On the Space of Functions with Growths Tempered by a Modulus of Continuity and Its Applications

Abstract

We are going to study the space of real functions defined on a bounded metric space and having growths tempered by a modulus of continuity. We prove also a sufficient condition for the relative compactness in the mentioned function space. Using that condition and the classical Schauder fixed point theorem, we show the existence theorem for some quadratic integral equations of Fredholm type in the space of functions satisfying the Hölder condition. An example illustrating the mentioned existence result is also included.

1. Introduction

The principal aim of the paper is to discuss the function space consisting of functions with growths tempered by a given modulus of continuity. Functions belonging to such a space have their moduli of continuity in some sense proportional to a given modulus of continuity. We assume here that the mentioned function space consists of real functions defined on a bounded metric space and, as we pointed out above, having growths tempered by a given modulus of continuity. Typical examples of function spaces of such a type are the space of functions satisfying the Lipschitz condition and the space of Hölder functions, that is, the space of functions satisfying the Hölder condition with a given exponent belonging to the interval (0,1].

It is a surprising fact that those spaces, although they are frequently used in mathematical investigations, were not intensively studied and described in mathematical literature (cf. [1–4]). It is probably caused by the fact that the norm in the mentioned function space is not convenient for use in comparison with the norm in the classical space of real functions being continuous on a compact set. On the other hand, there are no known convenient sufficient conditions for relative compactness of bounded subsets of the function space in question.

It is also worthwhile mentioning that it is very difficult to encounter some applications of the space of functions with growths tempered by a modulus of continuity in the theory of operator equations (functional, differential, integral, etc.), (cf. [5, 6]).

In this paper we are going to discuss in detail the above mentioned function space consisting of real functions having growths tempered by a given modulus of continuity. We describe a general form of such a space and we point out some important particular cases of spaces of the described type.

Apart from this, we indicate some essential properties of the considered function space. Namely, we show that such a space is not a closed subspace of the space of all real continuous functions with the supremum norm, with respect to this norm. The consequence of this property is the fact that the norm introduced in the space of functions with tempered growths is not equivalent to the classical supremum norm. This causes that it is not possible to use the supremum norm in all considerations conducted in the mentioned space of functions with tempered growths.

The main result proved in the paper is a sufficient condition for relative compactness of bounded subset of the discussed function space. We also point out several particular cases of that result, especially in the space of functions satisfying the Hölder condition.

The last part of the paper contains an application of the obtained results in proving a theorem on the existence of a solution of a quadratic integral equation in the space of functions satisfying the Hölder condition. An example showing the applicability of that theorem to a concrete quadratic integral equation is also included.

2. The Space of Functions with Tempered Moduli of Continuity

In this section we discuss the space of real functions defined on a given bounded metric space and having the growths tempered by a given modulus of continuity. Moreover, we will also study some important particular cases of that space.

To this end denote by ℝ the set of all real numbers and put ℝ₊=[0,∞). A function 𝜔∶ℝ₊→ℝ₊ is said to be a modulus of continuity if 𝜔(0)=0,𝜔(𝜀)>0 for 𝜀>0, and 𝜔 is nondecreasing on ℝ₊.

We will also assume (but not always) that the modulus of continuity 𝜔=𝜔(𝜀) is continuous at 𝜀=0, that is, 𝜔(𝜀)→0 as 𝜀→0.

In order to clarify the concept of the modulus of continuity, let us assume that (𝑋,𝑑) is a given bounded metric space. Denote by 𝐶(𝑋) the space of all real functions defined and continuous on the metric space 𝑋. For 𝑥∈𝐶(𝑋) and for an arbitrary fixed number 𝜀>0 let us define the quantity 𝜈(𝑥,𝜀) by the formula 𝜈(𝑥,𝜀)=sup{|𝑥(𝑢)−𝑥(𝑣)|∶𝑢,𝑣∈𝑋,𝑑(𝑢,𝑣)⩽𝜀}. (1)

The function 𝜀→𝜈(𝑥,𝜀) is called the modulus of continuity of the function 𝑥. Observe that in order to define this concept it is sufficient to consider the space 𝐵(𝑋) consisting of real functions defined and bounded on 𝑋.

Observe that if 𝑥∈𝐵(𝑋), then the modulus of continuity of 𝑥, that is, the function 𝜀→𝜈(𝑥,𝜀), is the modulus of continuity in the above defined sense. Obviously, this modulus is continuous at 𝜀=0 if and only if 𝑥 is uniformly continuous on 𝑋.

In what follows let us fix a modulus of continuity 𝜔=𝜔(𝜀) and assume, as before, that (𝑋,𝑑) is a given bounded metric space. Denote by 𝐶_𝜔(𝑋) the set of all real functions defined on 𝑋 such that their growths are tempered by the modulus of continuity 𝜔. More precisely, a function 𝑥=𝑥(𝑢) belongs to the set 𝐶_𝜔(𝑋) provided 𝑥∶𝑋→ℝ and there exists a constant 𝐾_𝑥>0 such that |𝑥(𝑢)−𝑥(𝑣)|⩽𝐾_𝑥𝜔(𝑑(𝑢,𝑣)) (2) for all 𝑢,𝑣∈𝑋. In other words, we have that 𝑥∈𝐶_𝜔(𝑋) if and only if the quantity |𝑥(𝑢)(𝑣sup−𝑥)|𝜔(𝑑(𝑢,𝑣))∶𝑢,𝑣∈𝑋,𝑢≠𝑣 (3) is finite.

It is easy to check that the set 𝐶_𝜔(𝑋) forms a linear space over the field of real numbers ℝ. Obviously, 𝐶_𝜔(𝑋) is a linear subspace of the space 𝐶(𝑋).

In the sequel, let us notice that if 𝑥∈𝐶_𝜔(𝑋), then in view of (2) there exists a constant 𝐾_𝑥>0 such that 𝜈(𝑥,𝜀)⩽𝐾_𝑥𝜔(𝜀) (4) for any 𝜀>0. Obviously the converse implication is also true.

Thus, we can say equivalently that the set 𝐶_𝜔(𝑋) consists of all real functions defined on  X such that the moduli of continuity of those functions are tempered by the given modulus of continuity 𝜔=𝜔(𝜀).

Now, fix arbitrarily an element 𝑢₀∈𝑋. For an arbitrary function 𝑥∈𝐶_𝜔(𝑋) we define the quantity ‖𝑥‖ by the formula ‖𝑥‖=𝑥𝑢₀|𝑥(𝑢)(𝑣+sup−𝑥)|𝜔(𝑑(.𝑢,𝑣))∶𝑢,𝑣∈𝑋,𝑢≠𝑣 (5)

Notice that ||𝑥||<∞ for any 𝑥∈𝐶_𝜔(𝑋). Moreover, it can be shown that ‖⋅‖ is a norm in the space 𝐶_𝜔(𝑋); that is, 𝐶_𝜔(𝑋) is a normed space with the norm defined by (5).

We show that the norm defined by (5) is complete. To this end, take a Cauchy sequence (𝑥_𝑛) in the space 𝐶_𝜔(𝑋). Fix arbitrarily a number 𝜀>0 and denote 𝐷=max{1,𝜔(diam𝑋)}, where diam𝑋 denotes the diameter of the metric space 𝑋. Then, we can find a natural number 𝑛₀ such that for 𝑛,𝑚⩾𝑛₀, we have that ‖𝑥_𝑛−𝑥_𝑚‖⩽𝜀/2𝐷 or, by (5), 𝑥_𝑛𝑢₀−𝑥_𝑚𝑢₀𝑥+sup^𝑛(𝑢)−𝑥_𝑚(𝑢)−𝑥_𝑛(𝑣)−𝑥_𝑚(𝑣)𝜔(𝑑(⩽𝜀𝑢,𝑣))∶𝑢,𝑣∈𝑋,𝑢≠𝑣.2𝐷 (6) Further, let us notice that from (6) we infer that 𝑥_𝑛𝑢₀−𝑥_𝑚𝑢₀⩽𝜀⩽𝜀2𝐷2. (7) This means that the real sequence {𝑥_𝑛(𝑢₀)} satisfies the Cauchy condition in ℝ with natural metric. Hence we obtain that this sequence is convergent to a real number, say 𝑥(𝑢₀), that is, lim_𝑛→∞𝑥_𝑛(𝑢₀)=𝑥(𝑢₀).

Next, keeping in mind (6) again we deduce that 𝑥sup^𝑛(𝑢)−𝑥_𝑚(𝑢)−𝑥_𝑛(𝑣)−𝑥_𝑚(𝑣)𝜔(𝑑(⩽𝜀𝑢,𝑣))∶𝑢,𝑣∈𝑋,𝑢≠𝑣.2𝐷 (8)

Putting in the above inequality 𝑣=𝑢₀, we get that for each 𝑢∈𝑋,𝑢≠𝑢₀, the following estimate holds 𝑥_𝑛(𝑢)−𝑥_𝑚(𝑢)−𝑥_𝑛𝑢₀−𝑥_𝑚𝑢₀𝜔𝑑𝑢,𝑢₀⩽𝜀2𝐷. (9)

This yields the inequality 𝑥_𝑛(𝑢)−𝑥_𝑚(𝑢)−𝑥_𝑛𝑢₀−𝑥_𝑚𝑢₀⩽𝜀𝜔𝑑2𝐷𝑢,𝑢₀ (10)

which is valid for 𝑢∈𝑋 and 𝑚,𝑛∈ℕ,𝑛,𝑚⩾𝑛₀ (ℕ denotes the set of natural numbers).

