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Balanced fractional opial inequalities
Chaos, Solitons and Fractals 42 (2009) 1523–1528
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Balanced fractional opial inequalities George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 11 March 2009
Here we present Lp , p > 1, fractional Opial type inequalities subject to high order boundary conditions. They involve the right and left Caputo, Riemann–Liouville fractional derivatives. These derivatives are blended together into the balanced Caputo, Riemann–Liouville, respectively, fractional derivative. This balanced fractional derivative is introduced here for the first time. We give applications to a special case. Ó 2009 Elsevier Ltd. All rights reserved.
1. Preliminaries This article is inspired by the famous theorem of Opial [11], 1960, which follows Theorem 1. Let xðtÞ 2 C 1 ð½0; hÞ be such that xð0Þ ¼ xðhÞ ¼ 0, and xðtÞ > 0 in ð0; hÞ. Then
Z
h
jxðtÞx0 ðt Þjdt 6
0
In (1), the constant
( xðtÞ ¼
h 4
h 4
Z
h
2
ðx0 ðtÞÞ dt:
ð1Þ
0
is the best possible. Inequality (1) holds as equality for the optimal function
ct;
0 6 t 6 h=2;
c ðh t Þ
h 2
6 t 6 h;
where c > 0 is an arbitrary constant. To prove easier Theorem 1, Beesack [5] proved the following well-known Opial type inequality which is used very commonly. This is another inspiration to our work. Theorem 2. Let xðtÞ be absolutely continuous in [0, a], and xð0Þ ¼ 0. Then
Z 0
a
jxðtÞx0 ðt Þjdt 6
a 2
Z
a
2
ðx0 ðtÞÞ dt:
ð2Þ
0
Inequality (2) is sharp, it is attained by xðtÞ ¼ ct, c > 0 is an arbitrary constant. Opial type inequalities are used a lot in proving uniqueness of solutions to differential equations, also to give upper bounds to their solutions. By themselves have made a great subject of intensive research and there exists a great literature about them. Typical and great sources on them are the monographs [1,2]. We need (see also [3,7–9,12]). E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.047
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G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
Definition 3. Let f 2 AC m ð½a; bÞ (space of functions from ½a; b into R with m 1 derivative absolutely continuous function on ½a; b), m 2 N, where m ¼ dae, a > 0 (de the ceiling of the number). We define the right Caputo fractional derivative of order a > 0, by
Dab f ðxÞ ¼
Z
ð1Þm Cðm aÞ
b
ðf xÞma1 f ðmÞ ðfÞdf:
ð3Þ
x
We set D0b f ðxÞ ¼ f ðxÞ, 8x 2 ½a; b. Note 4. Let f 2 AC m ð½a; bÞ, m ¼ dae, with a > 0, then f ðm1Þ 2 ACð½a; bÞ, which implies that f ðmÞ exists a.e. on ½a; b and that f ðmÞ 2 L1 ð½a; bÞ. Consequently if f 2 AC m ð½a; bÞ, then Dab f ðxÞ exists a.e. on ½a; b and Dab f 2 L1 ð½a; bÞ, see [3]. Observe that when a ¼ m 2 N, then m ðmÞ Dm ðxÞ; 8x 2 ½a; b: b f ðxÞ ¼ ð1Þ f
ð4Þ
We continue with the right Caputo fractional Taylor formula with integral remainder, see [3]. Theorem 5. Let f 2 AC m ð½a; bÞ, x 2 ½a; b, a > 0, m ¼ dae. Then
f ð xÞ ¼
Z b m1 ðkÞ X f ðbÞ 1 k ðx bÞ þ ðf xÞa1 Dab f ðfÞdf: k! CðaÞ x k¼0
ð5Þ
We need also (see [6], p. 38). Definition 6. Let f 2 AC m ð½a; bÞ; m 2 N, where m ¼ dae, a > 0. We define the left Caputo fractional derivative of order a > 0, by
Daa f ðxÞ ¼
1 Cðm aÞ
Z
x
ðx tÞma1 f ðmÞ ðt Þdt;
ð6Þ
a
8x 2 ½a; b. We set D0a f ðxÞ ¼ f ðxÞ, 8x 2 ½a; b. Again here Daa f exists a.e. on ½a; b and Daa f 2 L1 ð½a; bÞ, see [6], p. 13, 37 and 38. When a ¼ m 2 N, then ðm Þ Dm ðxÞ; 8x 2 ½a; b: a f ðxÞ ¼ f
ð7Þ
We continue with the left Caputo fractional Taylor formula with integral remainder, see [6], p. 40. Theorem 7. Let f 2 AC m ð½a; bÞ, m 2 N, where m ¼ dae, a > 0, x 2 ½a; b. Then
f ð xÞ ¼
Z x m1 ðkÞ X f ðaÞ 1 ðx sÞa1 Daa f ðsÞds: ðx aÞk þ k! C ð a Þ a k¼0
ð8Þ
Above C is the gamma function,
CðaÞ ¼
Z
1
et t a1 dt; a > 0:
0
We introduce the following new balanced Caputo fractional derivative. Definition 8. Let f 2 AC m ð½a; bÞ; m 2 N, m ¼ dae, a > 0, x 2 ½a; b. We define
( a
D f ðxÞ :¼
6 x 6 b; Dab f ðxÞ; for aþb 2 Daa f ðxÞ;
fora 6 x < aþb : 2
ð9Þ
In this article we establish Lp , p > 1, Opial type inequalities involving the balanced Caputo fractional derivative subject to high order boundary conditions, more precisely by assuming that
f ðkÞ ðaÞ ¼ f ðkÞ ðbÞ ¼ 0;
k ¼ 0; 1; . . . ; m 1:
We extend our results to Riemann–Liouville fractional derivatives.
