A Quadratic Integral Equation in the Space of Functions with Tempered Moduli of Continuity†

J. Appl. Math. & Informatics Vol. 33(2015), No. 3 - 4, pp. 351 - 363 http://dx.doi.org/10.14317/jami.2015.351

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

SHAN PENG, JINRONG WANG∗, FULAI CHEN

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.

AMS Mathematics Subject Classiﬁcation : 26A33, 26A51, 26D15. Key words and phrases : Quadratic integral equation, Tempered moduli of continuity, Schauder ﬁxed point theorem.

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 mono- graphs [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

Received July 25, 2014. Revised November 2, 2014. Accepted November 4, 2014. ∗Corresponding author. † The first and second authors acknowledge the support by the National Natural Science Foundation of China (11201091), Outstanding Scientific and Technological Innova- tion Talent Award of Education Department of Guizhou Province and Doctor Project of Guizhou Normal College (13BS010). The third author acknowledges the National Natural Science Foundation of Hunan Province under Grant 13JJ3120. ⃝c 2015 Korean SIGCAM and KSCAM. 351 352 Shan Peng, JinRong Wang, Fulai Chen 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, Bana´sand 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 Höldercondition. Fur- ther, Caballero et al. [22] study the solvability of a quadratic integral equation of Fredholm type in Hölderspaces. The aim of the paper is to investigate the existence of solutions of the following integral equation of Fredholm type ∫ b x(t) = f(t, x(t)) + x(t) k(t, τ)x(τ)dτ, t ∈ [a, b], (1) a 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 ω,g Hx > 0 such that | − | ≤ ω,g − x(t) x(s) Hx ω(g(t) g(s)) (2) 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]. ∈ ω,g For x Cω,g[a, b], we denote Hx be the least possible constant for which inequality (2) is satisfied. More precisely, we set { } |x(t) − x(s)| Hω,g = sup : t, s ∈ [a, b], t > s . x ω(g(t) − g(s)) Next, the space C [a, b] can be equipped with the norm ω,g { } |x(t) − x(s)| ∥x∥ = |x(a)| + sup : t, s ∈ [a, b], t > s , ω,g ω(g(t) − g(s)) A quadratic integral equation in the space of functions 353 for x ∈ Cω,g[a, b]. Then (Cω,g[a, b], ∥ · ∥ω,g) is a Banach space. Inspired by the properties of the space of Hölderfunctions 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

∥x∥∞ ≤ max{1, ω(g(b) − g(a))}∥x∥ω,g.

Proof. For any x ∈ Cω,g[a, b] and t ∈ [a, b] we obtain |x(t)| ≤ |x(t) − x(a)| + |x(a)| ≤ {| − | ∈ } | | sup x(t) x{(s) : t, s [a, b] + x(a) } |x(t) − x(s)| = |x(a)| + sup · ω(g(t) − g(s)) : t, s ∈ [a, b], t > s ω(g(t) − g(s)) { } |x(t) − x(s)| ≤ |x(a)| + ω(g(b) − g(a)) sup : t, s ∈ [a, b], t > s ω(g(t) − g(s)) ≤ { − } max{ 1, ω(g(b) {g(a)) }} |x(t) − x(s)| × |x(a)| + sup : t, s ∈ [a, b], t > s ω(g(t) − g(s))

≤ max{1, ω(g(b) − g(a))}∥x∥ω,g.

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 ⊂ ⊂ Cω2,g[a, b] Cω1,g[a, b] C[a, b]. ∈ Moreover, for any x Cω2,g[a, b] the following inequality holds ∥ ∥ ≤ { }∥ ∥ x ω1,g max 1,G x ω2,g. ∈ Proof. For any x Cω2,g[a, b], we obtain | − | ≤ ω2,g − ≤ ω2,g − x(t) x(s) Hx ω2(g(t) g(s)) GHx ω1(g(t) g(s)). This shows that x ∈ C [a, b] and hence we infer that inclusions hold. Further, ω1,g { } |x(t) − x(s)| ∥x∥ = |x(a)| + sup : t, s ∈ [a, b], t > s ω1,g ω (g(t) − g(s)) {1 } |x(t) − x(s)| ≤ |x(a)| + G sup : t, s ∈ [a, b], t > s ω2(g(t) − g(s)) ≤ { }∥ ∥ max 1,G x ω2,g.