From that last inequality we get 𝑥_𝑛(𝑢)−𝑥_𝑚(𝑢)−𝑥_𝑛𝑢₀−𝑥_𝑚𝑢₀⩽𝜀𝜔𝑑2𝐷𝑢,𝑢₀⩽𝜀2. (11)

Hence we obtain 𝑥_𝑛(𝑢)−𝑥_𝑚(𝑢)⩽𝑥_𝑛𝑢₀−𝑥_𝑚𝑢₀+𝜀2 (12) for an arbitrary 𝑢∈𝑋 and 𝑛,𝑚⩾𝑛₀.

Linking the above inequality and (7), we deduce that 𝑥_𝑛(𝑢)−𝑥_𝑚(𝑢)⩽𝜀 (13) for 𝑢∈𝑋 and for 𝑚,𝑛∈ℕ,𝑛,𝑚⩾𝑛₀. But this fact means that (𝑥_𝑛(𝑢)) is a real Cauchy sequence for arbitrarily fixed 𝑢∈𝑋. Thus this sequence is convergent in ℝ. Denote its limit by 𝑥(𝑢), that is, 𝑥(𝑢)=lim_𝑛→∞𝑥_𝑛(𝑢). (14)

Passing in (13) with 𝑚→∞, we obtain 𝑥_𝑛(𝑢)−𝑥(𝑢)⩽𝜀 (15)

for any 𝑢∈𝑋 and for 𝑛∈ℕ,𝑛⩾𝑛₀. This means that the sequence (𝑥_𝑛) converges uniformly to the function 𝑥 on the metric space 𝑋.

Now, we show that the sequence (𝑥_𝑛) converges to the function 𝑥 in the sense of norm (5). To this end we prove first that 𝑥∈𝐶_𝜔(𝑋). Indeed, taking into account the fact that (𝑥_𝑛) is a Cauchy sequence in the space 𝐶_𝜔(𝑋), we infer that (𝑥_𝑛) is bounded in 𝐶_𝜔(𝑋); that is, there exists a constant 𝑀>0 such that ‖𝑥_𝑛‖⩽𝑀 for 𝑛=1,2,…. Hence, in view of (5) we have 𝑥sup^𝑛(𝑢)−𝑥_𝑛(𝑣)𝜔(𝑑(𝑢,𝑣))∶𝑢,𝑣∈𝑋,𝑢≠𝑣⩽𝑀. (16)

This implies that for arbitrarily fixed 𝑢,𝑣∈𝑋,𝑢≠𝑣 and for 𝑛∈ℕ, we get 𝑥_𝑛(𝑢)−𝑥_𝑛(𝑣)𝜔(𝑑(𝑢,𝑣))⩽𝑀, (17) and, consequently, 𝑥_𝑛(𝑢)−𝑥_𝑛(𝑣)⩽𝑀𝜔(𝑑(𝑢,𝑣)) (18) for all 𝑢,𝑣∈𝑋 and for 𝑛∈ℕ.

Now, fix arbitrarily 𝑢,𝑣∈𝑋. Then, passing in the above inequality with 𝑛→∞, we obtain |𝑥(𝑢)−𝑥(𝑣)|⩽𝑀𝜔(𝑑(𝑢,𝑣)) (19) for all 𝑢,𝑣∈𝑋. But this means that 𝑥∈𝐶_𝜔(𝑋).

In what follows, using the fact that the sequence (𝑥_𝑛) satisfies inequality (6), we derive that for arbitrary 𝑢,𝑣∈𝑋,𝑢≠𝑣 and for arbitrary 𝑛,𝑚∈ℕ,𝑛,𝑚⩾𝑛₀, the following inequality holds 𝑥_𝑛(𝑢)−𝑥_𝑚(𝑢)−𝑥_𝑛(𝑣)−𝑥_𝑚(𝑣)𝜔(𝑑(⩽𝜀𝑢,𝑣)).2𝐷 (20)

Fixing 𝑢,𝑣 and passing with 𝑚→∞, we get 𝑥_𝑛(𝑢)−𝑥(𝑢)−𝑥_𝑛(𝑣)−𝑥(𝑣)𝜔(𝑑(⩽𝜀𝑢,𝑣)).2𝐷 (21)

Hence, in virtue of the arbitrariness of the choice of points 𝑢,𝑣∈𝑋,𝑢≠𝑣, we have the estimate 𝑥sup^𝑛(𝑢)−𝑥(𝑢)−𝑥_𝑛(𝑣)−𝑥(𝑣)𝜔(𝑑(⩽𝜀𝑢,𝑣))∶𝑢,𝑣∈𝑋,𝑢≠𝑣2𝐷 (22) being valid for 𝑛⩾𝑛₀.

In a similar way, keeping in mind inequality (6), we infer that 𝑥_𝑛𝑢₀−𝑥_𝑚𝑢₀⩽𝜀2𝐷 (23) for 𝑛,𝑚∈ℕ,𝑛,𝑚⩾𝑛₀. Hence, for 𝑚→∞, we obtain 𝑥_𝑛𝑢₀−𝑥𝑢₀⩽𝜀.2𝐷 (24)Now, combining (22), (24), and (5), we obtain that 𝑥_𝑛⩽𝜀−𝑥𝐷⩽𝜀 (25) for all 𝑛∈ℕ,𝑛⩾𝑛₀. This means that the sequence (𝑥_𝑛) is convergent in the space 𝐶_𝜔(𝑋) to the function 𝑥. Thus, 𝐶_𝜔(𝑋) is a Banach space.

Now, we provide a few examples.

Example 1. Take 𝜔_𝐿(𝜀)=𝜀 for 𝜀∈ℝ₊. Obviously 𝜔_𝐿 is a modulus of continuity. Moreover, notice that 𝑥∈𝐶_𝜔𝐿(𝑋) if and only if there exists a constant 𝐿_𝑥>0 such that |𝑥(𝑢)−𝑥(𝑣)|⩽𝐿_𝑥𝑑(𝑢,𝑣) (26) for 𝑢,𝑣∈𝑋. This means that the space 𝐶_𝜔𝐿(𝑋) consists of functions 𝑥∶𝑋→ℝ satisfying the Lipschitz condition. The norm in the space 𝐶_𝜔𝐿(𝑋) has the form ‖𝑥‖_𝐿=𝑥𝑢₀|𝑥(𝑢)(𝑣+sup−𝑥)|𝑑().𝑢,𝑣∶𝑢,𝑣∈𝑋,𝑢≠𝑣 (27)

Example 2. Let 𝜔_𝐻 denote the modulus of continuity which corresponds to the Hölder condition. This means that 𝜔_𝐻(𝜀)=𝜀^𝛼 for 𝜀⩾0, where 𝛼 is a fixed number from the interval (0,1].
Observe that 𝑥∈𝐶_𝜔𝐻(𝑋) if and only if there exists a constant 𝐻_𝑥>0 such that |𝑥(𝑢)−𝑥(𝑣)|⩽𝐻_𝑥(𝑑(𝑢,𝑣))^𝛼 (28) for all 𝑢,𝑣∈𝑋. In other words, the function 𝑥 satisfies the Hölder condition with a constant 𝐻_𝑥 and with an exponent 𝛼. Notice that the norm in the space 𝐶_𝜔𝐻(𝑋) has the from ‖𝑥‖_𝐻=𝑥𝑢₀|𝑥(𝑢)(𝑣+sup−𝑥)|(𝑑(𝑢,𝑣))^𝛼∶𝑢,𝑣∈𝑋,𝑢≠𝑣. (29)

Example 3. Now, let us take into account the modulus of continuity 𝜔 having the form 𝜔(𝜀)=0for𝜀=0,1for𝜀>0. (30)
Note that 𝑥∈𝐶_𝜔(𝑋) if and only if there exists a constant 𝐾_𝑥>0 such that |𝑥(𝑢)−𝑥(𝑣)|⩽𝐾_𝑥 for 𝑢,𝑣∈𝑋. Obviously, this condition is equivalent to the boundedness of the function 𝑥. Indeed, fix 𝑢₀∈𝑋. Then, for any 𝑢∈𝑋 we have |𝑥(𝑢)|−𝑥𝑢₀⩽𝑥(𝑢)−𝑥𝑢₀⩽𝐾_𝑥. (31)
This yields |𝑥(𝑢)|⩽𝑥𝑢₀+𝐾_𝑥<∞ (32) for 𝑢∈𝑋. The converse implication is also obvious. Thus we can write that 𝐶_𝜔(𝑋)=𝐵(𝑋); that is, 𝐶_𝜔(𝑋) contains all functions defined and bounded on 𝑋. Further, observe that the norm in the space 𝐶_𝜔(𝑋) has now the form ‖𝑥‖=𝑥𝑢₀+sup_{𝑢,𝑣∈𝑋}|𝑥(𝑢)−𝑥(𝑣)|. (33)
It is easy to show that this norm is equivalent to the classical supremum norm ‖⋅‖_∞ in the space 𝐵(𝑋). Indeed, we have ‖𝑥‖_∞⩽‖𝑥‖⩽3‖𝑥‖_∞ (34) for any function 𝑥∈𝐵(𝑋), where ||𝑥||_∞=sup_𝑢∈𝑋|𝑥(𝑢)|.

