ n=1 |(Fnx)(t)− (Fnx)(s)| ≤ ∞∑ n=1 |fn(t, x1(t), x2(t), . . .)− fn(s, x1(t), x2(t), . . .)| + ∞∑ n=1 |fn(s, x1(t), x2(t), . . .)− fn(s, x1(s), x2(s), . . .)| ≤ ε+ ∞∑ n=1 l(‖x‖BC1 ) n+q∑ i=n−p |xi(t)− xi(s)| = ε+ l(‖x‖BC1 ) ∞∑ n=1 [ |xn−p(t)− xn−p(s)| + |xn−p+1(t)− xn−p+1(s)|+ . . .+ |xn+q(t)− xn+q(s)| ] ≤ ε+ l(‖x‖BC1) [ ∞∑ n=1 |xn−p(t)− xn−p(s)| + ∞∑ n=1 |xn−p+1(t)− xn−p+1(s)|+ . . . 8 A. CHLEBOWICZ EJDE-2021/61 + ∞∑ n=1 |xn+q(t)− xn+q(s)| ] ≤ ε+ l(‖x‖BC1)(p+ q + 1) ∞∑ n=1 |xn(t)− xn(s)| ≤ ε+ l(‖x‖BC1 (p+ q + 1)‖x(t)− x(s)‖l1 . Observe that in the above calculations we assumed that n ≥ p + 1. Obviously in the case when 1 ≤ n ≤ p we obtain also the same estimate. Now, choosing the number δ > 0 in such a way that both assumption (v) and condition (3.3) are satisfied we obtain ‖(Fx)(t)− (Fx)(s)‖l1 ≤ ε+ (p+ q + 1)l(‖x‖BC1 )ε. Hence we infer that the function Fx is continuous on the interval [0, T ]. Based on the arbitrariness of T we obtain the continuity of the function Fx on the interval R+. Combining the above established facts we conclude that the operator F transforms the space BC1 into itself. Next, we intend to show that the operator V transforms the space BC1 into itself. Thus, let us take an arbitrary function x = x(t) = (xn(t)) ∈ BC1. At first we show the boundedness of the function V x on the interval R+. Indeed, for an arbitrary number t ∈ R+, in virtue of assumptions (iii) and (viii), we obtain ∞∑ n=1 |(Vnx)(t)| ≤ ∞∑ n=1 ∫ t 0 |kn(t, s)| |gn(s, x1(s), x2(s), . . .)|ds ≤ ∞∑ n=1 ∫ t 0 |kn(t, s)|Gds = G ∞∑ n=1 ∫ t 0 |kn(t, s)|ds ≤ GK1. (3.4) Hence we infer that ‖(V x)(t)‖l1 ≤ GK1. This means that the function V x is bounded on R+. To prove the continuity of the function V x on the interval R+ let us fix ε > 0 and T > 0. Next, let us choose a number δ > 0 according to (3.3) and t1, t2 ∈ [0, T ) such that |t2 − t1| ≤ δ. Without loss of generality we may assume that t2 > t1. Then, keeping in mind assumptions (iv) and (viii) and applying the Lebesgue monotone convergence theorem [21], we obtain the following estimates: ∞∑ n=1 |(Vnx)(t2)− (Vnx)(t1)| ≤ ∞∑ n=1 | ∫ t2 0 kn(t2, s)gn(s, x1(s), x2(s), . . .) ds − ∫ t2 0 kn(t1, s)gn(s, x1(s), x2(s), . . .) ds| EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 9 + ∞∑ n=1 | ∫ t2 0 kn(t1, s)gn(s, x1(s), x2(s), . . .) ds − ∫ t1 0 kn(t1, s)gn(s, x1(s), x2(s), . . .) ds| ≤ ∞∑ n=1 ∫ t2 0 |kn(t2, s)− kn(t1, s)| |gn(s, x1(s), x2(s), . . .)|ds + ∞∑ n=1 ∫ t2 t1 |kn(t1, s)| |gn(s, x1(s), x2(s), . . .)|ds ≤ ∞∑ n=1 ∫ t2 0 ωTk (δ)|gn(s, x1(s), x2(s), . . .)|ds+ ∞∑ n=1 ∫ t2 t1 K2 |gn(s, x1(s), x2(s), . . .)|ds ≤ ωTk (δ) ∞∑ n=1 ∫ t2 0 |gn(s, x1(s), x2(s), . . .)|ds+K2 ∞∑ n=1 ∫ t2 t1 |gn(s, x1(s), x2(s), . . .)|ds = ωTk (δ) ∫ t2 0 ∞∑ n=1 |gn(s, x1(s), x2(s), . . .)|ds+K2 ∫ t2 t1 ∞∑ n=1 |gn(s, x1(s), x2(s), . . .)