ω2(ϵ) Remark 2.1. In particular, if limϵ→0 = 0 then the above imbedding rela- ω1(ϵ) ∈ ∥ ∥ ≤ { }∥ ∥ tions 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. 354 Shan Peng, JinRong Wang, Fulai Chen

Theorem 2.4 (see Theorem 5 [21]). Assume that ω1, ω2 are moduli of continuity ω2(ϵ) being continuous at zero and such that limϵ→0 = 0. Further, assume that ω1(ϵ) (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).

ω2(ϵ) ω2,g Theorem 2.5. Suppose that limϵ→0 = 0. Denote B = {x ∈ Cω ,g[a, b]: ω1(ϵ) r 2 ∥ ∥ ≤ } ω2,g x ω2,g r . Then Br is compact in the space Cω1,g[a, b].

ω2,g Proof. By Theorem 2.4, since Br is a bounded subset in Cω2,g[a, b], it is a ⊂ ω2,g relatively compact subset of Cω1,g[a, b]. Suppose that (xn) Br and → ∥ · ∥ xn x (according to ω1,g) ∈ ∈ N with x Cω1,g[a, b]. This means that for ε > 0 we can ﬁnd n0 such that ∥ − ∥ ≤ xn x ω1,g ε, for any n ≥ n0, or, equivalently | − | xn(a){ x(a) } |x (t) − x(t) − (x (s) − x(s))| + sup n n : t, s ∈ [a, b], t > s ≤ ε, (3) ω1(g(t) − g(s)) for any n ≥ n0. This implies that xn(a) → x(a). Moreover, if in (3) we put s = a, then we get { } |x (t) − x(t) − (x (a) − x(a))| sup n n : t, s ∈ [a, b], t > s < ε, ω1(g(t) − g(s)) for any n ≥ n0. The last inequality implies that

|xn(t) − x(t) − (xn(a) − x(a))| < εω1(g(t) − g(s)) ≤ εω1(g(b) − g(a)), (4) 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

< εω1(g(b) − g(a)) + ε. Consequently,

∥xn − x∥∞ → 0. (5)

∈ ω2,g Next, we will prove that x Br . ⊂ ω2,g ⊂ In fact, since (xn) Br Cω2,g[a, b], we have that |x (t) − x (s)| n n ≤ r, ω2(g(t) − g(s)) for any t, s ∈ [a, b] with t > s, and, accordingly,

|xn(t) − xn(s)| ≤ rω2(g(t) − g(s)), A quadratic integral equation in the space of functions 355 for any t, s ∈ [a, b]. Letting in the above inequality with n → ∞ and taking into account (5), we deduce that

|x(t) − x(s)| ≤ rω2(g(t) − g(s)), for any t, s ∈ [a, b]. Hence we get |x(t) − x(s)| ≤ r, ω2(g(t) − g(s)) ∈ ∈ ω2,g ω2,g for any t, s [a, b], and this means that x Br . This proves that Br is a ∈ ω2,g closed subset of Cω1,g[a, b]. Thus, x Br 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

|f(t, x) − f(t, y)| ≤ k1|x − y|, 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 |f(t, x(s)) − f(s, x(s))| ≤ k2|x(s)|. ω2(g(t) − g(s))

(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 | − | ≤ − k(t, τ) k(s, τ) Kω2 ω2(g(t) g(s)), for any t, s, τ ∈ [a, b]. (H3) The following inequality is satisfied (2K + K (b − a))max2{1, ω (g(b) − g(a))}r2 [ ω2 2 ]

+ k1 + (k1 + k2) max{1, ω2(g(b) − g(a))} − 1 r + k + |a|k1 < 0, (6) {∫ } b | | ∈ where K = sup a k(t, τ) dτ : t [a, b] . z Consider the operator deﬁned on Cω2,g[a, b] by ∫ b (zx)(t) = f(t, x(t)) + x(t) k(t, τ)x(τ)dτ, t ∈ [a, b]. a z Lemma 3.1. The operator maps Cω2,g[a, b] into itself. 356 Shan Peng, JinRong Wang, Fulai Chen