The examples of the functions 𝑥(𝑢)=𝑢−1 and 𝑥(𝑢)=𝑢 in the space 𝐵([0,2]) with 𝑢₀=0 show that the equality signs in (34) are attained.

3. Some Properties of the Space of Hölder Functions

In order to simplify the considerations in this section, we restrict ourselves to spaces of real functions defined on a fixed interval [𝑎,𝑏]. In this way we will consider the space 𝐶[𝑎,𝑏] consisting of functions 𝑥∶[𝑎,𝑏]→ℝ, which are continuous on [𝑎,𝑏] and equipped with the norm ||𝑥||_∞=sup{|𝑥(𝑡)|∶𝑡∈[𝑎,𝑏]}.

Further, if 𝛼(0<𝛼⩽1) is a fixed number, then the symbol 𝐻_𝛼[𝑎,𝑏] will denote the space 𝐶_𝜔𝐻([𝑎,𝑏]) (see Example 2) with 𝜔_𝐻=𝜀^𝛼; that is, 𝐻_𝛼[𝑎,𝑏] is the collection of all real functions 𝑥 defined on [𝑎,𝑏] and satisfying the Hölder condition. More precisely, 𝑥∈𝐻_𝛼[𝑎,𝑏] if there exists a constant 𝐻^𝛼_𝑥 such that |𝑥(𝑡)−𝑥(𝑠)|⩽𝐻^𝛼_𝑥|𝑡−𝑠|^𝛼 (35) for all 𝑡,𝑠∈[𝑎,𝑏].

Observe that 𝐻_𝛼[𝑎,𝑏] forms a linear subspace of the linear space 𝐶[𝑎,𝑏].

In what follows, for 𝑥∈𝐻_𝛼[𝑎,𝑏], the symbol 𝐻^𝛼_𝑥 will denote the least possible constant for which inequality (35) is satisfied. In other words, we can write 𝐻^𝛼_𝑥|𝑥(𝑡)(𝑠=sup−𝑥)||𝑡−𝑠|^𝛼∶𝑡,𝑠∈[𝑎,𝑏],𝑡≠𝑠. (36)

Notice that in the case 𝛼=1, the linear space 𝐻₁[𝑎,𝑏] coincides with the space of real functions defined on [𝑎,𝑏] and satisfying the Lipschitz condition, that is, |𝑥(𝑡)−𝑥(𝑠)|⩽𝐿_𝑥|𝑡−𝑠| (37) for some constant 𝐿_𝑥>0 and for all 𝑡,𝑠∈[𝑎,𝑏]. This space will be denoted by the symbol 𝐿𝑖𝑝[𝑎,𝑏].

Our principal aim is to show that the spaces 𝐻_𝛼[𝑎,𝑏] (for 𝛼∈(0,1]) do not form closed subspaces of the space 𝐶[𝑎,𝑏] with respect to the norm ‖⋅‖_∞. In order to prove this fact, observe that the space 𝐻_𝛼[𝑎,𝑏] can be normed in a natural way (cf. Example 2) if for 𝑥∈𝐻_𝛼[𝑎,𝑏] we put ‖𝑥‖_𝛼=|𝑥(𝑎|𝑥(𝑡)(𝑠)|+sup−𝑥)||𝑡−𝑠|^𝛼∶𝑡,𝑠∈[𝑎,𝑏],𝑡≠𝑠. (38)

Observe that in view of (36), the above definition can be written in the from ‖𝑥‖_𝛼=|𝑥(𝑎)|+𝐻^𝛼_𝑥. (39) Obviously, the space 𝐻_𝛼[𝑎,𝑏] with the norm (38) is the Banach space (cf. Section 2).

Further on let us notice that for an arbitrarily fixed 𝑥∈𝐻_𝛼[𝑎,𝑏] and for an arbitrary 𝑡∈[𝑎,𝑏] we obtain |𝑥(𝑡⩽|𝑥(𝑡)(𝑎+|𝑥(𝑎⩽|𝑥(𝑎𝑥(𝑡)(𝑎[⩽|𝑥(𝑎𝑥(𝑡)(𝑠[=|𝑥(𝑎|𝑥(𝑡)(𝑠)|−𝑥)|)|)|+sup{|−𝑥)|∶𝑡∈𝑎,𝑏]})|+sup{|−𝑥)|∶𝑡,𝑠∈𝑎,𝑏]})|+sup−𝑥)||𝑡−𝑠|^𝛼|𝑡−𝑠|^𝛼[]⩽|𝑥(𝑎|𝑥(𝑡)(𝑠∶𝑡,𝑠∈𝑎,𝑏,𝑡≠𝑠)|+sup−𝑥)||𝑡−𝑠|^𝛼[]×()∶𝑡,𝑠∈𝑎,𝑏,𝑡≠𝑠𝑏−𝑎^𝛼⩽max1,(𝑏−𝑎)^𝛼𝑥(𝑎|𝑥(𝑡)(𝑠|)|+sup−𝑥)||𝑡−𝑠|^𝛼∶𝑡,𝑠∈[𝑎,𝑏],𝑡≠𝑠}. (40)

Hence we get ‖𝑥‖_∞⩽max1,(𝑏−𝑎)^𝛼‖𝑥‖_𝛼. (41)

The above inequality means that the norm ‖⋅‖_∞ is dominated by the norm ‖⋅‖_𝛼.

If the space 𝐻_𝛼[𝑎,𝑏] would be a closed subspace of 𝐶[𝑎,𝑏] with respect to the norm ‖⋅‖_∞, then this would mean that 𝐻_𝛼[𝑎,𝑏] (as the linear space) is a Banach space with respect to the norm ‖⋅‖_∞. In other words, the space 𝐻_𝛼[𝑎,𝑏] would be a Banach space both with respect to the norm ‖⋅‖_∞ and with respect to the norm ‖⋅‖_𝛼. Hence, keeping in mind (41) and taking into account the theorem on the equivalence of dominated norms [4], we would infer that the norms ‖⋅‖_∞ and ‖⋅‖_𝛼 are equivalent in the space 𝐻_𝛼[𝑎,𝑏].

We show that such a conclusion is not true.

To this end fix a number 𝛼∈(0,1] and consider the sequence (𝑥_𝑛) of real functions defined on the interval [0,1] by the formula 𝑥_𝑛(𝑡)=𝑛^𝛼𝑡^𝛼1for𝑡∈0,𝑛,11for𝑡∈𝑛,1, (42) for 𝑛=1,2,…. Observe that (𝑥_𝑛)⊂𝐶[0,1] and ||𝑥_𝑛||_∞=1 for 𝑛=1,2,…. On the other hand, we have 𝑥_𝑛(𝑡)−𝑥_𝑛(0)𝑡^𝛼=𝑛^𝛼𝑡^𝛼𝑡^𝛼=𝑛^𝛼,for𝑡∈(0,1]. (43)

This implies that ||𝑥_𝑛||_𝛼=𝑛^𝛼, for 𝑛=1,2,…. Hence we get 𝑥_𝑛∞𝑥_𝑛𝛼=1𝑛^𝛼 (44)

which shows that ||𝑥_𝑛||_∞/||𝑥_𝑛||_𝛼→0 as 𝑛→∞. Thus the norms ‖⋅‖_∞ and ‖⋅‖_𝛼 are not equivalent in the space 𝐻_𝛼[0,1]. Hence we infer that 𝐻_𝛼[0,1] is not closed subspace of the space 𝐶[0,1] with respect to the norm ‖⋅‖_∞. Obviously the same assertion is true if we replace the interval [0,1] by an arbitrary interval [𝑎,𝑏].

In what follows, let us observe that for 0<𝛼<𝛾⩽1 the following inclusions hold: 𝐻_𝛾[𝑎,𝑏]⊂𝐻_𝛼[𝑎,𝑏]⊂𝐶[𝑎,𝑏]. (45)

Particularly, taking into account that for 𝛾=1, we have 𝐻₁[𝑎,𝑏]=𝐿𝑖𝑝[𝑎,𝑏], (46)

which yields 𝐿𝑖𝑝[𝑎,𝑏]⊂𝐻_𝛾[𝑎,𝑏]⊂𝐻_𝛼[𝑎,𝑏]⊂𝐶[𝑎,𝑏] (47)

for 0<𝛼<𝛾<1.

Indeed, if 𝑥∈𝐻_𝛾[𝑎,𝑏], we obtain |𝑥(𝑡)−𝑥(𝑠)|⩽𝐻^𝛾_𝑥|𝑡−𝑠|^𝛾=𝐻^𝛾_𝑥|𝑡−𝑠|^𝛼⋅|𝑡−𝑠|^𝛾−𝛼⩽(𝑏−𝑎)^𝛾−𝛼𝐻^𝛾_𝑥|𝑡−𝑠|^𝛼. (48) This shows that 𝑥∈𝐻_𝛼[𝑎,𝑏] and hence we infer that inclusions (47) hold.

