RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES FOR SEVERAL FUNCTIONS WITH APPLICATIONS

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1 Communications in Alied Analysis 12 28, no. 4, RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES FOR SEVERAL FUNCTIONS WITH APPLICATIONS GEORGE A. ANASTASSIOU Deartment of Mathematical Sciences, University of Memhis Memhis, TN U.S.A. ABSTRACT. A large variety of very general L 1 form Oial tye inequalities 15] is resented involving Riemann-Liouville fractional derivatives 5], 12], 13], 14] of several functions in different orders and owers. From the established results derive several other articular results of secial interest. Alications of some of these secial inequalities are given in roving uniqueness of solution and in giving uer boun to solutions of initial value fractional roblems involving a very general system of several fractional differential equations. Uer boun to various Riemann-Liouville fractional derivatives of the solutions that are involved in the above systems are given too. 2 AMS Subject Classification. 26A33, 26D1, 26D15, 34A12, 34A99 Key Wor and Phrases. Oial tye inequality, Riemann-Liouville fractional derivative, system of fractional differential equations, uniqueness of solution, uer bound of solution. INTRODUCTION Oial inequalities aeared for the first time in 15] and then many authors dealt with them in different directions and for various cases. For a comlete recent account on the activity of this field see 3], and still it remains a very active area of research. One of their main attractions to these inequalities is their alications, esecially to roving uniqueness and uer boun of solution of initial value roblems in differential equations. The author was the first to resent Oial inequalities involving fractional derivatives of functions in 4], 5] with alications to fractional differential equations. See also 8], 9]. Fractional derivatives come u naturally in a number of fiel, esecially in Physics, see the recent book 11]. To name a few toics such as, fractional Kinetics of Hamiltonian Chaotic systems, Polymer Physics and Rheology, Regular variation in Thermodynamics, Biohysics, fractional time evolution, fractal time series, etc. One there deals also with stochastic fractional-difference equations and fractional diffusion equations. Great alications of these can be found in the study of DNA sequences. Other fractional differential equations arise in the study of susensions, coming from Received October 1, $15. c Dynamic Publishers, Inc.

2 378 G. A. ANASTASSIOU the fluid dynamical modeling of certain blood flow henomena. An excellent account in the study of fractional differential equations is in the recent book 16]. The study of fractional calculus started from 1695 by L Hosital and Leibniz, also continued later by J. Fourier in 1822 and Abel in 1823, and continues to our days in an increased fashion due to its many alications and necessity to deal with fractional henomena and structures. In this aer the author is greatly motivated and insired by the very imortant aers 1], 2]. Of course there the authors are dealing with other kin of derivative. Here the author continues his study of Riemann-Liouville fractional Oial inequalities now involving several different functions and roduces a wide variety of corresonding results with imortant alications to systems of several fractional differential equations. This article is a generalization of the author s earlier article 6]. We start in Section 1 with Background, we continue in Section 2 with the main results and we finish in Section 3 with alications. To give an idea to the reader of the kind of inequalities we are dealing with, briefly we mention a simle one x M D γ f j w D ν f j w dw { x ν γ } 2Γν γ ν γ D ν f j w dw 2, 2ν 2γ 1 x, for functions f j L 1, x, j = 1,...,M N; ν > γ etc. Here D β f stan for the Riemann-Liouville fractional derivative of f of order β. Furthermore one system of fractional differential equations we are dealing with briefly is of the form D ν f j t = F j t, {D γ i f 1 t} r, {Dγ i f 2 t} r,..., {D γ i f M t} r, all t a, b], j = 1,...,M; D ν k f j = α kj R, k = 1,..., ν] + 1. Here ν] is the integral art of ν. We need 1. BACKGROUND Definition 1 see 1], 13], 14]. Let α R + {}. For any f L 1, x; x R + {}, the Riemann-Liouville fractional integral of f of order α is defined by J α fs := 1 Γα s s t α 1 ftdt, s, x], 1

