On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations

Agnieszka Chlebowicz; Beata Rzepka

doi:10.1515/anona-2025-0105

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On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations

Agnieszka Chlebowicz and Beata Rzepka

Published/Copyright: August 29, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

In this article, we deal with the solvability of an infinite system of Volterra-Hammerstein-Stieltjes integral equations in the space of continuous and bounded functions defined on R + with values in the sequence space l 1 . We expand a known existence result for such an infinite system of integral equations and prove a theorem for a larger class of infinite systems of integral equations of this type.

Keywords: space of continuous and bounded functions defined on the real half-axis; space l 1 ; measure of noncompactness; fixed point theorem of Darbo type

MSC 2010: 47H08; 45G15

1 Introduction and preliminaries

For the past two decades, there have been several articles on differential and integral equations of fractional order [1,11,15,18]. It turned out that such equations can be considered as integral equations of Stieltjes type [2,10]. This approach made it possible to start a completely new research direction, which was initiated in [7,8] and developed in [3]. The article [3] is devoted to investigate conditions that guarantee the solvability of infinite system of Volterra-Hammerstein-Stieltjes integral equations in the space of bounded and continuous functions defined on R + with values in the space c 0 of real sequences converging to zero.

In this article, we cover a topic that was sparked in [3]. Our main result is also concerned with the solvability of the infinite system of integral equations of Volterra-Hammerstein-Stieltjes type, but in this case, we are looking for solutions in the space of bounded and continuous functions defined on R + with values in the space l 1 of absolutely summable real sequences. Since l 1 ⊂ c 0 such considerations are justified.

The main tools used in this article are measure of noncompactness (see [1,4–6,21,22]) and the fixed point theorem of Darbo type [9,14]. We will briefly describe these notions.

The first measure of noncompactness was introduced by Kuratowski [19] in 1930. He used the known fact that a bounded subset A of a Banach space E is relatively compact if and only if for each ε > 0 , one can find finitely many sets of diameter less or equal to ε , which cover A and defined a function

α ( A ) = inf { ε > 0 : A can be covered by a finite family of sets A 1 , A 2 , … , A m such that diam A i ≤ ε for i = 1 , 2 , … , m } .

The measure α ( X ) is called the Kuratowski measure of noncompactness. The similar measure of noncompactness is the Hausdorff measure of noncompactness, which was introduced in [16,17] by the following definition:

χ ( A ) = inf { ε > 0 : A has a finite ε -net in E } .

Functions α and χ take the value zero only for relatively compact sets, hence their name “measure of noncompactness.” Moreover, they are invariant under the passage to the convex hull (which is crucial in the proof of Darbo’s fixed point theorem) and share many “good” properties such as set additivity, algebraic additivity, monotonicity, and algebraic homogeneity. Therefore, it seems that the best way of dealing with measures of noncompactness is an axiomatic approach. The first who chose this way was Sadovskii [20] in 1972 and Banaś and Goebel [5] in 1980.

To introduce an axiomatic definition of a measure of noncompactness, we need some preliminaries. Thus, we will use the standard symbols R and N for the set of real and natural numbers, respectively. Moreover, we put R + = [ 0 , ∞ ) . Next, we assume that ( E , ‖ ⋅ ‖ ) is a Banach space with zero vector θ . In this space, we denote by B ( x , r ) the closed ball of radius r around x and we use the shortcut B r to denote the ball B ( θ , r ) . If X and Y are the subsets of E , then we use the standard symbols X + Y , λ X ( λ ∈ R ) to denote the algebraic operations on subsets of E . Next, we denote by X ¯ the closure of X and by Conv X the closed convex hull of the set X ⊂ E . In E , we define the following families:

M E = { A ⊂ E : A is nonempty and bounded } , N E = { A ∈ M E : A is relatively compact } .

Then, we can give the definition of a measure of noncompactness (cf. [5]).

Definition 1.1

A function μ : M E → R + is called a measure of noncompactness in E if it satisfies the following conditions:

The family ker μ = { X ∈ M E : μ ( X ) = 0 } is nonempty and ker μ ⊂ N E .
X ⊂ Y ⇒ μ ( X ) ≤ μ ( Y ) .
μ ( X ¯ ) = μ ( X ) .
μ ( Conv X ) = μ ( X ) .
μ ( λ X + ( 1 − λ ) Y ) ≤ λ μ ( X ) + ( 1 − λ ) μ ( Y ) for λ ∈ [ 0 , 1 ] .
If ( X n ) is a sequence of closed sets from M E such that X n + 1 ⊂ X n for n = 1 , 2 , … and if lim n → ∞ μ ( X n ) = 0 , then the set X ∞ = ⋂ n = 1 ∞ X n is nonempty.

Remark 1.2

We can note (see [5]) that X ∞ ∈ ker μ .

Next, let us consider the Banach space of all bounded and continuous functions x : R + → E and denote this space by BC ( R + , E ) . It is normed by the classical supremum norm

‖ x ‖ ∞ = sup { ‖ x ( t ) ‖ E : t ∈ R + } ,

for x ∈ BC ( R + , E ) . Assume also that χ is the Hausdorff measure of noncompactness in the Banach space E . Now, we describe the construction of a measure of noncompactness in the space BC ( R + , E ) introduced in article [4,13]. To this end, assume that X is any nonempty and bounded subset of the space BC ( R + , E ) . Fix numbers ε > 0 and T > 0 . For x ∈ X , we denote by ω T ( x , ε ) the modulus of continuity of the function x on the interval [ 0 , T ] defined by the formula

ω T ( x , ε ) = sup { ‖ x ( t ) − x ( s ) ‖ E : t , s ∈ [ 0 , T ] , ∣ t − s ∣ ≤ ε } .

Next, we put

ω T ( X , ε ) = sup { ω T ( x , ε ) : x ∈ X } , ω 0 T ( X ) = lim ε → 0 ω T ( X , ε ) ,

and

(1) ω 0 ( X ) = lim T → ∞ ω 0 T ( X ) .

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 ) ∈ M E . For T > 0 , we define

χ ¯ T ( X ) = sup { χ ( X ( t ) ) : t ∈ [ 0 , T ] }

and

(2) χ ¯ ∞ ( X ) = lim T → ∞ χ ¯ T ( X ) .

For a fixed T > 0 , we define

a T ( X ) = sup x ∈ X { sup { ‖ x ( t ) ‖ E : t ≥ T } } ,

and next, we put

(3) a ∞ ( X ) = lim T → ∞ a T ( X ) .

Finally, gathering (1)–(3) we define

(4) χ a ( X ) = ω 0 ( X ) + χ ¯ ∞ ( X ) + a ∞ ( X ) .

It can be shown [4] that the function χ a defined on the family M BC ( R + , E ) by formula (4) is a measure of noncompactness in the space BC ( R + , E ) .