Further, for 𝑥∈𝐻_𝛾[𝑎,𝑏] and for arbitrarily fixed 𝑡,𝑠∈[𝑎,𝑏],𝑡≠𝑠, we derive |𝑥(𝑡)−𝑥(𝑠)||𝑡−𝑠|^𝛼=|𝑥(𝑡)−𝑥(𝑠)||𝑡−𝑠|^𝛾|𝑡−𝑠|^𝛾−𝛼⩽|𝑥(𝑡)−𝑥(𝑠)||𝑡−𝑠|^𝛾(𝑏−𝑎)^𝛾−𝛼. (49)

Hence we get ‖𝑥‖_𝛼=|𝑥(𝑎|𝑥(𝑡)(𝑠)|+sup−𝑥)||𝑡−𝑠|^𝛼[]⩽|𝑥(𝑎|𝑥(𝑡)(𝑠∶𝑡,𝑠∈𝑎,𝑏,𝑡≠𝑠)|+sup−𝑥)||𝑡−𝑠|^𝛾∶𝑡,𝑠∈[𝑎,𝑏],𝑡≠𝑠(𝑏−𝑎)^𝛾−𝛼⩽max1,(𝑏−𝑎)^𝛾−𝛼‖𝑥‖_𝛾 (50) for 𝑥∈𝐻_𝛾[𝑎,𝑏]. Thus the norm ||⋅||_𝛼 is dominated by the norm ||⋅||_𝛾.

We show that the inverse assertion is not true. To this end, take the function sequence (𝑥_𝑛)⊂𝐻_𝛾[0,1], where 𝑥_𝑛 is defined by the formula 𝑥_𝑛(𝑡)=𝑛^𝛾𝑡^𝛾1if𝑡∈0,𝑛,11if𝑡∈𝑛,1 (51) for 𝑛=1,2,…. It is easily seen that 𝑥_𝑛𝛾=𝑥_𝑛(0𝑥)+sup^𝑛(𝑡)−𝑥_𝑛(𝑠)|𝑡−𝑠|^𝛾[]𝑛∶𝑡,𝑠∈0,1,𝑡≠𝑠=sup^𝛾𝑡^𝛾𝑡^𝛾1∶𝑡∈0,𝑛=𝑛^𝛾 (52) for =1,2,…. On the other hand, we have 𝑥_𝑛𝛼𝑛=sup^𝛾𝑡^𝛾𝑡^𝛼1∶𝑡∈0,𝑛=sup𝑛^𝛾𝑡^𝛾−𝛼1∶𝑡∈0,𝑛=𝑛^𝛼 (53) for 𝑛=1,2,…. Hence we get 𝑥_𝑛𝛼𝑥_𝑛𝛾=1𝑛^𝛾−𝛼 (54) which yields that ||𝑥_𝑛||_𝛼/||𝑥_𝑛||_𝛾→0 as 𝑛→∞. This shows that the norm ‖⋅‖_𝛾 is not dominated by the norm ‖⋅‖_𝛼. Thus these norms are not equivalent.

4. A Sufficient Condition for Relative Compactness in the Space 𝐶_𝜔(𝑋)

This section is devoted to present a criterion for relative compactness in the space 𝐶_𝜔(𝑋) of functions with growths tempered by a given modulus of continuity 𝜔=𝜔(𝜀). More precisely, we will discuss a sufficient condition for relative compactness in 𝐶_𝜔(𝑋).

It is worthwhile mentioning that as far as we know, there are no known conditions of such a type (e.g., cf. [1–4]).

Let us recall that the complete description of the space 𝐶_𝜔(𝑋) was given in Section 2.

Theorem 4. Assume that 𝜔=𝜔(𝜀) is a given modulus of continuity such that 𝜔(𝜀)→0 as 𝜀→0, and assume that 𝑋 is a compact metric space. Let 𝐴 be a bounded subset of the space 𝐶_𝜔(𝑋) such that functions belonging to 𝐴 are equicontinuous with respect to the modulus of continuity 𝜔; that is, the following condition is satisfied: ∀_𝜀>0∃_𝛿>0∀_𝑥∈𝐴∀_{𝑢,𝑣∈𝑋𝑢≠𝑣}𝑑()|𝑥(𝑢)(𝑣𝑢,𝑣⩽𝛿⇒−𝑥)|𝜔(𝑑(.𝑢,𝑣))⩽𝜀 (55) Then the set 𝐴 is relatively compact in the space 𝐶_𝜔(𝑋).