2. Results We present our main result. Theorem 9. Let f 2 AC m ð½a; bÞ; m 2 N, m ¼ dae, a > 0. Assume
ð10Þ
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G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
f ðkÞ ðaÞ ¼ f ðkÞ ðbÞ ¼ 0; 1 1 p; q > 1 : þ ¼ 1; p q
k ¼ 0; 1; . . . ; m 1; 1 and a > : q
(i) Case of 1 < q 6 2. Then
Z
b
jf ðxÞjDa f ðxÞdx 6
a
pða1Þþ2 p
2ðaþpÞ ðb aÞð 1
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
b
a D f ðxÞq dx
!2=q :
ð11Þ
:
ð12Þ
a
(ii) Case of q > 2. Then
Z
b
jf ðxÞjDa f ðxÞdx 6
a
pða1Þþ2 p
2ðaþqÞ ðb aÞð 1
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
(iii) When p ¼ q ¼ 2, a > 12, then
Z
b
jf ðxÞjDa f ðxÞdx 6
a
2ðaþ2Þ ðb aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðaÞ 2að2a 1Þ 1
a
Z
b
a
2
b
a D f ðxÞq dx
!2=q
a
!
D f ðxÞ dx :
ð13Þ
a
Remark 10. Let us say that a ¼ 1, then by (13) we obtain
Z a
b
ðb aÞ jf ðxÞjjf ðxÞjdx 6 4 0
Z
!
b 0
2
ðf ðxÞÞ dx ;
ð14Þ
a
that is reproving and recovering Opial’s inequality (1), see [11], see also Olech’s result [10]. Proof of Theorem 9. Let x 2 ½a; b. We have by assumption f ðkÞ ðaÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 and Theorem 7 that
f ðxÞ ¼
1 CðaÞ
Z
x
a
ðx sÞa1 Daa f ðsÞds;
ð15Þ
and by assumption f ðkÞ ðbÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 and Theorem 5 that
f ðxÞ ¼
1 CðaÞ
Z
b
x
ðs xÞa1 Dab f ðsÞds:
ð16Þ
Using Hölder’s inequality on (15) we get
j f ð xÞ j 6
1 CðaÞ
Z a
x
ðx sÞa1 Daa f ðsÞds 6 pða1Þþ1
1 ðx a Þ p ¼ CðaÞ ðpða 1Þ þ 1Þ1=p
Z a
x
1 CðaÞ
Z
a D f ðsÞq ds a
x
p 1=p Z ðx sÞa1 ds
a
a
x
a D f ðsÞq ds a
1=q
1=q :
ð17Þ
Set
zðxÞ :¼
Z a
x
a D f ðsÞq ds; a
ðzðaÞ ¼ 0Þ:
Then
q z0 ðxÞ ¼ Daa f ðxÞ ; and
a D f ðxÞ ¼ ðz0 ðxÞÞ1=q ; a
all a 6 x 6 b:
Therefore by (17) we have
jf ðxÞjDaa f ðxÞ 6 all a 6 x 6 x.
