Research Article A New Fractional Integral Inequality with Singularity and Its Application

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1 Abstract and Applied Analysis Volume 212, Article ID 93798, 12 pages doi:1.1155/212/93798 Research Article A New Fractional Integral Inequality with Singularity and Its Application Qiong-Xiang Kong 1 and Xiao-Li Ding 2 1 Department of Building Environment and Services Engineering, Xi an Jiaotong University, Xi an, Shaanxi 7149, China 2 Department of Mathematical Sciences, Xi an Jiaotong University, Xi an, Shaanxi 7149, China Correspondence should be addressed to Xiao-Li Ding, dingding65@126.com Received 3 March 212; Revised 3 April 212; Accepted 3 April 212 Academic Editor: Bashir Ahmad Copyright q 212 Q.-X. Kong and X.-L. Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove an integral inequality with singularity, which complements some known results. This inequality enables us to study the dependence of the solution on the initial condition to a fractional differential equation in the weighted space. 1. Introduction Integral inequalities provide an excellent tool for the properties of solutions to differential equations, such as boundedness, existence, uniqueness, and stability e.g., see 1 1. For this reason, the study of integral inequalities has been emphasized by many authors. For example, in 1919, Gronwall in 11 proved a remarkable inequality which can be described by the following. Suppose that x t satisfies the relation x t h t k s x s ds, t t t<t, 1.1 where all the functions involved are continuous on the interval t,t, T,andk t. Consider ( ) x t h t h s k s exp k τ dτ ds, t t<t. 1.2 t s

2 2 Abstract and Applied Analysis The inequality has attracted and continues to attract considerable attention in the literature. In 27, Ye et al. 12 reported an integral inequality with singular kernel. The inequality can be stated as follows. If β>, a t is a nonnegative and locally integrable on t<t, g t is a nonnegative, nondecreasing continuous function on t<t,andg t M, where T, M is a positive constant. Further suppose that u t is nonnegative and locally integrable on t<twith u t a t g t t s β 1 u s ds, t<t. 1.3 Then u t a t n 1 ( g t Γ ( β )) n Γ ( nβ ) t s nβ 1 a s ds, t<t. 1.4 Besides the above-mentioned inequalities, there are still many inequalities e.g., see But in the analysis of the dependence of the solution on the initial condition of a fractional differential equation in the weighted space, the bounds provided by the existing inequalities are not adequate. So it is natural and necessary to seek new inequality in order to obtain our desired results. In this paper, we present a new integral inequality, and then apply our inequality to investigate the dependence of the solution on the initial condition of a fractional differential equations in the weighted space. 2. An Integral Inequality In this section, our main aim is to establish an integral inequality with singularity. Before proceeding, we give some useful definitions and lemmas. Definition 2.1 see 14, 16. The gamma function is defined by Γ z e t t z 1 dt, z >. Definition 2.2 see 14, 16. The beta function is defined by B z, w 1 1 t z 1 t w 1 dt, z, w >. The beta function is connected with gamma function by the following relation 3, 14 : B z, w Γ z Γ w, z,w >. 2.1 Γ z w Lemma 2.3 see 14. Let z>, a, b R. Then the quotient expansion of two gamma functions at infinity can be represented as follows: ( Γ z a za b 1 O Γ z b ( 1 z )), z. 2.2

