Dynamic Systems and Applications ( 383-394 IMPROVEMENTS OF COMPOSITION RULE FOR THE CANAVATI FRACTIONAL DERIVATIVES AND APPLICATIONS TO OPIAL-TYPE INEQUALITIES M ANDRIĆ, J PEČARIĆ, AND I PERIĆ Faculty of Civil Engineering and Architecture, University of Split, Split, Croatia Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia Faculty of Food Technology and Biotechnology, University of Zagreb, Zagreb, Croatia ABSTRACT This paper gives improvements of a composition rule for the Canavati fractional derivatives and presents improvements and weighted versions of Opial-type inequalities involving the Canavati fractional derivatives AMS (MOS Subject Classification 6A33, 6D5 INTRODUCTION AND PRELIMINARIES This work, motivated by the monograph of Anastassiou [], presents improvements of a composition rule for the Canavati fractional derivatives which relax restrictions on the orders of fractional derivatives in the composition rule See Section Also, we will give versions of Opial-type inequalities known for Riemann-Liouville fractional derivatives, and we will present improvements of some Opial-type inequalities for the Canavati fractional derivatives which have the general form b a ( b w (t (D µ f(t (D µ f(t dt K a w (t (D ν f(t p p dt where w and w are weight functions, and D γ f denotes the Canavati fractional derivative of f of order γ First, following [5], we survey some facts about fractional derivatives Let g C([, ] Let ν >, n = [ν], [ ] the integral part, and ν = ν n, ν < Define the Riemann-Liouville fractional integral of g of order ν by (J ν g(x = Γ(ν (x t ν g(t dt, x [, ], The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants 83--37 (first author, 7-7889-888 (second author and 58-7889-5 (third author Received January 4, 56-76 $5 c Dynamic Publishers, Inc,
384 M ANDRIĆ, J PEČARIĆ, AND I PERIĆ where Γ is the gamma function Γ(ν = e t t ν dt Define the subspace C ν ([, ] of C n ([, ] as C ν ([, ] = {g C n ([, ] : J ν g (n C ([, ]} For g C ν ([, ] the Canavati ν-fractional derivative of g is defined by D ν g = DJ ν g (n, where D = d/dx Since we will compare our results with ones from [] we will now give a more general definition of the Riemann-Liouville fractional integral Let [a, b] R and x, x [a, b] such that x x where x is fixed For f C([a, b] the generalized Riemann-Liouville fractional integral of f of order ν is given by (J x ν f(x = Γ(ν x (x t ν f(t dt, x [x, b] Analogously, define the subspace C ν x ([a, b] of C n ([a, b] as C ν x ([a, b] = {f C n ([a, b] : J x ν f(n C ([x, b]} For f C ν x ([a, b] the generalized Canvati ν-fractional derivative of f over [x, b] is given by Notice that exists for f C ν x ([a, b] (J x ν f(n (x = D ν x f = DJ x ν f(n x (x t ν f (n (t dt Γ( ν x Lemma ([5], [] (i D n x f = f (n for n N (ii Let f C ν x ([a, b], ν > and f (i (x =, i =,,, n, n = [ν] Then That is, ( f(x = Γ(ν for all x [a, b] with x x f(x = (J x ν Dν x f(x x (x t ν (D ν x f(t dt, Lemma ([5], [9] Let f C([a, b], µ, ν > Then J x µ (Jx ν f = Jx µ+ν(f in [] The proof of the composition rule contained in the following lemma can be found Lemma 3 Let γ, ν be such that ν γ Let f C ν x ([a, b] be such that f (i (x =, i =,,, n Then ( (D γ x f(x = Γ(ν γ x (x t ν γ (D ν x f(t dt, hence (D γ x f(x = (J x ν γd ν x f(x and is continuous in x on [x, b]