3 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 379 and the Riemann-Liouville fractional derivative of f of order α by m D α 1 d s fs := s t m α 1 ftdt, 2 Γm a where m := α] + 1, ] is the integral art. In addition, we set D f := f := J f, J α f = D α f if α >, D α f := J α f, if < α 1. If α N, then D α f = f α the ordinary derivative. Definition 2 1]. We say that f L 1, x has an L fractional derivative D α f in, x], x R + {}, iff D α k f C, x], k = 1,..., m := α] + 1; α R + {}, and D α 1 f AC, x] absolutely continuous functions and D α f L, x. We need Lemma 3 1]. Let α R +, β > α, let f L 1, x, x R + {}, have an L fractional derivative D β f in, x], and let D β k f = for k = 1,..., β] + 1. Then D α fs = 1 Γβ α s s t β α 1 D β ftdt, s, x]. 3 Clearly here D α f AC, x] for β α 1 and in C, x] for β α, 1, hence D α f L, x and D α f L 1, x. then and 2. MAIN RESULTS Here we use a lot the following basic inequalities. Let α 1, α 2,...,α n, n N, Our first result follows next a r ar n a a n r, r 1, 4 a r a r n n 1 r a a n r, r 1. 5 Theorem 4. Let α 1, α 2 R +, β > α 1, α 2 and let f j L 1, x, j = 1,..., M N, x R + {} have, resectively, L fractional derivatives D β f j in, x], and let D β k f j =, for k = 1,..., β] + 1; j = 1,...,M. Consider also t > and 1 qt, with all t,, qt L t, x. Let λ β > and λ α1, λ α2, such that λ β <, where > 1. Set P i s := s s t β α i 1 1 t 1/ 1 dt, i = 1, 2; s x, 6 As := qsp 1s λα 1 1 P 2 s λα 2 1 Γβ α 1 λα 1Γβ α 2 λα 2 A x := As / λβ s λ β/ λβ /, 7, 8

4 38 G. A. ANASTASSIOU and M 1 λα δ1 1 +λβ/, if λ α1 + λ β, := 2 λα 1 +λ β 1, if λ α1 + λ β. Call ϕ 1 x := A x λα2 = λ α1 + λ β If λ α2 =, we obtain that, qs f j s λα 1 D β f j s λ β λβ / λβ 9 1 x δ1 ϕ M 1x s D β f j s ] λα 1 +λ β. 11 Proof. By Theorem 4 of 7] we obtain qs f j s λα 1 D β f j s λ β + f j+1 s λα 1 D β f j+1 s ] λ β λβ / λ β A x λα2 = λ α1 + λ β δ 1 s D β f j s + D β f j+1 s ] λα ] 1 +λ β, 12 j = 1, 2..., M 1, where 2 1 λα 1 +λβ/, if λ α1 + λ β, δ 1 := 1, if λ α1 + λ β. Hence by adding all the above we find { 1 qs D α 1 f j s λα 1 D β f j s λ β + ] } f j+1 s λα 1 D β f j+1 s λ β 13 M 1 δ 1 ϕ 1 x s D β f j s + D β f j+1 s ] ] λα 1 +λ β. 14 Also it hol Call qs f 1 s λα 1 D β f 1 s λ β + f M s λα 1 D β f M s λ β] δ 1 ϕ 1 x s D β f 1 s + D β f M s ] ] λα 1 +λ β. 15 1, if λ α1 + λ β, ε 1 =, if λ α1 + λ β. M 1 λα 1 +λ β 16

5 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 381 Adding 14 and 15, and using 4 and 5 we have We have roved 2 qs f j s λα 1 D β f j s λ β 17 { M 1 δ 1 ϕ 1 x s D β f j s + D β f j+1 s ] ] λα 1 +λ β + s D β f 1 s + D β f M s ] { δ 1 ε 1 ϕ 1 x s 2 D β f j s M qs f j s λα 1 D β f j s λ β ] } λα 1 +λ β } λα 1 +λ β. 18 { δ 1 2 λα 1 +λ β ] } λα 1 +λ β 1 ε 1 ϕ 1 x s D β f j s. 19 Clearly here we have δ 1 = δ 1 2 λα 1 +λ β 1 ε 1. 2 From 19 and 2 we derive 11. Next we give Theorem 5. All here as in Theorem 4. Denote 2 λα 2 /λ β 1, if λα2 λ β, δ 3 := 1, if λ α2 λ β, 1, if λ β + λ α2, ε 2 :=, if λ β + λ α2, M 1 λ β +λα and ϕ 2 x := A x λα1 = 2 λ λβ / β λβ δ λ β/ λ α2 + λ β