3 Abstract and Applied Analysis 3 Lemma 2.4. Let z>, a, b R. Then one has Γ z a ( Γ z b O z a b), z. 2.3 Proof. By Lemma 2.3, we have lim z Γ z a /Γ z b /z a b lim z 1 O 1/z 1, which proves that Γ z a /Γ z b O z a b as z. The proof of this lemma is completed. Based on Lemma 2.4, we can define a function. Definition 2.5. Let b>a>, ρ>. Then the following definition: F ρ,a,b z : c k z k, z R 2.4 k is well defined, where c is a positive constant, and c k 1 Γ kρ a /Γ kρ b c k. Proof. We only need to show that the series in 2.4 is uniformly convergent for z R. By Lemma 2.4, we know that c k 1 /c k Γ kρ a /Γ kρ b O kρ a b as k. Since b>a>, c k 1 /c k ask. This implies that the series in 2.4 is uniformly convergent for z R. It follows that the definition is well defined. Lemma 2.6. Let z, w >, t, s R and t / s. Then one has s t τ z 1 τ s w 1 dτ t s z w 1 Γ z Γ w Γ z w. 2.5 Proof. Making the substitution τ s ξ t s and combining the relation 2.1, weobtain s 1 t τ z 1 τ s w 1 dτ t s z w 1 1 ξ z 1 ξ w 1 dξ t s z w 1 B z, w t s z w 1 Γ z Γ w Γ z w. 2.6 The proof of this lemma is completed. Now we can state the integral inequality. Theorem 2.7. Let α, β, γ >, δ α γ 1 >, ν β γ 1 >, a>, and let b t be a nonnegative, nondecreasing continuous function on t<t, b t M, wheret, M is a positive constant. Further suppose that u t is nonnegative and t γ 1 u t is locally integrable on t<twith u t at α 1 b t t s β 1 s γ 1 u s ds, t<t. 2.7

4 4 Abstract and Applied Analysis Then one has u t at α 1 F ν,δ,δ β (Γ ( β ) b t t β), t<t. 2.8 Proof. For convenience, we define an operator Ru t b t t s β 1 s γ 1 u s ds. 2.9 Then 2.7 can be rewritten in the form u t at α 1 Ru t. 2.1 Since b t and u t are nonnegative, it is easy to induce that u t n ( R k at α 1) t k R n 1 u t, n N Let us prove that the following relation b t ( Γ ( β ) b t ) n 1 n 1 Γ iν t R n i 1 Γ ( iν β ) t s nν γ s γ 1 u s ds, <γ<1, u t ( ) nt Γ β b t n 1 γ 1 Γ ( nβ ) t s nβ 1 s γ 1 u s ds, γ 1, 2.12 holds for any n N, where i 1 1 1, and Rn u t asn for each t in t<t.

5 Abstract and Applied Analysis 5 Obviously, inequality 2.12 is valid for n 1, due to i Suppose that the inequality is satisfied for any fixed n N. Let us verify that it is also satisfied for n 1. We first prove the case <γ<1. According to the induction hypothesis and Lemma 2.6, we have R n 1 u t b t t s β 1 s γ 1 R n u s ds b 2 t ( Γ ( β ) b t ) n 1 Γ iν Γ ( iν β ) i 1 b 2 t ( Γ ( β ) b t ) n 1 n 1 Γ iν Γ ( iν β ) i 1 b 2 t ( Γ ( β ) b t ) n 1 n 1 Γ iν Γ ( iν β ) i 1 b 2 t ( Γ ( β ) b t ) n 1 n 1 Γ iν Γ ( iν β ) b t ( Γ ( β ) b t ) n n i 1 i 1 Γ iν Γ ( iν β ) s t s β 1 s γ 1 s τ nν γ τ γ 1 u τ dτds τ γ 1 u τ dτ τ γ 1 u τ dτ τ τ t s β 1 s γ 1 s τ nν γ ds t s β 1 s τ nν 1 ds τ γ 1 u τ t τ nν β 1 Γ( β ) Γ nν Γ ( nν β ) dτ t τ n 1 ν γ τ γ 1 u τ dτ, 2.13 which is estimated with the help of s γ 1 s τ γ 1, τ s, <γ< So, for the case <γ<1, inequality 2.12 is true for any n N. Now we prove the case γ 1. Similarly, according to the induction hypothesis and Lemma 2.6,weget R n 1 u t b t t s β 1 s γ 1 R n u s ds ( ) ns Γ β b s b t t s β 1 n 1 γ 1 s γ 1 Γ ( nβ ) s τ nβ 1 τ γ 1 u τ dτds ( ) n Γ β b t t b t Γ ( nβ ) τ γ 1 u τ dτ t s β 1 s n γ 1 s τ nβ 1 ds τ ( ) nt Γ β b t n γ 1 b t Γ ( nβ ) τ γ 1 u τ dτ t s β 1 s τ nβ 1 ds τ ( ) nt Γ β b t n γ 1 b t Γ ( nβ ) τ γ 1 u τ t τ nβ β 1 Γ( β ) Γ ( nβ ) Γ ( nβ β ) dτ s 2.15 ( Γ ( β ) b t ) n 1t n γ 1 Γ ( n 1 β ) τ γ 1 u τ t τ n 1 β 1 dτ,

