A QUADRATIC INTEGRAL EQUATION IN THE SPACE OF FUNCTIONS WITH TEMPERED MODULI OF CONTINUITY

Abstract

In this paper, we investigate existence of solutions to a class of quadratic integral equation of Fredholm type in the space of functions with tempered moduli of continuity. Two numerical examples are given to illustrate our results.

1. Introduction

Fractional integral and differential equations play increasingly important roles in the modeling of real world problems. Some problems in physics, mechanics and other fields can be described with the help of all kinds of fractional differential and integral equations. For more recent development on Riemann-Liouville, Caputo and Hadamard fractional calculus, the reader can refer to the monographs [1,2,3,4,5,6].

Quadratic integral equations arise naturally in applications of real world problems. For example, problems in the theory of radiative transfer in the theory of neutron transport and in the kinetic theory of gases lead to the quadratic equation [7,8,9,10]. There are many interesting existence results for all kinds of quadratic integral equations, one can refer to [11,12,13,14,15,16,17]. Our group extend to study the existence, local attractivity and stability of solutions to fractional version Urysohn type quadratic integral equations [18] and Erdélyi- Kober type quadratic integral equations [19] and Hadamard types quadratic integral equations [20] in the space of continuous functions.

Very recently, Banaś and Nalepa [21] study the space of real functions defined on a bounded metric space and having growths tempered by a modulus of continuity and derive the existence theorem for some quadratic integral equations of Fredholm type in the space of functions satisfying the Hölder condition. Further, Caballero et al. [22] study the solvability of a quadratic integral equation of Fredholm type in Hölder spaces.

The aim of the paper is to investigate the existence of solutions of the following integral equation of Fredholm type

in Cω,g[a, b] (see Section 2), where the functions f and k will be defined in the later.

By using a sufficient condition for the relative compactness in the space of functions with tempered moduli of continuity (see Theorem 2.5) and the classical Schauder fixed point theorem, we derive new existence result (see Theorem 3.5). Finally, two numerical examples are given to illustrate our results.

 

2. Preliminaries

Definition 2.1 (see Section 2 [21]). 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.

Let C[a, b] be the space of continuous functions on [a, b] equipped with ║x║∞ = sup{|x(t)| : t ∈ [a, b]} for x ∈ C[a, b]. We denote Cω,g[a, b] be the set of all real functions defined on [a, b] such that their growths are tempered by the modulus of continuity ω with respect to a function g. That is, there exists a constant such that

for all t, s ∈ [a, b] where g : [a, b] → ℝ is a monotonic function.

Without loss of generality, we suppose that the above g be a increasing function and g(t) − g(s) ≥ 0 for t ≥ s in the this paper. Obviously, Cω,g[a, b] is a linear subspace of C[a, b].

For x ∈ Cω,g[a, b], we denote be the least possible constant for which inequality (2) is satisfied. More precisely, we set

Next, the space Cω,g[a, b] can be equipped with the norm

for x ∈ Cω,g[a, b]. Then (Cω,g[a, b], ║۰║ω,g) is a Banach space.

Inspired by the properties of the space of Hölder functions in [21, see (41), (45)], we give the following sharp results.

Lemma 2.2. For any x ∈ Cω,g[a, b], the following inequality is satisfied

Proof. For any x ∈ Cω,g[a, b] and t ∈ [a, b] we obtain

Lemma 2.3. Suppose that ω2(g(t) − g(s)) ≤ Gω1(g(t) − g(s)) for t, s ∈ [a, b] where G > 0. Then we have

Moreover, for any x ∈ Cω2,g[a, b] the following inequality holds

Proof. For any x ∈ Cω2,g[a, b], we obtain

This shows that x ∈ Cω1,g[a, b] and hence we infer that inclusions hold. Further,

Remark 2.1. In particular, if then the above imbedding relations also hold and for any x ∈ Cω2,g[a, b], we have ║x║ω1,g ≤ max{1,M}║x║ω2,g = ║x║ω2,g, where M is a arbitrarily small positive number.

Theorem 2.4 (see Theorem 5 [21]). Assume that ω1, ω2 are moduli of continuity being continuous at zero and such that . Further, assume that (X, d) is a compact metric space. Then, if A is a bounded subset of the space Cω2,g(X) then A is relatively compact in the space Cω1,g(X).

Theorem 2.5. Suppose that . Denote . Then is compact in the space Cω1,g[a, b].

Proof. By Theorem 2.4, since is a bounded subset in Cω2,g[a, b], it is a relatively compact subset of Cω1,g[a, b]. Suppose that and

with x ∈ Cω1,g[a, b]. This means that for ε > 0 we can find n0 ∈ ℕ such that

for any n ≥ n0, or, equivalently

for any n ≥ n0.

