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On Some Fractional Quadratic Integral Inequalities KYUNGPOOK Math. J. 60(2020), 211-222 https://doi.org/10.5666/KMJ.2020.60.1.211 pISSN 1225-6951 eISSN 0454-8124 c Kyungpook Mathematical Journal On Some Fractional Quadratic Integral Inequalities Ahmed M. A. El-Sayed Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt e-mail : [email protected] Hind H. G. Hashem∗ Department of Mathematics, Faculty of Science, Qassim University, Buraidah, Saudi Arabia e-mail : [email protected] Abstract. Integral inequalities provide a very useful and handy tool for the study of qualitative as well as quantitative properties of solutions of differential and integral equa- tions. The main object of this work is to generalize some integral inequalities of quadratic type not only for integer order but also for arbitrary (fractional) order. We also study some inequalities of Pachpatte type. 1. Introduction and Preliminaries In the theory of differential and integral equations, Gronwall's lemma has been widely used in various applications since its first appearance in the article by Bell- man in 1943. In this article the author gave a fundamental lemma, known as Gronwall-Bellman Lemma, which is used to study the stability and asymptotic be- havior of solutions of differential equations. Gronwall's lemma has seen several generalizations to various forms [3, 11, 28]. The literature on these inequalities and their applications is vast; see [1, 2, 4] and the references given therein [19, 20, 23]. In addition, as the theory of calculus on time scales has developed over the last few years, some Gronwall-type integral inequalities on time scales have been established by many authors [17, 21, 24]. In this work, we shall study some fractional integral inequalities which can be used to prove the uniqueness of solutions for differential and integral equations of * Corresponding Author. Received September 5, 2018; revised March 17, 2019; accepted March 18, 2019. 2010 Mathematics Subject Classification: 26A33, 26E30, 26D10. Key words and phrases: fractional-order integral inequality, fractional quadratic integral equations, Bellman-Gronwall's inequality, inequality of Pachpatte type. 211 212 A. M. A. El-Sayed and H. H. G. Hashem fractional-order. These inequalities are similar to Bellman-Gronwall type inequali- ties which play a fundamental role in the qualitative as well as quantitative study of differential equations. Let L1 = L1[a; b] be the class of Lebesgue integrable functions on [a; b] with the standard norm. Now, we shall introduce the definitions of the fractional-order integral operators (see [18, 25, 26, 27]). Let β be a positive real number. Definition 1.1. The left-sided fractional integral of order β of the function f is defined on [a; b] by Z t β−1 β (t − s) (1.1) Ia+f(t) = f(s) ds; t > a a Γ(β) β and when a = 0; we have I0+f(t); t > 0. Definition 1.2. The right-sided fractional integral of order β of the function f is defined on [a; b] by Z b β−1 β (s − t) (1.2) Ib−f(t) = f(s) ds; t < b t Γ(β) β and when b = 0; we have I0−f(t); t < 0: For further properties of fractional calculus see [18, 25, 26, 27]. 