Therefore, if we want to apply the aforementioned construction to obtain the measure of noncompactness in BC ( R + , E ) for the given space E , we must know, besides the norm in the space E , the formula for the Hausdorff measure of noncompactness in E . It is known that in some Banach spaces ( c 0 , l p for 1 ≤ p < ∞ , C ( [ a , b ] ) ), we know such formulas. In this article, we will work in the space BC ( R + , E ) for E = l 1 . The space BC ( R + , l 1 ) we denote by BC 1 . The space l 1 is the space of all real sequences ( x n ) satisfying the condition ∑ n = 1 ∞ ∣ x n ∣ < ∞ . The norm in l 1 is given by

‖ ( x n ) ‖ l 1 = ∑ n = 1 ∞ ∣ x n ∣ .

Hence, the norm in the space BC 1 is defined in the following way:

‖ x ‖ BC 1 = sup { ‖ x ( t ) ‖ l 1 : t ∈ R + } = sup ∑ n = 1 ∞ ∣ x n ( t ) ∣ : t ∈ R + ,

where x ( t ) = ( x n ( t ) ) ∈ BC 1 . The exact formula for Hausdorff measure of noncompactness χ in l 1 is as follows:

(5) χ ( X ) = lim n → ∞ sup ∑ i = n ∞ ∣ x i ∣ : x = ( x k ) ∈ X .

The knowledge of formula (5) will allow us to obtain a formula for the measure of noncompactness in the space BC 1 . This is extremely important, since the use of the measure of noncompactness in the space BC 1 in our proof of main result of this article requires that this measure can be represented by some alternative explicit formula. In this case, based on formula (5), we can express the measure of noncompactness χ a in the space BC 1 by formula (4), where the components ω 0 ( X ) , χ ¯ ∞ ( X ) , and a ∞ ( X ) are represented in the following way (cf. [4]):

(6) ω 0 ( X ) = lim T → ∞ lim ε → 0 sup x = ( x n ) ∈ X sup ∑ n = 1 ∞ ∣ x n ( t ) − x n ( s ) ∣ : t , s ∈ [ 0 , T ] , ∣ t − s ∣ ≤ ε ,

(7) χ ¯ ∞ ( X ) = lim T → ∞ sup t ∈ [ 0 , T ] lim n → ∞ sup x = ( x k ) ∈ X ∑ i = n ∞ ∣ x i ( t ) ∣ ,

(8) a ∞ ( X ) = lim T → ∞ sup x = ( x n ) ∈ X sup t ≥ T ∑ n = 1 ∞ ∣ x n ( t ) ∣ .

The second tool used in the proof of our main result is the fixed point theorem of Darbo type [5,14]. Fundamental to this theorem is the concept of a measure of noncompactness.

Theorem 1.3

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 ( Q X ) ≤ k μ ( X ) for an arbitrary nonempty subset X of Ω . Then, the operator Q has at least one fixed point in the set Ω .

Moreover, in this article, we will work with the Stieltjes integrals of the form ∫ a b x ( s ) d s g ( t , s ) , where g : [ a , b ] × [ a , b ] → R and d s denotes the integration with respect to the variable s . If we have a function of two variables u ( t , s ) = u : [ a , b ] × [ c , d ] → R , then by ⋁ t = p q u ( t , s ) we denote the variation of the function t ↦ u ( t , s ) on the interval [ p , q ] ⊂ [ a , b ] , where s is fixed, s ∈ [ c , d ] . Analogously, we define ⋁ s = p q u ( t , s ) .

2 Main result

In this section, we study the existence of solutions in BC 1 for the infinite system of nonlinear quadratic integral equations of Volterra-Hammerstein-Stieltjes type of the form

(9) x n ( t ) = a n ( t ) + f n ( t , x 1 ( t ) , x 2 ( t ) , … ) ∫ 0 t k n ( t , s ) g n ( s , x 1 ( s ) , x 2 ( s ) , … ) d s H n ( t , s ) .

System (9) contains as a special case system of integral equations considered in [12].

At the beginning, we define three operators F , V , and Q on the space BC 1 :

(10) ( F x ) ( t ) = ( ( F n x ) ( t ) ) = ( f n ( t , x ( t ) ) ) = ( f n ( t , x 1 ( t ) , x 2 ( t ) , … ) ) ,

(11) ( V x ) ( t ) = ( ( V n x ) ( t ) ) = ∫ 0 t k n ( t , s ) g n ( s , x 1 ( s ) , x 2 ( s ) , … ) d s H n ( t , s ) ,

(12) ( Q x ) ( t ) = ( ( Q n x ) ( t ) ) = ( a n ( t ) + ( F n x ) ( t ) ( V n x ) ( t ) ) ,

for an arbitrary function x = ( x n ) ∈ BC 1 and for t ∈ R + .

Now, we prove the lemma concerning the operator F . So, assume that

The function ∑ n = 1 ∞ f n is defined on the set R + × l 1 and takes real values. Moreover, the function t ↦ ∑ n = 1 ∞ f n ( t , x 1 , x 2 , … ) is continuous on R + uniformly with respect to x = ( x n ) ∈ l 1 , i.e.,
∀ t 1 ∈ R + ∀ ε > 0 ∃ δ > 0 ∀ ( x i ) ∈ l 1 ∀ t 2 ∈ R + ( ∣ t 2 − t 1 ∣ ≤ δ ⇒ ∑ n = 1 ∞ ∣ f n ( t 2 , x 1 , x 2 , … ) − f n ( t 1 , x 1 , x 2 , … ) ∣ ≤ ε ) .
There exists a nondecreasing and continuous at 0 function l : R + → R + such that for any r > 0 and for all x = ( x i ) , y = ( y i ) ∈ l 1 with ‖ x ‖ l 1 ≤ r , ‖ y ‖ l 1 ≤ r and for t ∈ R + , n ∈ N , the following inequality is satisfied:
∣ f n ( t , x 1 , x 2 , … ) − f n ( t , y 1 , y 2 , … ) ∣ ≤ l ( r ) ∣ x n − y n ∣ .
Assume that ( f ¯ n ( t ) ) ∈ BC 1 and lim t → ∞ ∑ n = 1 ∞ f ¯ n ( t ) = 0 , where f ¯ n ( t ) = ∣ f n ( t , 0 , 0 , … ) ∣ .

We infer from (iii) that the constant F ¯ = sup ∑ n = 1 ∞ f ¯ n ( t ) : t ∈ R + is finite.

Lemma 2.1

Assume that conditions (i)–(iii) hold. Then, the operator F defined by (10) transforms the space BC 1 into itself and is continuous on every ball B r ⊂ BC 1 .