THE CANAVATI FRACTIONAL DERIVATIVES AND OPIAL-TYPE INEQUALITIES 385 Our first goal is to improve this composition rule for the Canavati fractional derivatives Using the Laplace transform we prove that one doesn t need all vanishing derivatives of the function f at point x and that condition ν γ can be relaxed This will be used in all presented Opial-type inequalities involving the Canavati fractional derivatives Next, we give two estimations in an Opial-type inequality motivated by Pang and Agarwal s extension [8, Theorem ] of an inequality due to Fink [6] given for classical derivatives (here Theorem 3 and Theorem 3 Comparison of the obtained constants is also given Also, applying differently Hölder s inequality we obtain an improvement of the next weighted Opial-type inequality for the Canavati fractional derivatives (see [] Theorem 4 ([, Theorem 35] Let µ, µ, ν be such that ν µ, ν µ and f C ν x ([a, b] with f (i (x =, i =,,, n, n := [ν] Here x, x [a, b], with x x Let w be a nonnegative continuous function on [a, b] Denote Then where K is given by ( Q := w(τ dτ x ( w(τ (D µ x f(τ (D µ x f(τ dτ K ((Dx ν f(t dt, x x K = Q (x x ν µ µ 3 6Γ(ν µ Γ(ν µ (ν µ 5 6 6 (ν µ 5 6 6 (4ν µ µ 7 3 Finally, let us emphasize in the next remark that generalized Riemann-Liouville fractional integral and generalized Canavati fractional derivative can, by simple substitutions, be reduced to the non-generalized case Remark 5 Note that generalized Riemann-Liouville fractional integral and generalized Canavati fractional derivative depend only on interval [x, b] It is easy to see that using transformation L : [, ] [x, b], L(t = (b x t + x (or inverse transformation, the generalized Rieman-Liouville fractional integral and the generalized Canavati fractional derivative can be expressed using non-generalized cases For example (J x ν f(x = (b x ν (J ν f ( x x b x where f(t = (b x t + x Therefore, all our results stated for non-generalized fractional derivative have an interpretation for the generalized fractional derivative and vice versa,
386 M ANDRIĆ, J PEČARIĆ, AND I PERIĆ MAIN RESULT We relax some conditions in composition rule for the Canavati fractional derivative given in Lemma 3 Theorem Let ν > γ >, n = [ν], m = [γ] Let f C ν ([, ] be such that f (i ( = for i = m, m +,,n Then (i ( f C γ ([, ] (ii ( (D γ f(x = for every x [, ] Γ(ν γ (x t ν γ (D ν f(t dt Proof Set ν = ν n, γ = γ m To prove (i, suppose first m = n (which gives ν > γ We have (3 J γ f (n = J ν γ J ν f (n = J ν γ+ ( DJ ν f (n, where the last equality in (3 follows using integration by parts and J ν f (n ( = Since DJ ν f (n C ([, ] and ν γ + > it follows J γ f (n C ([, ] by [5, Proposition ] If m < n, then using f (m ( = = f (n ( = and integration by parts it easily follows that f (m = J n m f (n We have J γ f (m = J γ J n m f (n = J γ+n m f (n The result again follows from [5, Proposition ] since γ + n m > and f (n C ([, ] We will prove ( using the Laplace transform Set g = f (m Now ( can be written as (4 (DJ γ g(x = ( J ν γ DJ ν g (n m (x = g (n m ( = where x [, ] and g( = g ( = = g (n m ( = Define auxiliary function h : [, R with (5 h(x = { n m k= g(x, x [, ] g (k ( k! (x k, x Obviously h C n m ([,, h( = h ( = = h (n m ( = Also h has polynomial growth at, so the Laplace transform of h exists The identity (4 will follow if we prove that d Γ( γ dx (x t γ h(t dt