|ds ≤ ωTk (δ) ∫ t2 0 ‖(gx)(s)‖l1ds+K2 ∫ t2 t1 ‖(gx)(s)‖l1ds ≤ ωTk (δ)GT +K2Gδ, (3.5) where ωTk (δ) denotes a common modulus of continuity of the function sequence t→ kn(t, s) on the interval [0, T ] (according to assumption (ii)). Obviously ωTk (δ)→ 0 as δ → 0. Hence, on the basis of (3.5) we obtain the following estimate ‖(V x)(t2)− (V x)(t1)‖l1 ≤ ωTk (δ)GT +K2Gδ, which allows us to infer that the function V x is continuous on [0, T ). The number T was arbitrarily chosen so we deduce that the function V x is continuous on the interval R+. Finally, combining the established properties of the function V x we deduce that the operator V maps the space BC1 into itself. Next, we are going to show that the operator Q transforms the space BC1 into itself. To prove this fact let us notice that the space BC1 can be treated as the Banach algebra with respect to the coordinatewise multiplication of sequences of functions. Thus, keeping in mind the definition of the operator Q and taking into account that the functions Fx and V x are continuous on R+ we infer that the function Qx is also continuous on R+. In a similar way we prove the boundedness of the function Qx on R+. Finally, we conclude that the operator Q transforms the space BC1 into BC1. Next, in view of estimates (3.2), (3.4) and assumption (i) for arbitrarily fixed t ∈ R+ we obtain ∞∑ n=1 |(Qnx)(t)| ≤ ∞∑ n=1 |an(t)|+ ∞∑ n=1 [ |(Fnx)(t)‖(Vnx)(t)| ] ≤ A+ ( ∞∑ n=1 |(Fnx)(t)| )( ∞∑ n=1 |(Vnx)(t)| ) ≤ A+ ‖(Fx)(t)‖l1‖(V x)(t)‖l1 10 A. CHLEBOWICZ EJDE-2021/61 ≤ A+ ‖Fx‖BC1 ‖V x‖BC1 ≤ A+ [ F + (p+ 1)m(‖x‖BC1 )‖x‖BC1 ] GK1. Hence, we obtain the inequality ‖Qx‖BC1 = A+ FGK1 + (p+ 1)GK1m(‖x‖BC1 )‖x‖BC1 . From the above estimate and assumption (ix) we deduce that there exists a number r0 > 0 such that the operator Q transforms the ball Br0 into itself. Next we prove that the operator Q is continuous on the ball Br0 . To this end fix a number ε > 0 and an element x ∈ Br0 . Next, take an arbitrary function y ∈ Br0 such that ‖x − y‖BC1 ≤ ε. Fix arbitrarily number t ∈ R+. In view of assumption (vi) we obtain the following estimates: ∞∑ n=1 |(Fnx)(t)− (Fny)(t)| = ∞∑ n=1 |fn(t, x1(t), x2(t), . . .)− fn(t, y1(t), y2(t), . . .)| ≤ ∞∑ n=1 l(r0) n+q∑ i=n−p |xi(t)− yi(t)| = ∞∑ n=1 l(r0) [ |xn−p(t)− yn−p(t)|+ |xn−p+1(t)− yn−p+1(t)|+ . . . + |xn+q(t)− yn+q(t)| ] ≤ l(r0) [ ∞∑ n=1 |xn−p(t)− yn−p(t)|+ ∞∑ n=1 |xn−p+1(t)− yn−p+1(t)|+ . . . + ∞∑ n=1 |xn+q(t)− yn+q(t)| ] ≤ l(r0)(p+ q + 1) ∞∑ n=1 |xn(t)− yn(t)| ≤ l(r0)(p+ q + 1) ‖x(t)− y(t)‖l1 . (3.6) Consequently ‖Fx− Fy‖BC1 ≤ l(r0)(p+ q + 1)ε. This means that the operator F is continuous on the ball Br0 . Furthermore, we consider the function δ = δ(ε) defined for ε > 0 in the following way δ(ε) = sup{|gn(t, x)− gn(t, y)| : x, y ∈ l1, ‖x− y‖l1 ≤ ε, t ∈ R+, n ∈ N}. Then, by assumption (viii) we deduce