Proof

Take a function x = x ( t ) = ( x n ( t ) ) ∈ BC 1 . We will show that F x ∈ BC 1 that is F x : R + → l 1 and is bounded and continuous. From assumptions ( i i ) and (iii) for any t ∈ R + , we have

∑ n = 1 ∞ ∣ ( F n x ) ( t ) ∣ = ∑ n = 1 ∞ ∣ f n ( t , x 1 ( t ) , x 2 ( t ) , … ) ∣ ≤ ∑ n = 1 ∞ ∣ f n ( t , x 1 ( t ) , x 2 ( t ) , … ) − f n ( t , 0 , 0 , … ) ∣ + ∑ n = 1 ∞ ∣ f n ( t , 0 , 0 , … ) ∣ ≤ ∑ n = 1 ∞ l ( ‖ x ( t ) ‖ l 1 ) ∣ x n ( t ) ∣ + ∑ n = 1 ∞ f ¯ n ( t ) = l ( ‖ x ( t ) ‖ l 1 ) ‖ x ( t ) ‖ l 1 + ∑ n = 1 ∞ f ¯ n ( t ) ,

so we obtain ( F x ) ( t ) ∈ l 1 for any t ∈ R + . Moreover, we can write

(13) ‖ F x ‖ BC 1 ≤ l ( ‖ x ‖ BC 1 ) ‖ x ‖ BC 1 + F ¯ < ∞ .

Therefore F x is bounded on R + .

The function x : R + → l 1 is continuous so we have

(14) ∀ t 1 ∈ R + ∀ ε > 0 ∃ δ > 0 ∀ t 2 ∈ R + ( ∣ t 2 − t 1 ∣ ≤ δ ⇒ ‖ x ( t 2 ) − x ( t 1 ) ‖ l 1 ≤ ε ) .

Take t 1 ∈ R + , ε > 0 and choose δ 1 > 0 according to the aforementioned condition and δ 2 > 0 according to the assumption ( i ) . Denote δ = min { δ 1 , δ 2 } . For t 1 , t 2 ∈ R + such that ∣ t 2 − t 1 ∣ ≤ δ and from assumptions ( i ) and ( i i ) , we obtain

∑ n = 1 ∞ ∣ ( F n x ) ( t 2 ) − ( F n x ) ( t 1 ) ∣ ≤ ∑ n = 1 ∞ ∣ f n ( t 2 , x 1 ( t 2 ) , x 2 ( t 2 ) , … ) − f n ( t 1 , x 1 ( t 2 ) , x 2 ( t 2 ) , … ) ∣ + ∑ n = 1 ∞ ∣ f n ( t 1 , x 1 ( t 2 ) , x 2 ( t 2 ) , … ) − f n ( t 1 , x 1 ( t 1 ) , x 2 ( t 1 ) , … ) ∣ ≤ ε + ∑ n = 1 ∞ l ( ‖ x ‖ BC 1 ) ∣ x n ( t 2 ) − x n ( t 1 ) ∣ = ε + l ( ‖ x ‖ BC 1 ) ∑ n = 1 ∞ ∣ x n ( t 2 ) − x n ( t 1 ) ∣ = ε + l ( ‖ x ‖ BC 1 ) ‖ x ( t 2 ) − x ( t 1 ) ‖ l 1 ≤ ε ( 1 + l ( ‖ x ‖ BC 1 ) ) ,

so we have

‖ ( F x ) ( t 2 ) − ( F x ) ( t 1 ) ‖ l 1 ≤ ε ( 1 + l ( ‖ x ‖ BC 1 ) ) .

Therefore, F x is continuous on R + .

Now, we will show that F is continuous on B r ⊂ BC 1 for any r > 0 . Fix ε > 0 and take x , y ∈ B r such that ‖ x − y ‖ BC 1 ≤ ε . Then, for t ∈ R + we have

∑ n = 1 ∞ ∣ ( F n x ) ( t ) − ( F n y ) ( t ) ∣ = ∑ n = 1 ∞ ∣ f n ( t , x 1 ( t ) , x 2 ( t ) , … ) − f n ( t , y 1 ( t ) , y 2 ( t ) , … ) ∣ ≤ ∑ n = 1 ∞ l ( r ) ∣ x n ( t ) − y n ( t ) ∣ = l ( r ) ‖ x ( t ) − y ( t ) ‖ l 1 ,

which gives

‖ F x − F y ‖ BC 1 ≤ l ( r ) ε .

Therefore, F is continuous on the ball B r ⊂ BC 1 .□

Now, we accept the undermentioned conditions:

The functions k n ( t , s ) = k n : R + 2 → R are continuous ( n ∈ N ) . Moreover, the functions t ↦ k n ( t , s ) are locally equicontinuous on R + uniformly with respect to s ∈ R + , i.e.,
∀ T > 0 ∀ ε > 0 ∃ δ > 0 ∀ n ∈ N ∀ t 1 , t 2 ∈ [ 0 , T ] ∀ s ∈ R + ( ∣ t 2 − t 1 ∣ ≤ δ ⇒ ∣ k n ( t 2 , s ) − k n ( t 1 , s ) ∣ ≤ ε ) .
The sequence ( k n ( t , s ) ) is equibounded on R + 2 , i.e., there exists a constant K > 0 such that ∣ k n ( t , s ) ∣ ≤ K for t , s ∈ R + and n ∈ N .
The functions g n = g n ( t , x 1 , x 2 , … ) map the set R + × R ∞ into R for n ∈ N . The operator g defined by ( g x ) ( t ) = ( g n ( t , x ) ) = ( g 1 ( t , x ) , g 2 ( t , x ) , … ) maps the set R + × l 1 into l 1 and is bounded by a positive constant G , i.e.,
∀ x ∈ l 1 ∀ t ∈ R + ‖ ( g x ) ( t ) ‖ l 1 ≤ G .
Moreover, the family of functions { ( g x ) ( t ) } t ∈ R + is equicontinuous at every point of the space l 1 , i.e., for any arbitrarily fixed x ∈ l 1 and for a given ε > 0 , there exists δ > 0 such that
‖ ( g y ) ( t ) − ( g x ) ( t ) ‖ l 1 ≤ ε ,
for every t ∈ R + and for any y ∈ l 1 such that ‖ y ( t ) − x ( t ) ‖ l 1 ≤ δ .
For each T > 0 and ε > 0 , there exists δ > 0 such that for all t 1 , t 2 ∈ [ 0 , T ] with t 1 < t 2 , t 2 − t 1 ≤ δ , the following inequality is satisfied:
∑ n = 1 ∞ ⋁ p = t 1 t 2 H n ( t 2 , p ) ≤ ε .
For any fixed t > 0 and for each n ∈ N , the function s ↦ H n ( t , s ) is of bounded variation on the interval [ 0 , t ] .
For every ε > 0 , there exists δ > 0 such that for all t 1 , t 2 ∈ R + with t 1 < t 2 , t 2 − t 1 ≤ δ , the following inequality holds:
∑ n = 1 ∞ ⋁ p = 0 t 1 ( H n ( t 2 , p ) − H n ( t 1 , p ) ) ≤ ε .
Moreover, there exists a constant W > 0 such that for any t ∈ R + ,
∑ n = 1 ∞ ⋁ p = 0 t H n ( t , p ) ≤ W .

The next lemma concerns the operator V .

Lemma 2.2

Assume that assumptions (iv)–(ix) hold. Then, the operator V defined by (11) maps the space BC 1 into itself and is continuous on every ball B r ⊂ BC 1 .