Proof. At first, let us observe that since 𝐴 is bounded, then there exists a constant 𝑀>0 such that for an arbitrary 𝑥∈𝐴 the following inequality holds: ‖𝑥‖=𝑥𝑢₀|𝑥(𝑢)(𝑣+sup−𝑥)|𝜔(𝑑(∶𝑢,𝑣))𝑢,𝑣∈𝑋,𝑢≠𝑣⩽𝑀, (56) where 𝑢₀ is a fixed element of the metric space 𝑋 (cf. (5)).
Particularly, we conclude that 𝑥𝑢₀⩽𝑀 (57) for an arbitrary 𝑥∈𝐴. Moreover, on the basis of (56) we infer that for any 𝑥∈𝐴 and for arbitrary 𝑢,𝑣∈𝑋,𝑢≠𝑣, the following inequality holds: |𝑥(𝑢)−𝑥(𝑣)|𝜔(𝑑(𝑢,𝑣))⩽𝑀. (58)
The above inequality yields |𝑥(𝑢)−𝑥(𝑣)|⩽𝑀𝜔(𝑑(𝑢,𝑣)) (59) for an arbitrary function 𝑥∈𝐴 and for all 𝑢,𝑣∈𝑋.
Since lim_𝜀→0𝜔(𝜀)=0, from (59) we deduce that all functions belonging to the set 𝐴 are equicontinuous on the set 𝑋. Apart from that, from (59) we have that for an arbitrary 𝑥∈𝐴 the inequality |𝑥(𝑢)−𝑥(𝑣)|⩽𝑀𝜔(diam𝑋) (60) holds for all 𝑢,𝑣∈𝑋. Putting in the above inequality 𝑣=𝑢₀, we infer that for arbitrary 𝑥∈𝐴 and 𝑢∈𝑋 the following estimate is satisfied: |𝑥(𝑢)|−𝑥𝑢₀⩽𝑥(𝑢)−𝑥𝑢₀⩽𝑀𝜔(diam𝑋). (61)
This implies that |𝑥(𝑢)|⩽𝑥𝑢₀+𝑀𝜔(diam𝑋) (62) for 𝑥∈𝐴 and 𝑢∈𝑋. This means that functions from the set 𝐴 are equibounded.
Now, let us consider an arbitrary sequence (𝑥_𝑛) in the set 𝐴. Taking into account the above established facts, we infer that functions of the sequence (𝑥_𝑛) are equibounded and equicontinuous on the set 𝑋. Hence, in view of Ascoli-Arzéla criterion we derive that there exists a subsequence of the sequence (𝑥_𝑛) which is uniformly convergent on the set 𝑋 to a function 𝑥=𝑥(𝑢). To avoid complicated notation, we will denote the mentioned subsequence of the sequence (𝑥_𝑛) by the same symbol (𝑥_𝑛). Observe that the function 𝑥=𝑥(𝑢) is continuous on the set 𝑋.
In what follows, we show that 𝑥∈𝐶_𝜔(𝑋). To this end observe that from the fact that functions belonging to the sequence (𝑥_𝑛) satisfy inequality (59), we have that 𝑥_𝑛(𝑢)−𝑥_𝑛(𝑣)⩽𝑀𝜔(𝑑(𝑢,𝑣)) (63)
for an arbitrary 𝑛∈ℕ and for 𝑢,𝑣∈𝑋. Since lim_𝑛→∞𝑥_𝑛(𝑧)=𝑥(𝑧) for an arbitrary element 𝑧∈𝑋, thus keeping in mind the continuity of the absolute value and other standard facts from mathematical analysis, we deduce from the last inequality that |𝑥(𝑢)−𝑥(𝑣)|⩽𝑀𝜔(𝑑(𝑢,𝑣)) (64) for arbitrary 𝑢,𝑣∈𝑋.
Similarly, on the basis of (57), we have 𝑥_𝑛𝑢₀⩽𝑀 (65) for 𝑛=1,2…. Hence we obtain that 𝑥𝑢₀⩽𝑀. (66)
Combining (64) and (66) we derive the following estimate: 𝑥𝑢₀+|𝑥(𝑢)−𝑥(𝑣)|𝜔(𝑑(𝑢,𝑣))⩽2𝑀 (67) for arbitrary 𝑢,𝑣∈𝑋,𝑢≠𝑣. This implies that 𝑥𝑢₀|𝑥(𝑢)(𝑣+sup−𝑥)|𝜔(𝑑(𝑢,𝑣))∶𝑢,𝑣∈𝑋,𝑢≠𝑣⩽2𝑀 (68) which shows that 𝑥∈𝐶_𝜔(𝑋).
Further on we show that the sequence (𝑥_𝑛) is convergent to the function 𝑥 in the sense of the norm of the space 𝐶_𝜔(𝑋).
To this end for convenience, let us denote 𝑋²₀=)},𝑋{(𝑢,𝑣∈𝑋×𝑋∶𝑢≠𝑣_2𝛿=(𝑢,𝑣)∈𝑋²₀(),𝑋∶𝑑𝑢,𝑣≤𝛿_2𝛿=𝑋²₀⧵𝑋²_𝛿, (69) where 𝛿>0 is a fixed number.
Obviously we have that 𝑋²₀=𝑋²_𝛿∪𝑋²_𝛿 and the sets 𝑋²_𝛿,𝑋²_𝛿 are disjoint.
Further, for an arbitrarily fixed natural number 𝑛, we obtain 𝑥_𝑛−𝑥=𝑥_𝑛𝑢₀−𝑥𝑢₀+sup_{()𝑢,𝑣∈𝑋20}𝑥_𝑛(𝑢)−𝑥(𝑢)−𝑥_𝑛(𝑣)+𝑥(𝑣)𝜔(𝑑(=𝑥𝑢,𝑣))_𝑛𝑢₀−𝑥𝑢₀+maxsup_{()∈𝑢,𝑣}𝑋²_𝛿𝑥_𝑛(𝑢)−𝑥(𝑢)−𝑥_𝑛(𝑣)+𝑥(𝑣)𝜔(𝑑(,𝑢,𝑣))sup_{(𝑢,𝑣)∈𝑋2𝛿}𝑥_𝑛(𝑢)−𝑥(𝑢)−𝑥_𝑛(𝑣)+𝑥(𝑣)𝜔(𝑑(.𝑢,𝑣)) (70)
Next, fix arbitrarily a number 𝜀>0. Let 𝛿>0 denote a number corresponding to 𝜀/4 according to the assumption about equicontinuity of functions from the set 𝐴 with respect to the modulus of continuity 𝜔=𝜔(𝜀).
Now, denote by 𝛽(𝜀) the number defined as follows: 𝛽(𝜀)𝜀=min2,𝜀𝜔(𝛿)4. (71)
Keeping in mind the fact that functions of the sequence (𝑥_𝑛) are uniformly convergent on the set 𝑋 to a function 𝑥, let us choose a natural number 𝑛₀ such that 𝑥_𝑛(𝑤)−𝑥(𝑤)≤𝛽(𝜀) (72) for all 𝑛⩾𝑛₀ and for any 𝑤∈𝑋.
Further, let us observe that 𝑑(𝑢,𝑣)>𝛿 for (𝑢,𝑣)∈𝑋²_𝛿. Hence we infer that 𝜔(𝑑(𝑢,𝑣))⩾𝜔(𝛿). Thus, for an arbitrary 𝑛⩾𝑛₀, in view of (71) and (72) we get sup_{()∈𝑢,𝑣}𝑋²_𝛿𝑥_𝑛(𝑢)−𝑥(𝑢)−𝑥_𝑛(𝑣)+𝑥(𝑣)𝜔(𝑑(𝑢,𝑣))⩽sup_{()∈𝑢,𝑣}𝑋²_𝛿𝑥_𝑛(𝑢)−𝑥(𝑢)+𝑥_𝑛(𝑣)−𝑥(𝑣)𝜔(𝑑(𝑢,𝑣))⩽sup_{()∈𝑢,𝑣}𝑋²_𝛿𝑥_𝑛(𝑢)−𝑥(𝑢)+𝑥_𝑛(𝑣)−𝑥(𝑣)𝜔(𝛿)𝛽(𝜀)⩽2𝜔(𝛿)≤𝜀2. (73)
Next, taking into account the fact that the functions of the sequence (𝑥_𝑛) belong to the set 𝐴, in virtue of the choice of the number 𝛿 to the number 𝜀/4, we infer that for an arbitrary pair (𝑢,𝑣)∈𝑋²_𝛿 and for an arbitrary natural number 𝑛 the following inequality is satisfied: 𝑥_𝑛(𝑢)−𝑥_𝑛(𝑣)𝜔(𝑑(⩽𝜀𝑢,𝑣))4. (74)
Hence we obtain 𝑥_𝑛(𝑢)−𝑥_𝑛(𝑣⩽𝜀)4𝜔(𝑑(𝑢,𝑣)) (75) for (𝑢,𝑣)∈𝑋 such that 𝑑(𝑢,𝑣)⩽𝛿.
Letting in the above inequality with 𝑛→∞, we get |𝑥(𝑢)(𝑣⩽𝜀−𝑥)|4𝜔(𝑑(𝑢,𝑣)) (76) for (𝑢,𝑣)∈𝑋 such that 𝑑(𝑢,𝑣)⩽𝛿. Consequently, we obtain |𝑥(𝑢)−𝑥(𝑣)|𝜔(𝑑(⩽𝜀𝑢,𝑣))4 (77) for arbitrary (𝑢,𝑣)∈𝑋²_𝛿.
Now, in view of (74) and (77), we derive the following estimates: sup_{()𝑢,𝑣∈𝑋2𝛿}𝑥_𝑛(𝑢)−𝑥(𝑢)−𝑥_𝑛(𝑣)+𝑥(𝑣)𝜔(𝑑(𝑢,𝑣))⩽sup_{()𝑢,𝑣∈𝑋2𝛿}𝑥_𝑛(𝑢)−𝑥_𝑛(𝑣)+|𝑥(𝑢)−𝑥(𝑣)|𝜔(𝑑(⩽𝜀𝑢,𝑣))2. (78)
Combining (70) with inequalities (72), (73), and (78), we obtain that ||𝑥_𝑛−𝑥||⩽𝜀 for 𝑛∈ℕ,𝑛⩾𝑛₀. This means that the sequence (𝑥_𝑛) is convergent to the function 𝑥 with respect to the norm of the space 𝐶_𝜔(𝑋). Finally, we conclude that the set 𝐴 is relatively compact in the space 𝐶_𝜔(𝑋) and the proof is complete.

Now, based on Theorem 4, we prove a manageable and handy sufficient condition for relative compactness in the space 𝐶_𝜔(𝑋).

Theorem 5. Assume that 𝜔₁, 𝜔₂ are moduli of continuity being continuous at zero and such that 𝜔₂(𝜀)=𝑜(𝜔₁(𝜀)) as 𝜀→0, that is, lim_𝜀→0𝜔₂(𝜀)𝜔₁(𝜀)=0. (79)
Further, assume that (𝑋,𝑑) is a compact metric space. Then, if 𝐴 is a bounded subset of the space 𝐶_𝜔2(𝑋) then 𝐴 is relatively compact in the space 𝐶_𝜔1(𝑋).

Proof. Since 𝐴 is bounded in the space 𝐶_𝜔2(𝑋), then there exists a constant 𝑀>0 such that ||𝑥||⩽𝑀 for any 𝑥∈𝐴, or equivalently 𝑥𝑢₀|𝑥(𝑢)(𝑣+sup−𝑥)|𝜔₂(𝑑(𝑢,𝑣))∶𝑢,𝑣∈𝑋,𝑢≠𝑣⩽𝑀. (80)
Hence we infer that |𝑥(𝑢)−𝑥(𝑣)|𝜔₂(𝑑(𝑢,𝑣))⩽𝑀 (81) for all 𝑢,𝑣∈𝑋,𝑢≠𝑣.
Further, fix an arbitrary number 𝑃>0. According to our assumptions there exists 𝛿₀>0 such that 𝜔₂(𝑡)/𝜔₁(𝑡)⩽𝑃 for 𝑡∈(0,𝛿₀]. This yields that 𝜔₂(𝑡)⩽𝑃𝜔₁(𝑡) (82) for 𝑡∈[0,𝛿₀].
Linking (81) and (82), we deduce that |𝑥(𝑢)−𝑥(𝑣)|⩽𝑃𝑀𝜔₁(𝑑(𝑢,𝑣)) (83) for all 𝑢,𝑣∈𝑋 such that 𝑑(𝑢,𝑣)⩽𝛿₀.
Now, take arbitrary 𝑢,𝑣∈𝑋 such that 𝑑(𝑢,𝑣)⩾𝛿₀. Then we get |𝑥(𝑢)−𝑥(𝑣)|𝜔₁(𝑑(⩽|𝑥(𝑢)(𝑣𝑢,𝑣))−𝑥)|𝜔₁𝛿₀=|𝑥(𝑢)−𝑥(𝑣)|𝜔₂(𝑑(⋅𝜔𝑢,𝑣))²(𝑑(𝑢,𝑣))𝜔₁𝛿₀. (84)
Hence, in view of (81), we obtain |𝑥(𝑢)−𝑥(𝑣)|𝜔₁(𝑑(𝜔𝑢,𝑣))⩽𝑀²(diam𝑋)𝜔₁𝛿₀ (85) for all 𝑢,𝑣∈𝑋 such that 𝑑(𝑢,𝑣)⩾𝛿₀.
Finally, joining (81), (82), and (85), we deduce that the set 𝐴 is bounded in the space 𝐶_𝜔1(𝑋).
In what follows, let us fix arbitrarily a number 𝜀₁>0. Keeping in mind the assumption requiring that 𝜔₂(𝑡)=𝑜(𝜔₁(𝑡)) as 𝑡→0, we can find 𝛿>0 such that 𝜔₂(𝑡)𝜔₁(𝑡)⩽𝜀₁ (86) provided 0<𝑡⩽𝛿.
Further, fix an arbitrary number 𝜀>0. Then, according to (86), for the number 𝜀₁=𝜀/𝑀 let us choose 𝛿>0. Then, for 𝑢,𝑣∈𝑋 such that 𝑢≠𝑣 and 𝑑(𝑢,𝑣)⩽𝛿 we get 𝜔₂(𝑑(𝑢,𝑣))𝜔₁(𝑑(⩽𝜀𝑢,𝑣))𝑀. (87)
Hence we obtain 𝜔₂(𝑑(⩽𝜀𝑢,𝑣))𝑀𝜔₁(𝑑(𝑢,𝑣)) (88) for 𝑢,𝑣∈𝑋,𝑢≠𝑣 and 𝑑(𝑢,𝑣)⩽𝛿.
Next, taking 𝑢,𝑣∈𝑋 such that 𝑢≠𝑣, 𝑑(𝑢,𝑣)⩽𝛿 and keeping in mind (81), we derive the following estimate: |𝑥(𝑢)−𝑥(𝑣)|⩽𝑀𝜔₂(𝑑(𝑢,𝑣))⩽𝜀𝜔₁(𝑑(𝑢,𝑣)). (89)
Hence we have |𝑥(𝑢)−𝑥(𝑣)|𝜔₁(𝑑(𝑢,𝑣))⩽𝜀 (90) for 𝑢,𝑣∈𝑋 such that 𝑢≠𝑣 and 𝑑(𝑢,𝑣)⩽𝛿.
Finally, in view of Theorem 4 we conclude that the set 𝐴 is relatively compact in the space 𝐶_𝜔1(𝑋). This completes the proof.