6 382 G. A. ANASTASSIOU If λ α1 =, then it hol qs {{ M 1 D α 2 f j+1 s λα 2 D β f j s λ β ] } f j s λα 2 D β f j+1 s λ β + f M s λα 2 D β f 1 s λ β f 1 s λα 2 D β f M s λ β] } 2 λ β +λα 2 ε 2 ϕ 2 x { ] s D β f j s Proof. From Theorem 5 of 7] we have } λ β +λα qs f j+1 s λα 2 D β f j s λ β f j s λα 2 D β f j+1 s λ β] ϕ 2 x s D β f j s + D β f j+1 s ] λ β +λα 2, 25 for j = 1,...,M 1. Hence by adding all of the above we get 1 qs D α 2 f j+1 s λα 2 D β f j s λ β ] f j s λα 2 D β f j+1 s λ β M 1 ϕ 2 x s D β f j s + D β f j+1 s ] λ β +λα Similarly it hol qs f M s λα 2 D β f 1 s λ β f 1 s λα 2 D β f M s λ β] ϕ 2 x s λ β +λα 2 D β f 1 s + D β f M s ]. 27 Adding 26 and 27 and using 4, 5 we derive 24. It follows the general case Theorem 6. All here as in Theorem 4. Denote 2 λα 1 +λα 2 /λβ 1, if λ α1 + λ α2 λ β, γ 1 := 1, if λ α1 + λ α2 λ β, 28 and 1, if λ α1 + λ α2 + λ β, γ 2 := 2 1 λα 1 +λα 2 +λβ/, if λ α1 + λ α2 + λ β. 29

7 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 383 Set ϕ 3 x := A x and Then qs λ β λ α1 + λ α2 λ α1 + λ α2 + λ β λ β λ λ β α 1 γ λ β 1, if λ α1 + λ α2 + λ β, ε 3 := M 1 λα 1 +λα 2 +λ β, if λ α1 + λ α2 + λ β. M 1 f j s λα 1 D α 2 f j+1 s λα 2 D β f j s λ β ] f j s λα 2 D α 1 f j+1 s λα 1 D β f j+1 s λ β + f 1 s λα 1 D α 2 f M s λα 2 D β f 1 s λ β ] γ 1 λ α2 λ β, 3 31 f 1 s λα 2 D α 1 f M s λα 1 D β f M s λ β] ] 2 λα 1 +λα 2 +λ β ε 3 ϕ 3 x { x M s D β f j s ] Proof. From Theorem 6 of 7] and by adding we get M 1 Also it hol qs f j s λα 1 D α 2 f j+1 s λα 2 D β f j s λ β f j s λα 2 D α 1 f j+1 s λα 1 D β f j+1 s λ β ϕ 3 x M 1 qs ] } λα 1 +λα 2 +λ β. 32 λα1 +λ α2 +λ β / s D β f j s + D β f j+1 s. 33 f 1 s λα 1 D α 2 f M s λα 2 D β f 1 s λ β f 1 s λα 2 D α 1 f M s λα 1 D β f M s λ β ϕ 3 x s D β f 1 s + D β f M s λα 1 +λα 2 +λ β. 34 Adding 33 and 34, along with 4, 5 we derive 32. We continue with ]

8 384 G. A. ANASTASSIOU Theorem 7. Let β > α 1 + 1, α 1 R + and let f j L 1, x, j = 1,..., M N, x R + {} have, resectively, L fractional derivatives D β f j, in, x], and let D β k f j =, for k = 1,..., β] + 1; j = 1,...,M. Consider also t > and qt, with t, 1 t, qt L, x. Let λ α, < λ α+1 < 1, and > 1. and Denote Lx := 2 2 λα/λα+1 1, if λ α λ α+1, θ 3 := 1, if λ α λ α+1, P 1 x := 35 1 λα+1 λα+1 qs 1/1 λα+1 θ3 λ α+1, 36 λ α + λ α+1 x s β α 1 1/ 1 s 1/ 1, 37 and Tx := Lx P 1 x 1 Γβ α 1 λα+λ α+1, 38 ω 1 := 2 1 λα+λ α+1, 39 Φx := Tx ω 1. 4 Also ut 1, if λ α + λ α+1, ε 4 :=, if λ α + λ α+1. M 1 λα+λ α+1 41 Then qs {{ M 1 D α 1 f j s λα D α1+1 f j+1 s λ α+1 + f j+1 s λα D α1+1 f j s ]} λ α+1 + f 1 s λα +1 f M s λ α+1 + f M s λα +1 f 1 s λ α+1 ]} 2 λα+λ α+1 ε 4 Φx s D β f j s Proof. From Theorem 8 of 7] we get qs Φx M 1 M 1 ] λα+λ α D α 1 f j s λα +1 f j+1 s λ α+1 + f j+1 s λα +1 f j s λ α+1 ] ] λα+λ α+1 s D β f j s + D β f j+1 s. 43