Proof

Take any function x = x ( t ) = ( x n ( t ) ) ∈ BC 1 . At first, we will show that V x : R + → l 1 is bounded and continuous. Indeed, for fixed number t ∈ R + from assumptions (v), (vi), (viii) and (ix), we obtain

∑ n = 1 ∞ ∣ ( V n x ) ( t ) ∣ ≤ ∑ n = 1 ∞ ∫ 0 t ∣ k n ( t , s ) ∣ ∣ g n ( s , x 1 ( s ) , x 2 ( s ) , … ) ∣ d s ⋁ p = 0 s H n ( t , p ) ≤ ∑ n = 1 ∞ ∫ 0 t ∣ k n ( t , s ) ∣ ‖ ( g x ) ( s ) ‖ l 1 d s ⋁ p = 0 s H n ( t , p ) ≤ K G ∑ n = 1 ∞ ∫ 0 t d s ⋁ p = 0 s H n ( t , p ) ≤ K G ∑ n = 1 ∞ ⋁ p = 0 t H n ( t , p ) ≤ K G W ,

and we obtain

‖ ( V x ) ( t ) ‖ l 1 ≤ K G W .

This implies that ( V x ) ( t ) ∈ l 1 . Moreover, we have

(15) ‖ V x ‖ BC 1 = sup { ‖ ( V x ) ( t ) ‖ l 1 : t ∈ R + } ≤ KGW < ∞ .

Therefore V x is bounded on R + .

To prove the continuity of V x on R + , let us fix ε > 0 , T > 0 , and t 1 ∈ [ 0 , T ] . Next, choose δ > 0 according to (14) and t 2 ∈ [ 0 , T ] such that ∣ t 2 − t 1 ∣ ≤ δ . Of course, we may assume that t 2 > t 1 . Then, using assumptions (v) and (vi), we obtain

∑ n = 1 ∞ ∣ ( V n x ) ( t 2 ) − ( V n x ) ( t 1 ) ∣ ≤ ∑ n = 1 ∞ ∫ 0 t 2 k n ( t 2 , s ) g n ( s , x 1 ( s ) , … ) d s H n ( t 2 , s ) − ∫ 0 t 1 k n ( t 2 , s ) g n ( s , x 1 ( s ) , … ) d s H n ( t 2 , s ) + ∑ n = 1 ∞ ∫ 0 t 1 k n ( t 2 , s ) g n ( s , x 1 ( s ) , … ) d s H n ( t 2 , s ) − ∫ 0 t 1 k n ( t 2 , s ) g n ( s , x 1 ( s ) , … ) d s H n ( t 1 , s ) + ∑ n = 1 ∞ ∫ 0 t 1 k n ( t 2 , s ) g n ( s , x 1 ( s ) , … ) d s H n ( t 1 , s ) − ∫ 0 t 1 k n ( t 1 , s ) g n ( s , x 1 ( s ) , … ) d s H n ( t 1 , s ) ≤ ∑ n = 1 ∞ ∫ t 1 t 2 k n ( t 2 , s ) g n ( s , x 1 ( s ) , … ) d s H n ( t 2 , s ) + ∑ n = 1 ∞ ∫ 0 t 1 k n ( t 2 , s ) g n ( s , x 1 ( s ) , … ) d s ( H n ( t 2 , s ) − H n ( t 1 , s ) ) + ∑ n = 1 ∞ ∫ 0 t 1 ( k n ( t 2 , s ) − k n ( t 1 , s ) ) g n ( s , x 1 ( s ) , … ) d s H n ( t 1 , s ) ≤ ∑ n = 1 ∞ ∫ t 1 t 2 ∣ k n ( t 2 , s ) ∣ ∣ g n ( s , x 1 ( s ) , … ) ∣ d s ⋁ p = t 1 s H n ( t 2 , p ) + ∑ n = 1 ∞ ∫ 0 t 1 ∣ k n ( t 2 , s ) ∣ ∣ g n ( s , x 1 ( s ) , … ) ∣ d s ⋁ p = 0 s ( H n ( t 2 , p ) − H n ( t 1 , p ) ) + ∑ n = 1 ∞ ∫ 0 t 1 ∣ k n ( t 2 , s ) − k n ( t 1 , s ) ∣ ∣ g n ( s , x 1 ( s ) , … ) ∣ d s ⋁ p = 0 s H n ( t 1 , p ) ≤ K G ∑ n = 1 ∞ ⋁ p = t 1 t 2 H n ( t 2 , p ) + K G ∑ n = 1 ∞ ⋁ p = 0 t 1 ( H n ( t 2 , p ) − H n ( t 1 , p ) ) + ω k T ( δ ) G ∑ n = 1 ∞ ⋁ p = 0 t 1 H n ( t 1 , p ) ,

where

ω k T ( δ ) = sup { ∣ k n ( t 2 , s ) − k n ( t 1 , s ) ∣ : t 2 , t 1 , s ∈ [ 0 , T ] , ∣ t 2 − t 1 ∣ ≤ δ , n ∈ N }

is a common modulus of continuity of the function sequence t ↦ k n ( t , s ) on the interval [ 0 , T ] (according to assumption (iv)). Obviously, ω k T ( δ ) → 0 as δ → 0 for each T > 0 .

Denote

M T ( δ ) = sup ∑ n = 1 ∞ ⋁ p = 0 t 1 ( H n ( t 2 , p ) − H n ( t 1 , p ) ) : t 2 ∈ ( t 1 , T ] , ∣ t 2 − t 1 ∣ ≤ δ , N T ( δ ) = sup ∑ n = 1 ∞ ⋁ p = t 1 t 2 H n ( t 2 , p ) : t 2 ∈ ( t 1 , T ] , ∣ t 2 − t 1 ∣ ≤ δ ,

then we can write

(16) ∑ n = 1 ∞ ∣ ( V n x ) ( t 2 ) − ( V n x ) ( t 1 ) ∣ ≤ K G N T ( δ ) + K G M T ( δ ) + ω k T ( δ ) G W

and hence, we obtain

‖ ( V x ) ( t 2 ) − ( V x ) ( t 1 ) ‖ l 1 ≤ K G ( N T ( δ ) + M T ( δ ) ) + ω k T ( δ ) G W .

From assumptions (ix) and (vii) and the property of the function ε ↦ ω k T ( ε ) , the last inequality implies that the function V x is continuous on the interval [ 0 , T ] for any T > 0 . This gives that the function V x is continuous on R + . Finally, we infer that the operator V maps the space BC 1 into itself.