Example 6. In order to illustrate the applicability of Theorem 5, let us consider two moduli of continuity of Hölder type having the form 𝜔₁(𝜀)=𝜀^𝛼,𝜔₂(𝜀)=𝜀^𝛽, where 0<𝛼<𝛽⩽1 (cf. Section 3). Then we have lim_𝜀→0𝜔₂(𝜀)𝜔₁(𝜀)=lim_𝜀→0𝜀^𝛽−𝛼=0. (91)
This shows that the moduli of continuity 𝜔₁(𝜀) and 𝜔₂(𝜀) satisfy the conditions of Theorem 5.
Thus, if we assume that a set 𝐴 is bounded in the space 𝐶_𝜔2([𝑎,𝑏]) or 𝐶_𝜔2(𝑋), then 𝐴 is relatively compact in the space 𝐶_𝜔1([𝑎,𝑏]) or 𝐶_𝜔1(𝑋), respectively, provided (𝑋,𝑑) is a compact metric space.
In other words, if there exist numbers 𝛼, 𝛽 with 0<𝛼<1 and 𝛼<𝛽⩽1 such that 𝐴 is a bounded set in the Hölder space with the exponent 𝛽, that is, there exists a constant 𝑀>0 such that |𝑥(𝑢)−𝑥(𝑣)|⩽𝑀(𝑑(𝑢,𝑣))^𝛽 (92) for any 𝑥∈𝐴 and for all 𝑢,𝑣∈𝑋, then the set 𝐴 is relatively compact in the Hölder space with the exponent 𝛼.

5. Application to a Quadratic Integral Equation

In this final section we are going to present an application of the results obtained in previous sections to derive an existence result for a quadratic integral equation of Fredholm type. Equations of such a type occur naturally in connection with some problems investigated in the theories of radiative transfer, neutron transport, and the kinetic theory of gases [7–11]. Those integral equations are known as Chandrasekhar quadratic integral equations (cf. [10]).

Let us pay attention to some important facts connected with our considerations.

First of all, notice that the study of the existence of solutions of functional, differential, and integral equations is very complicated in the Lipschitz and Hölder function spaces which were considered in Section 3. To justify such an opinion, let us recall a result due to Matkowski [6] asserting that the so-called superposition operator generated by the function 𝑓=𝑓(𝑡,𝑥) maps the space 𝐿𝑖𝑝[𝑎,𝑏] into itself and is Lipschitzian (with respect to the norm in the space 𝐿𝑖𝑝[𝑎,𝑏] described in Example 1) if and only if the function 𝑓 is linear; that is, there exist functions 𝑝,𝑞∈𝐿𝑖𝑝[𝑎,𝑏] such that 𝑓(𝑡,𝑥)=𝑝(𝑡)+𝑞(𝑡)𝑥 (93) for all 𝑡∈[𝑎,𝑏] and 𝑥∈ℝ. Other results in this direction can be found in [5], for instance.

This shows that the investigations on the existence of solutions of operator equations in the space 𝐶_𝜔(𝑋) consisting of functions with growths tempered by a modulus of continuity are rather a delicate matter.

As we indicate above, the object of the study conducted in this section is the quadratic integral equation of Fredholm type having the form 𝑥(𝑡)=𝑝(𝑡)+𝑥(𝑡)^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏, (94) where 𝑡∈[𝑎,𝑏].

In our further considerations we will assume that 𝛽 is a fixed number in the interval (0,1].

Now, we formulate the assumptions under which we will study (94). (i)The function 𝑝=𝑝(𝑡) belongs to the Hölder space 𝐻_𝛽[𝑎,𝑏].(ii)𝑘∶[𝑎,𝑏]×[𝑎,𝑏]→ℝ is a continuous function such that it satisfies the Hölder condition with the exponent 𝛽 with respect to the first variable; that is, there exists a constant 𝑘_𝛽>0 such that |𝑘(𝑡,𝜏)−𝑘(𝑠,𝜏)|⩽𝑘_𝛽|𝑡−𝑠|^𝛽 (95) for all 𝑡,𝑠,𝜏∈[𝑎,𝑏].

In what follows, based on the above assumptions, we can define the finite constant 𝐾 by putting 𝐾=sup^𝑏_𝑎|𝑘(𝑡,𝜏)|𝑑𝜏∶𝑡∈[𝑎,𝑏]. (96)

Now, we are prepared to present our last assumption.(iii)The following inequality holds: 𝑝_𝛽max1,(𝑏−𝑎)^𝛽22𝐾+𝑘_𝛽(<1𝑏−𝑎)4, (97) where ||𝑝||_𝛽 denotes the norm of the function 𝑝 in the space 𝐻_𝛽[𝑎,𝑏].

Now, we can formulate our existence result concerning (94).

Theorem 7. Under assumptions (𝑖)−(𝑖𝑖𝑖) (94) has at least one solution belonging to the space 𝐻_𝛼[𝑎,𝑏], where 𝛼 is arbitrarily fixed number such that 0<𝛼<𝛽.

