Fractional Sobolev Spaces via Riemann-Liouville Derivatives

Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 128043, 15 pages http://dx.doi.org/10.1155/2013/128043

Research Article Fractional Sobolev Spaces via Riemann-Liouville Derivatives Dariusz Idczak and StanisBaw Walczak Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland Correspondence should be addressed to Dariusz Idczak; [email protected] Received 31 July 2013; Accepted 19 October 2013 Academic Editor: Ismat Beg Copyright © 2013 D. Idczak and S. Walczak. 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. Using Riemann-Liouville derivatives, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones. Next, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and compactness of some imbeddings. An application to boundary value problems is given as well.

1. Introduction Let 1 ≤ 𝑝 < ∞. Classical Sobolev space 𝑊1,𝑝 = 𝑊1,𝑝 ((𝑎, 𝑏), R𝑚 ) of order one on an open bounded interval (𝑎, 𝑏) ⊂ R is defined by (cf. [1]) 𝑏

𝑊1,𝑝 = {𝑢 ∈ 𝐿𝑝 ; ∃ 𝑝 ∀ ∞ ∫ 𝑢 (𝑡) 𝜑(1) (𝑡) 𝑑𝑡 𝑔∈𝐿 𝜑∈𝐶𝑐

𝑎

𝑏

(1)

= − ∫ 𝑔 (𝑡) 𝜑 (𝑡) 𝑑𝑡} , 𝑎

𝑝

𝑝

Theorem 1. Let 𝑛 ∈ N. Then, 𝑢 ∈ 𝑊𝑛,𝑝 if and only if there exist functions 𝑔1 , . . . , 𝑔𝑛 ∈ 𝐿𝑝 such that 𝑏

𝑏

𝑎

𝑎

∫ 𝑢 (𝑡) 𝜑(𝑗) (𝑡) 𝑑𝑡 = (−1)𝑗 ∫ 𝑔𝑗 (𝑡) 𝜑 (𝑡) 𝑑𝑡

for 𝜑 ∈ 𝐶𝑐∞ and 𝑗 = 1, . . . , 𝑛, where 𝜑(𝑗) denotes the classical derivative of 𝜑 of order 𝑗. In such a case there exists an absolutely continuous function 𝑢̃ : [𝑎, 𝑏] → R𝑚 such that 𝑢 = 𝑢̃ a.e. on (𝑎, 𝑏), which has absolutely continuous classical derivatives 𝑢(1) , . . . , 𝑢(𝑛−1) , the derivative 𝑢(𝑛) = (𝑢(𝑛−1) )(1) ∈ 𝐿𝑝 , and 𝑔1 = 𝑢(1) , . . . , 𝑔𝑛 = 𝑢(𝑛) .

𝑚

where 𝐿 = 𝐿 ((𝑎, 𝑏), R ) is the space of functions 𝑔 : (𝑎, 𝑏) → R𝑚 that are integrable with power 𝑝, 𝐶𝑐∞ = 𝐶𝑐∞ ((𝑎, 𝑏), R𝑚 ) is the set of smooth functions 𝜑 : (𝑎, 𝑏) → R𝑚 with compact support supp 𝜑 ⊂ (𝑎, 𝑏), and 𝜑(1) is the classical derivative of 𝜑. The function 𝑔 satisfying the above condition is denoted by 𝑢󸀠 and called the weak derivative of 𝑢 of order one. Sobolev space 𝑊𝑛,𝑝 = 𝑊𝑛,𝑝 ((𝑎, 𝑏), R𝑚 ) of order 𝑛 > 1 is defined by (cf. [1])

𝑛,𝑝

The spaces 𝑊

can be characterized as follows.