Now, fix r > 0 . We will prove that V is continuous on the ball B r ⊂ BC 1 . Take ε > 0 and x , y ∈ B r such that ‖ x − y ‖ BC 1 ≤ ε . For any fixed t ∈ R + from assumptions (v) and (ix), we obtain

∑ n = 1 ∞ ∣ ( V n x ) ( t ) − ( V n y ) ( t ) ∣ = ∑ n = 1 ∞ ∫ 0 t k n ( t , s ) g n ( s , x 1 ( s ) , … ) d s H n ( t , s ) − ∫ 0 t k n ( t , s ) g n ( s , y 1 ( s ) , … ) d s H n ( t , s ) ≤ ∑ n = 1 ∞ ∫ 0 t ∣ k n ( t , s ) ∣ ∣ g n ( s , x 1 ( s ) , … ) − g n ( s , y 1 ( s ) , … ) ∣ d s ⋁ p = 0 s H n ( t , p ) ≤ K δ ( ε ) ∑ n = 1 ∞ ⋁ p = 0 t H n ( t , p ) ≤ K δ ( ε ) W ,

where δ ( ε ) is defined for ε > 0 as follows:

δ ( ε ) = sup { ∣ g n ( t , x ) − g n ( t , y ) ∣ : x , y ∈ l 1 , ‖ x ( t ) − y ( t ) ‖ l 1 ≤ ε , t ∈ R + , n ∈ N } ,

so from assumption (vi), we have δ ( ε ) → 0 as ε → 0 . Then, we obtain

‖ V x − V y ‖ BC 1 ≤ K W δ ( ε ) ,

which proves the continuity of the operator V on the ball B r .□

Now, we give our final assumptions and prove the main result of this article.

Assume that ( a n ( t ) ) ∈ BC 1 and lim t → ∞ ∑ n = 1 ∞ ∣ a n ( t ) ∣ = 0 . So we may define the finite constant
A = sup ∑ n = 1 ∞ ∣ a n ( t ) ∣ : t ∈ R + .
There exists a positive solution r 0 of the inequality
A + K G W ( F ¯ + r l ( r ) ) ≤ r such that K G W l ( r 0 ) < 1 .

We will need the following simple lemma (cf. [12]) that follows on the Cauchy condition for real sequences.

Lemma 2.3

If ( a n ) ∈ l 1 , then lim n → ∞ ∑ i = n ∞ ∣ a i ∣ = 0 .

Theorem 2.4

Under assumptions (i)–(xi), infinite system of integral equations (9) has at least one solution x ( t ) = ( x n ( t ) ) in the space BC 1 .

Proof

To prove the existence of a solution for (9) in the space BC 1 , we will use Theorem 1.3. At the beginning, we will show that the operator Q defined by (12) maps the space BC 1 into itself. Let us note that the space BC 1 can be treated as the Banach algebra with respect to the coordinatewise multiplication of sequences of functions. Thus, from Lemmas 2.1 and 2.2, we conclude that the function Q x is continuous on R + . In a similar way, we prove the boundedness of the function Q x on R + . Finally, we conclude that the operator Q maps the space BC 1 into BC 1 .

Next, using estimates (13) and (15) and assumption (x), we obtain for fixed t ∈ R +

∑ n = 1 ∞ ∣ ( Q n x ) ( t ) ∣ ≤ ∑ n = 1 ∞ ∣ a n ( t ) ∣ + ∑ n = 1 ∞ ( ∣ ( F n x ) ( t ) ∣ ∣ ( V n x ) ( t ) ∣ ) ≤ ∑ n = 1 ∞ ∣ a n ( t ) ∣ + ∑ n = 1 ∞ ∣ ( F n x ) ( t ) ∣ ∑ n = 1 ∞ ∣ ( V n x ) ( t ) ∣ = ∑ n = 1 ∞ ∣ a n ( t ) ∣ + ‖ ( F x ) ( t ) ‖ l 1 ‖ ( V x ) ( t ) ‖ l 1 ≤ A + ‖ F x ‖ BC 1 ‖ V x ‖ BC 1 .

Hence, we obtain

‖ Q x ‖ BC 1 ≤ A + ( l ( ‖ x ‖ BC 1 ) ‖ x ‖ BC 1 + F ¯ ) K G W .

Using the aforementioned estimate and assumption (xi), we infer that there exists a number r 0 > 0 such that the operator Q maps the ball B r 0 ⊂ BC 1 into itself.

It follows from Lemmas 2.1 and 2.2 that the operators F and V are continuous on the ball B r 0 . So we deduce that Q is also continuous on B r 0 .

We will now study the behavior of the operators F , V , and Q with respect to the measure of noncompactness χ a defined by (4). Take a nonempty subset X ⊂ B r 0 and fix numbers ε > 0 and T > 0 . Choose a function x = x ( t ) = ( x n ( t ) ) ∈ X , then for t 1 , t 2 ∈ [ 0 , T ] such that ∣ t 2 − t 1 ∣ ≤ ε (we can assume that t 1 < t 2 ) from assumptions (i) and (ii), we obtain

∑ n = 1 ∞ ∣ ( F n x ) ( t 2 ) − ( F n x ) ( t 1 ) ∣ ≤ ∑ n = 1 ∞ ∣ f n ( t 2 , x 1 ( t 2 ) , x 2 ( t 2 ) , … ) − f n ( t 1 , x 1 ( t 2 ) , x 2 ( t 2 ) , … ) ∣ + ∑ n = 1 ∞ ∣ f n ( t 1 , x 1 ( t 2 ) , x 2 ( t 2 ) , … ) − f n ( t 1 , x 1 ( t 1 ) , x 2 ( t 1 ) , … ) ∣ ≤ ω f T ( ε ) + l ( r 0 ) ∑ n = 1 ∞ ∣ x n ( t 2 ) − x n ( t 1 ) ∣ ,

where we denoted

ω f T ( ε ) = sup ∑ n = 1 ∞ ∣ f n ( t 2 , x 1 , x 2 , … ) − f n ( t 1 , x 1 , x 2 , … ) ∣ : t 2 , t 1 ∈ [ 0 , T ] , ∣ t 2 − t 1 ∣ ≤ ε , ( x k ) ∈ l 1 .

From assumption (i), we have ω f T ( ε ) → 0 as ε → 0 , so finally, we have

(17) ω T ( F x , ε ) ≤ ω f T ( ε ) + l ( r 0 ) ω T ( x , ε ) .

In a similar way as above (cf. estimate (16)), we obtain the inequality

(18) ω T ( V x , ε ) ≤ K G ( N T ( ε ) + M T ( ε ) ) + G W ω k T ( ε ) .

Finally, based on (13) and (15), we obtain

‖ ( Q x ) ( t 2 ) − ( Q x ) ( t 1 ) ‖ l 1 ≤ ‖ a ( t 2 ) − a ( t 1 ) ‖ l 1 + ‖ ( F x ) ( t 2 ) ( V x ) ( t 2 ) − ( V x ) ( t 2 ) ( F x ) ( t 1 ) ‖ l 1 + ‖ ( V x ) ( t 2 ) ( F x ) ( t 1 ) − ( F x ) ( t 1 ) ( V x ) ( t 1 ) ‖ l 1 = ‖ a ( t 2 ) − a ( t 1 ) ‖ l 1 + ‖ ( V x ) ( t 2 ) ‖ l 1 ‖ ( F x ) ( t 2 ) − ( F x ) ( t 1 ) ‖ l 1 + ‖ ( F x ) ( t 1 ) ‖ l 1 ‖ ( V x ) ( t 2 ) − ( V x ) ( t 1 ) ‖ l 1 ≤ ω T ( a , ε ) + K G W ω T ( F x , ε ) + ( F ¯ + r 0 l ( r 0 ) ) ω T ( V x , ε ) ,

where ( a ( t ) ) denotes ( a n ( t ) ) . Now, from (17), (18), and the aforementioned inequality, we obtain

ω T ( Q x , ε ) ≤ ω T ( a , ε ) + K G W ω f T ( ε ) + K G W l ( r 0 ) ω T ( x , ε ) + ( K G ( N T ( ε ) + M T ( ε ) ) + G W ω k T ( ε ) ) ( F ¯ + r 0 l ( r 0 ) ) .