pða1Þþ1
1 ðx aÞ p 1=q ðzðxÞz0 ðxÞÞ ; CðaÞ ðpða 1Þ þ 1Þ1=p
ð18Þ
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G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
Next working similarly with (16) we obtain
jf ðxÞj 6
1 CðaÞ
Z x
b
ðs xÞa1 Dab f ðsÞds 6 Z
pða1Þþ1 p
¼ Set
kðxÞ :¼
1 ð b xÞ CðaÞ ðpða 1Þ þ 1Þ1=p Z
b
x
a D f ðsÞq ds ¼ b
Z
x
x b
b
Z
1 CðaÞ
b
p ðs xÞa1 ds
!1=p Z
x
a D f ðsÞq ds b
a D f ðsÞq ds; b
b
x
a D f ðsÞq ds b
!1=q
!1=q ð19Þ
:
ðkðbÞ ¼ 0Þ:
Then
q k0 ðxÞ ¼ Dab f ðxÞ and
a D f ðxÞ ¼ ðk0 ðxÞÞ1=q ; b
all a 6 x 6 b:
Therefore by (19) we have
jf ðxÞjDab f ðxÞ 6
pða1Þþ1
1 ðb xÞ p 1=q ðkðxÞk0 ðxÞÞ ; CðaÞ ðpða 1Þ þ 1Þ1=p
ð20Þ
all x 6 x 6 b. Next we integrate (18) over ½a; x to obtain
Z
x
a
jf ðxÞjDaa f ðxÞdx 6 6
¼
¼
Z
1
CðaÞðpða 1Þ þ 1Þ
1=p
x
pða1Þþ1 p
ðx aÞ
Z
1
CðaÞðpða 1Þ þ 1Þ1=p
x
ðx aÞpða1Þþ1 dx
1=q
1=p Z
a
1
dx
CðaÞðpða 1Þ þ 1Þ
x
zðxÞz0 ðxÞdx
1=q
a
ðx aÞ 1=p
21=q ðx aÞ
ðzðxÞz0 ðxÞÞ
a
pða1Þþ2 p
ðpða 1Þ þ 2Þ
2=q
zðxÞ
1=p
21=q
Z
pða1Þþ2 p
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
a D f ðxÞq dx a
x
a
2=q :
ð21Þ
2=q a D f ðxÞq dx ; a
ð22Þ
So we have proved
Z
x
a
jf ðxÞjDaa f ðxÞdx 6
21=q ðx aÞ
Z
pða1Þþ2 p
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
x
a
for all a 6 x 6 b. By (22) we get
Z
aþb 2
a
jf ðxÞjDaa f ðxÞdx 6
ðb aÞ
ðpða1Þþ2Þ p
ðpða1Þþ2Þ 1 þq p
2 ½
Z
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
aþb 2
a
a D f ðxÞq dx a
!2=q ð23Þ
:
Similarly we integrate (20) over ½x; b to obtain
Z
b
x
jf ðxÞjDab f ðxÞdx 6
6
¼
Z
1
CðaÞðpða 1Þ þ 1Þ
1=p
b
ð b xÞ
pða1Þþ1 p
ðkðxÞk0 ðxÞÞ
1=q
dx
x
Z
1
CðaÞðpða 1Þ þ 1Þ1=p
b
ðb xÞ
pða1Þþ1
dx
!1=p Z
x
kðxÞk ðxÞdx
ðb xÞ
ðkðxÞÞ2=q
CðaÞðpða 1Þ þ 1Þ1=p ðpða 1Þ þ 2Þ1=p
21=q
x
b
jf ðxÞjDab f ðxÞdx 6
for all a 6 x 6 b.
pða1Þþ2 p
21=q ðb xÞ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
Z
b x
ð24Þ
:
We have proved that
Z
0
x pða1Þþ2 p
1
!1=q
b
a D f ðxÞq dx b
!2=q ;
ð25Þ
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G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
By (25) we get
Z
b
aþb 2
jf ðxÞjDab f ðxÞdx 6
ðpða1Þþ2Þ p
ðb aÞ
2½
ðpða1Þþ2Þ 1 þq p
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
Adding (23) and (26) we get
Z
b
jf ðxÞjDa f ðxÞdx 6
a
Z
pða1Þþ2 p
2ðaþpÞ ðb aÞð 1
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
2 4
b
aþb 2
Z
aþb 2
a
a D f ðxÞq dx b
a D f ðxÞq dx a
!2=q ð26Þ
:
!2=q þ
Z
b aþb 2
a D f ðxÞq dx b
!2=q 3 5 ¼: ðÞ: ð27Þ
Assume 1 < q 6 2, then Therefore we get
2 q
P 1.
2ðaþpÞ ðb aÞð 1
ðÞ 6
pða1Þþ2 p
"Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p Z
pða1Þþ2 1 2ðaþpÞ ðb aÞð p Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
b
aþb 2
a
a D f ðxÞq dx þ a
Z
b
aþb 2
!2=q a D f ðxÞq dx :
a D f ðxÞq dx b
#2=q ¼
ð28Þ
ð29Þ
a
So for 1 < q 6 2 we have proved (11). Assume now q > 2, then 0 < 2q < 1. Therefore we get pða1Þþ2 p
2ðaþpÞ ðb aÞð 1
ðÞ 6
¼
"Z
Þ 212q
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p 2ð
pða1Þþ2 p
Þ ðb aÞð
aþ1q
a
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ
aþb 2
1=p
b
a D f ðxÞq dx þ a
a D f ðxÞq dx
Z
b
aþb 2
!2=q
a D f ðxÞq dx b
#2=q
ð30Þ
:
a
So when q > 2 we have established (12). (iii) The case of p ¼ q ¼ 2, see (13), is obvious, it derives from (11) immediately. We need (see [3,6–8], p. 22).