that δ(ε) → 0 as ε → 0. Take ε > 0 and choose arbitrary x, y ∈ Br0 such that ‖x−y‖BC1 ≤ ε. For a fixed t ∈ R+ we obtain ∞∑ n=1 |(Vnx)(t)− (Vny)(t)| ≤ ∞∑ n=1 ∫ t 0 |kn(t, s)‖gn ( s, x1(s), x2(s), . . . ) − gn ( s, y1(s), y2(s), . . . ) |ds EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 11 ≤ ∞∑ n=1 ∫ t 0 |kn(t, s)|δ(ε) ds ≤ K1δ(ε). This implies the estimate ‖V x− V y‖BC1 ≤ K1δ(ε) and proves the continuity of the operator V on the ball Br0 . Finally, linking the continuity of the operators F and V on the ball Br0 and taking into account the representation of the operator Q given at the beginning of the proof, we infer that Q is continuous on Br0 . Further, we study the behaviour of the operators F , V and Q with respect to measure of noncompactness χa defined by (2.5) (cf. also the components of the measure χa defined by formulas (2.2)-(2.4)). To this end take a nonempty subset X of the ball Br0 . Next, fix arbitrary numbers ε > 0 and T > 0 and choose a function x = x(t) = (xn(t)) ∈ X. Then, for t, s ∈ [0, T ] such that |t − s| ≤ ε, making use of assumptions (v) and (vi), we obtain ∞∑ n=1 |(Fnx)(t)− (Fnx)(s)| ≤ ∞∑ n=1 |fn(t, x1(t), x2(t), . . .)− fn(s, x1(t), x2(t), . . .)| + ∞∑ n=1 |fn(s, x1(t), x2(t), . . .)− fn(s, x1(s), x2(s), . . .)| ≤ ωT1 (F, ε) + l(r0)(p+ q + 1) ∞∑ n=1 |xn(t)− xn(s)| ≤ ωT1 (F, ε) + l(r0)(p+ q + 1)ωT (x, ε), where we denoted ωT1 (F, ε) = sup { ∞∑ n=1 |fn(t, x1, x2, . . .)− fn(s, x1, x2, . . .)| : t, s ∈ [0, T ], |t− s| ≤ ε, x ∈ Br0 } . Obviously, in view of assumption (v) we have that ωT1 (F, ε) → 0 as ε → 0. From the above estimate we obtain ωT (Fx, ε) ≤ ωT1 (F, ε) + l(r0)(p+ q + 1)ωT (x, ε). (3.7) In a similar way as above (cf. estimate (3.5)) we obtain the inequality ωT (V x, ε) ≤ GTωTk (ε) +GK2ε. (3.8) Finally, take ε > 0 and T > 0. Then, for an arbitrary function x ∈ X and for t, s ∈ [0, T ] such that |t− s| ≤ ε based on (3.7), (3.8), (3.2) and (3.4) we obtain ‖(Qx)(t)− (Qx)(s)‖l1 ≤ ‖a(t)− a(s)‖l1 + ‖(Fx)(t)(V x)(t)− (Fx)(s)(V x)(s)‖l1 ≤ ‖a(t)− a(s)‖l1 + ‖(Fx)(t)(V x)(t)− (V x)(t)(Fx)(s)‖l1 + ‖(V x)(t)(Fx)(s)− (Fx)(s)(V x)(s)‖l1 ≤ ‖a(t)− a(s)‖l1 + ‖(V x)(t)‖l1‖(Fx)(t)− (Fx)(s)‖l1 + ‖(Fx)(s)‖l1‖(V x)(t)− (V x)(s)‖l1 ≤ ωT (a, ε) +GK1ω T (Fx, ε) 12 A. CHLEBOWICZ EJDE-2021/61 + [ F + (p+ 1)r0m(r0) ] ωT (V x, ε) ≤ ωT (a, ε) +GK1 { ωT1 (F, ε) + l(r0)(p+ q + 1)ωT (x, ε) } + [ F + (p+ 1) r0m(r0) ] { K2Gε+GTωTk (ε) } , where ωT1 (F, ε) and ωTk (ε) were defined previously. Now, from the above inequality, we derive ωT (QX, ε) ≤ ωT (a, ε) +GK1 { ωT1 (F, ε) + l(r0)(p+ q + 1)ωT (X, ε) } + [ F + (p+ 1)r0m(r0) ]{ K2Gε+GTωTk (ε) } . (3.9) From assumption (i) we deduce that ωT (a, ε) → 0 as ε → 0. Moreover, in view of assumption (v) we have that ωT1 (F, ε) → 0 as ε → 0. Similarly, keeping in mind assumption (ii) we conclude that ωTk (ε)→ 0 