On a Riemann-Liouville Generalized Taylor's ... - Semantic Scholar

Journal of Mathematical Analysis and Applications 231, 255–265 (1999) Article ID jmaa.1998.6224, available online at http://www.idealibrary.com on

On a Riemann–Liouville Generalized Taylor’s Formula* J. J. Trujillo Departamento de An´ alisis Matem´ atica, Universidad de La Laguna, ES-38271 La Laguna, Tenerife, Spain E-mail: [email protected]

M. Rivero Departamento de Matem´ atica Fundamental, Universidad de La Laguna, ES-38271 La Laguna, Tenerife, Spain E-mail: [email protected]

and B. Bonilla Departamento de An´ alisis Matem´ atica, Universidad de La Laguna, ES-38271 La Laguna, Tenerife, Spain E-mail: [email protected] Submitted by H. M. Srivastava Received April 30, 1998

In this paper, a generalized Taylor’s formula of the kind f x =

n X j=0

aj x − aj+1α−1 + Tn x;

where aj ∈ , x > a, 0 ≤ α ≤ 1, is established. Such expression is precisely the classical Taylor’s formula in the particular case α = 1. In addition, detailed expressions for Tn x and aj , involving the Riemann–Liouville fractional derivative, and some applications are also given. © 1999 Academic Press Key Words: Generalized Taylor’s formula, Riemann–Liouville operator, fractional calculus, fractional differential equations. * Paper partially supported by DGICYT and by DGUI of G.A.CC. 255 0022-247X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.

256

trujillo, rivero, and bonilla 1. INTRODUCTION

The ordinary Taylor’s formula has been generalized by many authors. Riemann [6], had already written a formal version of the generalized Taylor’s series: f x + h =

∞ X

hm+r Dm+r f x; a 0m + r + 1 m=−∞

1:1

where for α ≤ 0; Dαa f x = Ia−α f x is the Riemann–Liouville fractional integral of order −α. This fractional integral operator is defined for β ∈ + , a ∈ and x > a as follows: 1 Zx x − tβ−1 f tdt; 1:2 Iaβ f x = 0β a and for n ∈ N and n − 1 < α ≤ n is Dαa f x = Dn Ian−α f x, where D denotes the classical derivative. Moreover, Ia0 f x = f x. The proof of the validity of such an expansion for certain classes of functions was undertaken by Hardy [4], both for finite and infinite a. Afterwards, Watanabe [8] obtained the following relation: f x =

n−1 X

x − x0 α+k Dα+k f x0 + Rn;m 0α + k + 1 a k=−m

1:3

with m < α, x > x0 ≥ a, and: Dα+n Rn;m = Ixα+n a f x + 0

Z x0 1 x − t−α−m−1 Dα−m−1 f tdt: a 0α − m a

On the other hand, a variant of the generalized Taylor’s series was given by Dzherbashyan and Nersesyan [3]. For f having all of the required continuous derivatives, they obtained: Zx 1 Dαk f 0 αk x − tαm −1 Dαm f tdt: x + 01 + α 01 + α 0 k m k=0 1:4 where x > 0, α0 , α1 ; : : : ; αm is an increasing sequence of real numbers such 1−α −α 1+α that 0 < αk − αk−1 ≤ 1, k = 1; : : : ; m and Dαk f = I0 k k−1 D0 k−1 f . In this paper, under certain conditions for f and α ∈ 0; 1, the following generalized Taylor’s formula: f x =

m−1 X

f x =

n X

cj

0j + 1α j=0

x − aj+1α−1 + Rn x; a

1:5

on a riemann–liouville generalized taylor’s formula

257

is obtained, with n+1α

Da f ξ x − an+1α ; Rn x; a = 0n + 1α + 1 + cj = 0αx − a1−α Djα a f xa ;

a≤ξ≤x

∀j = 0; : : : ; n

and the sequential fractional derivative is denoted by n

^

α α Dnα a = Da · · · · · · Da ;

1:6

where n ∈ , according to the definition introduced by Miller and Ross [5]. Also a generalized mean value theorem and some applications of above generalized Taylor’s formula are given. 2. DEFINITIONS AND PROPERTIES Let be an real interval and α ∈ 0; 1. Definition 2.1. Let f a Lebesgue measurable function in , α ∈ 0; 1 and x0 ∈ . f is called α-continuous in x0 if there exists λ ∈ 0; 1 − α for which gx = x − x0 λ f x is a continuous function in x0 . Moreover, f is called 1-continuous in x0 if it is continuous in x0 . As usualy it is said that “f is a α-continuous function on if f is α-continuous for every x in ,” and it is denoted: Cα = f ∈ F x f is α-continuous in ; and so C1 = C. Definition 2.2. α if