Definition 11. Let a > 0, m ¼ dae, f 2 AC m ð½a; bÞ. We define the right Riemann–Liouville fractional derivative by
Dab f ðxÞ :¼
m Z b ð1Þm d ðt xÞma1 f ðt Þdt; Cðm aÞ dx x
ð31Þ
D0b f ðxÞ :¼ IðxÞðthe identity operatorÞ: We also define the left Riemann–Liouville fractional derivative by
Daaþ f ðxÞ :¼
m Z x 1 d ðx tÞma1 f ðt Þdt; Cðm aÞ dx a
ð32Þ
D0aþ f ðxÞ :¼ IðxÞ: We further define the new balanced Riemann–Liouville fractional derivative
( a
D f ðxÞ :¼
Dab f ðxÞ; for a
aþb 2
6 x 6 b;
Daþ f ðxÞ; for a 6 x < aþb : 2
ð33Þ
Remark 12. Let now f 2 C m ð½a; bÞ, m ¼ dae, a > 0. In [4] we have proved that Dab f ðxÞ, Daa f ðxÞ are continuous functions in x 2 ½a; b. Of course C m ð½a; bÞ AC m ð½a; bÞ, so that f 2 AC m ð½a; bÞ. Thus by Theorem 9 of [3], we obtain that also Dab f ðxÞ exists and continuous for every x 2 ½a; b. Furthermore if f ðkÞ ðbÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 we get
Dab f ðxÞ ¼ Dab f ðxÞ;
ð34Þ
8x 2 ½a; b. Similarly, by [6], p. 39, we get that Daaþ f ðxÞ exists and continuous in x 2 ½a; b. Furthermore if f ðkÞ ðaÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 we get
Daaþ f ðxÞ ¼ Daa f ðxÞ;
8x 2 ½a; b.
ð35Þ
1528
G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
So if f ðkÞ ðaÞ ¼ f ðkÞ ðbÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 we obtain that
Da f ðxÞ ¼ Da f ðxÞ;
ð36Þ
8x 2 ½a; b. So by Theorem 9 we obtain the corresponding results for the balanced Riemann–Liouville fractional derivative Theorem 13. Let f 2 C m ð½a; bÞ, m 2 N, m ¼ dae, a > 0. Assume f ðkÞ ðaÞ ¼ f ðkÞ ðbÞ ¼ 0, k ¼ 0; 1; . . . ; m 1; p, q > 1 : 1p þ 1q ¼ 1, and a > 1q. (i) Case of 1 < q 6 2. Then
Z
pða1Þþ2 p
2ðaþpÞ ðb aÞð 1
b
a
jf ðxÞjjD f ðxÞjdx 6
a
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
b
!2=q a
q
a
q
jD f ðxÞj dx
:
ð37Þ
:
ð38Þ
a
(ii) Case of q > 2. Then
Z
pða1Þþ2 p
2ðaþqÞ ðb aÞð 1
b
a
jf ðxÞjjD f ðxÞjdx 6
a
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
b
!2=q jD f ðxÞj dx
a
(iii) When p ¼ q ¼ 2; a > 12, then
Z
b
2ðaþ2Þ ðb aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðaÞ 2að2a 1Þ 1
jf ðxÞjjDa f ðxÞjdx 6
a
a
Z
b
! 2
ðDa f ðxÞÞ dx :
ð39Þ
a
3. Conclusion According to the monographs [1,2], our presented method of involving balanced fractional derivatives into Opial type inequalities, subject to boundary conditions, could be expanded to all possible directions, by producing interesting results and applications. Especially all these results proved here, and similar to be proved, are expected to have wide applications to fractional differential equations. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Agarwal RP, Pang PYH. Opial inequalities with applications in differential and difference equations. Dordrecht, London: Kluwer; 1995. Anastassiou GA. Fractional differentiation inequalities, research monograph, accepted. Berlin, NY: Springer; 2009. Anastassiou GA. On right fractional calculus. Chaos Soliton Fract 2009;42:365–76. Anastassiou GA. Fractional Korovkin Theory. Chaos Soliton Fract 2009, accepted for publication. Beesack PR. On an integral inequality of Z. Opial. Trans Am Math Soc 1962;104:470–5. Kai Diethelm. Fractional differential equations. Available from: http://www.tu-bs.de/~diethelm/lehre/f-dgl02/fde-skript.ps.gz. El-Sayed AMA, Gaber M. On the finite Caputo and finite Riesz derivatives. Electron J Theor Phys 2006;3(12):81–95. Frederico GS, Torres DFM. Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int Math Forum 2008;3(10):479–93. Gorenflo R, Mainardi F. Essentials of fractional calculus, Maphysto Center 2000. Available from: http://www.maphysto.dk/oldpages/events/ LevyCAC2000/MainardiNotes/fm2k0a.ps. [10] Olech C. A simple proof of a certain result of Z. Opial. Ann Polon Math 1960;8:61–3. [11] Opial Z. Sur une inegalite. Ann Polon Math 1960;8:29–32. [12] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives, theory and applications. Amsterdam: Gordon and Breach; 1993 [English translation from the Russian, integrals and derivatives of fractional order and some of their applications. Minsk: Nauka i Tekhnika; 1987].