(2)

(4)

Remark 2. One shows that 𝑔1 = 𝑢󸀠 , 𝑔2 = (𝑢󸀠 )󸀠 , . . ., 𝑔𝑛 = (⋅ ⋅ ⋅ (𝑢󸀠 )󸀠 ⋅ ⋅ ⋅ )󸀠 (𝑛 times). Remark 3. In our paper, we will identify functions defined on (𝑎, 𝑏) and [𝑎, 𝑏] ((𝑎, 𝑏], [𝑎, 𝑏), resp.) that are equal a.e. on (𝑎, 𝑏). Each of the functions 𝑔𝑗 , 𝑗 = 1, . . . , 𝑛, is denoted by 𝐷𝑗 𝑢 and called the weak derivative of 𝑢 of order 𝑗. The space 𝑊𝑛,𝑝 endowed with a norm 𝑛

𝑊𝑛,𝑝 = {𝑢 ∈ 𝑊𝑛−1,𝑝 ; 𝑢󸀠 ∈ 𝑊𝑛−1,𝑝 } .

(3)

󵄩 󵄩𝑝 𝑝 ‖𝑢‖𝑊𝑛,𝑝 = ∑󵄩󵄩󵄩󵄩𝐷𝑖 𝑢󵄩󵄩󵄩󵄩𝐿𝑝 ,

𝑢 ∈ 𝑊𝑛,𝑝 ,

(5)

𝑖=0

where 𝐷0 𝑢 = 𝑢, has many useful properties such as completeness, reflexivity (for 1 < 𝑝 < ∞), and separability.

2 Moreover, some imbeddings of these spaces are compact (cf. [1]). In the last years, many papers and books on fractional calculus and its applications have appeared. Most of them concern fractional differential equations, including calculus of variations and optimal control. In the classical (positive integer) case the fundamental role in this field is played by the mentioned Sobolev spaces. To our best knowledge, there are no “fractional” Sobolev spaces based on the notion of fractional derivative in Riemann-Liouville sense, which seems to be the most used in the theory of fractional differential equations. Our aim is to give some systematic basics for applications of fractional calculus to differential equations. More precisely, we extend the above definitions, with the aid of the Riemann-Liouville derivatives, to the case of noninteger positive (fractional) order 𝛼, derive a fractional counterpart of Theorem 1, and prove the basic properties of the introduced spaces. When 𝛼 = 𝑛 ∈ N, the obtained results reduce to the classical ones. In the literature, some generalizations of Sobolev spaces to noninteger orders, on a domain Ω ⊂ R𝑛 , are known (cf. [2]): Gagliardo spaces 𝑊𝛼,𝑝 (Ω), Besov spaces 𝐵𝛼,𝑝 (Ω), and Nikolskii spaces 𝐻𝛼,𝑝 (Ω). They have been introduced with the aid of approaches different from ours and their comparison with our spaces (in the case of Ω = (𝑎, 𝑏)) is an open problem. Let us point that only Gagliardo spaces coincide with the classical Sobolev spaces when 𝛼 = 𝑛. The paper is organized as follows. In the second section, we recall some basic notions and facts from the fractional calculus including a characterization of functions possessing the left (right) Riemann-Liouville derivatives. In the third section, we derive some special cases of the fractional theorem on the integration by parts. In the fourth section, we define the fractional Sobolev spaces of any order 𝛼 > 0 and characterize them. In the fifth section, we derive a fractional counterpart of Theorem 1, define the weak fractional derivatives of order 𝛼 > 0, and show that they coincide with the Riemann-Liouville derivatives. In the sixth section, we introduce two norms in the fractional Sobolev spaces and prove their equivalence. In the seventh section, we derive completeness, reflexivity, and separability of the introduced spaces. In the eighth section, we prove compactness of some imbeddings. In the ninth section, we present some applications of the obtained results to fractional boundary value problems using a variational approach. In the paper, we limit ourselves to the left fractional Sobolev spaces, but in an analogous way one can define right spaces and derive their appropriate properties.