Proof. At the beginning, let us notice that in view of inclusions (47) we have that if a function 𝑓 satisfies the Hölder condition with the exponent 𝛽, then it satisfies this condition with the exponent 𝛼. Thus, from assumptions (i) and (ii) it follows that there exists a constant 𝑘_𝛼 such that |𝑘(𝑡,𝜏)−𝑘(𝑠,𝜏)|⩽𝑘_𝛼|𝑡−𝑠|^𝛼,𝑝(𝑡)−𝑝(𝑠)⩽𝐻^𝛼_𝑝|𝑡−𝑠|^𝛼 (98) for all 𝑡,𝑠,𝜏∈[𝑎,𝑏], where the constant 𝐻^𝛼_𝑝 associated with the function 𝑝 is defined by (36).
Now, let us consider the operator 𝐹 defined on the space 𝐻_𝛽[𝑎,𝑏] by the formula (𝐹𝑥)(𝑡)=𝑝(𝑡)+𝑥(𝑡)^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏, (99) for 𝑡∈[𝑎,𝑏].
Further, take an arbitrary function 𝑥∈𝐻_𝛽[𝑎,𝑏]. Then, for arbitrarily fixed 𝑡,𝑠∈[𝑎,𝑏], in view of assumptions (i) and (ii) we obtain |(𝐹𝑥)(𝑡)−(𝐹𝑥)(𝑠)||𝑡−𝑠|^𝛽⩽𝑝(𝑡)−𝑝(𝑠)|𝑡−𝑠|^𝛽+𝑥(𝑡)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏−𝑥(𝑠)∫^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)𝑑𝜏|𝑡−𝑠|^𝛽⩽𝐻^𝛽_𝑝+𝑥(𝑡)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏−𝑥(𝑠)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏|𝑡−𝑠|^𝛽+𝑥(𝑠)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏−𝑥(𝑠)∫^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)𝑑𝜏|𝑡−𝑠|^𝛽⩽𝐻^𝛽_𝑝+|𝑥(𝑡)−𝑥(𝑠)||𝑡−𝑠|^𝛽𝑏𝑎|𝑘(𝑥(𝜏+|𝑥(𝑠∫𝑡,𝜏)||)|𝑑𝜏)|^𝑏𝑎|𝑘(𝑡,𝜏)𝑥(𝜏)−𝑘(𝑠,𝜏)||𝑥(𝜏)|𝑑𝜏|𝑡−𝑠|^𝛽⩽𝐻^𝛽_𝑝+𝐻^𝛽_𝑥^𝑏_𝑎|𝑘(𝑡,𝜏)|‖𝑥‖_∞+‖𝑥‖𝑑𝜏_∞‖𝑥‖_∞∫^𝑏_𝑎𝑘_𝛽|𝑡−𝑠|^𝛽𝑑𝜏|𝑡−𝑠|^𝛽⩽𝐻^𝛽_𝑝+𝐻^𝛽_𝑥‖𝑥‖_∞𝐾+‖𝑥‖²_∞𝑘_𝛽(𝑏−𝑎). (100)
Hence, in view of the inequalities ||𝑥||_∞⩽max{1,(𝑏−𝑎)^𝛽}‖𝑥‖_𝛽,𝐻^𝛽_𝑥⩽‖𝑥‖_𝛽 (cf. (39) and (41)), we derive the following estimate: ||𝐹𝑥||_𝛽⩽|(𝐹𝑥)(𝑎)|+𝐻^𝛽_𝑝+𝐾‖𝑥‖_∞𝐻^𝛽_𝑥+𝑘_𝛽(𝑏−𝑎)‖𝑥‖²_∞⩽𝑝_𝛽+𝐾‖𝑥‖²_∞+𝐾‖𝑥‖_∞‖𝑥‖_𝛽+𝑘_𝛽(𝑏−𝑎)max1,(𝑏−𝑎)^𝛽2‖𝑥‖²_𝛽⩽𝑝_𝛽+2𝐾max1,(𝑏−𝑎)^𝛽2‖𝑥‖²_𝛽+𝑘_𝛽(𝑏−𝑎)max1,(𝑏−𝑎)^𝛽2‖𝑥‖²_𝛽⩽𝑝_𝛽+max1,(𝑏−𝑎)^𝛽22𝐾+𝑘_𝛽(𝑏−𝑎)‖𝑥‖²_𝛽. (101)
This shows that the operator 𝐹 transforms the space 𝐻_𝛽[𝑎,𝑏] into itself. Moreover, in view of assumption (iii), we infer that 𝐹 transforms into itself the ball 𝐵_𝛽 (in the space 𝐻_𝛽[𝑎,𝑏]) centered at the zero function 𝜃 and with radius 𝑟₀, where 𝑟₀ is an arbitrary number from the interval [𝑟₁,𝑟₂], while 𝑟₁=1−1−4𝛾𝑝_𝛽,𝑟2𝛾₂=1+1−4𝛾𝑝_𝛽,2𝛾 (102) and 𝛾=(max{1,(𝑏−𝑎)^𝛽})²(2𝐾+𝑘_𝛽(𝑏−𝑎)).
Now, observe that in view of (47) the ball 𝐵_𝛽 is contained in the space 𝐻_𝛼[𝑎,𝑏]. Further, keeping in mind the fact established in Example 6, we conclude that the set 𝐵_𝛽 is relatively compact in the space 𝐻_𝛼[𝑎,𝑏]. Moreover, taking into account estimate (50), we deduce that 𝐵_𝛽 is also closed in the space 𝐻_𝛼[𝑎,𝑏].
Thus, gathering the above obtained facts, we infer that the set 𝐵_𝛽 is a compact and (obviously) convex subset of the space 𝐻_𝛼[𝑎,𝑏].
In what follows, we will treat the ball 𝐵_𝛽 as a subset of the space 𝐻_𝛼[𝑎,𝑏].
We now show that the operator 𝐹 is continuous on the space 𝐻_𝛼[𝑎,𝑏].
To this end, fix arbitrarily 𝑥∈𝐻_𝛼[𝑎,𝑏] and a number 𝛿>0. Assume that 𝑦∈𝐻_𝛼[𝑎,𝑏] is an arbitrary function such that ||𝑥−𝑦||_𝛼⩽𝛿. Then, for arbitrary 𝑡,𝑠∈[𝑎,𝑏] we obtain (𝐹𝑥)(𝑡)−𝐹𝑦(𝑡)−(𝐹𝑥)(𝑠)−𝐹𝑦(𝑠)|𝑡−𝑠|^𝛼=𝑥(𝑡)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏−𝑦(𝑡)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼−𝑥(𝑠)∫^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)𝑑𝜏−𝑦(𝑠)∫^𝑏_𝑎𝑘(𝑠,𝜏)𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼⩽𝑥(𝑡)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏−𝑦(𝑡)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼+𝑦(𝑡)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏−𝑦(𝑡)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼−𝑥(𝑠)∫^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)𝑑𝜏−𝑦(𝑠)∫^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼−𝑦(𝑠)∫^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)𝑑𝜏−𝑦(𝑠)∫^𝑏_𝑎𝑘(𝑠,𝜏)𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼=𝑥(𝑡)−𝑦(𝑡)^𝑏_𝑎𝑘()𝑥(𝜏)(𝑡)𝑡,𝜏𝑑𝜏+𝑦^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)−𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼−1−𝑥(𝑠)−𝑦(𝑠)^𝑏_𝑎𝑘()𝑥(𝜏)(𝑠)𝑠,𝜏𝑑𝜏+𝑦^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)−𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼−1⩽𝑥(𝑡)−𝑦(𝑡)−𝑥(𝑠)−𝑦(𝑠)∫^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼+𝑥(𝑠)−𝑦(𝑠)∫^𝑏_𝑎[𝑘(𝑡,𝜏)−𝑘(𝑠,𝜏)]𝑥(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼+𝑦(𝑡)^𝑏_𝑎𝑘(𝑥(𝜏)(𝜏(𝑠)𝑡,𝜏)−𝑦)𝑑𝜏−𝑦^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)−𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼−1⩽𝑥(𝑡)−𝑦(𝑡)−𝑥(𝑠)−𝑦(𝑠)|𝑡−𝑠|^𝛼‖𝑥‖_∞^𝑏_𝑎|𝑘(+𝑥(𝑠)(𝑠−𝑥(𝑎)(𝑎+𝑥(𝑎)(𝑎𝑥‖𝑡,𝜏)|𝑑𝜏−𝑦)−𝑦)−𝑦)‖_∞∫^𝑏_𝑎|𝑘(𝑡,𝜏)−𝑘(𝑠,𝜏)|𝑑𝜏|𝑡−𝑠|^𝛼+𝑦(𝑡)^𝑏_𝑎𝑘(𝑥(𝜏)(𝜏(𝑠)𝑡,𝜏)−𝑦)𝑑𝜏−𝑦^𝑏_𝑎𝑘(𝑡,𝜏)𝑥(𝜏)−𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼−1+𝑦(𝑠)^𝑏_𝑎𝑘(𝑥(𝜏)(𝜏(𝑠)𝑡,𝜏)−𝑦)𝑑𝜏−𝑦^𝑏_𝑎𝑘(𝑠,𝜏)𝑥(𝜏)−𝑦(𝜏)𝑑𝜏|𝑡−𝑠|^𝛼−1⩽𝑥−𝑦_𝛼‖𝑥‖_∞sup^𝑏_𝑎|𝑘(𝑡,𝜏)|𝑑𝜏∶𝑡∈[𝑎,𝑏]+sup_{[]𝑡,𝑠∈𝑎,𝑏}𝑥(𝑡)(𝑡−𝑥(𝑠)(𝑠×‖𝑥‖−𝑦)−𝑦)_∞𝑏𝑎𝑘_𝛽|𝑡−𝑠|^𝛽|𝑡−𝑠|^𝛼+𝑥(𝑎)(𝑎𝑥‖𝑑𝜏−𝑦)‖_∞∫^𝑏_𝑎𝑘_𝛽|𝑡−𝑠|^𝛽|𝑡−𝑠|^𝛼+𝑦(𝑡)(𝑠𝑑𝜏−𝑦)|𝑡−𝑠|^𝛼𝑏_𝑎|𝑘(𝑥(𝜏)(𝜏+𝑦(𝑠𝑡,𝜏)|−𝑦)𝑑𝜏)^𝑏_𝑎|𝑘(𝑡,𝜏)−𝑘(𝑠,𝜏)||𝑡−𝑠|^𝛼𝑥(𝜏)(𝜏‖𝑥‖−𝑦)𝑑𝜏⩽𝐾_∞𝑥−𝑦_𝛼+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}‖𝑥‖_∞𝑥(𝑡)(𝑡−𝑥(𝑠)(𝑠×sup−𝑦)−𝑦)|𝑡−𝑠|^𝛼×|𝑡−𝑠|^𝛼[]+𝑥(𝑎)(𝑎𝑥‖∶𝑡,𝑠∈𝑎,𝑏,𝑡≠𝑠−𝑦)‖_∞𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}+𝐻^𝛼_𝑦|𝑡−𝑠|^𝛼|𝑡−𝑠|^𝛼𝑏_𝑎|𝑘(×𝑡,𝜏)|sup_{[]𝑡∈𝑎,𝑏}𝑥(𝑡)(𝑡−𝑥(𝑎)(𝑎+𝑥(𝑎)(𝑎+𝑦−𝑦)−𝑦)−𝑦)𝑑𝜏_∞×^𝑏_𝑎𝑘_𝛽|𝑡−𝑠|^𝛽|𝑡−𝑠|^𝛼sup_{[]𝑡∈𝑎,𝑏}𝑥(𝑡)(𝑡−𝑥(𝑎)(𝑎+𝑥(𝑎)(𝑎‖𝑥‖−𝑦)−𝑦)−𝑦)𝑑𝜏⩽𝐾_∞𝑥−𝑦_𝛼+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}‖𝑥‖_∞𝑥(𝑡)(𝑡−𝑥(𝑠)(𝑠×sup−𝑦)−𝑦)|𝑡−𝑠|^𝛼∶𝑡,𝑠∈[𝑎,𝑏],𝑡≠𝑠(𝑏−𝑎)^𝛼+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}‖𝑥‖_∞𝑥(𝑎)−𝑦(𝑎)+𝐻^𝛼_𝑦𝑥(𝑡)(𝑡−𝑥(𝑠)(𝑠∶[sup−𝑦)−𝑦)𝑡,𝑠∈𝑎,𝑏]}^𝑏_𝑎|𝑘(𝑡,𝜏)|𝑑𝜏+𝐻^𝛼_𝑦𝑥(𝑎)−𝑦(𝑎)^𝑏_𝑎|𝑘(𝑡,𝜏)|𝑑𝜏+𝑘_𝛽𝑦_∞(𝑏−𝑎)^{𝛽−𝛼+1}𝑥(𝑡)(𝑡−𝑥(𝑠)(𝑠∶[×sup−𝑦)−𝑦)𝑡,𝑠∈𝑎,𝑏]}+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}𝑦_∞𝑥(𝑎)(𝑎‖𝑥‖−𝑦)⩽𝐾_∞𝑥−𝑦_𝛼+𝑘_𝛽(𝑏−𝑎)^𝛽+1‖𝑥‖_∞𝑥−𝑦_𝛼+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}‖𝑥‖_∞𝑥(𝑎)−𝑦(𝑎)+𝐻^𝛼_𝑦𝑥(𝑡)(𝑡−𝑥(𝑠)(𝑠𝐾sup−𝑦)−𝑦)|𝑡−𝑠|^𝛼∶𝑡,𝑠∈[𝑎,𝑏],𝑡≠𝑠(𝑏−𝑎)^𝛼+𝐻^𝛼_𝑦𝐾𝑥(𝑎)−𝑦(𝑎)+𝑘_𝛽𝑦_∞(𝑏−𝑎)^{𝛽−𝛼+1}𝑥(𝑡)(𝑡−𝑥(𝑠)(𝑠×sup−𝑦)−𝑦)|𝑡−𝑠|^𝛼∶𝑡,𝑠∈[𝑎,𝑏],𝑡≠𝑠(𝑏−𝑎)^𝛼+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}𝑦_∞𝑥(𝑎)(𝑎‖𝑥‖−𝑦)⩽𝐾_∞𝑥−𝑦_𝛼+𝑘_𝛽(𝑏−𝑎)^𝛽+1‖𝑥‖_∞𝑥−𝑦_𝛼+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}‖𝑥‖_∞𝑥(𝑎)−𝑦(𝑎)+𝐻^𝛼_𝑦𝐾(𝑏−𝑎)^𝛼𝑥−𝑦_𝛼+𝐻^𝛼_𝑦𝐾𝑥(𝑎)−𝑦(𝑎)+𝑘_𝛽𝑦_∞(𝑏−𝑎)^𝛽+1𝑥−𝑦_𝛼+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}𝑦_∞𝑥(𝑎)−𝑦(𝑎). (103)
Further, let us observe that we have the following estimates: 𝑎)−(𝑎=𝑝(𝑎)(𝑎)(𝐹𝑥)(𝐹𝑦)+𝑥^𝑏_𝑎𝑘()𝑥(𝜏)(𝑎)(𝑎)𝑎,𝜏𝑑𝜏−𝑝−𝑦^𝑏_𝑎𝑘()𝑦(𝜏)=𝑥(𝑎)𝑎,𝜏𝑑𝜏_𝑏𝑎𝑘(𝑎,𝜏)𝑥(𝜏)𝑑𝜏−𝑥(𝑎)^𝑏_𝑎𝑘()𝑦(𝜏)(𝑎)𝑎,𝜏𝑑𝜏+𝑥^𝑏_𝑎𝑘()𝑦(𝜏)(𝑎)𝑎,𝜏𝑑𝜏−𝑦^𝑏_𝑎𝑘()𝑦(𝜏)⩽𝑥(𝑎)𝑎,𝜏𝑑𝜏_𝑏𝑎𝑘()𝑥(𝜏)(𝜏)+𝑥(𝑎)(𝑎)𝑎,𝜏−𝑦𝑑𝜏−𝑦^𝑏_𝑎𝑘()𝑦(𝜏)⩽|𝑥(𝑎𝑎,𝜏𝑑𝜏)|^𝑏_𝑎|𝑘(𝑥(𝜏)(𝜏+𝑥(𝑎)(𝑎𝑎,𝜏)|−𝑦)𝑑𝜏−𝑦)^𝑏_𝑎|𝑘(𝑦(𝜏⩽‖𝑥‖𝑎,𝜏)|)𝑑𝜏_∞𝑥−𝑦_∞^𝑏_𝑎|𝑘(+𝑎,𝜏)|𝑑𝜏𝑥−𝑦_𝛼𝑦_∞^𝑏_𝑎|𝑘(‖𝑥‖𝑎,𝜏)|𝑑𝜏⩽𝐾_∞𝑥−𝑦_∞+𝑥−𝑦_𝛼𝑦_∞. (104)
Moreover, we have that ||𝑦||_𝛼⩽||𝑥||_𝛼+𝛿 and from (39) we derive the following estimate: 𝐻^𝛼_𝑦=𝑦_𝛼−𝑦(𝑎)⩽𝑦_𝛼. (105)
Combining the above established facts and (41), in view of (103) and (104), we obtain following estimates: 𝑓𝑥−𝑓𝑦_𝛼⩽𝐾max1,(𝑏−𝑎)^𝛼22‖𝑥‖_𝛼𝛿++𝛿𝐾+𝑘_𝛽(𝑏−𝑎)^𝛽+1‖𝑥‖_∞𝛿+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}‖𝑥‖_∞𝛿+𝐾(𝑏−𝑎)^𝛼‖𝑥‖_𝛼𝛿‖𝑥‖+𝛿+𝐾_𝛼+𝛿𝛿+𝑘_𝛽(𝑏−𝑎)^𝛽+1max1,(𝑏−𝑎)^𝛼‖𝑥‖_𝛼+𝛿𝛿+𝑘_𝛽(𝑏−𝑎)^{𝛽−𝛼+1}max1,(𝑏−𝑎)^𝛼‖𝑥‖_𝛼+𝛿𝛿. (106)
This shows that the operator 𝐹 is continuous at the point 𝑥∈𝐻_𝛼[𝑎,𝑏]. Since 𝑥 was chosen arbitrarily, we deduce that 𝐹 is continuous on the Hölder space 𝐻_𝛼[𝑎,𝑏]. Particularly, the operator 𝐹 maps continuously the ball 𝐵_𝛽 into itself. Thus, taking into account the fact that the set 𝐵_𝛽 is compact in the space 𝐻_𝛼[𝑎,𝑏] and applying the classical Schauder fixed point principle, we complete the proof.