(6 THE CANAVATI FRACTIONAL DERIVATIVES AND OPIAL-TYPE INEQUALITIES 387 t = Γ(ν γγ( ν (x t ν γ d dt (t y ν h (n m (y dy dt holds for every x Using standard properties of the Laplace transform we have ( d x L (x t γ h(t dt (s Γ( γ dx ( x = Γ( γ s L (x t γ h(t dt (s s (7 = Γ( γ L ( x γ (sl(h(s = s γ L(h(s On the other hand we have ( L Γ( νγ(ν γ (8 = = (x t ν γ d dt Γ( νγ(ν γ L ( x ν γ (sl s γ ν Γ( ν s L ( x ν (sl ( h (n m (s t ( d dt = s γ ν s s ν sn m L(h(s = s γ L(h(s (t y ν h (n m (y dy dt (s t (t y ν h (n m (y dy dt (s Using (7 and (8 it follows that both sides of (6 have the same Laplace transform and since both sides are continuous function, we conclude that equality holds in (6 for every x This completes the proof of the theorem 3 OPIAL-TYPE INEQUALITIES The first theorem is an Opial-type inequality due to Fink who proved it for ordinary derivatives [6] Although our proof is similar to the one given in [8, Theorem ], we sketch a proof for the reader s convenience In [, Section 5] the result is given for Riemann Liouville s fractional derivatives without a discussion of the best possible cases Using different technique we prove another estimation of the same type of inequality and compare obtained estimations Theorem 3 Let /p + /q = with p, q > Let ν > µ µ +, n = [ν] and m = [µ ] Let f C ν ([, ] be such that f (i ( = for i = m, m +,,n and let x [, ] Then (3 where A is given by (D µ f(τ (D µ f(τ dτ A x ν µ µ + q ( (D ν f(τ p p dτ, A = p Γ(ν µ Γ(ν µ + [q(ν µ + ] q [q(ν µ µ + ] q
388 M ANDRIĆ, J PEČARIĆ, AND I PERIĆ Inequality (3 is sharp in the case µ = µ + and the equality is attained for (3 f(s = Γ(ν s (s t ν (x t q p (ν µ dt Proof Set α j = ν µ j, j =, Notice that α α Let t s x Then (see [6] (33 [ (τ t α + (τ sα + + (τ sα + (τ ] (x t α (x s α + tα + dτ α + In the following calculation we abbreviate c := [Γ(ν µ Γ(ν µ ], c := [Γ(ν µ + Γ(ν µ ], For τ [, x], by (, we have c 3 := q(ν µ +, ǫ := ν µ µ + q (D µ j f(τ = Γ(ν µ j (τ t α i + (D ν f(t dt Using this representation, the auxiliary inequality (33, and Hölder s inequality, we obtain (34 (35 (36 (D µ f(τ (D µ f(τ dτ ( c = c (D ν f(t ( (D ν f(t (τ t α + dt { t (D ν f(s (D ν f(s (τ s α + ds dτ ( } [(τ t α + (τ sα + + (τ sα + (τ tα + ] dτ ds dt c c = c c /q 3 c c /q 3 ( (D ν f(t (D ν f(s (x t α (x s α+ ds dt t ( ( (D ν f(t (x t α (D ν f(s p p x ds (x s q(α+ ds t ( (D ν f(t (x t ǫ (D ν f(s p ds ( ( (D ν f(t p (D ν f(s p ds { ( = c c /q 3 (ǫq + /q x (ǫq+/q x t t t p dt ( p x dt (x t ǫq q dt } p (D ν f(t p dt q dt