∈ ∈ 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

|(zx)(t) − (zx)(s)| ω (g(t) − g(s)) 2 ∫ ∫ b b |f(t, x(t)) − f(s, x(s))| |x(t) k(t, τ)x(τ)dτ − x(s) k(t, τ)x(τ)dτ| ≤ + a a ω (g(t) − g(s)) ω (g(t) − g(s)) 2 ∫ ∫ 2 |x(s) b k(t, τ)x(τ)dτ − x(s) b k(s, τ)x(τ)dτ| + a a ω2(g(t) − g(s)) | − | | − | ≤ f(t, x(t)) f(t, x(s)) + f(t, x(s)) f(s, x(s)) ω (g(t) − g(s)) 2 ∫ ∫ b |x(t) − x(s)| b |x(s)| |k(t, τ) − k(s, τ)||x(τ)|dτ | || | a + − k(t, τ) x(τ) dτ + − ω2(g(t) g(s)) a ω2(g(t) g(s)) k |x(t) − x(s)| |f(t, x(s)) − f(s, x(s))| ≤ 1 + ω2(g(t) − g(s)) ω2(g(t) − g(s)) ∫ ∫ b b |x(t) − x(s)| ∥x∥∞∥x∥∞ |k(t, τ) − k(s, τ)|dτ ∥ ∥ | | a + − x ∞ k(t, τ) dτ + − ω2(g(t) g(s)) a ω2(g(t) g(s)) | − | ω ,g x(t) x(s) ≤ k H 2 + k |x(s)| + K∥x∥∞ 1 x 2 ω (g(t) − g(s)) ∫ 2 2 b ∥x∥∞ Kω ω2(g(t) − g(s))dτ + a 2 ω2(g(t) − g(s)) ≤ ω2,g ∥ ∥ ∥ ∥ ω2,g − ∥ ∥2 k1Hx + k2 x ∞ + K x ∞Hx + Kω2 (b a) x ∞. ∥ ∥ ≤ { − }∥ ∥ ω2,g ≤ By Lemma 2.2, since x ∞ max 1, ω2(g(b) g(a)) x ω2,g and, as Hx ∥ ∥ x ω2,g, we infer that |(zx)(t) − (zx)(s)|

ω2(g(t) − g(s)) ≤ { − } ∥ ∥ (k1 + k2 max 1, ω2(g(b) g(a)) ) x ω2,g +(K + K (b − a))max2{1, ω (g(b) − g(a))}∥x∥2 . ω2 2 ω2,g z This proves that the operator maps Cω2,g[a, b] into itself. Lemma 3.2. Let Bω2,g = {x ∈ C [a, b]: ∥x∥ ≤ r } where r > 0 satisfy- r0 ω2,g ω2,g 0 0 ing the inequality (6). Then z : Bω2,g → Bω2,g. r0 r0

∈ ω2,g Proof. For any x Br , one has 0 ∫ b ∥z ∥ ≤ | | | | | || | x ω2,g f(a, x(a)) + x(a) k(a, τ) x(τ) dτ a { − } ∥ ∥ + (k1 + k2 max 1, ω2(g(b) g(a)) ) x ω2,g +(K + K (b − a))max2{1, ω (g(b) − g(a))}∥x∥2 ω2 2 ω2,g A quadratic integral equation in the space of functions 357

2 ≤ |f(a, x(a)) − f(a, a)| + |f(a, a)| + K∥x∥∞ { − } ∥ ∥ + (k1 + k2 max 1, ω2(g(b) g(a)) ) x ω2,g +(K + K (b − a))max2{1, ω (g(b) − g(a))}∥x∥2 ω2 2 ω2,g ≤ | − | { − } ∥ ∥ k1 x(a) a + k + (k1 + k2 max 1, ω2(g(b) g(a)) ) x ω2,g +(2K + K (b − a))max2{1, ω (f(b) − f(a))}∥x∥2 ω2 2 ω2,g ≤ | | ∥ ∥ { − } ∥ ∥ a k1 + k1 x ∞ + k + (k1 + k2 max 1, ω2(g(b) g(a)) ) x ω2,g 2 2 +(2K + Kω (b − a))max {1, ω2(g(b) − g(a))}∥x∥ [2 ω2,g] | | { − } ∥ ∥ = k + a k1 + k1 + (k1 + k2) max 1, ω2(g(b) g(a)) x ω2,g