We take the supremum over x ∈ X and obtain

(19) ω T ( Q X , ε ) ≤ ω T ( a , ε ) + K G W ω f T ( ε ) + K G W l ( r 0 ) ω T ( X , ε ) + K G ( N T ( ε ) + M T ( ε ) ) ( F ¯ + r 0 l ( r 0 ) ) + G W ( F ¯ + r 0 l ( r 0 ) ) ω k T ( ε ) .

Passing with ε → 0 , we get ω T ( a , ε ) → 0 (from assumption (x)), ω f T ( ε ) → 0 (from assumption ( i ) ), ω k T ( ε ) → 0 (from assumption (iv)), M T ( ε ) → 0 (from assumption (ix)) and N T ( ε ) → 0 (from assumption (vii)). Therefore, from (19), we have

ω 0 T ( Q X ) ≤ K G W l ( r 0 ) ω 0 T ( X ) .

Passing with T → ∞ , we obtain

(20) ω 0 ( Q X ) ≤ K G W l ( r 0 ) ω 0 ( X ) ,

where ω 0 ( X ) is defined by formula (6).

Now, we will estimate the second component of the measure of noncompactness χ a defined by (4). Take a function x ∈ X and T > 0 . Then, as for estimates (13) and (15), we obtain for t ∈ [ 0 , T ] :

∑ i = n ∞ ∣ ( Q i x ) ( t ) ∣ ≤ ∑ i = n ∞ ∣ a i ( t ) ∣ + ∑ i = n ∞ ∣ ( F i x ) ( t ) ( V i x ) ( t ) ∣ ≤ ∑ i = n ∞ ∣ a i ( t ) ∣ + ∑ i = n ∞ ∣ ( F i x ) ( t ) ∣ ∑ i = n ∞ ∣ ( V i x ) ( t ) ∣ ≤ ∑ i = n ∞ ∣ a i ( t ) ∣ + ∑ i = n ∞ f i ¯ ( t ) + l ( ‖ x ( t ) ‖ l 1 ) ∑ i = n ∞ ∣ x i ( t ) ∣ K G W .

We take the supremum over all x = ( x k ) ∈ X ⊂ B r 0 . Hence, we derive

sup x = ( x k ) ∈ X ∑ i = n ∞ ∣ ( Q i x ) ( t ) ∣ ≤ ∑ i = n ∞ ∣ a i ( t ) ∣ + K G W ∑ i = n ∞ f ¯ i ( t ) + K G W l ( r 0 ) sup x = ( x k ) ∈ X ∑ i = n ∞ ∣ x i ( t ) ∣ .

Passing with n → ∞ , from assumption (iii), Lemma 2.3, and formula (5), we obtain

χ ( ( Q X ) ( t ) ) ≤ K G W l ( r 0 ) χ ( X ( t ) ) .

Take the supremum over t ∈ [ 0 , T ] and next pass with T → ∞ , then we have

(21) χ ¯ ∞ ( Q X ) ≤ K G W l ( r 0 ) χ ¯ ∞ ( X ) ,

where χ ¯ ∞ ( X ) is defined by formula (7).

Now, we will estimate the last component a ∞ ( X ) defined by (8) of the measure of noncompactness χ a ( X ) . Similarly as above, assume that X is a nonempty subset of the ball B r 0 . Fix a function x ∈ X , a number T > 0 and t ≥ T . Then, in view of (13) and (15), we obtain

∑ n = 1 ∞ ∣ ( Q n x ) ( t ) ∣ ≤ ∑ n = 1 ∞ ∣ a n ( t ) ∣ + ∑ n = 1 ∞ f ¯ n ( t ) + l ( r 0 ) ∑ n = 1 ∞ ∣ x n ( t ) ∣ K G W .

Hence,

‖ ( Q x ) ( t ) ‖ l 1 ≤ ∑ n = 1 ∞ ∣ a n ( t ) ∣ + K G W ∑ n = 1 ∞ f ¯ n ( t ) + K G W l ( r 0 ) ‖ x ( t ) ‖ l 1 .

Taking supremum over t ≥ T and x = ( x k ) ∈ X , we have

sup x = ( x k ) ∈ X { sup t ≥ T ‖ ( Q x ) ( t ) ‖ l 1 } ≤ sup t ≥ T ∑ n = 1 ∞ ∣ a n ( t ) ∣ + K G W sup t ≥ T ∑ n = 1 ∞ f ¯ n ( t ) + K G W l ( r 0 ) sup x = ( x k ) ∈ X { sup t ≥ T ‖ x ( t ) ‖ l 1 } .

Then, passing with T → ∞ , from assumptions (x) and (iii), we obtain

(22) a ∞ ( Q X ) ≤ K G W l ( r 0 ) a ∞ ( X ) .

Finally, from (20), (21), (22), and (4), we infer that

(23) χ a ( Q X ) ≤ K G W l ( r 0 ) χ a ( X ) .

Now, on the basis of Theorem 1.3, estimate (23), and the second inequality from assumption (xi), we conclude that infinite system of integral equations (9) has at least one solution in the space BC 1 . The proof is complete.□

3 Example

In this section, we will present an example to illustrate the result from Theorem 2.4. Let us consider the following infinite system of nonlinear quadratic integral equations of Volterra-Hammerstein type:

(24) x n ( t ) = α 5 n 2 + t 2 + β t n 2 + n 2 t 2 + γ n 2 sin x n ( t ) 1 + x n 2 ( t ) + x n 2 ( t ) × ∫ 0 t s 1 + 4 n ( t + s ) arctan x n ( s ) ( 1 + t ) 2 + n 4 x n 2 ( s ) 1 n 4 + t 2 + s 2 d s ,

where t ∈ R + , n ∈ N , and α , β , γ are the positive constants. System (24) is an example of the infinite system of nonlinear quadratic integral equations of Volterra-Hammerstein-Stieltjes (9), where

(25) a n ( t ) = α 5 n 2 + t 2 ,

(26) f n ( t , x 1 , x 2 , … ) = β t n 2 + n 2 t 2 + γ n 2 sin x n 1 + x n 2 + x n 2 ,

(27) k n ( t , s ) = s 1 + 4 n ( t + s ) ,

(28) g n ( t , x 1 , x 2 , … ) = arctan x n ( 1 + t ) 2 + n 4 x n 2 ,

where n ∈ N , and t , s ∈ R + and α , β , and γ are the positive constants. We have also

(29) H n ( t , s ) = 1 n 4 + t 2 arctan s n 4 + t 2

because

d s H n ( t , s ) = ∂ H n ( t , s ) ∂ s d s = 1 n 4 + t 2 ⋅ 1 1 + s 2 n 4 + t 2 ⋅ 1 n 4 + t 2 d s = 1 n 4 + t 2 + s 2 d s .