Let a ∈ . The function f is called a-singular of order f x = k < ∞ and k 6= 0: x→a x − aα−1 lim

Let α ∈ + , a ∈ , E an interval, E ⊂ , such that a ≤ x, ∀x ∈ E. Then we write α a Iα E = f ∈ F x Ia f x exists and it is finite ∀x ∈ E a Dα E

= f ∈ F x Dαa f x exists and it is finite ∀x ∈ E ;

where F stands for the set of real functions of a single real variable with domain in . Some properties of the Riemann–Liouville derivative and integral operators will be extensively used. They are collected in the next two propositions.

258

trujillo, rivero, and bonilla

Proposition 2.1. (i)

Let α ∈ 0; 1; a; b ⊂ . Then

If f ∈ Ca; b and f ∈ a Iα a; b, then Dαa Iaα f x = f x;

(ii)

∀x ∈ a; b:

2:1

If f , Dαa f ∈ Ca; b and Dαa f ∈ a Iα a; b, then Iaα Dαa f x = f x + k

x − aα−1 ; 0α

∀x ∈ a; b;

2:2

where k = −0αx − a1−α f xa+ = −Ia1−α f xa+ :

2:3

Proof. (i) It follows from its corresponding ordinary case α = 1. (ii) First, it is assumed that Iaα Dαa f x = f x + 9x, where 9x is a suitable function. From this Dαa 9x = 0 and then 9x = kx − aα−1 /0α. Therefore k = −0α limx−→a+ x − a1−α f x − Iaα Dαa f x = −0αx − a1−α f xa+ : Proposition 2.2. Let α ∈ 0; 1, m ∈ and f a function. If one of the following conditions is satisfied, (i) f ∈ La; b and m + 1α ≥ 1. (ii) f ∈ Cγ a; b with 0 ≤ 1 − m + 1α ≤ γ ≤ 1. (iii) f ∈ Cγ a; b with 0 ≤ 1 − m + 1α ≤ γ ≤ 1 and it is a-singular of order α, then the relation: m+1α

Ia

f x = Iaα Iamα f x = Iamα Iaα f x; ∀x ∈ a; b

2:4

holds true. Proof.

See Bonilla–Trujillo–Rivero [2] and Samko–Kilbas–Marichev [7]. 3. A GENERALIZED MEAN VALUE THEOREM

Theorem 3.1. Let α ∈ 0; 1 and f ∈ Ca; b such that Dαa f ∈ Ca; b. Then f x = x − a1−α f xa+ x − aα−1 + Dαa f ξ with a ≤ ξ ≤ x:

x − aα ; 0α + 1

∀x ∈ a; b 3:1


259

Proof. Since Iaα Dαa f x =

1 Zx x − tα−1 Dαa f tdt; 0α a

by using the integral mean value theorem, we have Iaα Dαa f x = Dαa f ξ

x − aα ; 0α + 1

where a ≤ ξ ≤ x. Now (3.1) is obtained from (2.2). Remark 3.1. (i) If b = a + h and f ∈ Ca; a + h such that Dαa f ∈ Ca; a + h, then f a + h = h1−α f xa+ hα−1 + Dαa f ξ

hα ; 0α + 1

with a ≤ ξ ≤ a + h. (ii) If f ∈ La; b and Dαa f ∈ Ca; b then (3.1) holds almost everywhere in a; b. (iii)

Another variant of the above theorem can be given as follows:

f x = Ia1−α f xa+ x − aα−1 + Dαa f ξ

x − aα ; ∀x ∈ a; b; 0α + 1

with a ≤ ξ ≤ x, under the same conditions for f as those in Theorem 3.1. Corollary 3.1.

Let α ∈ 0; 1 and g ∈ Ca; b such that Dαa x − aα−1 gx ∈ Ca; b:

Then gx = ga+ +

Dαa x − aα−1 gxξ x − a; 0α + 1

∀x ∈ a; b

3:2

for some ξ, a ≤ ξ ≤ x. Proof. The function f x = x − aα−1 gx satisfies the conditions of the above theorem. So f x = ga+ x − aα−1 + and (3.2) is obtained.