9 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 385 Similarly it hol qs f 1 s λα +1 f M s λ α+1 + f M s λα +1 f 1 s λ α+1 ] Φx ] λα+λ α+1 s D β f 1 s + D β f M s. 44 Adding 43 and 44, along with 4, 5 we obtain 42. Next it comes Theorem 8. All as in Theorem 4. Consider the secial case of λ α2 = λ α1 + λ β. Denote Then qs λβ / λβ Tx := A x 2 2λα 1 3λβ/, 45 λ α1 + λ β 1, if 2λ α1 + λ β, ε 5 := 46, if 2λ α1 + λ β. { {M 1 M 1 2λα 1 +λ β f j s λα 1 f j+1 s λα 1 +λ β D β f j s λ β f j s λα 1 +λ β f j+1 s λα 1 + D β f j+1 s ]} λ β f 1 s λα 1 f M s λα 1 +λ β D β f 1 s λ β f 1 s λα 1 +λ β f M s λα 1 D β f M s λ β] } 2 2λα 1 +λ β x M ε 3 Tx s D β f j s ] 2λα 1 +λ β. 47 Proof. Based on Theorem 9 of 7]. The rest as in the roof of Theorem 7. Next we give secial case of the above theorems. Corollary 9 to Theorem 4, λ α2 =, t = qt = 1. It hol x M f j s λα 1 D β f j s λ β δ 1 ϕ 1x x M D β f j s ] ] λα 1 +λ β. 48 In 48, A x λα2 = of ϕ 1 x is given in 7], Corollary 1, equation 123.

10 386 G. A. ANASTASSIOU Corollary 1 to Theorem 4, λ α2 =, t = qt = 1, λ α1 = λ β = 1, = 2. In detail, let α 1 R +, β > α 1 and let f j L 1, x, j = 1,..., M N, x R + {}, have, resectively, L fractional derivatives D β f j in, x], and let D β k f j = for k = 1,...,β] + 1; j = 1,...,M. Then M f j s D β f j s { x β α 1 x 2Γβ α 1 M D β f j s ] } β α 1 2β 2α1 1 Proof. Bared on our Corollary 9 and Corollary 11 of 7], see inequality 13 there. Corollary 11 to Theorem 5, λ α1 =, t = qt = 1. It hol + {{ M 1 f j+1 s λα 2 D β f j s λ β f j s λα 2 D β f j+1 s ] } λ β f M s λα 2 D β f 1 s λ β f 1 s λα 2 D β f M s β] } λ 2 λ β +λα 2 ε 2 ϕ 2 x { x M D β f j s ] } λ β +λα 2. 5 In 5, A x λα1 = of ϕ 2x is given in 7], Corollary 12, equation 137 there. Corollary 12 to Theorem 5, λ α1 =, t = qt = 1, λ α2 = λ β = 1, = 2. In detail, let α 2 R +, β > α 2 and let f j L 1, x, j = 1,..., M N, x R + {}, have, resectively, L fractional derivatives D β f j in, x], and let D β k f j =, for k = 1,...,β] + 1; j = 1,..., M. Then {{ M 1 D α 2 f j+1 s D β f j s + D α 2 f j s D β f j+1 s ]} + f M s D β f 1 s + D α 2 f 1 s D β f M s ]} { 2 x β α 2 } Γβ α 2 D β f j s ] β α 2 2β 2α2 1 Proof. From Corollary 11 and Corollary 13 of 7], esecially equation 146 there.