as ε→ 0. Thus, in view of the above established facts and estimate (3.9) we obtain ωT0 (QX) ≤ GK1(p+ q + 1)l(r0)ωT0 (X). Consequently, we obtain ω0(QX) ≤ GK1(p+ q + 1)l(r0)ω0(X). (3.10) In what follows we estimate the second component expressed by (2.6) of mea- sure of noncompactness χa defined by (2.5). More precisely, we will estimate the expression χ∞(QX) for an arbitrarily fixed nonempty subset X of the ball Br0 . To this end, fix a function x ∈ X and a number T > 0. Then, for t ∈ [0, T ] and for an arbitrary natural number n, arguing similarly as in estimates (3.2) and (3.4), we obtain ∞∑ i=n |(Qix)(t)| ≤ ∞∑ i=n |ai(t)|+ ∞∑ i=n |(Fix)(t)(Vix)(t)| ≤ ∞∑ i=n |ai(t)|+ [ ∞∑ i=n |(Fix)(t)| ][ ∞∑ i=n |(Vix)(t)| ] ≤ ∞∑ i=n |ai(t)|+ [ ∞∑ i=n fi(t) + (p+ 1)m(‖x(t)‖l1) ∞∑ i=n |xi(t)| ] GK1 ≤ ∞∑ i=n |ai(t)|+ [ ∞∑ i=n fi(t) + (p+ 1)m(r0) ∞∑ i=n |xi(t)| ] GK1. Hence, we derive sup x=(xi)∈X { ∞∑ i=n |(Qix)(t) } ≤ ∞∑ i=n |ai(t)|+GK1 ∞∑ i=n f i(t) +GK1(p+ 1)m(r0) sup x=(xi)∈X { ∞∑ i=n |xi(t)| } . Passing to the limit as n→∞ and using assumptions (i), (vii) and Lemma 3.1 we obtain χ∞(QX) ≤ GK1(p+ 1)m(r0)χ∞(X), (3.11) where χ∞(X) is defined by formula (2.3). Next we estimate the last component a∞(X) of the measure of noncompactness χa(X). Similarly as above, let us assume that X is a nonempty subset of the ball EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 13 Br0 . Fix a function x ∈ X and a number T > 0. Next, take an arbitrary number t ≥ T . Then, in view of (3.2) and (3.4), we obtain ∞∑ n=1 |(Qnx)(t)| ≤ ∞∑ n=1 |an(t)|+ (p+ 1)l(r0)GK1 ∞∑ n=1 |xn(t)|+GK1 ∞∑ n=1 fn(t). Hence, taking supremum over t ≥ T and x = (xn) ∈ X, we obtain sup x=(xn)∈X { sup t≥T { ∞∑ n=1 |(Qnx)(t)| }} ≤ sup t≥T { ∞∑ n=1 |an(t)| } + (p+ 1)l(r0)GK1 sup x=(xn)∈X { sup t≥T { ∞∑ n=1 |xn(t)| }} +GK1 sup t≥T { ∞∑ n=1 |fn(t)| } . Further, as T →∞ and applying assumptions (i) and (vii) we obtain the inequality a∞(QX) ≤ (p+ 1)l(r0)GK1a∞(X). (3.12) Finally, combining (3.10), (3.11), (3.12) and keeping in mind formula (2.5), we obtain the inequality χa(QX) ≤ GK1(p+ q + 1) max{l(r0),m(r0)}χa(X). (3.13) Now, in view of Theorem 2.2, estimate (3.13) and the second inequality from as- sumption (ix) we conclude that there exists at least one solution x(t) = (xn(t)) of infinite system of integral equations (3.1) belonging to the ball Br0 in the space BC1 = BC(R+, l1). The proof is complete. � 4. An example In this section we illustrate our main result obtained in Theorem 3.2 by an exam- ple. We consider the infinite system of nonlinear quadratic Volterra-Hammerstein integral equations x1(t) = αe−2t + ( β 1 + t2 + γ x3(t) 1 + x21(t) ) × ∫ t 0 1 (1 + s2)[1 + (t+ s)2] arctan s 12 [ x1(s) 1 + x21(s) + x2(s) 1 + 22x22(s) ] ds, x2(t) = αe−2t + ( β 2(22 + t2) + γ x3(t) 1 + x21(t) + γ x4(t) 1 + x22(t) ) × ∫ t 0 1 (22 + s2)[1 + (t+ s)2] arctan s 22 [ x2(s) 1 + 22x22(s) + x3(s) 1 + 32x23(s) ] ds, . . . (4.1) xn(t) = αe−2t tn−1 (n− 1)! + ( β n(n2 + t2) + γ xn(t) 1 + x2n−2(t) + γ xn+1(t) 1 + x2n−1(t) + γ xn+2(t) 1 + x2n(t) ) × ∫ t 0 1 (n2 + s2)[1 + (t+ s)2] arctan s n2 [ xn(s) 1 + n2x2n(s) + xn+1(s) 1 + (n+ 1)2x2n+1(s) ] ds, 14 A. CHLEBOWICZ EJDE-2021/61 where t ∈ R+, n ∈ N, n ≥ 3 and α, β, γ are positive constants. Observe that infinite system of integral equations (4.1) is a special case of system (3.1) if we put an(t) = αe−2t tn−1 (n− 1)! , (4.2) fn(t, x1, x2, . . .) = β n(n2 + t2) + γ xn 1 + x2n−2 + γ xn+1 1 + x2n−1 + γ xn+2 1 + x2n , (4.3) kn(t, s) = 1 (n2 + s2)[1 + (t+ s)2] , (4.4) gn(t, x1, x2, . . .) = arctan t n2 [ xn 1 + n2x2n + xn+1 1 + (n+ 1)2x2n+1 ] , (4.5) where n ∈ N, and t, s ∈ R+ in formulas (4.2), (4.4) and (4.5), while in formula (4.3) n ≥ 3 and t, s ∈ R+. In the case n = 1 and n = 2 formula (4.3) should be replaced by the following ones: f1(t, x1, x2, . . .) = β 1 + t2 + γ x3 1 + x21 , (4.6) f2(t, x1, x2, . . .) = β 2(22 + t2) + γ x3 1 + x21 + γ x4 1 + x22 . (4.7) Notice that in our further investigations we will always consider formula (4.3) in order to express the function fn(t, x1, x2, . . .). Now we show that infinite system of integral equations(4.1) has a solution in the Banach space BC1. Our goal is to show that the components of the infinite system of integral equations (4.1) defined by formulas (4.2)-(4.5) satisfy assumptions of Theorem 3.2. At the beginning observe that the sequence of functions (an(t)) belongs the space BC1. Indeed, for arbitrarily fixed t ∈ R+ we have ∞∑ n=1 an(t) = αe−2t ∞∑ n=1 tn−1 (n− 1)! = αe−2tet = αe−t. Obviously, we have that lim t→∞ ∞∑ n=1 an(t) = lim t→∞ αe−t = 0. This shows that assumption (i) is satisfied. We obtain A = sup { ∞∑ n=1 |an(t)| : t ∈ R+ } = α. Further, for arbitrarily fixed t1, t2 ∈ R+, s ∈ R+ and for n ∈ N, we obtain |kn(t2, s)− kn(t1, s)| = 1 n2 + s2 ∣∣∣ 1 1 + (t2 + s)2 − 1 1 + (t1 + s)2 ∣∣∣ ≤ 1 n2 |(t2 + s)2 − (t1 + s)2| [1 + (t2 + s)2][1 + (t1 + s)2] ≤ |t2 − t1|(t2 + t1 + 2s) [1 + (t2 + s)2][1 + (t1 + s)2] ≤ 1 n2 |t2 − t1| [ t2 + s 1 + (t2 + s)2 + t1 + s 1 + (t1 + s)2 ] EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 15 ≤ 1 n2 |t2 − t1| (1 2 + 1 2 ) ≤ |t2 − t1|. Hence we see that the functions kn(t, s) are even equicontinuous on the set R+ uniformly with respect to s ∈ R+. Thus the functions kn(t, s) satisfy assumption (ii). Also, observe that ∞∑ n=1 ∫ t 0 |kn(t, s)|ds = ∞∑ n=1 ∫ t 0 1 n2 + s2 · 1 1 + (t+ s)2 ds ≤ ∞∑ n=1 1 n2 ∫ t 0 1 1 + (t+ s)2 ds = ∞∑ n=1 1 n2 ∫ 2t t du 1 + u2 ≤ ∞∑ n=1 1 n2 arctan 2t ≤ π2 6 · π 2 = π3 12 . From the above estimate we deduce that the functions kn(t, s) satisfy assumption (iii) with the constant K1 = π3/12. Next, taking arbitrary t, s ∈ R+ and n ∈ N, we