+(2K + K (b − a))max2{1, ω (g(b) − g(a))}∥x∥2 . ω2 2 ω2,g Consequently, from above it follows that z transforms the ball Bω2,g = {x ∈ r0 C [a, b]: ∥x∥ ≤ r } into itself, for any r ∈ [r , r ]; i.e., z : Bω2,g → Bω2,g, ω2,g ω2,g 0 0 1 2 r0 r0 where r1 ≤ r0 ≤ r2.

Lemma 3.3. Bω2,g is a compact subset in C [a, b]. r0 ω1,g Proof. According to Theorem 2.5, we can know Bω2,g is a compact subset in r0 Cω1,g[a, b].

Lemma 3.4. The operator z is continuous on Bω2,g, where we consider the r0 norm ∥ · ∥ in Bω2,g. ω1,g r0 Proof. To do this, we ﬁx x ∈ Bω2,g and ε > 0. Suppose that y ∈ Bω2,g and r0 r0 ∥ − ∥ ≤ ε x y ω1,g δ, where δ is a positive number such that δ < 2ρ where ρ = max{ρ1, ρ2}, ρ1, ρ2 is deﬁned below. Then, for any t, s ∈ [a, b] with t > s, we have |[(zx)(t) − (zy)(t)] − [(zx)(s) − (zy)(s)]| − ω1(g(t) g(s))

k1|x(t) − y(t)| − k1|x(s) − y(s)| ≤ ω1(g(t) − g(s)) ∫ ∫ b − b [x(t) a k(t, τ)x(τ)dτ y(t) a k(t, τ)x(τ)dτ] + ω (g(t) − g(s)) ∫ 1 ∫ [y(t) b k(t, τ)x(τ)dτ − y(t) b k(t, τ)y(τ)dτ] + a a ω (g(t) − g(s)) ∫ 1 ∫ [x(s) b k(s, τ)x(τ)dτ − y(s) b k(s, τ)x(τ)dτ] − a a ω1(g(t) − g(s)) ∫ ∫ b − b [y(s) a k(s, τ)x(τ)dτ y(s) a k(s, τ)y(τ)dτ] − ω1(g(t) − g(s)) 358 Shan Peng, JinRong Wang, Fulai Chen

|x(t) − y(t)| − |x(s) − y(s)| = k 1 ω (g(t) − g(s)) 1 ∫ b 1 + (x(t) − y(t)) k(t, τ)x(τ)dτ ω (g(t) − g(s)) 1 ∫ a b +y(t) k(t, τ)(x(τ) − y(τ))dτ a ∫ ∫ b b