11 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 387 Corollary 13 to Theorem 6, λ α1 = λ α2 = λ β = 1, = 3, t = qt = 1. It hol M 1 D α 1 f j s f j+1 s D β f j s + D α 2 f j s f j+1 s D β f j+1 s ] + f 1 s f M s D β f 1 s + D α 2 f 1 s f M s D β f M s ]] 2 ϕ 3 x D β f j s ]] Here ϕ 3 := A x, 53 where in this secial case, A x = 4 x 2β α 1 α 2 Γβ α 1 Γβ α 2 33β 3α 1 13β 3α 2 12β α 1 α 2 ] 2/3. 54 Proof. From Theorem 6 and Corollary 14 of 7], see there equation 151 which is here 54. Corollary 14 to Theorem 7, λ α = 1, λ α+1 = 1/2, = 3/2, t = qt = 1. In detail: let β > α 1 +1, α 1 R + and let f j L 1, x, j = 1,...,M N, x R + {} have, resectively, L fractional derivatives D β f j in, x], and let D β k f j =, for k = 1,...,β] + 1; j = 1,..., M. Set Then + {{ M 1 Φ 2 x 3β 3α x := 3β 3α1 2 Γβ α 1 3/2. 55 ] } f j s +1 f j+1 s + f j+1 s +1 f j s f 1 s +1 f M s + f M s +1 f 1 s ] } ] 2Φ x D β f j s 3/2. 56 Proof. Based on Theorem 7 here, and Corollary 15 of 7], see there equation 161 which is here 55.

12 388 G. A. ANASTASSIOU Corollary 15 to Theorem 8, here = 2λ α1 + λ β > 1, t = qt = 1. It hol {{ 1 f j s λα 1 f j+1 s λα 1 +λ β D β f j s λ β f j s λα 1 +λ β f j+1 s λα 1 + D β f j+1 s ] } λ β f 1 s λα 1 f M s λα 1 +λ β D β f 1 s λ β f 1 s λα 1 +λ β f M s λα 1 D β f M s λ β] } 2 Tx D β f j s ] 2λα 1 +λ β. 57 Here Tx in 57 is given by 45 and in detail by Tx of 7], see there Corollary 16 and equations Corollary 16 to Theorem 8, = 4, λ α1 = λ β = 1, t = qt = 1. It hol {{ 1 f j s D α 2 f j+1 s 2 D β f j s + D α 2 f j s 2 f j+1 s D β f j+1 s ] } + f 1 s D α 2 f M s 2 D β f 1 s + D α 2 f 1 s 2 f M s D β f M s ] } 2 Tx D β f j s ] Here in 58 we have that Tx = T x of Corollary 17 of 7], for that see there equations Next we resent the L case. Theorem 17. Let α 1, α 2 R +, β > α 1, α 2 and let f j L 1, x, j = 1,..., M N, x R + {}, have, resectively, L fractional derivatives D β f j in, x], and let D β k f j = for k = 1,...,β] + 1; j = 1,...,M. Consider s, s L, x. Let λ α1, λ α2, λ β. Set { s x βλα 1 α 1λ α1 +βλ α2 α 2 λ α2 } +1 ρx :=. Γβ α λα 1Γβ α λα 2βλ α1 α 1 λ α1 + βλ α2 α 2 λ α2 + 1] 59

13 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 389 Then s { {M 1 f j s λα 1 f j+1 s λα 2 D β f j s λ β f j s λα 2 f j+1 s λα 1 f 1 s λα 2 f M s λα 1 { M { ρx D β f j 2λ α1 +λ β Proof. Based on Theorem 18 of 7]. Similarly we give D β f j+1 s ]} λ β + f 1 s λα 1 f M s λα 2 D β f 1 s λ β D β f M s λ β] } Theorem 18 as in Theorem 17, λ α2 =. It hol s f j s λα 1 D β f j s λ β + } } D β f j 2λ α2. 6 x βλα 1 α 1λ α1 +1 M s D β f j λ α1 +λ β. 61 βλ α1 α 1 λ α1 + 1Γβ α λα 1 Proof. Based on Theorem 19 of 7]. It follows Theorem 19 as in Theorem 17, λ α2 = λ α1 + λ β. It hol {{ 1 s f j s λα 1 f j+1 s λα 1 +λ β D β f j s λ β f j s λα 1 +λ β f j+1 s λα 1 + D β f j+1 s ] } λ β f 1 s λα 1 f M s λα 1 +λ β D β f 1 s λ β f 1 s λα 1 +λ β f M s λα 1 D β f M s λ β] } 2 x 2βλα 1 α 1λ α1 +βλ β α 2 λ α1 α 2 λ β +1 s 2βλ α1 α 1 λ α1 + βλ β α 2 λ α1 α 2 λ β + 1Γβ α λα 1Γβ α λα 1 +λ β M D β f j 2λ α1 +λ β. 62 Proof. By Theorem 2 of 7].