Now, we will show that the functions defined by (25)–(29) satisfy the assumptions of Theorem 2.4. At the beginning, we will verify the first three assumptions about the functions f n = f n ( t , x 1 , x 2 , … ) defined by (26). Taking x = ( x n ) ∈ l 1 , we can easily obtain that the first part of assumption ( i ) is satisfied, because from the standard Weierstrass test follows that the series ∑ n = 1 ∞ f n ( t , x 1 , x 2 , … ) is uniformly convergent on the set R + × l 1 . The second part of assumption ( i ) is satisfied on the basis of the behavior of the function z ↦ z 1 + z 2 for z ∈ R + .

Now, fix arbitrarily a number r > 0 and take x = ( x n ) , y = ( y n ) ∈ l 1 such that ‖ x ‖ l 1 = ∑ n = 1 ∞ ∣ x n ∣ ≤ r , ‖ y ‖ l 1 = ∑ n = 1 ∞ ∣ y n ∣ ≤ r . Then, for a fixed natural number n and t ∈ R + we derive the following estimate:

∣ f n ( t , x 1 , x 2 , … ) − f n ( t , y 1 , y 2 , … ) ∣ ≤ γ ( 2 r + 1 ) ∣ x n − y n ∣ .

Hence, the functions f n , n ∈ N , satisfy assumption ( i i ) with the nondecreasing function

(30) l ( r ) = ( 2 r + 1 ) γ .

Next, for t ∈ R + and n ∈ N , let us consider the function

f ¯ n ( t ) = ∣ f n ( t , 0 , 0 , … ) ∣ = β t n 2 + n 2 t 2 .

For t 1 , t 2 ∈ R + , we obtain

∑ n = 1 ∞ ∣ f ¯ n ( t 2 ) − f ¯ n ( t 1 ) ∣ ≤ β π 2 6 ∣ t 2 − t 1 ∣ ,

and for t ∈ R +

∑ n = 1 ∞ ∣ f ¯ n ( t ) ∣ ≤ β π 2 12 ,

which indicates that ( f ¯ n ( t ) ) ∈ BC 1 . Obviously, lim t → ∞ ∑ n = 1 ∞ f ¯ n ( t ) = lim t → ∞ β π 2 6 t 1 + t 2 = 0 and we conclude that assumption (iii) is satisfied. Moreover, we have

F ¯ = sup ∑ n = 1 ∞ f ¯ n ( t ) : t ∈ R + = β π 2 12 .

The functions k n = k n ( t , s ) , defined by (27), are continuous on R + 2 . Furthermore, for arbitrary t 1 , t 2 , s ∈ R + and n ∈ N , we obtain

∣ k n ( t 2 , s ) − k n ( t 1 , s ) ∣ ≤ ∣ t 2 − t 1 ∣ ,

so the functions t ↦ k n ( t , s ) are equicontinuous on R + uniformly with respect to s ∈ R + . We have ∣ k n ( t , s ) ∣ ≤ 1 4 for t , s ∈ R + and n ∈ N ; therefore, the sequence ( k n ( t , s ) ) is equibounded on R + 2 . It means that assumptions (iv) and (v) are satisfied and K = 1 4 .

Let us note that the function g n = g n ( t , x 1 , x 2 , … ) defined by (28) on R + × R ∞ takes real values for n ∈ N . Now, fix arbitrarily x = ( x n ) ∈ l 1 , then for t ∈ R + , we obtain

‖ ( g x ) ( t ) ‖ l 1 = ∑ n = 1 ∞ ∣ g n ( t , x 1 , x 2 , … ) ∣ ≤ π 2 12 ;

hence, the operator g defined in assumption (vi) maps the set R + × l 1 into l 1 and is bounded by positive constant G = π 2 12 .

Furthermore, for any x = ( x n ) ∈ l 1 and for a given ε > 0 , there exists δ > 0 such that for every t ∈ R + and for any y = ( y n ) ∈ l 1 such that ‖ y ( t ) − x ( t ) ‖ l 1 ≤ δ , we have

‖ ( g y ) ( t ) − ( g x ) ( t ) ‖ l 1 = ∑ n = 1 ∞ ∣ g n ( t , y 1 , y 2 , … ) − g n ( t , x 1 , x 2 , … ) ∣ ≤ 5 4 ‖ y ( t ) − x ( t ) ‖ l 1 .

So, we obtain that the family of functions { ( g x ) ( t ) } t ∈ R + is equicontinuous at every point of the space l 1 , i.e., assumption (vi) is satisfied.

Next, we will check assumptions (vii), (viii), and (ix) for H n ( t , s ) . For fixed t ∈ R + and n ∈ N , we obtain that the function s ↦ H n ( t , s ) is nondecreasing on R + . It means that this function is nondecreasing on each interval [ 0 , t ] for any n ∈ N . Hence, we infer that assumption (vii) is satisfied, as for each T > 0 and ε > 0 and for all t 1 , t 2 ∈ [ 0 , T ] with t 1 < t 2 , t 2 − t 1 ≤ δ , we obtain

∑ n = 1 ∞ ⋁ p = t 1 t 2 H n ( t 2 , p ) = ∑ n = 1 ∞ ∣ H n ( t 2 , t 2 ) − H n ( t 2 , t 1 ) ∣ ≤ π 4 90 ∣ t 2 − t 1 ∣ .

We also conclude that for any fixed t > 0 and for each n ∈ N , the function s ↦ H n ( t , s ) is of bounded variation on the interval [ 0 , t ] , i.e., assumption (viii) is satisfied.

Note that for fixed t 1 , t 2 ∈ R + such that t 1 < t 2 and for fixed n ∈ N , the function s ↦ H n ( t 2 , s ) − H n ( t 1 , s ) , where s ∈ R + , is nonincreasing on R + . It means that this function is nonincreasing on each interval [ 0 , t ] for any n ∈ N . Using this property, we have assumption (ix). Namely, for every ε > 0 and for all t 1 , t 2 ∈ R + with t 1 < t 2 , t 2 − t 1 ≤ δ , the following inequality holds:

∑ n = 1 ∞ ⋁ p = 0 t 1 ( H n ( t 2 , p ) − H n ( t 1 , p ) ) ≤ ( π + 8 ) π 4 360 ∣ t 2 − t 1 ∣ .

Moreover, from the monotonicity of the functions s ↦ H n ( t , s ) , we have for any t ∈ R + ,

∑ n = 1 ∞ ⋁ p = 0 t H n ( t , p ) ≤ π 3 24 , i.e., W = π 3 24 .