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Balanced fractional opial inequalities George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 11 March 2009
Here we present Lp , p > 1, fractional Opial type inequalities subject to high order boundary conditions. They involve the right and left Caputo, Riemann–Liouville fractional derivatives. These derivatives are blended together into the balanced Caputo, Riemann–Liouville, respectively, fractional derivative. This balanced fractional derivative is introduced here for the first time. We give applications to a special case. Ó 2009 Elsevier Ltd. All rights reserved.
1. Preliminaries This article is inspired by the famous theorem of Opial [11], 1960, which follows Theorem 1. Let xðtÞ 2 C 1 ð½0; hÞ be such that xð0Þ ¼ xðhÞ ¼ 0, and xðtÞ > 0 in ð0; hÞ. Then
Z
h
jxðtÞx0 ðt Þjdt 6
0
In (1), the constant
( xðtÞ ¼
h 4
h 4
Z
h
2
ðx0 ðtÞÞ dt:
ð1Þ
0
is the best possible. Inequality (1) holds as equality for the optimal function
ct;
0 6 t 6 h=2;
c ðh t Þ
h 2
6 t 6 h;
where c > 0 is an arbitrary constant. To prove easier Theorem 1, Beesack [5] proved the following well-known Opial type inequality which is used very commonly. This is another inspiration to our work. Theorem 2. Let xðtÞ be absolutely continuous in [0, a], and xð0Þ ¼ 0. Then
Z 0
a
jxðtÞx0 ðt Þjdt 6
a 2
Z
a
2
ðx0 ðtÞÞ dt:
ð2Þ
0
Inequality (2) is sharp, it is attained by xðtÞ ¼ ct, c > 0 is an arbitrary constant. Opial type inequalities are used a lot in proving uniqueness of solutions to differential equations, also to give upper bounds to their solutions. By themselves have made a great subject of intensive research and there exists a great literature about them. Typical and great sources on them are the monographs [1,2]. We need (see also [3,7–9,12]). E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.047
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G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
Definition 3. Let f 2 AC m ð½a; bÞ (space of functions from ½a; b into R with m 1 derivative absolutely continuous function on ½a; b), m 2 N, where m ¼ dae, a > 0 (de the ceiling of the number). We define the right Caputo fractional derivative of order a > 0, by
Dab f ðxÞ ¼
Z
ð1Þm Cðm aÞ
b
ðf xÞma1 f ðmÞ ðfÞdf:
ð3Þ
x
We set D0b f ðxÞ ¼ f ðxÞ, 8x 2 ½a; b. Note 4. Let f 2 AC m ð½a; bÞ, m ¼ dae, with a > 0, then f ðm1Þ 2 ACð½a; bÞ, which implies that f ðmÞ exists a.e. on ½a; b and that f ðmÞ 2 L1 ð½a; bÞ. Consequently if f 2 AC m ð½a; bÞ, then Dab f ðxÞ exists a.e. on ½a; b and Dab f 2 L1 ð½a; bÞ, see [3]. Observe that when a ¼ m 2 N, then m ðmÞ Dm ðxÞ; 8x 2 ½a; b: b f ðxÞ ¼ ð1Þ f
ð4Þ
We continue with the right Caputo fractional Taylor formula with integral remainder, see [3]. Theorem 5. Let f 2 AC m ð½a; bÞ, x 2 ½a; b, a > 0, m ¼ dae. Then
f ð xÞ ¼
Z b m1 ðkÞ X f ðbÞ 1 k ðx bÞ þ ðf xÞa1 Dab f ðfÞdf: k! CðaÞ x k¼0
ð5Þ
We need also (see [6], p. 38). Definition 6. Let f 2 AC m ð½a; bÞ; m 2 N, where m ¼ dae, a > 0. We define the left Caputo fractional derivative of order a > 0, by
Daa f ðxÞ ¼
1 Cðm aÞ
Z
x
ðx tÞma1 f ðmÞ ðt Þdt;
ð6Þ
a
8x 2 ½a; b. We set D0a f ðxÞ ¼ f ðxÞ, 8x 2 ½a; b. Again here Daa f exists a.e. on ½a; b and Daa f 2 L1 ð½a; bÞ, see [6], p. 13, 37 and 38. When a ¼ m 2 N, then ðm Þ Dm ðxÞ; 8x 2 ½a; b: a f ðxÞ ¼ f
ð7Þ
We continue with the left Caputo fractional Taylor formula with integral remainder, see [6], p. 40. Theorem 7. Let f 2 AC m ð½a; bÞ, m 2 N, where m ¼ dae, a > 0, x 2 ½a; b. Then
f ð xÞ ¼
Z x m1 ðkÞ X f ðaÞ 1 ðx sÞa1 Daa f ðsÞds: ðx aÞk þ k! C ð a Þ a k¼0
ð8Þ
Above C is the gamma function,
CðaÞ ¼
Z
1
et t a1 dt; a > 0:
0
We introduce the following new balanced Caputo fractional derivative. Definition 8. Let f 2 AC m ð½a; bÞ; m 2 N, m ¼ dae, a > 0, x 2 ½a; b. We define
( a
D f ðxÞ :¼
6 x 6 b; Dab f ðxÞ; for aþb 2 Daa f ðxÞ;
fora 6 x < aþb : 2
ð9Þ
In this article we establish Lp , p > 1, Opial type inequalities involving the balanced Caputo fractional derivative subject to high order boundary conditions, more precisely by assuming that
f ðkÞ ðaÞ ¼ f ðkÞ ðbÞ ¼ 0;
k ¼ 0; 1; . . . ; m 1:
We extend our results to Riemann–Liouville fractional derivatives.