2. Preliminaries By 𝐴𝐶𝑛,1 = 𝐴𝐶𝑛,1 ([𝑎, 𝑏], R𝑚 ), where 𝑛 ∈ N, we denote the set of all functions 𝑓 : [𝑎, 𝑏] → R𝑚 that have a representative (a.e. on [𝑎, 𝑏]) which is absolutely continuous together with its classical derivatives of orders 1, . . . , 𝑛 − 1. Of course, such a function possesses also the classical derivative of order 𝑛, existing a.e. on [𝑎, 𝑏] and belonging to 𝐿1 . These classical derivatives are the weak derivatives 𝐷1 𝑓, . . . , 𝐷𝑛 𝑓 of 𝑓. It is

Journal of Function Spaces and Applications known (cf. [3, Lemma 2.4]) that 𝑓 ∈ 𝐴𝐶𝑛,1 if and only if there exist 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 ∈ R𝑚 and 𝜑 ∈ 𝐿1 such that 𝑛−1 𝑐 𝑓 (𝑡) = ∑ 𝑖 (𝑡 − 𝑎)𝑖 𝑖=0 𝑖! 𝑡

𝑡1

𝑡𝑛−1

𝑎

𝑎

𝑎

+ ∫ ∫ ⋅⋅⋅∫

𝜑 (𝜏) 𝑑𝜏𝑑𝑡𝑛−1 ⋅ ⋅ ⋅ 𝑑𝑡1 ,

𝑡 ∈ [𝑎, 𝑏] a.e. (6)

In such a case, 𝐷𝑖 𝑓 (𝑎) = 𝑐𝑖 ,

𝑖 = 0, . . . , 𝑛 − 1,

𝐷𝑛 𝑓 (𝑡) = 𝜑 (𝑡) ,

𝑡 ∈ [𝑎, 𝑏] a.e.

(7)

By 𝐴𝐶𝑛,𝑝 = 𝐴𝐶𝑛,𝑝 ([𝑎, 𝑏], R𝑚 ) we denote the set of all functions 𝑓 : [𝑎, 𝑏] → R𝑚 that have representation (6) with 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 ∈ R𝑚 and 𝜑 ∈ 𝐿𝑝 . It is known that 𝐴𝐶𝑛,𝑝 = 𝑊𝑛,𝑝 for 1 ≤ 𝑝 < ∞. Let 𝛼 > 0, and let ℎ ∈ 𝐿1 . By the left Riemann-Liouville fractional integral of ℎ on the interval [𝑎, 𝑏] we mean (cf. [3]) 𝛼 ℎ given by a function 𝐼𝑎+ 𝛼 (𝐼𝑎+ ℎ) (𝑡) =

𝑡 ℎ (𝜏) 1 𝑑𝜏, ∫ Γ (𝛼) 𝑎 (𝑡 − 𝜏)1−𝛼

𝑡 ∈ [𝑎, 𝑏] a.e.

(8)

Remark 4. The above integral exists and is finite a.e. on [𝑎, 𝑏]. In view of the convergence (cf. [3, Theorem 2.7]) 𝛼 lim (𝐼𝑎+ ℎ) (𝑡) = ℎ (𝑡) ,

𝛼 → 0+

𝑡 ∈ [𝑎, 𝑏] a.e.,

(9)

it is natural to put 0 ℎ) (𝑡) = ℎ (𝑡) , (𝐼𝑎+

𝑡 ∈ [𝑎, 𝑏] a.e.

(10)

Remark 5. If 𝛼 ≥ 1, the right side of (8) exists (and is finite) 𝛼 ℎ can be defined everywhere on everywhere on [𝑎, 𝑏]. So, 𝐼𝑎+ [𝑎, 𝑏]. In such a case it is continuous on [𝑎, 𝑏]. Remark 6. It is easy to see that if ℎ is essentially bounded on [𝑎, 𝑏] and 0 < 𝛼 < 1, then the right side of (8) is defined 𝛼 ℎ can and bounded everywhere on (𝑎, 𝑏]. So, in this case 𝐼𝑎+ be defined everywhere on (𝑎, 𝑏]. From [3, Theorem 3.6] it 𝛼 ℎ is equal everywhere on (𝑎, 𝑏] follows that in such a case 𝐼𝑎+ to a continuous function on [𝑎, 𝑏]. It is also known (cf. [3, 𝛼 ℎ ∈ 𝐿𝑝 . Theorem 2.6]) that if ℎ ∈ 𝐿𝑝 with 1 ≤ 𝑝 < ∞, then 𝐼𝑎+ 𝛼,1 = Let 𝑛 − 1 < 𝛼 ≤ 𝑛, where 𝑛 ∈ N. By 𝐴𝐶𝑎+ 𝛼,1 𝐴𝐶𝑎+ ([𝑎, 𝑏], R𝑚 ) we denote the set of all functions 𝑓 : [𝑎, 𝑏] → R𝑚 that have the representation 𝑛−1

