Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type

1 Introduction

Integral equations create a very significant 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 different 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 infinite 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 infinite 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₊, c₀) of functions defined, continuous and bounded on the real half-axis and with values in the sequence space c₀. In the present paper we prove a stronger result. Namely, we show that infinite system of integral equations of Volterra-Hammerstein type has at least one solution in the space BC(ℝ₊, l₁) consisting of all functions defined, continuous and bounded on the interval ℝ₊ with values in the sequence space l₁. Of course each such solution belongs to the space BC(ℝ₊, c₀) considered in [10].

Let us mention that paper [10] was the first one in which the study of solvability of infinite systems of integral equations defined on an unbounded interval was carried out. All known up to now results have been obtained in the space of functions defined on a bounded interval (see [11, 12, 13, 14], for example).

2 Notation, definitions 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₊ = [0, ∞).

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 B_r 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 definition 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.

The set ker μ defined 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 satisfies the following additional conditions:

(vii) μ(X + Y) ≤ μ(X) + μ(Y).

(viii) μ (λX) = |λ| μ(X) for λ ∈ ℝ.

Moreover, we say that a measure of noncompactness μ has maximum property if

(ix) μ(X ∪ Y) = max{μ(X), μ(Y)}.

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 first measure of noncompactness was defined in 1930 by Kuratowski [16] in the following way

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 defined the Hausdorff measure of noncompactness ([17, 18]):

It can be shown that the measure χ(X) is also regular and that both defined above measures α(X) and χ(X) are equivalent. But despite of these similarities it turns out that the Hausdorff 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 find explicit formulas describing the Hausdorff 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 Hausdorff measure of noncompactness in the space l₁ (cf. [15]). So, let us call to mind that the space l₁ 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 Hausdorff measure of noncompactness of any bounded set X ∈ 𝔐_l1:

To prove our main result we will also need a fixed point theorem involving a measure of noncompactness. The basic theorem in this subject is known fixed point theorem proved by Darbo [19]. We will use a modified version of Darbo theorem formulated below (cf. [15]).

3 Measures of noncompactness in the space BC(ℝ₊, l₁)

In [10] we investigated measures of noncompactness in the space BC(ℝ₊, E) consisting of all functions defined, continuous and bounded on ℝ₊ with values in a fixed 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 infinite dimensional space. Consider the Banach space BC(ℝ₊, E) equipped with the supremum norm ∥x∥_∞ defined in the standard way

For further purposes, for a fixed T > 0 consider the Banach space C_T = C([0, T], E) consisting of functions defined and continuous on the interval [0, T] with values in E and endowed with the norm

In [10] we defined three formulas for measures of noncompactness in the space BC(ℝ₊, E) and each such formula is a sum of three components. The first and the second component are the same in each formula and we start to present them. So, let us fix a set X ∈ 𝔐_BC(ℝ+,E) and numbers T > 0 and ε > 0. For any function x ∈ X we define the modulus of continuity ω^T(x, ε) of the function x on the interval [0, T] by the classical formula

Next, let us define

and finally, we put

Notice that both above limits exist (for details see [10]). The quantity ω₀(X) is the first component of each of mentioned earlier measures of noncompactness in BC(ℝ₊, E). Next, to obtain the second component, assume that y = y(X) is a measure of noncompactness in the space E. Fix number t ∈ ℝ₊ and denote by X(t) the so-called cross-section of the set X at the point t:

Further, for a fixed T > 0 let us put

and

Notice that the above limit exists (see [10]). The obtained quantity y_∞(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 infinity. We can do it in three ways and we will describe each of them. So, for a fixed T > 0 let us define

Next, notice that there exists the limit

and the quantity a_∞(X) is considered as the third component of the measure of noncompactness in the space BC(ℝ₊, E).

Instead of a_∞(X) we can take

where

or we can also define

where diam X(t) is understood as

And now let us consider the functions y_a, y_b, y_c defined on the family 𝔐_BC(ℝ+,E) as follows

In [10] we proved that under some assumptions on y the functions y_a, y_b, y_c are measures of noncompactness in the space BC(ℝ₊, E). More precisely, we have the following result.

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 y_∞ we replace the Hausdorff measure of noncompactness χ by an arbitrary regular measure of noncompactness μ equivalent to the Hausdorff measure χ[10].

Now, we are going to present formulas (3.7), (3.8) and (3.9) in the special case, for E = l₁. The space l₁ 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 infinite systems of integral equations (see Theorem 4.2).