have |kn(t, s)| = 1 n2 + s2 · 1 1 + (t+ s)2 ≤ 1 n2 ≤ 1. This shows that assumption (iv) is satisfied with the constant K2 = 1. Now we verify assumptions (v), (vi) and (vii) for the function fn(t, x1, x2, . . .) defined by (4.3). The cases n = 1 and n = 2 are easier to verify so we consider the case n ≥ 3. We have fn(t, x1, x2, . . .) = β n(n2 + t2) + γ xn 1 + x2n−2 + γ xn+1 1 + x2n−1 + γ xn+2 1 + x2n . Hence for an arbitrary t ∈ R+ and an arbitrary x = (xn) ∈ l1 the series ∞∑ n=1 fn(t, x1, x2, . . .) is uniformly convergent on the set R+×l1. This conclusion follows immediately from the standard Weierstrass test. Next, fix arbitrarily t, s ∈ R+ and x = (xn) ∈ l1. Then we obtain ∞∑ n=1 |fn(t, x1, x2, . . .)− fn(s, x1, x2, . . .)| = ∞∑ n=1 1 n ∣∣∣ β n2 + t2 − β n2 + s2 ∣∣∣ ≤ β ∞∑ n=1 1 n · |t2 − s2| (n2 + t2)(n2 + s2) 16 A. CHLEBOWICZ EJDE-2021/61 ≤ β ∞∑ n=1 1 n |t− s| [ t (n2 + t2)(n2 + s2) + s (n2 + t2)(n2 + s2) ] ≤ β ∞∑ n=1 |t− s|1 2 [ 1 n2 + s2 + 1 n2 + t2 ] ≤ β ∞∑ n=1 |t− s|1 2 · 2 n2 = β π2 6 |t− s|. This shows that the second part of assumption (v) is satisfied (even more, the function t → ∑∞ n=1 fn(t, x1, x2, . . .) is uniformly continuous on R+ uniformly with respect to x = (xn) ∈ l1). Further on, fix an arbitrary number r > 0 and take x = (xn), y = (yn) ∈ l1 such that ‖x‖l1 ≤ r, ‖y‖l1 ≤ r. Then, for arbitrarily fixed t ∈ R+ and for n ∈ N (n ≥ 3), we obtain |fn(t, x1, x2, . . .)− fn(t, y1, y2, . . .)| ≤ γ ∣∣∣ xn 1 + x2n−2 − yn 1 + y2n−2 ∣∣∣+ γ ∣∣∣ xn+1 1 + x2n−1 − yn+1 1 + y2n−1 ∣∣∣ + γ ∣∣∣ xn+2 1 + x2n − yn+2 1 + y2n ∣∣∣. (4.8) To estimate the right-hand side of (4.8), we consider numbers a, b, c, d such that |a| ≤ r, |b| ≤ r, |c| ≤ r and |d| ≤ r. We have∣∣∣ a 1 + b2 − c 1 + d2 ∣∣∣ = |a(1 + d2)− c(1 + b2)| (1 + b2)(1 + d2) = |a− c+ ad2 − cb2| (1 + b2)(1 + d2) ≤ |a− c|+ |ad 2 − cb2| (1 + b2)(1 + d2) ≤ |a− c|+ |ad 2 − cd2|+ |cd2 − cb2| (1 + b2)(1 + d2) ≤ |a− c|+ |a− c|d 2 + |c‖b− d|(|b|+ |d|) (1 + b2)(1 + d2) ≤ |a− c|+ |a− c| d2 (1 + b2)(1 + b2) + |c‖b− d| [ |b| (1 + b2)(1 + d2) + |d| (1 + b2)(1 + d2) ] ≤ |a− c|+ |a− c| 1 1 + b2 + |c‖b− d|+ (1 2 1 1 + d2 + 1 2 1 1 + b2 ) ≤ 2|a− c|+ r|b− d| ≤ max{2, r}(|a− c|+ |b− d|). EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 17 Using this inequality to estimate (4.8), we obtain |fn(t, x1.x2, . . .)− fn(t, y1, y2, . . .)| ≤ γ max {2, r} [|xn−2 − yn−2|+ |xn − yn|] + γ max {2, r} [ |xn−1 − yn−1|+ |xn+1 − yn+1| ] + γ max {2, r} [|xn − yn|+ |xn+2 − yn+2|] ≤ 2γ max {2, r} n+2∑ i=n−2 |xi − yi|. The above estimate shows that assumption (vi) is satisfied with p = q = 2 and l(r) = 2γmax{2, r}. Assume that t ∈ R+ and n is an arbitrarily fixed natural number n (n ≥ 3). Then for a fixed r > 0 and for x = (xn) ∈ l1 with ‖x‖l1 ≤ r we obtain |fn(t, x1, x2, . . .)