𝑐𝑖 (𝑡 − 𝑎)𝛼−𝑛+𝑖 𝑖=0 Γ (𝛼 − 𝑛 + 1 + 𝑖)

𝑓 (𝑡) = ∑

𝛼 + 𝐼𝑎+ 𝜑 (𝑡) ,

(11)

𝑡 ∈ [𝑎, 𝑏] a.e.,

𝑛,1 with 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 ∈ R𝑚 and 𝜑 ∈ 𝐿1 . Of course, 𝐴𝐶𝑎+ = 𝑛,1 𝐴𝐶 .

Journal of Function Spaces and Applications

3

We say that 𝑓 ∈ 𝐿1 possesses the left Riemann-Liouville 𝛼 𝑓 of order 𝛼 ∈ (𝑛 − 1, 𝑛], 𝑛 ∈ N, on the derivative 𝐷𝑎+ 𝑛−𝛼 interval [𝑎, 𝑏] if 𝐼𝑎+ 𝑓 ∈ 𝐴𝐶𝑛,1 . By this derivative we mean 𝑛 𝑛−𝛼 𝑛 𝑓 = 𝐷𝑛 𝑓. the derivative 𝐷 (𝐼𝑎+ 𝑓). Of course, 𝐷𝑎+ The next theorem can be deduced from [3, Corollary 2.1, Lemma 2.5 (b), Lemma 2.6 (b)] but, to our best knowledge, it has not been formulated by other authors. In [4] we give a direct proof of it in the case of 𝑛 = 1, in [5]—when 𝑛 ≥ 2. In the case of 𝛼 = 𝑛 it reduces to the theorem on the integral representation of type (6) of functions belonging to 𝐴𝐶𝑛,1 . Theorem 7. If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, and 𝑓 ∈ 𝐿1 , then 𝑓 𝛼 𝑓 of order 𝛼 on has the left Riemann-Liouville derivative 𝐷𝑎+ 𝛼,1 the interval [𝑎, 𝑏] if and only if 𝑓 ∈ 𝐴𝐶𝑎+ ; that is, 𝑓 has the representation (11). In such a case, 𝑛−𝛼 𝐷𝑖 (𝐼𝑎+ 𝑓) (𝑎) = 𝑐𝑖 , 𝛼 𝑓) (𝑡) (𝐷𝑎+

= 𝜑 (𝑡) ,

𝑖 − 0, . . . , 𝑛 − 1, 𝑡 ∈ [𝑎, 𝑏] a.e.

(12)

𝛼,𝑝

By 𝐴𝐶𝑎+ (1 ≤ 𝑝 < ∞) we denote the set of all functions 𝑓 : [𝑎, 𝑏] → R𝑚 possessing representation (11) with 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 ∈ R𝑚 , 𝜑 ∈ 𝐿𝑝 . Remark 8. It is easy to see that the above theorem implies the following one (for any 1 ≤ 𝑝 < ∞): 𝑓 has the left Riemann𝛼,𝑝 𝛼 𝑓 ∈ 𝐿𝑝 if and only if 𝑓 ∈ 𝐴𝐶𝑎+ ; that Liouville derivative 𝐷𝑎+ 𝑝 is, 𝑓 has the representation (11) with 𝜑 ∈ 𝐿 . Let 𝛼 > 0. By the right Riemann-Liouville fractional integral of ℎ ∈ 𝐿1 on the interval [𝑎, 𝑏] we mean a function 𝑏 ℎ (𝜏) 1 𝛼 ℎ) (𝑡) = 𝑑𝜏, (𝐼𝑏− ∫ Γ (𝛼) 𝑡 (𝜏 − 𝑡)1−𝛼

𝑡 ∈ [𝑎, 𝑏] a.e.

