Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β

1 Introduction and preliminaries

The study of functional integral equations is an important branch of non-linear analysis and has various applications in natural and applied sciences. Most of the problems, for instance, in mechanics and population dynamics can be formulated using integral equations. On the other hand, differential equations may be converted into integral equations and their solutions can be obtained.

The topic of solutions to functional equations has captured the attention of numerous researchers over an extensive period of time. Several approaches have been used to establish the existence of solutions of functional equations such as functional integral equations, differential equations, integro-differential equations, and fractional integro-differential equations [1–4].

Measures of non-compactness are important tool in fixed point theory, which is also crucial in the study of existence of solution of functional equations.

In 1930, Kuratowski [5] introduced the concept of measures of non-compactness, a function that determines the degree of non-compactness of a set. Later on, Banaś [6] generalized this concept into Banach space. In 1955, Darbo [7] used the Kuratowski measure of non-compactness to prove his famous fixed point theorem, which was a generalization of the classical Schauder’s fixed point theorem and Banach’s contraction principle. Darbo’s fixed point theorem and it is several generalizations such as Meir-Keeler fixed point theorem are used by many researchers to study the existence of solutions of integral equations and differential equations in Banach spaces.

The study of sequence spaces has also attracted the interest of many researchers. Kreyszig [8] has shown some sequence spaces, which are also Banach spaces in his book titled introductory functional analysis with applications. Banaś and Mursaleen [9] discussed measures of non-compactness in some classical sequence spaces. Recently, several studies have been conducted on the investigation of the solvability of functional integral equations in some Banach sequence spaces. We can refer [10–13] for some works on existence of solutions of functional integral equations in Banach sequence spaces. This work extends and generalizes the results of Malik and Jalal [10] in tempered sequence spaces c 0 β and ℓ 1 β .

Sometimes it is not suitable to consider the classical sequence spaces for studying the solvability of functional equations such as initial value differential equations and integral equations, since the solution might be out of such sequences. Therefore, it is important to consider the enlarged sequence spaces.

These spaces are defined by fixing a non-increasing sequence β = β n , where β n is a real positive sequence for n ∈ N . Such sequence β will be called the tempering sequence.

Let ω denote all sequences y = ( y k ) k = 1 ∞ real or complex such that β n y n → 0 as n → ∞ . We denote this sequence space by c 0 β . It is a Banach space under the norm

In a similar manner, we examine the space ℓ 1 β , which encompasses real (or complex) sequences ( y n ) with the property that ( β n y n ) converges to a finite value. This particular space is a Banach space, characterized by the norm

Banaś and Krajewska [2], proved that the pairs of Banach sequence spaces ( c 0 , c 0 β ) and ( ℓ 1 , ℓ 1 β ) are isometric. It is important to note that if we let β n = 1 , then we have c 0 β = c 0 and ℓ 1 β = ℓ 1 . In order to obtain the enlarged sequence spaces c 0 β and ℓ 1 β , we assume that the sequence β n → 0 as n → ∞ .

In this research article, we will investigate the existence of solutions for an infinite system of integral equations with a Hammerstein-type structure in three variables of the form

where ( ψ 1 , ψ 2 , ψ 3 ) ∈ [ s , t ] × [ s , t ] × [ s , t ] .

Consider R + = [ 0 , ∞ ) , and suppose that ( M , ∥ . ∥ ) is a real Banach space. If X is a non-empty subset of M , we use X ¯ and ConvX to represent the closure and convex closure of X , respectively. The family of all non-empty and bounded subsets of M is denoted as Q M , and its subfamily consisting of all relatively compact sets is denoted as N M . The space of continuously differentiable functions on I 3 = [ s , t ] × [ s , t ] × [ s , t ] is denoted as C 1 ( I 3 , R ) .

In this article, the Hausdorff measure of non-compactness will be used to establish our results on the existence of solutions of a system of integral equations of Hammerstein type in three variables.

The following is the axiomatic definition of measures of non-compactness.

If a measure of non-compactness satisfies the following additional conditions, then it is called a regular measure:

1. λ ( Y 1 ∪ Y 2 ) = max { λ ( Y 1 ) , λ ( Y 2 ) } ,

2. λ ( Y 1 + Y 2 ) ≤ λ ( Y 1 ) + λ ( Y 2 ) ,

3. λ ( k Y ) = ∣ k ∣ λ ( Y ) ,

4. ker λ = N M .

The following results obtained by Darbo [7] have been very useful for proving the existence of solutions of functional equations.

In 1969, Meir and Keeler [14] proved a very interesting fixed point, which is also a generalization of Banach’s contraction principle and Schauder’s fixed point theorem. Later on, Aghajan and Mursaleen [1] generalized the concept of Meir-Keeler contractions to Banach spaces and came up with the following results:

2 Hausdorff measure of non-compactness in Banach spaces c 0 β and ℓ 1 β

Banaś and Mursaleen [9] established the formula for the Hausdorff measure of non-compactness in the classical sequence spaces c 0 and ℓ 1 in one variable. We can consider publications [2,10,12,13] for the formulas of measures of non-compactness in some sequence spaces. In this section, the Hausdorff measure of non-compactness in sequence spaces c 0 β and ℓ 1 β in three variables is formulated.

Let H be a bounded subset of the Banach space ( c 0 β , ∥ . ∥ c 0 β ), and the Hausdorff measure of non-compactness is given by

where v ( ψ 1 , ψ 2 , ψ 3 ) = ( v j ( ψ 1 , ψ 2 , ψ 3 ) ) j = 1 ∞ ∈ c 0 β for each ( ψ 1 , ψ 2 , ψ 3 ) ∈ I 3 and H ∈ Q c 0 β .