6 6 Abstract and Applied Analysis which is calculated with the help of s n γ 1 t n γ 1, s t, γ 1, n N So, for the case γ 1, inequality 2.12 is true for any n N. Based on this analysis, we conclude that inequality 2.12 holds for any n N. Next, we show that R n u t asn. Now, we go back to inequality For the case < γ < 1, we denote K n t, s B n t s nν γ, where B n b t Γ β b t n 1 n 1 i 1 Γ iν /Γ iν β.notethat B 1 b t, B n 1 B n Γ ( β ) Γ nν b t Γ ( nν β ) Since b t M, by Lemma 2.4, weobtainb n 1 /B n asn. This implies that K n t, s asn. It follows that R n u t asn for the case <γ<1. For the case γ 1, we denote K n t, s B n t s nβ 1, where B n Γ β b t n t n 1 γ 1 /Γ nβ. Note that B 1 b t, Γ ( β ) b t t γ 1 Γ nβ B n Γ ( nβ β ). B n Using the same arguments as above, we know that K n t, s asn. It follows that R n u t asn for the case γ 1. So, it has R n u t asn for the two cases <γ<1andγ 1. This, together with 2.11, leadstou t k Rk at α 1 t. Finally, we show that ( R k at α 1) t a ( Γ ( β ) b t ) k ck t α 1 t kν, k N, 2.19 where c 1, c k k 1 i Γ iν δ /Γ iν δ β, k N. Obviously, inequality 2.19 is true for k. Suppose that the inequality is satisfied for any fixed k N. Let us verify that it is also satisfied for k 1. According to the induction hypothesis and Lemma 2.6, weobtain ( R k 1 at α 1) t b t t s β 1 s γ 1( R k as α 1) s ds a ( Γ ( β ) b t ) k ck b t t s β 1 s kν δ 1 ds a ( Γ ( β ) b t ) k ck b t t kν δ β 1 Γ( β ) Γ kν δ Γ ( kν δ β ) 2.2 a ( Γ ( β ) b t ) k 1 ck 1 t α 1 t k 1 ν.

7 Abstract and Applied Analysis 7 This proves that inequality 2.19 is satisfied for any k N. In other words, we have proved that u t a ( Γ ( β ) b t ) k ck t α 1 t kν, k 2.21 where c 1, c k k 1 i Γ iν δ /Γ iν δ β, k N. By virtue of Definition 2.5, we can arrive at inequality 2.8 and the proof of this theorem is completed. For the case b t b> in Theorem 2.7, we can obtain the following corollary, which can be found in 17. Corollary 2.8. Let α, β, γ >, δ α γ 1 >, ν β γ 1 >, a, b >. And suppose that u t is nonnegative and t γ 1 u t is locally integrable on t<t (T )with u t at α 1 b t s β 1 s γ 1 u s ds, t<t Then one has u t at α 1 F ν,δ,δ β (Γ ( β ) bt β), t<t For α γ 1 in Theorem 2.7, we can arrive at the following corollary, which can be found in 12. Corollary 2.9. Let β, a>, b t be a nonnegative, nondecreasing continuous function on t<t, b t M, wheret, M is a positive constant. And suppose that u t is nonnegative and locally integrable on t<twith u t a b t t s β 1 u s ds, t<t Then one has u t ae β (Γ ( β ) b t t β), t<t Application In this section, we will apply our established result to study the dependence of the solution on the initial condition of a fractional differential equation with the Riemann-Liouville derivative. For the reader s convenience, we first recall several definitions of the Reimann- Liouville integral and derivative. From now on, we assume that T is a finite positive constant, that is, T /.