(13)

Theorem 10. If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, and 𝑓 ∈ 𝐿1 , then 𝑓 𝛼 𝑓 of order 𝛼 on has the right Riemann-Liouville derivative 𝐷𝑏− 𝛼,1 the interval [𝑎, 𝑏] if and only if 𝑓 ∈ 𝐴𝐶𝑏− 𝑓; that is, 𝑓 has the representation (15). In such a case, 𝑛−𝛼 𝐷𝑖 (𝐼𝑏− 𝑓) (𝑏) = 𝑑𝑖 , 𝛼 𝑓) (𝑡) = 𝜓 (𝑡) , (𝐷𝑏−

𝑡 ∈ [𝑎, 𝑏] a.e.

Remark 11. As in the case of left Riemann-Liouville derivative, the following theorem holds true for any 1 ≤ 𝑝 < ∞: 𝑓 𝛼 𝑓 ∈ 𝐿𝑝 if and has the right Riemann-Liouville derivative 𝐷𝑏− 𝛼,𝑝 only if 𝑓 ∈ 𝐴𝐶𝑏− ; that is, 𝑓 has the representation (15) with 𝜓 ∈ 𝐿𝑝 . In the next section, we will use the following two theorems (cf. [6, 7]). Theorem 12. If 𝛼, 𝛽 > 0, then 𝛼,1 𝛽,1 ⊂ 𝐴𝐶𝑎+ 𝐴𝐶𝑎+

𝑛−1

𝑓 (𝑡) = ∑ (−1)𝑖 𝑖=0

+

𝑑𝑖 (𝑏 − 𝑡)𝛼−𝑛+𝑖 Γ (𝛼 − 𝑛 + 1 + 𝑖)

𝛼 𝐼𝑏− 𝜓 (𝜏) ,

(15)

𝑡 ∈ [𝑎, 𝑏] a.e.,

with 𝑑0 , 𝑑1 , . . . , 𝑑𝑛−1 ∈ R𝑚 and 𝜓 ∈ 𝐿1 . It is easy to see that 𝑛,1 = 𝐴𝐶𝑛,1 . 𝐴𝐶𝑏− We say that 𝑓 ∈ 𝐿1 possesses the right Riemann-Liouville 𝛼 𝑓 of order 𝛼 ∈ (𝑛 − 1, 𝑛], 𝑛 ∈ N, on the derivative 𝐷𝑏− 𝑛−𝛼 interval [𝑎, 𝑏] if 𝐼𝑏− 𝑓 ∈ 𝐴𝐶𝑛,1 . By this derivative we mean 1−𝛼 𝑛 the function (−1)𝑛 𝐷𝑛 (𝐼𝑏− 𝑓). Of course, 𝐷𝑏− 𝑓 = (−1)𝑛 𝐷𝑛 𝑓. We also have the following.

(17)

if and only if ∞

𝛼 ∈ ⋃ [𝛽 + 𝑖, 1 + 𝑖]

𝛽 ∈ (0, 1] ,

(18)

𝑖=0

or 𝛽 ∈ (1, ∞) ,

𝛼 = 𝛽 + 𝑖,

𝑖 = 0, 1, . . . .

(19)

Theorem 13. (a) If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 0 < 𝛽 < 𝛼 − 𝑛 + 1, 𝛼,1 and 𝑥 ∈ 𝐴𝐶𝑎+ is of the form (11), then 𝑡 ∈ [𝑎, 𝑏] a.e.,

(20)

where 𝜓 ∈ 𝐿1 is given by 𝑛−1

𝑐𝑖 (𝑡 − 𝑎)𝛼−𝑛−𝛽+𝑖 𝑖=0 Γ (𝛼 − 𝑛 + 1 − 𝛽 + 𝑖)

𝛼 𝜑 has the properties analogous to those Remark 9. Clearly, 𝐼𝑏− described in Remarks 5 and 6. 𝛼,1 𝛼,1 = 𝐴𝐶𝑏− ([𝑎, 𝑏], R𝑚 ) we denote the set of all By 𝐴𝐶𝑏− functions 𝑓 : [𝑎, 𝑏] → R𝑚 that have the representation

(16)

𝛼,𝑝

𝛽 𝜓 (𝑡) , 𝑥 (𝑡) = 𝐼𝑎+

(14)

𝑡 ∈ [𝑎, 𝑏] a.e.