2. Results We present our main result. Theorem 9. Let f 2 AC m ð½a; bÞ; m 2 N, m ¼ dae, a > 0. Assume
ð10Þ
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G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
f ðkÞ ðaÞ ¼ f ðkÞ ðbÞ ¼ 0; 1 1 p; q > 1 : þ ¼ 1; p q
k ¼ 0; 1; . . . ; m 1; 1 and a > : q
(i) Case of 1 < q 6 2. Then
Z
b
jf ðxÞjDa f ðxÞdx 6
a
pða1Þþ2 p
2ðaþpÞ ðb aÞð 1
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
b
a D f ðxÞq dx
!2=q :
ð11Þ
:
ð12Þ
a
(ii) Case of q > 2. Then
Z
b
jf ðxÞjDa f ðxÞdx 6
a
pða1Þþ2 p
2ðaþqÞ ðb aÞð 1
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
(iii) When p ¼ q ¼ 2, a > 12, then
Z
b
jf ðxÞjDa f ðxÞdx 6
a
2ðaþ2Þ ðb aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðaÞ 2að2a 1Þ 1
a
Z
b
a
2
b
a D f ðxÞq dx
!2=q
a
!
D f ðxÞ dx :
ð13Þ
a
Remark 10. Let us say that a ¼ 1, then by (13) we obtain
Z a
b
ðb aÞ jf ðxÞjjf ðxÞjdx 6 4 0
Z
!
b 0
2
ðf ðxÞÞ dx ;
ð14Þ
a
that is reproving and recovering Opial’s inequality (1), see [11], see also Olech’s result [10]. Proof of Theorem 9. Let x 2 ½a; b. We have by assumption f ðkÞ ðaÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 and Theorem 7 that
f ðxÞ ¼
1 CðaÞ
Z
x
a
ðx sÞa1 Daa f ðsÞds;
ð15Þ
and by assumption f ðkÞ ðbÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 and Theorem 5 that
f ðxÞ ¼
1 CðaÞ
Z
b
x
ðs xÞa1 Dab f ðsÞds:
ð16Þ
Using Hölder’s inequality on (15) we get
j f ð xÞ j 6
1 CðaÞ
Z a
x
ðx sÞa1 Daa f ðsÞds 6 pða1Þþ1
1 ðx a Þ p ¼ CðaÞ ðpða 1Þ þ 1Þ1=p
Z a
x
1 CðaÞ
Z
a D f ðsÞq ds a
x
p 1=p Z ðx sÞa1 ds
a
a
x
a D f ðsÞq ds a
1=q
1=q :
ð17Þ
Set
zðxÞ :¼
Z a
x
a D f ðsÞq ds; a
ðzðaÞ ¼ 0Þ:
Then
q z0 ðxÞ ¼ Daa f ðxÞ ; and
a D f ðxÞ ¼ ðz0 ðxÞÞ1=q ; a
all a 6 x 6 b:
Therefore by (17) we have
jf ðxÞjDaa f ðxÞ 6 all a 6 x 6 x.