14 39 G. A. ANASTASSIOU We continue with Theorem 2 as in Theorem 17, λ β =, λ α1 = λ α2. It hol {{ 1 ] } s f j s λα 1 f j+1 s λα 1 f j s λα 1 f j+1 s λα 1 + f 1 s λα 1 f M s λα 1 f 1 s λα 1 f M s λα 1] } 2 ρ x M ] D β f j 2λ α1. 63 Here we have ρ x 2βλα 1 α 1λ α1 α 2 λ α1 +1 s x := 2β λ α1 α 1 λ α1 α 2 λ α1 + 1Γβ α λα 1Γβ α λα 1 Proof. Based on Theorem 21 of 7]. Next we give Theorem 21 as in Theorem 17, λ α1 =, λ α2 = λ β. It hol {{ 1 s f j+1 s λα 2 D β f j s λα 2 f j s λα 2 D β f j+1 s ] } λα 2 + f M s λα 2 D β f 1 s λα 2 f 1 s λα 2 D β f M s 2] } λα x βλα 2 α 2λ α2 +1 s 2 βλ α2 α 2 λ α2 + 1Γβ α λα 2 Proof. Based on Theorem 22 of 7]. Some secial cases follow. M D β f j 2λ α Corollary 22 to Theorem 2, all as in Theorem 17, λ β =, λ α1 = λ α2, α 2 = α It hol + s {{ M 1 f j s λα 1 D α 1 +1 f j+1 s λα 1 + D α 1 +1 f j s ] } λα 1 f j+1 s λα 1 f 1 s λα 1 D α 1 +1 f M s λα 1 + D α 1 +1 f 1 s λα 1 f M s 1] } λα x 2βλα 1 2α 1λ α1 λ α1 +1 s 2 2βλ α1 2α 1 λ α1 λ α1 + 1β α 1 λα 1Γβ α 1 2λα 1 M D β f j 2λ α1 ]. 66

15 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 391 Proof. Based on Corollary 23 of 7]. Corollary 23 to Corollary 22. In detail: Let α 1 R +, β > α and let f j L 1, x, j = 1,..., M N, x R + {}, have, resectively, L fractional derivatives D β f j in, x], and let D β k f j =, for k = 1,...,β] + 1; j = 1,...,M. Then {{ 1 D α 1 f j s D α1+1 f j+1 s + D α1+1 f j s f j+1 s ]} + f 1 s +1 f M s + +1 f 1 s f M s ]} x 2 β α 1 β α 1 2 Γβ α 1 2 Proof. Based on Corollary 24 of 7]. M D β f j Corollary 24 to Corollary 23. It hol x M f j s D α1+1 f j s x 2 β α 1 M D β f 2β α 1 2 Γβ α 1 2 j Proof. Based on inequality 27 of 7]. We resent our first alication. 3. APPLICATIONS Theorem 25. Let α i R +, β > α i, i = 1,..., r N, and let f j L 1, x, j = 1,...,M N; x R + {}, have, resectively, L fractional derivatives D β f j in, x], and let D β k f j = α kj R for k = 1,..., β] + 1; j = 1,...,M. Furthermore for j = 1,..., M, we have that D β f j s = F j s, {D α i f j s} r,m,, 69 all s, x]. Here F j s, z 1, z 2,..., z M are continuous for z 1, z 2,..., z M R r M, bounded for s, x], and satisfy the Lischitz condition F j t; x 11, x 12,...,x 1r ; x 21, x 22,...,x 2r ; x 31,...,x 3r ;...,1,..., r F j t, x 11, x 12,...,x 1r; x 21, x 22,...,x 2r; x 31,...,x 3r;...,x M1,..., x Mr r M q l,i,j t x li x li, 7