By 𝐴𝐶𝑏− , 1 < 𝑝 < ∞, we denote the set of all functions 𝑓 : [𝑎, 𝑏] → R𝑚 possessing representation (15) with 𝑑0 , 𝑑1 , . . . , 𝑑𝑛−1 ∈ R𝑚 , 𝜓 ∈ 𝐿𝑝 .

Similarly, as in the case of left integral, we put 0 (𝐼𝑏− ℎ) (𝑡) = ℎ (𝑡) ,

𝑖 = 0, . . . , 𝑛 − 1,

𝜓 (𝑡) = ∑

𝛼−𝛽 + 𝐼𝑎+ 𝜑 (𝑡) ,

(21)

𝑡 ∈ [𝑎, 𝑏] a.e.

(b) If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, 𝛽 = 𝛼 − 𝑗, 𝑗 ∈ 𝛼,1 are of the form (11), then {1, . . . , 𝑛 − 1}, and 𝑥 ∈ 𝐴𝐶𝑎+ (𝑛−𝑗)−1

𝑥 (𝑡) = ∑ 𝑖=0

𝑑𝑖 (𝑡 − 𝑎)𝛽−(𝑛−𝑗)+𝑖 Γ (𝛽 − (𝑛 − 𝑗) + 1 + 𝑖)

𝛽 + 𝐼𝑎+ 𝜓 (𝑡) ,

(22)

𝑡 ∈ [𝑎, 𝑏] a.e.,

where 𝑑𝑖 = 𝑐𝑖 ,

𝑖 = 0, . . . , (𝑛 − 𝑗) − 1,

(23)

𝑗−1

𝑖 𝑗 𝜓 (𝑡) = ∑𝐼𝑎+ 𝑐𝑛−𝑗+𝑖 (𝑡) + 𝐼𝑎+ 𝜑 (𝑡) , 𝑖=0

𝑡 ∈ [𝑎, 𝑏] a.e.

(24)

4


3. Some Special Cases of Integration by Parts Below, 𝐶 = 𝐶([𝑎, 𝑏], R𝑚 ) is the set of continuous functions 𝑓 : [𝑎, 𝑏] → R𝑚 and 𝐶𝑛 = 𝐶𝑛 ([𝑎, 𝑏], R𝑚 ) is the set of functions 𝑓 : [𝑎, 𝑏] → R𝑚 such that 𝑓(𝑖) ∈ 𝐶, 𝑖 = 0, . . . , 𝑛 (by 𝑓(0) we mean the function 𝑓). Of course, these classical derivatives coincide with the weak ones 𝐷0 𝑓, 𝐷1 𝑓, . . . , 𝐷𝑛 𝑓. Lemma 14. If 𝑛 − 1 < 𝛼 < 𝑛, 𝑛 ∈ N, 𝑓 ∈ 𝐶𝑛 , and

𝛼,1 be of the form (11) with 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 ∈ Proof. Let 𝑓 ∈ 𝐴𝐶𝑎+ 𝑚 1 R and 𝜑 ∈ 𝐿 . We have (Lemma 14 and Remark 9) 𝑏

𝛼 𝑔) (𝑡) 𝑑𝑡 ∫ 𝑓 (𝑡) (𝐷𝑏− 𝑎

𝑏

𝑛−𝛼 𝑛 (𝐷𝑏− 𝑔)) (𝑡) 𝑑𝑡 = ∫ 𝑓 (𝑡) (𝐼𝑏− 𝑎

𝐷 𝑓 (𝑏) = 0,

𝑖 = 0, . . . , 𝑛 − 1,

(25)

𝑎

𝑛−𝛼 𝑛 × (𝐼𝑏− (𝐷𝑏− 𝑔)) (𝑡) 𝑑𝑡 𝑛−1

𝑛−𝛼 𝑓) (𝑏) = 0, 𝐷𝑖 (𝐼𝑏−

=

𝑐𝑖 𝛼 𝜑 (𝜏)) (𝑡 − 𝑎)𝛼−𝑛+𝑖 + 𝐼𝑎+ Γ − 𝑛 + 1 + 𝑖) (𝛼 𝑖=0

= ∫ (∑

𝑛−𝛼 then 𝐼𝑏− 𝑓 ∈ 𝐶𝑛 ,

𝛼 𝑓 𝐷𝑏−

𝑛−1

𝑏

𝑖

𝑛−𝛼 𝐼𝑏−

𝑛 (𝐷𝑏− 𝑓)

=

𝑖 = 0, . . . , 𝑛 − 1, 𝑛−𝛼 𝐼𝑏−

𝑛

𝑛

((−1) 𝐷 𝑓) ∈ 𝐶.