pða1Þþ1
1 ðx aÞ p 1=q ðzðxÞz0 ðxÞÞ ; CðaÞ ðpða 1Þ þ 1Þ1=p
ð18Þ
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G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
Next working similarly with (16) we obtain
jf ðxÞj 6
1 CðaÞ
Z x
b
ðs xÞa1 Dab f ðsÞds 6 Z
pða1Þþ1 p
¼ Set
kðxÞ :¼
1 ð b xÞ CðaÞ ðpða 1Þ þ 1Þ1=p Z
b
x
a D f ðsÞq ds ¼ b
Z
x
x b
b
Z
1 CðaÞ
b
p ðs xÞa1 ds
!1=p Z
x
a D f ðsÞq ds b
a D f ðsÞq ds; b
b
x
a D f ðsÞq ds b
!1=q
!1=q ð19Þ
:
ðkðbÞ ¼ 0Þ:
Then
q k0 ðxÞ ¼ Dab f ðxÞ and
a D f ðxÞ ¼ ðk0 ðxÞÞ1=q ; b
all a 6 x 6 b:
Therefore by (19) we have
jf ðxÞjDab f ðxÞ 6
pða1Þþ1
1 ðb xÞ p 1=q ðkðxÞk0 ðxÞÞ ; CðaÞ ðpða 1Þ þ 1Þ1=p
ð20Þ
all x 6 x 6 b. Next we integrate (18) over ½a; x to obtain
Z
x
a
jf ðxÞjDaa f ðxÞdx 6 6
¼
¼
Z
1
CðaÞðpða 1Þ þ 1Þ
1=p
x
pða1Þþ1 p
ðx aÞ
Z
1
CðaÞðpða 1Þ þ 1Þ1=p
x
ðx aÞpða1Þþ1 dx
1=q
1=p Z
a
1
dx
CðaÞðpða 1Þ þ 1Þ
x
zðxÞz0 ðxÞdx
1=q
a
ðx aÞ 1=p
21=q ðx aÞ
ðzðxÞz0 ðxÞÞ
a
pða1Þþ2 p
ðpða 1Þ þ 2Þ
2=q
zðxÞ
1=p
21=q
Z
pða1Þþ2 p
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
a D f ðxÞq dx a
x
a
2=q :
ð21Þ
2=q a D f ðxÞq dx ; a
ð22Þ
So we have proved
Z
x
a
jf ðxÞjDaa f ðxÞdx 6
21=q ðx aÞ
Z
pða1Þþ2 p
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
x
a
for all a 6 x 6 b. By (22) we get
Z
aþb 2
a
jf ðxÞjDaa f ðxÞdx 6
ðb aÞ
ðpða1Þþ2Þ p
ðpða1Þþ2Þ 1 þq p
2 ½
Z
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
aþb 2
a
a D f ðxÞq dx a
!2=q ð23Þ
:
Similarly we integrate (20) over ½x; b to obtain
Z
b
x
jf ðxÞjDab f ðxÞdx 6
6
¼
Z
1
CðaÞðpða 1Þ þ 1Þ
1=p
b
ð b xÞ
pða1Þþ1 p
ðkðxÞk0 ðxÞÞ
1=q
dx
x
Z
1
CðaÞðpða 1Þ þ 1Þ1=p
b
ðb xÞ
pða1Þþ1
dx
!1=p Z
x
kðxÞk ðxÞdx
ðb xÞ
ðkðxÞÞ2=q
CðaÞðpða 1Þ þ 1Þ1=p ðpða 1Þ þ 2Þ1=p
21=q
x
b
jf ðxÞjDab f ðxÞdx 6
for all a 6 x 6 b.
pða1Þþ2 p
21=q ðb xÞ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
Z
b x
ð24Þ
:
We have proved that
Z
0
x pða1Þþ2 p
1
!1=q
b
a D f ðxÞq dx b
!2=q ;
ð25Þ
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G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
By (25) we get
Z
b
aþb 2
jf ðxÞjDab f ðxÞdx 6
ðpða1Þþ2Þ p
ðb aÞ
2½
ðpða1Þþ2Þ 1 þq p
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
Adding (23) and (26) we get
Z
b
jf ðxÞjDa f ðxÞdx 6
a
Z
pða1Þþ2 p
2ðaþpÞ ðb aÞð 1
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
2 4
b
aþb 2
Z
aþb 2
a
a D f ðxÞq dx b
a D f ðxÞq dx a
!2=q ð26Þ
:
!2=q þ
Z
b aþb 2
a D f ðxÞq dx b
!2=q 3 5 ¼: ðÞ: ð27Þ
Assume 1 < q 6 2, then Therefore we get
2 q
P 1.
2ðaþpÞ ðb aÞð 1
ðÞ 6
pða1Þþ2 p
"Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p Z
pða1Þþ2 1 2ðaþpÞ ðb aÞð p Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
b
aþb 2
a
a D f ðxÞq dx þ a
Z
b
aþb 2
!2=q a D f ðxÞq dx :
a D f ðxÞq dx b
#2=q ¼
ð28Þ
ð29Þ
a
So for 1 < q 6 2 we have proved (11). Assume now q > 2, then 0 < 2q < 1. Therefore we get pða1Þþ2 p
2ðaþpÞ ðb aÞð 1
ðÞ 6
¼
"Z
Þ 212q
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p 2ð
pða1Þþ2 p
Þ ðb aÞð
aþ1q
a
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ
aþb 2
1=p
b
a D f ðxÞq dx þ a
a D f ðxÞq dx
Z
b
aþb 2
!2=q
a D f ðxÞq dx b
#2=q
ð30Þ
:
a
So when q > 2 we have established (12). (iii) The case of p ¼ q ¼ 2, see (13), is obvious, it derives from (11) immediately. We need (see [3,6–8], p. 22).