| ≤ β n(n2 + t2) + γ |xn| 1 + x2n−2 + γ |xn+1| 1 + x2n−1 + γ |xn+2| 1 + x2n ≤ β n(n2 + t2) + γ(|xn|+ |xn+1|+ |xn+2|) = β n(n2 + t2) + γ n+2∑ i=n |xi|. Thus we see that assumption (vii) is satisfied with m(r) = γ and fn(t) = β n(n2+t2) for n = 1, 2, . . .. Observe that for t > 0 we have ∞∑ n=1 fn(t) = β ∞∑ n=1 1 n(n2 + t2) ≤ β ∞∑ n=1 1 nt2 = β t ∞∑ n=1 1 n2 = βπ2 6t . This implies that limt→∞ ∑∞ n=1 fn(t) = 0 and shows that the remaining part of assumption (vii) is also satisfied. Obviously, we have that sup { ∞∑ n=1 fn(t) : t ∈ R+ } ≤ β ∞∑ n=1 1 n3 ≤ β π 2 6 . Thus, we can accept that F = β π 2 6 , where F is the constant defined as F = sup {∑∞ n=1 fn(t) : t ∈ R+ } . To verify assumption (viii) let us first notice that for a fixed natural number n the function gn = gn(t, x1, x2, . . .) defined on the set R+ × R∞ by formula (4.5) takes real values for n = 1, 2, . . .. Next, we fix arbitrarily x = (xn) ∈ l1. Then, for 18 A. CHLEBOWICZ EJDE-2021/61 t ∈ R+ we obtain ∞∑ n=1 |gn(t, x1, x2, . . .) ≤ ∞∑ n=1 arctan t n2 [ |xn| 1 + n2x2n + |xn+1| 1 + (n+ 1)2x2n+1 ] ≤ π 2 ∞∑ n=1 1 n2 ( 1 2 + 1 4 ) = 3π 8 ∞∑ n=1 1 n2 = π3 16 . (4.9) It follows from the above estimate that the operator g defined in assumption (viii) transforms the set R+× l1 into l1. Moreover, on the basis of (4.9) we infer that the operator g is bounded on R+× l1 and ‖(gx)(t)‖l1 ≤ π3/16. This means that we can accept in our situation that G = π3/16, where G is the constant from assumption (viii). Now we take a number ε > 0 and choose arbitrary elements x = (xn), y = (yn) ∈ l1 such that ‖x− y‖l1 ≤ ε. Then, for a fixed t ∈ R+ we obtain ‖(gy)(t)− (gx)(t)‖l1 = ∞∑ n=1 |gn(t, y1, y2, . . .)− gn(t, x1, x2, . . .)| = ∞∑ n=1 arctan t n2 ∣∣∣ yn 1 + n2y2n + yn+1 1 + (n+ 1)2y2n+1 − xn 1 + n2x2n − xn+1 1 + (n+ 1)2x2n+1 ∣∣∣ = π 2 ∞∑ n=1 1 n2 [∣∣∣ yn 1 + n2y2n − xn 1 + n2x2n ∣∣∣ + ∣∣∣ yn+1 1 + (n+ 1)2y2n+1 − xn+1 1 + (n+ 1)2x2n+1 ∣∣∣] ≤ π 2 ∞∑ n=1 1 n2 [ |yn + n2ynx 2 n − xn − n2xny2n| (1 + n2y2n)(1 + n2x2n) + |yn+1 + (n+ 1)2yn+1x 2 n+1 − xn+1 − (n+ 1)2xn+1y 2 n+1| (1 + (n+ 1)2y2n+1)(1 + (n+ 1)2x2n+1) ] ≤ π 2 ∞∑ n=1 1 n2 [ |yn − xn|+ n2|xn‖yn‖yn − xn| (1 + n2y2n)(1 + n2x2n) + |yn+1 − xn+1|+ (n+ 1)2|xn+1‖yn+1‖yn+1 − xn+1| (1 + (n+ 1)2y2n+1)(1 + (n+ 1)2x2n+1) ] ≤ π 2 ∞∑ n=1 1 n2 { |yn − xn| [ 1 (1 + n2y2n)(1 + n2x2n) + n|yn| 1 + n2y2n · n|xn| 1 + n2x2n ] + |yn+1 − xn+1| [ 1 (1 + (n+ 1)2y2n+1)(1 + (n+ 1)2x2n+1) + (n+ 1)|yn+1| 1 + (n+ 1)2y2n+1 · (n+ 1)|xn+1| 1 + (n+ 1)2x2n+1 ]} EJDE-2021/61 EXISTENCE OF SOLUTIONS TO SYSTEMS OF INTEGRAL EQUATIONS 19 ≤ π 2 ∞∑ n=1 1 n2 [ |yn − xn|(1 + 1 2 · 1 2 ) + |yn+1 − xn+1|(1 + 1 2 · 1 2 ) ] ≤ 5π 8 ∞∑ n=1 1 n2 [ |yn − xn|+ |yn+1 − xn+1 ] ≤ 5π 8 ∞∑ n=1 1 n2 ∞∑ k=1 |yk − xk| = 5π3 48 ‖y − x‖l1 ≤ 5π3 8 ε. This proves the last part of assumption (viii). Finally, let us consider the first inequality from assumption (ix). Gather all values of the