Dαa x − aα−1 gxξ x − aα 0α + 1

260


Corollary 3.2. Let α ∈ 0; 1. Let g be a continuous function on a; b, differentiable in the ordinary sense when x > a and such that Dαa x − aα−1 gx ∈ Ca; b. Then Dαa x − aα−1 gxa+ = 0α + 1Dga+ :

3:3

Proof. Applying the above theorem to the function f x = x − aα−1 gx, we obtain Dαa f ξ = =

0α + 1 f x − x − a1−α f xa+ x − aα−1 x − aα 0α + 1 gx − ga+ : x − a

Now, (3.3) is obtained by taking limits when x → a+ . Corollary 3.3. Let α ∈ 0; 1. Let g be a continuous function on a; b such that ga+ = gb and Dαa x − aα−1 gx ∈ Ca; b: Then there exists ξ, a ≤ ξ ≤ b, such that Dαa x − aα−1 gxξ = 0: Proof.

It follows from Corollary 3.1.

4. A GENERALIZATION OF TAYLOR’S FORMULA Proposition 4.1. such that (i) (ii)

Set α ∈ 0; 1 and m ∈ − 0. Let f be a function m+1α

Dmα a f and Da m+1α f Da

f are continuous in a; b,

∈a Iα a; b, m+1α

f is γ-continuous in (iii) If α < 1/2 and m + 1α < 1, then Da m+1α a, with 1 − m + 1α ≤ γ ≤ 1, or Da f is a-singular of order α. Then m+1α

Iamα Dmα a f x − Ia

m+1α

Da

f x = cm

x − am+1α−1 ; 0m + 1α

+ 1−α mα Da f a+ : where cm = 0αx − a1−α Dmα a f xa = Ia

Dαa f

∀x ∈ a; b: 4:1;

If m = 0, and f is a continuous function such that Dαa f ∈ Ca; b and ∈ a Iα a; b then, (4.1) also holds.


261

Proof. For m = 0 see Samko–Kilbas–Marichev [7]. For m > 0, we find from (2.4) that m+1α

Iamα Dmα a f − Ia

m+1α

Da

mα α α mα f = Iamα Dmα a f − Ia Ia Da Da f:

Now (4.1) follows from (1.6), (2.1) and (2.2). Theorem 4.1. Set α ∈ 0; 1, n ∈ . Let f be a continuous function in a; b satisfying the following conditions: jα

jα

(i)

∀j = 1; : : : ; n, Da f ∈ Ca; b and Da f ∈ a Iα a; b.

(ii)

Da

n+1α

f is continuous on a; b.

(iii) If α < 1/2 then, for each j ∈ N, 1 ≤ j ≤ n, such that j + 1α < 1, j+1α f x is γ-continuous in x = a for some γ, 1 − j + 1α ≤ γ ≤ 1, or Da a-singular of order α. Then, ∀x ∈ a; b, f x =

n c x − aj+1α−1 X j

0j + 1α

j=0

+ Rn x; a;

4:2

with n+1α

Rn x; a =

Da f ξ x − an+1α ; 0n + 1α + 1

a≤ξ≤x

4:3

and for each j ∈ N, 0 ≤ j ≤ n, + 1−α jα Da f a+ : cj = 0αx − a1−α Djα a f xa = Ia

Proof.

By using (4.1), for j = 0; : : : ; n, it follows that f x =

n c x − aj+1α−1 X j j=0

0j + 1α

n+1α

+ Ia

n+1α

Da

f x:

Applying the integral mean value theorem, we have Zx 1 n+1α n+1α n+1α Da f x = x − tn+1α−1 Da f tdt Ia 0αn + 1 a Zx n+1α f ξ x − tn+1α−1 dt = Da a

n+1α

= Da

f ξ

x − an+1α 0n + 1α + 1

with a ≤ ξ ≤ x, and so (4.2) is obtained.

4:4

262


Corollary 4.1. Set α ∈ 0; 1 and n ∈ . Let g be a continuous function on a; b such that the function f x = x − aα−1 gx satisfies the conditions of the above theorem. Then, ∀x ∈ a; b, gx =

n c x − ajα X j

0j + 1α j=0

+ R0n x; a

4:5

with R0 n x; a =

n+1α

Da

x − aα−1 gxξ x − anα+1 ; a ≤ ξ ≤ x 0n + 1α + 1

and where α−1 gxa+ cj = 0αx − a1−α Djα a x − a α−1 gxa+ = Ia1−α Djα a x − a

for each j ∈ N, 1 ≤ j ≤ n. Proof.