Definition 11. Let a > 0, m ¼ dae, f 2 AC m ð½a; bÞ. We define the right Riemann–Liouville fractional derivative by
Dab f ðxÞ :¼
m Z b ð1Þm d ðt xÞma1 f ðt Þdt; Cðm aÞ dx x
ð31Þ
D0b f ðxÞ :¼ IðxÞðthe identity operatorÞ: We also define the left Riemann–Liouville fractional derivative by
Daaþ f ðxÞ :¼
m Z x 1 d ðx tÞma1 f ðt Þdt; Cðm aÞ dx a
ð32Þ
D0aþ f ðxÞ :¼ IðxÞ: We further define the new balanced Riemann–Liouville fractional derivative
( a
D f ðxÞ :¼
Dab f ðxÞ; for a
aþb 2
6 x 6 b;
Daþ f ðxÞ; for a 6 x < aþb : 2
ð33Þ
Remark 12. Let now f 2 C m ð½a; bÞ, m ¼ dae, a > 0. In [4] we have proved that Dab f ðxÞ, Daa f ðxÞ are continuous functions in x 2 ½a; b. Of course C m ð½a; bÞ AC m ð½a; bÞ, so that f 2 AC m ð½a; bÞ. Thus by Theorem 9 of [3], we obtain that also Dab f ðxÞ exists and continuous for every x 2 ½a; b. Furthermore if f ðkÞ ðbÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 we get
Dab f ðxÞ ¼ Dab f ðxÞ;
ð34Þ
8x 2 ½a; b. Similarly, by [6], p. 39, we get that Daaþ f ðxÞ exists and continuous in x 2 ½a; b. Furthermore if f ðkÞ ðaÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 we get
Daaþ f ðxÞ ¼ Daa f ðxÞ;
8x 2 ½a; b.
ð35Þ
1528
G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 1523–1528
So if f ðkÞ ðaÞ ¼ f ðkÞ ðbÞ ¼ 0, k ¼ 0; 1; . . . ; m 1 we obtain that
Da f ðxÞ ¼ Da f ðxÞ;
ð36Þ
8x 2 ½a; b. So by Theorem 9 we obtain the corresponding results for the balanced Riemann–Liouville fractional derivative Theorem 13. Let f 2 C m ð½a; bÞ, m 2 N, m ¼ dae, a > 0. Assume f ðkÞ ðaÞ ¼ f ðkÞ ðbÞ ¼ 0, k ¼ 0; 1; . . . ; m 1; p, q > 1 : 1p þ 1q ¼ 1, and a > 1q. (i) Case of 1 < q 6 2. Then
Z
pða1Þþ2 p
2ðaþpÞ ðb aÞð 1
b
a
jf ðxÞjjD f ðxÞjdx 6
a
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
b
!2=q a
q
a
q
jD f ðxÞj dx
:
ð37Þ
:
ð38Þ
a
(ii) Case of q > 2. Then
Z
pða1Þþ2 p
2ðaþqÞ ðb aÞð 1
b
a
jf ðxÞjjD f ðxÞjdx 6
a
Z
Þ
CðaÞ½ðpða 1Þ þ 1Þðpða 1Þ þ 2Þ1=p
b
!2=q jD f ðxÞj dx
a
(iii) When p ¼ q ¼ 2; a > 12, then
Z
b
2ðaþ2Þ ðb aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðaÞ 2að2a 1Þ 1
jf ðxÞjjDa f ðxÞjdx 6
a
a
Z
b
! 2
ðDa f ðxÞÞ dx :
ð39Þ
a
3. Conclusion According to the monographs [1,2], our presented method of involving balanced fractional derivatives into Opial type inequalities, subject to boundary conditions, could be expanded to all possible directions, by producing interesting results and applications. Especially all these results proved here, and similar to be proved, are expected to have wide applications to fractional differential equations. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Agarwal RP, Pang PYH. Opial inequalities with applications in differential and difference equations. Dordrecht, London: Kluwer; 1995. Anastassiou GA. Fractional differentiation inequalities, research monograph, accepted. Berlin, NY: Springer; 2009. Anastassiou GA. On right fractional calculus. Chaos Soliton Fract 2009;42:365–76. Anastassiou GA. Fractional Korovkin Theory. Chaos Soliton Fract 2009, accepted for publication. Beesack PR. On an integral inequality of Z. Opial. Trans Am Math Soc 1962;104:470–5. Kai Diethelm. Fractional differential equations. Available from: http://www.tu-bs.de/~diethelm/lehre/f-dgl02/fde-skript.ps.gz. El-Sayed AMA, Gaber M. On the finite Caputo and finite Riesz derivatives. Electron J Theor Phys 2006;3(12):81–95. Frederico GS, Torres DFM. Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int Math Forum 2008;3(10):479–93. Gorenflo R, Mainardi F. Essentials of fractional calculus, Maphysto Center 2000. Available from: http://www.maphysto.dk/oldpages/events/ LevyCAC2000/MainardiNotes/fm2k0a.ps. [10] Olech C. A simple proof of a certain result of Z. Opial. Ann Polon Math 1960;8:61–3. [11] Opial Z. Sur une inegalite. Ann Polon Math 1960;8:29–32. [12] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives, theory and applications. Amsterdam: Gordon and Breach; 1993 [English translation from the Russian, integrals and derivatives of fractional order and some of their applications. Minsk: Nauka i Tekhnika; 1987].