THE CANAVATI FRACTIONAL DERIVATIVES AND OPIAL-TYPE INEQUALITIES 389 It is obvious that in the case α = α + we have equality in (33 Using equality condition for Hölder s inequality we have also equality in (35 for the function f given in (3 since obviously (D ν f(s p = (x s q(α + Straightforward calculation shows that for this function equality holds also in (36 Equality in (34 in this case is obvious Using a different technique we obtain yet another constant for the previous inequality: Theorem 3 Let /p + /q = with p, q > Let µ j >, ν > µ j + q, n = [ν] and m = min{[µ ], [µ ]} Let f C ν ([, ] be such that f (i ( = for i = m, m +,,n and let x [, ] Then (D µ f(τ (D µ f(τ dτ A x ν µ µ + q where A is given by A = q ( (D ν f(τ p p dτ, [q(ν µ µ + ] j= Γ(ν µ j [q(ν µ j + ] q Proof Set α j = ν µ j, j =, For τ [, x], by (, we have (D µ j f(τ = By Hölder s inequality we have (D µ j f(τ dτ j= j= Γ(α j + Γ(α j + j= j= Γ(α j + (qα j + q = A x ν µ µ + q ( τ (τ t α j (D ν f(t dt ( τ ( (τ t qα q τ j dt (D ν f(t p p dt dτ ( (D ν f(t p p dt (D ν f(t p p x dt τ α +α + q dτ Remark 33 The constants A and A from the two previous theorems are in general not comparable, but there are cases when we can do that Notice A = (ν µ + q q (ν µ + q q (ν µ µ + p q A p (ν µ (ν µ + q q
39 M ANDRIĆ, J PEČARIĆ, AND I PERIĆ We want to find cases when A < A Set ν µ = d, µ µ = d Then A < A is equivalent to (37 (d + d + q q (d + d + q q < (d + q q q d (d + q q If d is big enough, then the left side of (37 tends to zero, while the right side is independent of d Therefore, in this case A < A Let d =, that is µ = µ + (see the discussion of sharpness in Theorem 3 Then the reverse inequality in (37 is equivalent to ( qd + qd q + > + q, qd which is equivalent to inequality ( /q qd + q < qd, qd + + qd and this is a simple consequence of the Bernoulli inequality This is in accordance with Theorem 3 An illustrative case is p = q = In this case (37 is equivalent to That is, A < A is equivalent to (d d + ( d 4 d + d d + d > d > d = d + 4d + 4d 4 + 8d3 6d + 4d + (d Notice that lim d d = and lim d d = Roughly speaking, for d or d 5 estimation in Theorem 3 is better than estimation in Theorem 3, otherwise the opposite conclusion holds For example, for d = and d > + 73 6867 or for d = and d > 5+ 5 4 374 estimation in Theorem 3 is better than estimation in Theorem 3 We also consider a weighted Opial-type inequality for the generalized fractional derivative given in [] (here Theorem 4 Set C = Q (x x ν µ µ Γ(ν µ Γ(ν µ and recall that the constant in inequality from Theorem 4 is (38 K = C 3 6(ν µ 5 6 6 (ν µ 5 6 6 (4ν µ µ 7 3 Several applications of Hölder s inequality on different factors with different indices will improve this constant
THE CANAVATI FRACTIONAL DERIVATIVES AND OPIAL-TYPE INEQUALITIES 39 Theorem 34 Let µ j >, ν > µ j + 5 6, n = [ν] and m = min{[µ ], [µ ]} Let f C ν x ([a, b] be such that f (i (x = for i = m, m+,,n Here x, x [a, b] R, x x where x is fixed If w C([a, b] is a nonnegative function on [a, b], then w(τ (D µ x f(τ (D µ x f(τ dτ K (Dx ν f(t dt, x x where K is given by (39 K = C 3 6(ν µ 5 6 6 (ν µ 5 6 6 (4ν µ µ Proof Set α j = ν µ j, j =, For τ [x, x], by Remark 5 and (, we have (D µ j x f(τ = By Hölder s inequality we have w(τ (D µ j x f(τ dτ x ( j= x w(τ dτ τ (τ t α j (Dx ν Γ(α j + f(t dt x ( { Q x j= Γ(α j + x x j= ( τ j= Again, for p = 3 and q = 3 τ ( τ (τ t α j (Dx ν f(t dt dt x x and for p = 4 and q = 4 3 τ Therefore w(τ x we have x (τ t 3 α j (D ν x f(t 3 dt ( τ j= (D µ j x f(τ dτ Q j= Γ(α j + { x (τ x 4 3 j= Q j= Γ(α j + (6α j + 6 (D µ j x f(τ dτ x (τ t α j (D ν x f(t dt 3 ( τ x (τ t 6α j dt ( τ dτ} x (τ t 3 α j (D ν x f(t 3 dt 4 ( τ x (D ν x f(t dt ( } (τ x 6α j+ 3 dτ 6α j + x (D ν x f(t dt 3, 3 4 ( (Dx ν f(t dt (τ x α +α + dτ x x