16 392 G. A. ANASTASSIOU j = 1, 2,..., M, where all q l,i,j s are bounded on, x], 1 i r, l = 1,...,M. Call Assume here that φ x := W W := max { q l,i,j, l, j = 1, 2,..., M; i = 1,...,r} M 1 r 2 x β α i Γβ α i < β α i 2β 2αi 1 Then, if the system 69 has two airs of solutions f 1, f 2,...,f M and f 1, f 2,...,f M, we rove that f j = fj, j = 1, 2,..., M; that is we have uniqueness of solution for the fractional system 69. Proof. Assume that there are two M-tules of solutions f 1,...,f M and, f 1,...,f M satisfying the system 69. Set g j := f j f j, j = 1, 2,...,M. Then Dβ k g j =, k = 1,...,β] + 1; j = 1,...,M. It hol D β g j s = F j s, {D α i f j s} r,m, F j s, { D α i fj s} r,m. 73, Hence by 7 we have D β g j s r M g l,i,j t D α i g l s. 74 Thus The last imlies r M D β g j s W D α i g l s. 75 r M D β g j s 2 W D β g j s D α i g l s, 76 and M D β g j s 2 W r M M D β g j s D α i g l s. 77 Integrating 77 we observe I := D β g j s 2 { r M M W D β g j s D α i g l s }. 78

17 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 393 That is { r I W D α i g λ s D β g λ s + τ,m {1,...,M} τ m λ=1 D α i g m s D β g τ s + D α i g τ s D β g m s ] }. 79 Using Corollary 1 from here, and Corollary 13 of 7], we find { r x β α i I I W 2 Γβ α i β α i 2β 2αi 1 x β αi M 1I + 2 Γβ αi β α i 2β 2αi 1] }, 8 i.e we got that I φ x I. 81 If I then φ x 1, a contradiction by assumtion that φ x < 1, see 72. Therefore I =, imlying that M D β g j s 2 =, a.e. in, x], 82 i.e. and D β g j s 2 =, a.e. in, x], j = 1,...,M, 83 D β g j s =, a.e. in, x], j = 1,...,M. 84 By D β k g j =, k = 1,..., β] + 1; j = 1,..., M, and Lemma 3, aly 3 for α =, we find g j s, all s, x], all j = 1,..., M. The last imlies f j = fj, j = 1,...,M, over, x], that is roving the uniqueness of solution to the initial value roblem of this theorem. It follows another related alication. Theorem 26. Let α i R +, β > α i, i = 1,..., r N and let f j L 1, x, j = 1,...,M N; x R + {}, have, resectively, fractional derivatives D β f j in, x], that are absolutely continuous on, x], and let D β k f j = for k = 1,...,β] + 1; j = 1,...,M. Furthermore D β f j = A j R. Furthermore for s x we have holding the system of fractional differential equations D β f j s = Fj s, {D α i f j s} r,m,, { D β f j s } M, 85

18 394 G. A. ANASTASSIOU for j = 1,..., M. Here F j is Lebesgue measurable on, x] R r+1 M such that F j t; x 11, x 12,...,x 1r, x 1,r+1 ; x 21, x 22,..., x 2r, x 2,r+1 ; r M...1, 2,...,,r+1 q l,i,j t x li, 86 where q l,i,j, 1 i r; l, j = 1,...,M, are bounded on, x]. Call Also we set s x and Then W := max{ q l,i,j ; l, j = 1,...,M; i = 1,..., r}. 87 θs := ρ := M D β f λ s 2, 88 λ=1 M λ=1 A 2 λ, 89 Qs := W1 + r s β α i 2M 1 Γβ α i, 9 β α i 2β 2αi 1 Consequently we get χs := { ρ 1 + Qs e R s Qtdt s e R t Qydy dt]} 1/2. 91 θs χs, s x. 92 D β f j s χs, 93 s f j s 1 s t β 1 χtdt, 94 Γβ all s x, j = 1,...,M. Also it hol D α i f j s 1 Γβ α i all s x, j = 1,...,M, i = 1,..., r. s s t β α i 1 χtdt, 95 Proof. We observe that D β f j sd β f j s = D β f j s F j s, {D α i f j s} r,m,, { D β f j s } M, 96 j = 1,...,M, all s x. We then integrate 96, y x. y D β f j sd β f j s y = D β f j s F j s, {D α i f j s} r,m,, { D β f j s } M, 97