2. Quadratic Integral Inequalities The existence of solutions of nonlinear quadratic integral equations and some properties of their solutions have been well studied recently (see [5, 6, 7, 8, 9, 10] and [12, 13, 14, 15, 16]). Let α; β > 0: The existence of solutions of the quadratic integral equation of arbitrary (fractional) orders α and β α β (2.1) x(t) = h(t) + Ia+ f(t; x(t)) Ia+ g(t; x(t)); t > a have been studied in [16] and [14] under the following assumptions. (i) The functions f; g :[a; b] × R+ ! R+ satisfy the Carath`eodory condition (i.e. are measurable in t for all x 2 R+ and continuous in x for all t 2 [a; b]). Moreover, there exist two functions m1; m2 2 L1; and a positive constant k such that jf(t; x)j ≤ m1(t) + k x(t); jg(t; x)j ≤ m2(t); 8 (t; x) 2 [a; b] × R+: γ (ii) There exists a positive constant M2 such that Ia+m2(t) ≤ M2; γ < β: σ (iii) There exists a positive constant M1 such that Ia+m1(t) ≤ M1; σ < α: On Some Fractional Quadratic Integral Inequalities 213 (iv) h 2 L1: Now, consider the nonlinear quadratic integral inequality of fractional orders α β (2.2) x(t) ≤ h(t) + Ia+ f(t; x(t)) Ia+ g(t; x(t)); t > a; α; β > 0 Theorem 2.1. Let assumptions (i){(iv) be satisfied. Let x(t) 2 L1 and satisfies α+β−γ kM2b inequality (2.2) for almost all t 2 [a; b]. If Γ(β−γ+1)Γ(α+1) < 1; then 1 β−γ β−γ+α−σ X k M2 b M1 M2 b x(t) ≤ ( Iα )n h(t) + : Γ(β − γ + 1) a+ Γ(β − γ + 1)Γ(α − σ + 1) n = 0 Proof. Using assumptions (i)-(iv) and inequality (2.2), we have α β x(t) ≤ h(t) + Ia+ (m1(t) + k x(t)):Ia+ m2(t); t > a; α; β > 0 α−σ σ β−γ γ α β−γ γ x(t) ≤ h(t) + Ia+ Ia+m1(t):Ia+ Ia+m2(t) + k Ia+x(t):Ia+ Ia+m2(t); M M bβ−γ+α−σ kM bβ−γ x(t) ≤ h(t) + 1 2 + 2 Iα x(t); t > a; α > 0 Γ(β − γ + 1)Γ(α − σ + 1) Γ(β − γ + 1) a+ k M bβ−γ M M bβ−γ+α−σ ) (I − 2 Iα ) x(t) ≤ h(t) + 1 2 Γ(β − γ + 1) a+ Γ(β − γ + 1)Γ(α − σ + 1) Since β−γ Z b β−γ k M2 b α k M2 b α k Ia+ x(t) k = j Ia+ x(t) j dt Γ(β − γ + 1) a Γ(β − γ + 1) Z b k M bβ−γ Z t (t − s)α − 1 ≤ 2 jx(s)j ds dt a Γ(β − γ + 1) a Γ(α) k M bβ−γ Z b Z b (t − s)α − 1 ≤ 2 dt jx(s)j ds Γ(β − γ + 1) a s Γ(α) k M bα+β−γ Z b ≤ 2 jx(s)j ds Γ(β − γ + 1)Γ(α + 1) a α+β−γ k M2 b and Γ(β−γ+1)Γ(α+1) < 1, then k M bβ−γ k 2 Iα x(t) k < kx(t)k Γ(β − γ + 1) a+ k M bβ−γ and (I − 2 Iα )−1 exists. Γ(β − γ + 1) a+ Then 1 β−γ β−γ+α−σ X k M2 b M1 M2 b x(t) ≤ ( Iα )n h(t) + : 2 Γ(β − γ + 1) a+ Γ(β − γ + 1)Γ(α − σ + 1) n = 0 Remark 2.2. If we replace assumptions (ii) and (iii) by 214 A. M. A. El-Sayed and H. H. G. Hashem β (ii*) There exists a positive constant M2 such that Ia+m2(t) ≤ M2; α (iii*) There exists a positive constant M1 such that Ia+m1(t) ≤ M1; then we can obtain the following result. Theorem 2.3. Let assumptions (i), (ii*), (iii*) and(iv) be satisfied. Let x(t) 2 L1 α+β kM2b satisfies the inequality (2.2) for almost all t 2 [a; b]. If Γ(β+1)Γ(α+1) < 1; then 1 β β+α X k M2 b M1 M2 b x(t) ≤ ( Iα )n h(t) + : Γ(β + 1) a+ Γ(β + 1)Γ(α + 1) n = 0 Inequality (2.2) involves many integral inequalities of fractional order. So, some particular cases can be obtained as follows. Corollary 2.4. Let f; g :[a; b] × R+ ! R+ are continuous functions, and h 2 L1: If x(t) 2 L1 and satisfies the inequality (2.2) for almost all t 2 [a; b]. Then M M bβ+α x(t) ≤ h(t) + 1 2 ; Γ(β + 1)Γ(α + 1) where M1 = sup jg(t; x)j;M2 = sup jf(t; x)j: 8t2[a;b] 8t2[a;b] Corollary 2.5. Let assumptions (i){(iv) (with α = β; f = g) be satisfied. Let x(t) 2 L1 and satisfies the inequality α 2 x(t) ≤ h(t) + Ia+ f(t; x(t)) ; t > a; α > 0 2α−γ kM2b for almost all t 2 [a; b]. If Γ(α−γ+1)Γ(α+1) < 1; then 1 α−γ 2 2α−γ X k M2 b M b x(t) ≤ ( Iα )n h(t) + 2 : Γ(α − γ + 1) a+ Γ(α − γ + 1)Γ(α + 1) n = 0 Corollary 2.6. Let assumptions (i)-(iv) (with α = β; f = g and h = 0) be satisfied. Let x(t) 2 L1 and satisfies the inequality p α x(t) ≤ Ia+ f(t; x(t)); t > a; α > 0 2α−γ kM2b for almost all t 2 [a; b]. If Γ(α−γ+1)Γ(α+1) < 1; then 1 α−γ 2 2α−γ X k M2 b M b x(t) ≤ ( Iα )n 2 : Γ(α − γ + 1) a+ Γ(α − γ + 1)Γ(α + 1) n = 0 On Some Fractional Quadratic Integral Inequalities 215 Letting α; β ! 1; then we have Corollary 2.7. Let h(t); f(t; x) and g(t; x) satisfy the assumptions of Theorem 2.3.
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