Therefore, we will work in the Banach space

The fundamental fact in our investigations is that every function x ∈ BC(ℝ₊, l₁) can be regarded as a function sequence

for t ∈ ℝ₊, where for any fixed t the sequence (x_n(t)) is a real sequence being an element of the space l₁. 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 ω₀(X), χ_∞(X) and a_∞(X) of the measure of noncompactness χ_a(X) for any set X ∈ 𝔐_BC(ℝ+,l1). So, let us start with ω₀(X). Fix arbitrarily numbers T > 0 and ε > 0. For any x = x(t) = (x_n(t)) ∈ X we have

Hence, we get

and next

Finally, we obtain

Next, we are going to define the second component occuring in the formula for the measure χ_a(X). To this end let us assume that X ∈ 𝔐_BC(ℝ+,l1) and t ∈ ℝ₊ is arbitrarily fixed. Using (2.1) we have

Next, for a fixed T > 0 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 χ_a(X). Thus, fix an arbitrary number T > 0. 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₊, l₁), where ω₀(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₊, l₁), where

and

4 Theorem on the existence of solutions of infinite systems of integral equations on the real half-axis

Let us consider the following infinite system of nonlinear quadratic integral equations of the Volterra-Hammerstein type

for t ∈ ℝ₊ and for n = 1, 2, ….

In [10] we proved that system (4.1) has at least one solution in the space BC(ℝ₊, c₀) := {x : ℝ₊ → c₀ : 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₊, l₁). For the convenience, from now on the space BC(ℝ₊, l₁) will be denoted by BC₁.

At the beginning we provide a lemma which we will utilize in the proof of our main result.

In what follows we will investigate the solvability of system (4.1) in the space BC₁ under the below listed assumptions.

1. The function sequence (a_n(t)) is an element of the space BC₁ such that limt→∞∑n=1∞ a_n(t) = 0.

2. The functions k_n(t, s) = k_n : R+2 → ℝ are continuous on the set R+2 (n = 1, 2, …). Moreover, the functions t → k_n(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∀s∈R+∀t1,t2∈[0,T][|t2−t1|≤δ⇒|kn(t2,s)−kn(t1,s)|≤ε].

3. There exists a constant K₁ > 0 such that

   ∑n=1∞∫0t|kn(t,s)|ds≤K1

   for any t ∈ ℝ₊.

4. The sequence (k_n(t, s)) is equibounded on R+2 i.e., there exists a constant K₂ > 0 such that |k_n(t, s)| ≤ K₂ for t, s ∈ ℝ₊ and n = 1, 2, ….

5. The function ∑n=1∞ f_n is defined on the set ℝ₊ × ℝ^∞ and takes real values. Moreover, the function t → ∑n=1∞ f_n(t, x_n, x_n+1, …) is continuous on ℝ₊ uniformly with respect to x = (x_n) ∈ l₁ i.e., the following condition is satisfied

   ∀ε>0∀t0∈R+∃δ>0∀(xi)∈l1∀t∈R+[|t−t0|≤δ⇒∑n=1∞|fn(t,xn,xn+1,…)−fn(t0,xn,xn+1,…)|≤ε].

6. There exists a function l : ℝ₊ → ℝ₊ such that l is nondecreasing on ℝ₊, continuous at 0 and the following condition is satisfied

   |fn(t,xn,xn+1,…)−fn(t,yn,yn+1,…)|≤l(r)|xn−yn|

   for all x = (x_i), y = (y_i) ∈ l₁ such that ∥x∥_l1 ≤ r, ∥y∥_l1 ≤ r, for any t ∈ ℝ₊ and for n = 1, 2, ….

7. The sequence of functions (f_n), where f_n(t) = |f_n(t, 0, 0, …)| belongs to the space BC₁ and limt→∞∑n=1∞ f_n(t) = 0.

   Let us notice that on the basis of assumption (vii) we infer that there exists the finite constant F = sup { ∑n=1∞ f_n(t) : t ∈ ℝ₊}.

   Now, we can formulate our further assumptions concerning infinite system (4.1).

8. The function g_n is defined on the set ℝ₊ × ℝ^∞ and takes real values for n = 1, 2, …. Moreover, the operator g defined on the space ℝ₊ × l₁ by the equality

   (gx)(t)=(gn(t,x))=(g1(t,x),g2(t,x),…)

   transforms the space ℝ₊ × l₁ into l₁ and is such that the family of functions {(gx)(t)}_t∈ℝ+ is equicontinuous at every point of the space l₁ i.e., for each 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.

9. The operator g defined in assumption (viii) is bounded on the space ℝ₊ × l₁. More precisely, there exists a positive constant G such that ∥(gx)(t)∥_l1 ≤ G for any x ∈ l₁ and for each t ∈ ℝ₊.

10. There exists a positive solution r₀ of the inequality

    A+F¯G¯K1+G¯K1rl(r)≤r

    such that G K₁ l(r₀) < 1, where the constants F, G, K₁ were defined above and the constant A is defined in the following way

    A=sup{∑n=1∞|an(t)|:t∈R+}.

Now we can present our main result concerning the solvability of infinite system of integral equations (4.1).