This implies that xn(a) → x(a).

Moreover, if in (3) we put s = a, then we get

for any n ≥ n0.

The last inequality implies that

for any n ≥ n0 and for any t ∈ [a, b].

Therefore, for any n ≥ n0 and any t ∈ [a, b] and taking into account (3) and (4), we have

Consequently,

Next, we will prove that .

In fact, since , we have that

for any t, s ∈ [a, b] with t > s, and, accordingly,

for any t, s ∈ [a, b].

Letting in the above inequality with n → ∞ and taking into account (5), we deduce that

for any t, s ∈ [a, b].

Hence we get

for any t, s ∈ [a, b], and this means that . This proves that is a closed subset of Cω1,g[a, b]. Thus, is a compact subset of Cω1,g[a, b]. This finishes the proof. □

 

3. Main results

In this section, we will study the solvability of the equation (1) in Cω,g[a, b]. We will use the following assumptions:

(H1) f : [a, b] × ℝ → ℝ is a continuous function and there exists a positive number k1 such that

and set k = |f(a, a)|. Meanwhile, for any t, s ∈ [a, b] and t > s, there exists a positive constant k2 such that the inequality

(H2) k : [a, b] × [a, b] → ℝ is a continuous function satisfies the tempered by the modulus of continuity with respect to the first variable, that is, there exists a constant Kω2 such that

for any t, s, τ ∈ [a, b].

(H3) The following inequality is satisfied

where .

Consider the operator F defined on Cω2,g[a, b] by

Lemma 3.1. The operator F maps Cω2,g[a, b] into itself.

Proof. In fact, we take x ∈ Cω2,g[a, b] and t, s ∈ [a, b] with t > s. Then, by assumptions (H1)-(H3), we obtain

By Lemma 2.2, since ║x║∞ ≤ max{1, ω2(g(b) − g(a))}║x║ω2,g and, as , we infer that

This proves that the operator F maps Cω2,g[a, b] into itself. □

Lemma 3.2. Let where r0 > 0 satisfy-ing the inequality (6). Then .

Proof. For any , one has

Consequently, from above it follows that F transforms the ball into itself, for any r0 ∈ [r1, r2]; i.e., , where r1 ≤ r0 ≤ r2. □

Lemma 3.3. is a compact subset in C ω1,g[a, b].

Proof. According to Theorem 2.5, we can know is a compact subset in Cω1,g[a, b]. □

Lemma 3.4. The operator F is continuous on , where we consider the norm ║ㆍ║ω1,g in .

Proof. To do this, we fix and ε > 0. Suppose that and ║x − y║ω1,g ≤ δ, where δ is a positive number such that where ρ = max{ρ1, ρ2}, ρ1, ρ2 is defined below. Then, for any t, s ∈ [a, b] with t > s, we have

Define

Since ║y║ω1,g ≤ ║y║ω2,g (see Remark 2.1) and , from the above inequality we infer that

On the other hand,

where

which yields that

By (7) and (8), we have

This proves the operator F is continuous at the point for the norm ║ㆍ║ω1,g. □

Theorem 3.5. Under assumptions (H1)-(H3), the equation (1) has at least one solution in the space Cω1,g[a, b].

Proof. According to Lemma 3.1, Lemma 3.2, Lemma 3.3 and Lemma 3.4, the operator F is continuous at the point for the norm ║·║ω1,g. Since is compact in Cω2,g[a, b], applying the classical Schauder fixed point theorem we obtain the desired result. □

 

4. Examples

Now we make two examples illustrating the main results in the above section.

Example 4.1. Let us consider the quadratic integral equation

Set arctan x(t) and for t,τ ∈ [1,e]. It is easy to see that

which implies Kω2 = 1 and

So we can choose

Moreover, .

On the other hand,

so we can get , k = |f(1, 1)| = 0 and

so .

In what follows, the condition (H3) reduce to the inequality

Obviously, there exist a positive number satisfying these conditions. For example, one can choose r = 0.1.

Finally, applying Theorem 3.5, we conclude that the quadratic integral equation has at least one solution in the space C|ㆍ|α,ln.[1, e] and displayed in Fig.1.

Fig.1The solution of the equation (9).

Example 4.2. Consider another quadratic integral equation

Set . Obviously,

which gives

Then we choose

Moreover, .

On the other hand,

we can get and

so derive .

In what follows, the condition (H3) reduce to the inequality

The condition reduce to r < 0.1676. Obviously, there exist a positive number satisfying these conditions. For example, one can choose r = 0.16.

Finally, applying Theorem 3.5, we conclude that the quadratic integral equation has at least one solution in the space C|·|α,.[0, 1] and displayed in Fig.2.

Fig.2The solution of the equation (10).