It follows from the above theorem.

Remark 4.1. In a natural way, all functions f x satisfying the conditions of Theorem 4.1 could be expanded in a generalized Taylor’s series as follows: f x = x − aα−1

∞ c x − ajα X j j=0

0j + 1α

;

4:6

with cj given above, holds for all x ∈ a; b, where the series converges and lim Rn x; a = 0:

n→∞

The functions which can be expanded as in (4.6) will be called α-analytic in x = a. 5. APPLICATIONS 1.

Let us consider the fractional differential equation: Dα0 yx = λyx;

5:1

where 0 < α ≤ 1, λ a real number and x > 0. It is assumed that y(x) is 0-singular of order α, and continuous ∀x > 0.


263

jα

Since D0 yx = λj yx, ∀j ∈ , and ∀x > 0, xj+1α = 0: j→∞ 0j + 1α + 1 lim

We obtain yx =

∞ X j=0

1−α

where c0 = 0αx Then

cj

xj+1α−1 ; 0j + 1α

+

yx0 , cj = λj c0 . yx = c0 xα−1

∞ X

λj xjα 0j + 1α j=0

which converges ∀x > 0. If it is assumed that lim+ x1−α yx =

x→0

1 ; 0α

it may define the α-exponential function: α−1 eλx α =x

∞ X

λj

j=0

xjα = xα−1 Eα;α λxα 0j + 1α

5:2

where Eα;β z is the Mittag–Leffler function. The general solution of (5.1) is then, yx = keλx α . 2. Let us consider now the fractional differential equation D2α 0 yx = −yx

5:3

with 0 < α ≤ 1, x > 0 and D20 α = Dα0 Dα0 [see (1.6)]. Assuming that yx is 0-singular of order α and continuous ∀x > 0, then ∀n ∈ , 4n+1α

D4nα 0 yx = yx; D0 4n+3α yx D0

4n+2α

yx = Dα0 yx; D0

yx = −yx;

= −Dα0 yx

and using the generalized Taylor’s formula, we obtain ∞ X x2nα x2n+1α + c1 xα−1 : −1n 02n + 1α 02n + 2α n=0 n=0 5:4 If it is assumed that

yx = c0 xα−1 (A)

∞ X

−1n

x1−α yx0+ = 0

and

x1−α Dα0 yx0+ =

1 ; 0α

264


it may define, sinα x = xα−1

∞ X

−1n

n=0

x2n+1α = x2α−1 E2α;2α −x2α 02n + 2α

5:5

which converges ∀x > 0. (B)

If it is assumed that x1−α yx0+ =

1 and x1−α Dα0 yx0+ = 0 0α

it may similarly define: cosα x = xα−1

∞ X

−1n

n=0

x2nα = xα−1 E2α;α −x2α 02n + 1α

5:6

which converges ∀x > 0. The general solution of (5.2) is yx = c0 sinα x + c1 cosα x. The above functions satisfy the following relations: sinα x =

−ix eix eix + e−ix α − eα α ; cosα x = α 2i 2

and 2 2 ix −ix eix α = cosα x + isinα x; sinα x + cosα x = eα eα

just as in the ordinary case, where i is the complex imaginary unit (i2 = −1), using the notation of Euler, and eλx α is the natural extension of (5.2) to complex values of λ. ACKNOWLEDGEMENTS The authors would like to thank Professor H. M. Srivastava and R. Srivastava for their valuable ideas when to work in this subject was started, and also Professor A. A. Kilbas for his interesting comments about the preprint.

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4. G. H. Hardy, Riemann’s form of Taylor’s series, J. London Math. Soc. 20, (1945), 48–57. 5. K. S. Miller and B. Ross, “An introduction to the fractional calculus and fractional differential equations,” Wiley, New York, 1993. 6. B. Riemann, Versuch einer allgemeinen auffasung der integration und differentiation, Gesammelte Math. Werke und Wissenchaftlicher. Leipzig: Teubner (1876), 331–344. 7. S. G. Samko, A. A. Kilbas, and O. I. Marichev, “Fractional integrals and derivatives. Theory and applications,” Gordon and Breach, Reading, 1993. 8. Y. Watanabe, On some properties of fractional powers of linear operators, Proc. Japan Acad. 37 (1961), 273–275.