−(x(s) − y(s)) k(s, τ)x(τ)dτ − y(s) k(s, τ)(x(τ) − y(τ))dτ a a ∫ |(x(t) − y(t)) − (x(s) − y(s))| b ≤ k ∥x − y∥ + ∥x∥∞ |k(t, τ)|dτ 1 ω1,g − ω1(g(t) g(s)) a +[|(x(s) − y(s)) − (x(a) − y(a))| + |(x(a) − y(a))|]∥x∥∞ ∫ b | − | × k(t, τ) k(s, τ) − dτ a ω1(g(t) g(s)) ∫ ∫ b − − b − y(t) a k(t, τ)(x(τ) y(τ))dτ y(s) a k(t, τ)(x(τ) y(τ))dτ + − ω1(g(t) g(s)) ∫ ∫ b − − b − y(s) a k(t, τ)(x(τ) y(τ))dτ y(s) a k(s, τ)(x(τ) y(τ))dτ + ω1(g(t) − g(s)) ≤ k ∥x − y∥ + K∥x − y∥ ∥x∥∞ 1 { ω1,g ω1,g } ∫ b − Kω2 ω2(g(t) g(s)) + sup |(x(t) − y(t)) − (x(s) − y(s))| ∥x∥∞ dτ ω (f(t) − f(s)) ∫ a 1 b − Kω2 ω2(g(t) g(s)) +|(x(a) − y(a))|∥x∥∞ dτ ω (g(t) − g(s)) ∫ a 1 |y(t) − y(s)| b + |k(t, τ)||x(τ) − y(τ)|dτ ω (g(t) − g(s)) 1 ∫ a b | − | | | k(t, τ) k(s, τ) | − | + y(s) − x(τ) y(τ) dτ a ω1(g(t) g(s)) ≤ k ∥x − y∥ + K∥x − y∥ ∥x∥∞ 1 ω1,g {ω1,g } | − − − | − ∥ ∥ (x(t) y(t)) (x(s) y(s)) − +M(b a) x ∞Kω2 sup ω1(g(t) g(s)) ω1(g(t) − g(s)) ω1,g +M(b − a)Kω ∥x∥∞|(x(a) − y(a))| + KH ∥x − y∥∞ 2 ∫ y b K ω (g(t) − g(s)) ∥ ∥ ∥ − ∥ ω2 2 + y ∞ x y ∞ − dτ a ω1(g(t) g(s)) ≤ ∥ − ∥ ∥ − ∥ ∥ ∥ k1 x y ω1,g + K x y ω1,g x ∞ − ∥ ∥ − ∥ − ∥ +M(b a) x ∞Kω2 ω1(g(b) g(a)) x y ω1,g A quadratic integral equation in the space of functions 359

− ∥ ∥ ∥ − ∥ ∥ ∥ ∥ − ∥ +M(b a)Kω2 x ∞ x y ∞ + K y ω1,g x y ∞ − ∥ ∥ ∥ − ∥ +M(b a)Kω2 y ∞ x y ∞ ≤ ∥ − ∥ { − }∥ − ∥ ∥ ∥ k1 x y ω1,g + K max 1, ω2(g(b) g(a)) x y ω1,g x ω2,g − − { − }∥ − ∥ ∥ ∥ +M(b a)Kω2 ω1(g(b) g(a)) max 1, ω2(g(b) g(a)) x y ω1,g x ω2,g − { − } +M(b a)Kω2 max 1, ω2(g(b) g(a)) × { − }∥ − ∥ ∥ ∥ max 1, ω1(g(b) g(a)) x y ω1,g x ω2,g { − }∥ − ∥ ∥ ∥ +K max 1, ω1(g(b) g(a)) x y ω1,g y ω1,g − { − } +M(b a)Kω2 max 1, ω2(g(b) g(a)) × { − }∥ − ∥ ∥ ∥ max 1, ω1(g(b) g(a)) x y ω1,g y ω2,g. Deﬁne

ρ1 = k1 + K max{1, ω2(g(b) − g(a))}r0 − − { − } +M(b a)Kω2 ω1(g(b) g(a)) max 1, ω2(g(b) g(a)) r0 − { − } { − } +2M(b a)Kω2 max 1, ω2(g(b) g(a)) max 1, ω1(g(b) g(a)) r0

+K max{1, ω1(g(b) − g(a))}r0. Since ∥y∥ ≤ ∥y∥ (see Remark 2.1) and x, y ∈ Bω2,g, from the above ω1,g ω2,g r0 inequality we infer that |[(zx)(t) − (zy)(t)] − [(zx)(s) − (zy)(s)]| ε ≤ ρ1δ < . (7) ω1(g(t) − g(s)) 2 On the other hand, |(zx)(a) − (zy)(a)| ∫ ∫ b b

≤ |f(a, x(a)) − f(a, y(a))| + x(a) k(a, τ)x(τ)dτ − x(a) k(a, τ)y(τ)dτ ∫ ∫a a b b

+ x(a) k(a, τ)y(τ)dτ − y(a) k(a, τ)y(τ)dτ a ∫ a b

≤ k1|x(a) − y(a)| + x(a) k(a, τ)(x(τ) − y(τ))dτ ∫ a b

+ (x(a) − y(a)) k(a, τ)y(τ)dτ a ≤ k1∥x − y∥∞ + K∥x∥∞∥x − y∥∞ + K∥y∥∞∥x − y∥∞ ≤ { − }∥ − ∥ k1 max 1, ω1(g(b) g(a)) x y ω1,g { − } { − }∥ ∥ ∥ − ∥ +K max 1, ω2(g(b) g(a)) max 1, ω1(g(b) g(a)) x ω2,g x y ω1,g { − } { − }∥ ∥ ∥ − ∥ +K max 1, ω2(g(b) g(a)) max 1, ω1(g(b) g(a)) y ω2,g x y ω1,g