constants A,F ,G,K1, p, q and take into account the formulas for l(r) and m(r), then we see that the mentioned inequality has the form α+ β π8 1152 + γ π6 64 r ≤ r. (4.10) Thus, taking γ < 64/π6 we infer that any number r such that r ≥ 1 18 · 1152α+ π8β 64− π6γ satisfies inequality (4.10). Hence it follows that we can accept the value of r0 as r0 = 1 18 · 1152α+ π8β 64− π6γ . The second inequality from (ix) has the form 5π6 192 max{2γmax{2, r0}, γ} < 1 and we can choose γ so that the above inequality is satisfied. This proves that assumption (ix) is satisfied. Now we apply Theorem 3.2 and we obtain that infinite system of nonlinear integral equations 3.1 has at least one solution x = x(t) = (xn(t)) in the space BC1 = BC(R+, l1). Remark 4.1. Let us notice that the functions kn(t, s) and gn(t, x1, x2, . . .) in equa- tion (4.1) defined by formulas (4.4), (4.5) coincide with the functions kn(t, s) and gn(t, x1, x2, . . .) from Example 5.1 in [9] (n=1,2,. . . ). However, we repeated calcu- lations concerning those functions to make our paper complete. References [1] R. R. Akhmerov, M. I. Kamenski, A. S. Potapov, A. E. Rodkina, B. N. Sadovskii; Measures of Noncompactness and Condensing Operators, Birkhäuser Verlag, Basel, 1992. [2] J. Ayerbe Toledano, T. Dominguez Benavides. G. Lopez Acedo; Measures of Noncompactness in Metric Fixed Point Theorem, Birkhäuser Verlag, Basel, 1997. [3] J. Banaś, A. Chlebowicz; On solutions of an infinite system of nonlinear integral equations on the real half-axis, Banach J. Math. Anal., 13 (2019), 944–968. [4] J. Banaś, A. Chlebowicz, W. Woś; On measures of noncompactness in the space of functions defined on the half-axis with values in a Banach space, J. Math. Anal. 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Jalal; Infinite systems of integral equations in two variables of Hammerstein type in c0 and l1 spaces, Filomat, 33 (2019), 3441–3455. [19] M. Mursaleen, S. M. H. Rizvi; Solvability of infinite systems of second order differential equations in c0 and l1 by Meir-Keeler condensing operators, Proc. Amer. Math. Soc., 144 (2016), 4279–4289. [20] V. Radulescu, D. Repovš; Partial Differential Equations with Variable Exponents: Varia- tional Methods and Qualitative Analysis, CRC Press, Boca Raton, 2015. [21] W. Rudin; Real and Complex Analysis, McGraw-Hill, New York, 1987. [22] P. P. Zabrejko, A. I. Koshelev, M. A. Krasnosel’skii, S. G. Mikhlin, L. S. Rakovschik, J. Stetsenko; Integral Equations, Nordhoff, Leyden, 1975. Agnieszka Chlebowicz Department of Nonlinear Analysis, Rzeszów University of Technology, al. Powstańców Warszawy 8, 35-959 Rzeszów, Poland Email address: agnchleb@prz.edu.pl 1. Introduction 2. Preliminaries 3. Solutions of an infinite system of integral equations on the real half-axis 4. An example References