8 8 Abstract and Applied Analysis Definition 3.1 see 14, 16. Let<p<1. The Riemann-Liouville integral of order p is defined by ( I p x ) t 1 Γ ( p ) t s p 1 x s ds, t T. 3.1 Definition 3.2 see 14, 16. Let<p<1. The Riemann-Liouville derivative of order p is defined by D p 1 x t Γ ( 1 p ) d dt t s p x s ds, t T. 3.2 Now we consider the following initial value problem of the form D p x t f t, x t, lim I 1 p t x t x, <p<1, <t T, x R. 3.3 With regard to problem 3.3, the existence and uniqueness of the solution can be found in the book by Kilbas et al. 14. For the completeness of this paper, we give the existence and uniqueness of the solution to 3.3 in the weighted space C 1 p,t. The space C 1 p,t consists of all functions x C,T such that t 1 p x t C,T, which turns out to be a Banach space when endowed with the norm x 1 p max t T t 1 p x t. Theorem 3.3 see 14. Let <p<1, and f t, x :,T R R be a function such that for any x R, f t, x C 1 p,t. Further assume that for any t,t, x, y R, the following inequality f t, x f ( t, y ) L x y 3.4 holds, where L> is a constant. Then there exists a unique solution x t to problem 3.3 in the space C 1 p,t. Theorem 3.4. Let <p<1, and f t, x :,T R R be a function such that for any x R, f t, x C 1 p,t. Further assume that for any t,t, x, y R, the following inequality f t, x f ( t, y ) L x y 3.5 holds, where L> is a constant. Assume that x and y are the solutions of problem 3.3 and D p y t f ( t, y t ), lim I 1 p t y t y, <t T, y R, 3.6

9 Abstract and Applied Analysis 9 respectively. Then, for t T, one has x y Γ ( p ) F 2p 1,p,2p Lt, 1 <p<1, t T, 2 t 1 p x y x t y t Γ ( p ) F 2p q 1,p q,2p q L t, <p 1 2, t<1, x y Γ ( p ) F 2p q 1,p q,2p q L t, <p 1 2, 1 t T, 3.7 where q, q,l,l are positive constants such that 1 2p <q<log L/L t, <p 1 2, <t<1, } q > max {1 2p, log L/L, <p , 1 t T. t Proof. The proof is rather technical. We first prove the case 1/2 <p<1and t T. Since x t and y t are the solutions of 3.3 and 3.6, we have x t x t p 1 Γ ( p ) 1 Γ ( p ) t s p 1 f s, x s ds, 3.9 y t y t p 1 Γ ( p ) 1 Γ ( p ) t s p 1 f ( s, y s ) ds. 3.1 Subtracting 3.1 from 3.9 and using the Lipschitz condition 3.5,weobtain x t y t x y t p 1 Γ ( p ) L Γ ( p ) t s p 1 x s y s ds Taking into account that x t,y t C 1 p,t, we multiply at both sides of inequality 3.11 by t 1 p to get t 1 p x t y t x y Γ ( p ) Lt1 p Γ ( p ) t s p 1 s p 1 s 1 p x s y s ds Denote u t t 1 p x t y t. Then, 3.12 can be written as u t x y Γ ( p ) Lt1 p Γ ( p ) t s p 1 s p 1 u s ds. 3.13