≤ ρ2δ, where

ρ2 = k1 max{1, ω1(g(b) − g(a))} 360 Shan Peng, JinRong Wang, Fulai Chen

+2K max{1, ω2(g(b) − g(a))} max{1, ω1(g(b) − g(a))}r0. which yields that ε |(zx)(a) − (zy)(a)| ≤ ρ δ < . (8) 2 2 By (7) and (8), we have ∥z − z ∥ x y ω1,g | z − z | = ( x)({a) ( y)(a) } |[(zx)(t) − (zy)(t)] − [(zx)(s) − (zy)(s)]| + sup : t, s ∈ [a, b], t > s ω1(g(t) − g(s)) ε ε < + = ε. 2 2 This proves the operator z is continuous at the point x ∈ Bω2,g for the norm r0 ∥ · ∥ 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 z is continuous at the point x ∈ Bω2,g for the norm ∥·∥ . Since Bω2,g r0 ω1,g r0 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 ∫ 1 √ e √ x(τ) x(t) = ln t arctan x(t) + x(t) ln t + ln τ dτ, t ∈ [1, e]. (9) 100 1 τ √ √ 1 ln t+ln τ ∈ Set f(t, x(t)) = 100 ln t arctan x(t) and k(t, τ) = τ for t, τ [1, e]. It is easy to see that 1 |k(t, τ) − k(s, τ)| ≤ | ln t − ln s| 2 , which implies Kω2 = 1 and 1 g(t) = ln t, ω2(g(t) − g(s)) = | ln t − ln s| 2 , ω2(g(e) − g(1)) = 1. So we can choose 1 ω (g(t) − g(s)) = | ln t − ln s|α, 0 < α < . 1 2 ∫ √ √ { e | ln t+ln τ | ∈ } 2 − Moreover, K = sup 1 τ dτ : t [1, e] = 3 (2 2 1). On the other hand, 1 √ 1 |f(t, x(t)) − f(t, y(t))| = ln t| arctan x(t) − arctan y(t)| ≤ |x(t) − y(t)|, 100 100 A quadratic integral equation in the space of functions 361

1 | | so we can get k1 = 100 , k = f(1, 1) = 0 and √ √ | 1 ln t arctan x(s) − 1 ln s arctan x(s)| 1 1 100 100 ≤ | | ≤ | | 1 arctan x(s) x(s) , | ln t − ln s| 2 100 100

1 so k2 = 100 . In what follows, the condition (H3) reduce to the inequality ( √ ) 8 2 − 7 97 1 + e r2 − r + < 0. 3 100 100

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|·|α,ln ·[1, e] and displayed in Fig.1.

−7 x 10 8

x 4

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 t

Fig.1 The solution of the equation (9). Example 4.2. Consider another quadratic integral equation ∫ √ 1 √ 1 x(t) = 5 t + 1 sin x(t) + x(t) 5 3t2 + τx(τ)dτ, t ∈ [0, 1]. (10) 10 0 √ √ 1 5 5 2 ∈ Set f(t, x(t)) = 10 t + 1 sin x(t) and k(t, τ) = 3t + τ, t [0, 1]. Obviously, √ 2 2 1 5 1 |k(t, τ) − k(s, τ)| ≤ |3t − 3s | 5 ≤ 6|t − s| 5 , √ 5 1 − | − | 5 − which gives Kω2 = 6, g(t) = t, ω2(g(t) g(s)) = t s , ω2(g(1) g(0)) = 1. Then we choose 1 ω (g(t) − g(s)) = |t − s|α, 0 < α < . 1 5 ∫ √ 1 5 6 { | 2 | ∈ } { 5 2 5 |1 ∈ Moreover, K√= sup√ 0 3t + τ dτ : t [0, 1] = sup 6 (3t + τ) 0 : t } 5 5 − 5 [0, 1] = 6 (4 4 3 3). 362 Shan Peng, JinRong Wang, Fulai Chen