39 M ANDRIĆ, J PEČARIĆ, AND I PERIĆ = K x (D ν x f(t dt Theorem 35 Let µ j >, ν > µ j +, n = [ν] and m = min{[µ ], [µ ]} Let f C ν x ([a, b] be such that f (i (x = for i = m, m+,,n Here x, x [a, b] R, x x where x is fixed If w C([a, b] is a nonnegative function on [a, b], then w(τ (D µ x f(τ (D µ x f(τ dτ K 3 (Dx ν f(t dt, x x where K 3 is given by (3 K 3 = C (ν µ (ν µ (4ν µ µ Proof Write α j = ν µ j, j =, For τ [x, x], by Remark 5 and (, we have (D µ j x f(τ = By Hölder s inequalities we have τ (τ t α j (Dx ν Γ(α j + f(t dt x w(τ (D µ j x f(τ dτ x j= ( τ w(τ (τ t α j (Dx ν x Γ(α j= j + f(t dt dτ x x ( τ ( τ j= Γ(α w(τ (Dx ν j + f(t dt (τ t α j dt dτ x x j= x ( x j= Γ(α (D ν j + (α j + x f(t dt w(τ(τ x α +α + dτ x x ( x j= Γ(α (D j + (α j + x ν f(t dt x ( ( w(τ x dτ (τ x α +α + dτ x x x = K 3 (Dx ν f(t dt x Remark 36 Comparing three constants K i, that is (38, (39 and (3, we conclude (3 K 3 < K < K
THE CANAVATI FRACTIONAL DERIVATIVES AND OPIAL-TYPE INEQUALITIES 393 The second inequality is obvious The first inequality is equivalent to 3 6ν 6µ 5 ν µ 3 6ν 6µ 5 ν µ <, for ν µ j > 5/6, j =,, which holds since 3 3x 5 x < / for x > 5/3 Finally, we give a very general Opial-type inequality involving the Canavati fractional derivatives which is analogous to [8, Theorem 3] for ordinary derivatives The proof is omitted (see [4] Theorem 37 Let l N, µ j >, ν > µ j, n = [ν] and m = min{[µ j ] : j =,, l} Let f C ν ([, ] be such that f (i ( = for i = m, m +,, n Let w, w be continuous positive weight functions on [, x] where x [, ] Let r j >, r = l j= r j, s k > and s k + = for k =, Let p R be such that p > s s and k let σ = /s /p Suppose also ν > µ j + σ, j =,,l Denote ( ( Q = w (τ s s x r dτ, P = w (τ s s p dτ Then w (τ l (D µ j f(τ r j dτ j= ( P A P Q A xρ+ ( r p w (ττ ρ dτ w (t (D ν f(t p dt ( r s (ρs + s w (t (D ν f(t p dt where α j = ν µ j, ρ = Σ l j=α j r j + σr and A is given by A = σ rσ l j= Γ(α j + r j (αj + σ r jσ p, REFERENCES [] R P Agarwal, P Y H Pang, Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht, Boston, London, 995 [] G A Anastassiou, Fractional Differentiation Inequalities, Springer, 9 [3] G A Anastassiou, J J Koliha, J Pečarić, Opial inequalities for fractional derivatives, Dynamic Systems and Appl, ;395 46, [4] G A Anastassiou, J J Koliha, J Pečarić, Opial type L p -inequalities for fractional derivatives, InterJ Math & Math Sci, 3(;85 95, [5] J A Canavati, The Riemann-Liouville Integral, Nieuw Archief Voor Wiskunde, 5(;53 75, 987 [6] A M Fink, On Opial s inequality for f (n, Proc Amer Math Soc, 5;77 8, 99 [7] A A Kilbas, H M Srivastava, J J Trujillo, Theory and Applications of Fractional Deifferential Equations, Mathematics Studies 4, North Holland, 6
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