Now, we illustrate the above obtained result by an example.

Example 8. Let us consider the following quadratic integral equation: 𝑥(𝑡)=𝑞𝑡+𝑟+𝑥(𝑡)√¹₀³𝑚𝑡²+𝜏𝑥(𝜏)𝑑𝜏, (107) where 𝑡∈[0,1] and 𝑞,𝑟,𝑚 are positive constants.
Observe that (107) is a particular case of (94) if we put 𝑝(𝑡)=𝑞𝑡+𝑟,𝑘(𝑡,𝜏)=√³𝑚𝑡²+𝜏. Further, we get 𝑝(𝑡)−𝑝(𝑠)⩽√𝑞|𝑡−𝑠|^1/2. (108)
Moreover, using the inequality proved in [12], we obtain |𝑘(𝑡,𝜏)−𝑘(𝑠,𝜏)|=√³𝑚𝑡²+𝜏−√³𝑚𝑠²+𝜏⩽√³𝑚|𝑡−𝑠|^2/3. (109)
Thus, in our considerations related to Theorem 7 we can take 𝛽=1/2. Then obviously we have |𝑘(𝑡,𝜏)−𝑘(𝑠,𝜏)|⩽√³𝑚|𝑡−𝑠|^2/3=√³𝑚|𝑡−𝑠|^1/2|𝑡−𝑠|^1/6⩽√³𝑚|𝑡−𝑠|^1/2. (110)
Hence we infer that 𝑘_𝛽=𝑘_1/2=√³𝑚. Apart from this, we get 𝑝_1/2=𝑝(0𝑝(𝑡)(𝑠)+sup−𝑝)|𝑡−𝑠|^1/2[]=√∶𝑡,𝑠∈0,1,𝑡≠𝑠𝑟+√𝑞. (111)
This shows that functions 𝑝(𝑡) and 𝑘(𝑡,𝜏) involved in (107) satisfy assumptions (i) and (ii) of Theorem 7.
Further, we can calculate that 𝐾=sup¹₀|𝑘(𝑡,𝜏)|𝑑𝜏∶𝑡∈[0,1]=sup√¹₀³𝑚𝑡²[3+𝜏𝑑𝜏∶𝑡∈0,1]=sup4³𝑚𝑡²+1⁴−³𝑚𝑡²⁴[=3∶𝑡∈0,1]4³(𝑚+1)⁴−√³𝑚⁴. (112)
Hence we deduce that the inequality from assumption (iii) is satisfied provided √𝑞+√𝑟√³3𝑚+2³(𝑚+1)⁴−√³𝑚⁴<14. (113)
It is easy to check that the above inequality is satisfied if we put, for example, 𝑞=𝑟=1/216,𝑚=1/128.
Finally, applying Theorem 7, we conclude that (107) has at least one solution in the space 𝐻_𝛼[0,1] with 0<𝛼<1/2, provided inequality (113) is satisfied.