We will check now assumption (x) relating to the function a n ( t ) = α 5 n 2 + t 2 defined by (25). For t 1 , t 2 ∈ R + , we obtain

∑ n = 1 ∞ ∣ a n ( t 2 ) − a n ( t 1 ) ∣ ≤ α 5 5 ζ ( 3 ) ∣ t 2 − t 1 ∣ ,

where ζ denotes the Riemann zeta function and ζ ( 3 ) ≅ 1.202057 . Moreover, for t ∈ R + ,

∑ n = 1 ∞ ∣ a n ( t ) ∣ ≤ α π 2 30 ,

which gives that ( a n ( t ) ) ∈ BC 1 . Hence, lim n → ∞ ∑ i = n ∞ ∣ a i ( t ) ∣ = 0 . We also have

lim t → ∞ ∑ n = 1 ∞ ∣ a n ( t ) ∣ = lim t → ∞ α ∑ n = 1 ∞ 1 5 n 2 + t 2 = 0 .

Therefore, assumption (x) is satisfied. Moreover, we have A = sup ∑ n = 1 ∞ ∣ a n ( t ) ∣ : t ∈ R + = α π 2 30 .

Finally, taking into account the obtained constants F ¯ , K , G , W , and A and the function l ( r ) defined by (30), the first inequality from assumption (xi) has the form

(31) r 2 + 1 2 − 24 ⋅ 24 γ π 5 r + 96 α 5 γ π 3 + β π 2 24 γ ≤ 0 .

We show that there exists a positive solution r 0 of (31) such that the second inequality from assumption (xi)

(32) π 5 48 ⋅ 24 ( 2 r 0 + 1 ) γ < 1

is satisfied. Since α , β , and γ are the positive constants, we can analyze different cases.

Consider the case when α = 0.1 , β = 0.1 , and γ = 1 . Then, for example, the number r 0 = 1 3 satisfies both inequalities (31) and (32). It means that assumption (xi) is satisfied, and from Theorem 2.4, we obtain that the infinite system of Volterra-Hammerstein integral equations (24) has a solution belonging to the ball B 1 3 in the space BC 1 .

If we make only a small change in the value of the constant γ , i.e., if we take, for example, α = 0.1 , β = 0.1 , and γ = 1.5 , we can show that the number r 0 = 1 7 ≅ 0.142857 satisfies both inequalities (31) and (32). Then, applying Theorem 2.4, we obtain that system (24) has a solution belonging to the ball B 1 7 in the space BC 1 .

Funding information: The authors state no funding involved.
Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] R. P. Agarwal, M. Benchohra, and D. Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math. 55 (2009), 221–230, https://doi.org/10.1007/s00025-009-0434-5. Search in Google Scholar

[2] A. K. Ashirbayev, J. Banaś, and R. Bekmoldayeva, A unified approach to some classes of nonlinear integral equations, J. Funct. Spaces 306231 (2014), 9, https://doi.org/10.1155/2014/306231. Search in Google Scholar

[3] J. Banaś and R. Taktak, Measures of noncompactness in the study of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat 117 (2023), 95, https://doi.org/10.1007/s13398-023-01424-8. Search in Google Scholar

[4] J. Banaś and A. Chlebowicz, On solutions of an infinite system of nonlinear integral equations on the real half-axis, Banach J. Math. Anal. 13 (2019), no. 4, 944–968, https://doi.org/10.1215/17358787-2019-0008. Search in Google Scholar

[5] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker, New York, 1980. Search in Google Scholar

[6] J. Banaś and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 2014. 10.1007/978-81-322-1886-9Search in Google Scholar

[7] J. Banaś and B. Rzepka, On solutions of infinite systems of integral equations of Hammerstein type, J. Nonlinear Convex Anal. 18 (2017), no. 2, 261–278. http://www.yokohamapublishers.jp/online2/jncav18-2.html.Search in Google Scholar

[8] J. Banaś and B. Rzepka, The technique of Volterra-Stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders, Comput. Math. Appl. 64 (2012), 3108–3116, https://doi.org/10.1016/j.camwa.2012.03.006. Search in Google Scholar

[9] J. Banaś and W. Woś, Solvability of an infinite system of integral equations on the real half-axis, Adv. Nonlinear Anal. 10 (2021), 202–216, https://doi.org/10.1515/anona-2020-0114. Search in Google Scholar

[10] J. Banaś and T. Zaja̧c, A new approach to the theory of functional integral equations of fractional order, J. Math. Anal. Appl. 375 (2011), 375–387, https://doi.org/10.1016/j.jmaa.2010.09.004. Search in Google Scholar

[11] M. Benchohra, J. R. Graef, and S. Hamani, Existence results for boundary value problems with non-linear fractional differential equations, Appl. Anal. 87 (2008), 851–863, https://doi.org/10.1080/00036810802307579. Search in Google Scholar

[12] A. Chlebowicz, Existence of solutions to infinite systems of nonlinear integral equations on the real half-axis, Electron. J. Differential Equations 61 (2021), 1–20, https://doi.org/10.58997/ejde.2021.61. Search in Google Scholar

[13] A. Chlebowicz, Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type, Adv. Nonlinear Anal. 9 (2020), 1187–1204, https://doi.org/10.1515/anona-2020-0045. Search in Google Scholar

[14] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92. http://eudml.org/doc/106925.Search in Google Scholar

[15] M. A. Darwish and J. Henderson, Existence and asymptotic stability of solutions of a perturbed quadratic fractional integral equation, Fract. Calc. Appl. Anal. 12 (2009), 71–86, https://doi.org/10.4134/BKMS.2011.48.3.539. Search in Google Scholar

[16] L. S. Goldenštein, I. T. Gohberg, and A. S. Markus, Investigations of some properties of bounded linear operators with their q-norms, Ucen. Zap. Kishinevsk. Gos. Univ. 29 (1957), 29–36. Search in Google Scholar

[17] L. S. Goldenštein and A. S Markus, On the measure of non-compactness of bounded sets and of linear operators, Studies in Algebra and Math. Anal., Izdat. Karta Moldovenjaske, Kishinev (1965), 45–54. Search in Google Scholar

[18] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elseiver Science B. V., Amsterdam, 2006. Search in Google Scholar

[19] K. Kuratowski, Sur les espaces complets, Fundam. Math. 15 (1930), 301–309. https://eudml.org/doc/212357.10.4064/fm-15-1-301-309Search in Google Scholar

[20] B. N. Sadovskii, Limit-compact and condensing operators, Uspekhi Mat. Nauk 27 (1972), 81–146, https://doi.org/10.1070/RM1972v027n01ABEH001364. Search in Google Scholar

[21] T. Sathiyaraj, P. Balasubramaniam, and K. Ratnavelu, Controllability of non-instantaneous impulsive large-scale neutral fractional stochastic systems with Poisson jumps, Rend. Circ. Mat. Palermo II. Ser 74 (2025), 37, https://doi.org/10.1007/s12215-024-01139-8. Search in Google Scholar

[22] J. Zhu and D. Yan, Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions, Demonstr. Math. 57 (2024), no. 1, 20230137, https://doi.org/10.1515/dema-2023-0137. Search in Google Scholar

Received: 2025-03-26

Revised: 2025-05-21

Accepted: 2025-07-10

Published Online: 2025-08-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2025-0105

Keywords for this article

space of continuous and bounded functions defined on the real half-axis; space l 1; measure of noncompactness; fixed point theorem of Darbo type

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