19 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 395 Hence we obtain D β f j s 2 y 86 y 2 D β f j s F j 87 y r M ] W D β f j s D α i f l s r M y = W D α i f l s D β f j s. 98 Thus we have for j = 1,..., M that D β f j y 2 A 2 j + 2W r M y D α i f l s D β f j s. 99 Consequently it hol r M M y s θy ρ + 2W D α i f l s D β f j 1 { r { y M = ρ + 2W D α i f λ t D β f λ t dt + τ,m {1,...,M} τ m λ=1 y D α i f m t D β f τ t + D α i f τ t D β f m t } } dt. 11 Using Corollary 1 from here, and Corollary 13 of 7], we find { r { y β α i y θy ρ + 2 W 2 Γβ α i θtdt β α i 2β 2αi 1 y β α i y + 2 Γβ αi M 1 θtdt} }. 12 β α i 2β 2αi 1 Hence we have y θy ρ + Qy θtdt, all y x. 13 Here ρ, Qy, Q =, θy, all y x. As in the roof of Theorem 27 of 7], see also 5], we get 92, 93. Using Lemma 3, see 3, for α =, and 93 we obtain 94. Using again 3 and 93 we get 95. Finally we give a secialized alication. Theorem 27. Let α i R +, β > α i, i = 1,..., r N and f j L 1, x, j = 1,...,M N; x R + {}, have, resectively, fractional derivatives D β f j in, x], that are absolutely continuous on, x], and let D β µ f j = for µ = 1,...,β] + 1;

20 396 G. A. ANASTASSIOU j = 1,...,M. Furthermore D β f j = A j R. Furthermore for s x we have holding the system of fractional differential equations D β f j s = Fj s, {D α i f j s} r,m,, { D β f j s } M, 14 for j = 1,..., M. For fixed i {1,..., r} we assume that α i +1 = α i + 1, where α i, α i +1 {α 1,...,α r }. Call k := α i, γ := α i + 1, i.e. γ = k + 1. Here F j is Lebesgue measurable on, x] R r+1 M such that F j t, x 11, x 12,...,x 1r, x 1,r+1 ; x 21, x 22,...,x 2r, x 2,r+1 ; x 31, x 32,..., x 3r, x 3,r+1 ;...;1, 2,..., r,,r+1 { { M 1 } q l,1,j t x li x l+1,i +1 + q l,2,j t x l+1,i x l,i +1 } + q M,1,j t x 1i,i +1 + q M,2,j t i x 1,i +1, 15 where all q l,1,j, q l,2,j, are bounded over, x]. Put W := max { q l,1,j, q l,2,j } M l,. 16 Also set all s x, and θs := M D β f j s, s x, 17 ρ := M A j, Φ 2 s := 3β 3k 2 s 3β 3k 1 2 Γβ k 3/2, 18 Qs := 2MWΦ s, s x, 19 σ := Qs = 4MW x 3β 3k 1 2 3β 3k 2 Γβ k 3/2. 11 We assume that Call Then in articular we have xσ ρ < ] 4ρ 3/2 s σρ 2 s 2 ϕs := ρ + Qs 2 σ, all s x. 112 ρ s 2 θs ϕs, all s x, 113 D β f j s ϕs, j = 1,..., M, for all s x. 114

21 RIEMANN-LIOUVILLE FRACTIONAL OPIAL INEQUALITIES 397 Furthermore we get f j s 1 Γβ and, D α i 1 f j s Γβ α i j = 1,...,M; i = 1,...,r, all s x. s s t β 1 ϕtdt, 115 s s t β α i 1 ϕtdt, 116 Proof. Notice here that W > and σ >. Clearly, here D β f j are L fractional derivatives in, x] For s y x, by 14 we get that y y D β f j s = F j s, {D α i f j s} r,m,, { D β f j s } M. 117 That is Then we observe that D β f j y A j + + W y y D β f j y = A j + y F j s,... A j F j s, { { M 1 D α i fl s D α i +1 f l+1 s + D α i fl+1 s } D α i +1 f l s + D α i f1 s D α i +1 f M s + D α i fm s } D α i +1 f 1 s. 119 That is D β f j y A j + W y { {M 1 + D k f l+1 s } D k+1 f l s D k f l s D k+1 f l+1 s + D k f 1 s D k+1 f M s + D k f M s D k+1 f 1 s }. 12 By Corollary 14 we obtain s M D β f j s A j + 2Wφ s D β f l t 3/2 dt, 121 j = 1,...,M, all s x. Therefore by adding all of inequalities 121 we get s M θs ρ + 2MWφ s D β f l t dt 3/2, ρ + 2MWφ s s M 3/2 D β f l t dt, 123

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