On the other hand, √ 1 √ 5 2 |f(t, x(t)) − f(t, y(t))| = 5 t + 1| sin x(t) − sin y(t)| ≤ |x(t) − y(t)|, 10 10 √ 5 2 we can get k1 = , k = |f(0, 0)| = 0 and √ 10 √ | 1 5 t + 1 sin x(s) − 1 5 s + 1 sin x(s)| 1 1 10 10 ≤ | | ≤ | | 1 sin x(s) x(s) , |t − s| 5 10 10 1 so derive k2 = 10 . In what follows, the condition (H ) reduce to the inequality √ √ 3 √ √ (200 5 4 − 150 5 3 + 30 5 6)r2 + (6 5 2 − 27)r < 0. 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.

−4 x 10

x 2

0 0 0.2 0.4 0.6 0.8 1 t

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

References

1. D. Baleanu, J.A.T. Machado and A.C.-J. Luo, Fractional Dynamics and Control, Springer, 2012. 2. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006. 3. V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009. 4. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equa- tions, John Wiley, New York, 1993. 5. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 6. V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, HEP, 2010. A quadratic integral equation in the space of functions 363

7. K.M. Case, P.F. Zweifel, Linear Transport Theory, Addison Wesley, Reading, M. A, 1967. 8. S. Chandrasekhar, Radiative transfer, Dover Publications, New York, 1960. 9. S. Hu, M. Khavani, W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal., 34(1989), 261-266. 10. C.T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Int. Eq., 4(1982), 221-237. 11. J. Bana´s,M. Lecko and W.G. El-Sayed, Existence theorems of some quadratic integral equation, J. Math. Anal. Appl., 222(1998), 276-285. 12. J. Bana´s,J. Caballero, J. Rocha and K. Sadarangani, Monotonic solutions of a class of quadratic integral equations of Volterra type, Comput. Math. Appl., 49(2005), 943-952. 13. J. Caballero, J. Rocha and K. Sadarangani, On monotonic solutions of an integral equation of Volterra type, J. Comput. Appl. Math., 174(2005), 119-133. 14. M.A. Darwish, On solvability of some quadratic functional-integral equation in Banach algebras, Commun. Appl. Anal., 11(2007), 441-450. 15. M.A, Darwish and S.K. Ntouyas, On a quadratic fractional Hammerstein-Volterra integral equations with linear modification of the argument, Nonlinear Anal., 74(2011), 3510-3517. 16. M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl., 311(2005), 112-119. 17. R.P. Agarwal, J. Bana´s,K. Bana´sand D. O’Regan, Solvability of a quadratic Hammer- stein integral equation in the class of functions having limits at infinity, J. Int. Eq. Appl., 23(2011), 157-181. 18. J. Wang, X. Dong and Y. Zhou, Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations, Commun. Nonlinear Sci. Numer. Simul., 17(2012), 545-554. 19. J. Wang, X. Dong and Y. Zhou, Analysis of nonlinear integral equations with Erdélyi- Kober fractional operator, Commun. Nonlinear Sci. Numer. Simul., 17(2012), 3129-3139. 20. J. Wang, C. Zhu and Y. Zhou, Study on a quadratic Hadamard types fractional integral equation on an unbounded interval, Topological Methods in Nonlinear Analysis, 42(2013), 257-275. 21. J. Bana´sand R. Nalepa, On the space of functions with growths tempered by a modulus of continuity and its applications, Journal of Function Spaces and Applications, 2013(2013), Article ID 820437, 13 pages. 22. J. Caballero, M.A. Darwish and K. Sadarangani, Solvability of a quadratic integral equation of Fredholm type in Hölder spaces, Electronic Journal of Differential Equations, 2014(2014), No.31, 1-10.

Shan Peng Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China. e-mail: [email protected]

JinRong Wang Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China. Department of Mathematics, Guizhou Normal College, Guiyang, Guizhou 550018, P.R. China. e-mail: [email protected]

Fulai Chen Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, P.R. China. e-mail: [email protected]