The Hausdorff measure of non-compactness χ in the Banach space ℓ 1 β is defined as follows:

where v ( ψ 1 , ψ 2 , ψ 3 ) = ( v j ( ψ 1 , ψ 2 , ψ 3 ) ) j = 1 ∞ ∈ ℓ 1 β for each ( ψ 1 , ψ 2 , ψ 3 ) ∈ I 3 and H ∈ Q ℓ 1 β .

We will prove the existence of solutions for the infinite system of integral equation (1) in Banach spaces c 0 β and ℓ 1 β using the concept of Meir-Keeler condensing operators and the Hausdorff measure of non-compactness.

3 Existence of solution in the space c 0 β

Assume that the following conditions are satisfied:

1. The functions ( h i ) i = 1 ∞ defined on the set I 3 × R ∞ are continuous and have real values. The operator T : I 3 × c 0 β → c 0 β is defined on the space I 3 × c 0 β as

   ( ψ 1 , ψ 2 , ψ 3 , x ) → ( T x ) ( ψ 1 , ψ 2 , ψ 3 ) = ( β 1 h 1 ( ψ 1 , ψ 2 , ψ 3 ) , β 2 h 2 ( ψ 1 , ψ 2 , ψ 3 ) , … ) .

   The collection of functions { ( T x ) ( ψ 1 , ψ 2 , ψ 3 ) } ( ψ 1 , ψ 2 , ψ 3 ) ∈ I 3 exhibits equicontinuity at all points in the space c 0 β . This means that for any given value of ε > 0 , there exists a corresponding δ > 0 such that

   ∥ x − y ∥ c 0 β ≤ δ ⇒ ∥ ( T x ) ( ψ 1 , ψ 2 , ψ 3 ) − ( T y ) ( ψ 1 , ψ 2 , ψ 3 ) ∥ c 0 β ≤ ε .

2. For each fixed ( ψ 1 , ψ 2 , ψ 3 ) ∈ I 3 , x ( ψ 1 , ψ 2 , ψ 3 ) ∈ C 1 ( I 3 , c 0 β ) , the following inequality holds:

   ∣ h n ( ψ 1 , ψ 2 , ψ 3 ) , x ( ψ 1 , ψ 2 , ψ 3 ) ∣ ≤ p n ( ψ 1 , ψ 2 , ψ 3 ) + q n ( ψ 1 , ψ 2 , ψ 3 ) sup i ≥ n { ∣ β i v i ∣ } , n ∈ N ,

   where p i ( ψ 1 , ψ 2 , ψ 3 ) and q i ( ψ 1 , ψ 2 , ψ 3 ) represent the continuous real-valued functions defined on I 3 . Additionally, the sequence of functions ( q i ( ψ 1 , ψ 2 , ψ 3 ) ) i ∈ N is uniformly bounded on I 3 , while the sequence of functions ( p i ( ψ 1 , ψ 2 , ψ 3 ) ) i ∈ N uniformly converges on I 3 to a function that becomes identically zero on I 3 .

3. The functions K n : I 6 → R are continuous throughout the entire domain I 6 , where n belongs to the set of natural numbers N . Additionally, these functions K n ( ψ 1 , ψ 2 , ψ 3 , ς 1 , ς 2 , ς 3 ) exhibit equicontinuity with respect to the variables ( ψ 1 , ψ 2 , ψ 3 ) , meaning that for any ε > 0 , there exists a corresponding δ > 0 such that

   ∣ K n ( ψ 1 , ψ 2 , ψ 3 , ς 1 , ς 2 , ς 3 ) − K n ( ψ 1 ¯ , ψ 2 ¯ , ψ 3 ¯ , ς 1 ¯ , ς 2 ¯ , ς 3 ¯ ) ∣ < ε ,

   where ∣ ψ 1 − ψ 1 ¯ ∣ < δ , ∣ ψ 2 − ψ 2 ¯ ∣ < δ , and ∣ ψ 3 − ψ 3 ¯ ∣ < δ , for all ( ς 1 , ς 2 , ς 3 ) ∈ I 3 . In addition to that the function sequence ( K n ( ψ 1 , ψ 2 , ψ 3 , ς 1 , ς 2 , ς 3 ) ) is equibounded on the set I 3 , and there exists a constant

   K = sup { ∣ K n ( ψ 1 , ψ 2 , ψ 3 , ς 1 , ς 2 , ς 3 ) ∣ : ( ψ 1 , ψ 2 , ψ 3 ) , ( ς 1 , ς 2 , ς 3 ) ∈ I 3 , n ∈ N } < ∞ .

4. r n : I 3 → R is continuous; also, the sequence of functions ( r n ) uniformly converges to zero on I 3 . We define the finite constant ℛ as

   ℛ = sup { ∣ r n ( ψ 1 , ψ 2 , ψ 3 ) ∣ : ( ψ 1 , ψ 2 , ψ 3 ) ∈ I 3 , n ∈ N } .

   Considering condition ( i i ) , the following finite constants are defined:

   Q = sup { β n q n ( ψ 1 , ψ 2 , ψ 3 ) : ( ψ 1 , ψ 2 , ψ 3 ) ∈ I 3 , n ∈ R } , P = sup { β n p n ( ψ 1 , ψ 2 , ψ 3 ) : ( ψ 1 , ψ 2 , ψ 3 ) ∈ I 3 , n ∈ N } .