10 1 Abstract and Applied Analysis Putting a x y /Γ p, b t Lt 1 p /Γ p, α 1, β p, γ p, weseethatα, β, γ >, δ α γ 1 p>, ν β γ 1 2p 1 >, a>, and b t is a nondecreasing continuous function due to p, L >. So, applying Theorem 2.7 to 3.13,weobtain x y u t Γ ( p ) F 2p 1,p,2p Lt, 1 <p<1, t T Next, we prove the case <p 1/2 and t<1. Notice that the Lipschitz condition 3.5 holds for each t in t,t. Since t>andl>, we can always choose two positive constants q, L such that 1 2p <q<log L/L t, <t< Condition 3.15 means that 1 2p < q and L < L t q. That is to say, if the Lipschitz condition 3.5 holds for each t in t,t, then we can always choose two positive constants q, L such that the following condition 1 2p <q, f t, u f t, v L t q u v 3.16 holds for each t in t,t. Subtracting 3.1 from 3.9 and using condition 3.16, weobtain x x t y t y t p 1 Γ ( L ) p Γ ( p ) t s p 1 s q x s y s ds Multiplying t 1 p on both sides of 3.17, weget u t x y Γ ( p ) L t 1 p Γ ( p ) t s p 1 s p q 1 u s ds, 3.18 where u t is defined as before. Now, putting a x y /Γ p, b t L t 1 p /Γ p, α 1, β p, γ p q, weseethatα, β, γ >, δ α γ 1 p q>, ν β γ 1 2p q 1 q 1 2p >, a>, b t is a nondecreasing continuous function due to p, L >. So, applying Theorem 2.7 to 3.18, we have x y u t Γ ( p ) F 2p q 1,p q,2p q L t, <p 1, t< Finally, we prove the case <p 1/2 and1 t T. Since t>andl>, we can always choose two positive constants q, L such that } q > max {1 2p, log L/L, 1 t T. 3.2 t

11 Abstract and Applied Analysis 11 Condition 3.2 means that 1 2p <q and L<L t q. Using the same arguments as above, we can obtain that x y u t Γ ( p ) F ( 2p q 1,p q,2p q L t ), <p 1, 1 t T, where u t is defined as before. So the conclusion of this theorem is true. From the proof of Theorem 3.4, we can see that the integral inequality in Theorem 2.7 is very useful. This demonstrates that our investigation is meaningful. Acknowledgmets This work was supported by the National Natural Science Foundation of China and the International Science and Technology S&T Cooperation Program of China 21DFA147. References 1 D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, Mathematical Analysis and Applications, vol. 24, no. 2, pp , A. Chen, F. Chen, and S. Deng, On almost automorphic mild solutions for fractional semilinear initial value problems, Computers & Mathematics with Applications, vol. 59, no. 3, pp , L. Hu, Y. Ren, and R. Sakthivel, Existence and uniqueness of mild solutions for semilinear integrodifferential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum, vol. 79, no. 3, pp , Y.-L. Jiang, A general approach to waveform relaxation solutions of nonlinear differential-algebraic equations: the continuous-time and discrete-time cases, IEEE Transactions on Circuits and Systems, vol. 51, no. 9, pp , Y.-L. Jiang, R. M. M. Chen, and O. Wing, Convergence analysis of waveform relaxation for nonlinear differential-algebraic equations of index one, IEEE Transactions on Circuits and Systems, vol. 47, no. 11, pp , 2. 6 N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 7, pp , R. A. C. Ferreira, A discrete fractional Gronwall inequality, Proceedings of the American Mathematical Society, vol. 14, no. 5, pp , J. J. Nieto, Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Applied Mathematics Letters, vol. 23, no. 1, pp , H. Ye and J. Gao, Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay, Applied Mathematics and Computation, vol. 218, no. 8, pp , B. Ahmad, J. J. Nieto, and A. Alsaedi, Existence and uniqueness of solutions for nonlinear fractional dierential equations with non-separated type integral boundary conditions, Acta Mathematica Scientia, vol. 31, pp , T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of dierential equations, Annals of Mathematics, vol. 2, no. 4, pp , H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, Mathematical Analysis and Applications, vol. 328, no. 2, pp , Z. Denton and A. S. Vatsala, Fractional integral inequalities and applications, Computers & Mathematics with Applications, vol. 59, no. 3, pp , A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 24 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 26.

12 12 Abstract and Applied Analysis 15 V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Communications in Applied Analysis, vol. 11, no. 3-4, pp , I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 84 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.

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