(26)

𝛼−𝑛+1+𝑖 𝑛−𝛼 𝑛 = ∑ 𝑐𝑖 𝐼𝑏− (𝐼𝑏− 𝐷𝑏− 𝑔) (𝑎) 𝑖=0

𝑏

𝛼 𝑛 𝑛 + ∫ (𝐷𝑎+ 𝑓) (𝑡) (𝐼𝑏− 𝐷𝑏− 𝑔) (𝑡) 𝑑𝑡 𝑎

𝛼 Consequently, 𝑓 ∈ 𝐼𝑏− (𝐶).

𝑛−1

1+𝑖 𝑛 = ∑ 𝑐𝑖 𝐼𝑏− 𝐷𝑏− 𝑔 (𝑎)

Proof. If 𝑓 ∈ 𝐶𝑛 and

𝑖=0

𝐷𝑖 𝑓 (𝑏) = 0,

𝑖 = 0, . . . , 𝑛 − 1,

(27)

𝑏

𝛼 + ∫ (𝐷𝑎+ 𝑓) (𝑡) 𝑔 (𝑡) 𝑑𝑡 𝑎

then (cf. Remark 6 and [3, formulas (2.21) and (2.58)])

𝑏

𝛼 𝑓) (𝑡) 𝑔 (𝑡) 𝑑𝑡. = ∫ (𝐷𝑎+ 𝑎

𝑛−𝛼 𝑛−𝛼 𝑛 𝑛 𝐼𝑏− 𝑓 (𝑡) = 𝐼𝑏− 𝐼𝑏− (𝐷𝑏− 𝑓) (𝑡) 𝑛 𝑛−𝛼 𝑛 = 𝐼𝑏− 𝐼𝑏− (𝐷𝑏− 𝑓) (𝑡)

=

𝑛 𝑛−𝛼 𝐼𝑏− 𝐼𝑏−

𝑛

(28)

𝑛

((−1) 𝐷 𝑓) (𝑡)

𝑛−𝛼 𝑓 ∈ 𝐶𝑛 for all 𝑡 ∈ [𝑎, 𝑏]. This means (cf. Remark 6) that 𝐼𝑏− 𝑛 𝑛 𝛼 𝑛−𝛼 and 𝐷𝑏− 𝑓 = 𝐼𝑏− ((−1) 𝐷 𝑓) ∈ 𝐶. Equalities (28) imply also the equalities

𝐷

𝑖

𝑛−𝛼 (𝐼𝑏− 𝑓) (𝑏)

(32)

Existence of the first integral and the first equality follows from Lemma 14; in the third equality we used an integral form of a classical fractional theorem on the integration by parts (cf. [3, formula (2.20)]), in the fourth equality we used continuity of 𝐷𝑛 𝑔 and in the fifth equality we used the equalities 𝑏

= 0,

𝑖 = 0, . . . , 𝑛 − 1.

(29)

1 𝑛 𝑛 (𝐷𝑏− 𝑔) (𝑡) = ∫ (𝐷𝑏− 𝑔) (𝑡) 𝑑𝑡 𝐼𝑏− 𝑡

𝑏

𝛼 So (cf. Theorem 7), 𝑓 ∈ 𝐼𝑏− (𝐶).

= (−1)𝑛 ∫ (𝐷𝑛 𝑔) (𝑡) 𝑑𝑡

Remark 15. Analogous lemma can be obtained for the left 𝑛−𝛼 𝛼 𝑓 and derivative 𝐷𝑎+ 𝑓 (without the term (−1)𝑛 ). integral 𝐼𝑎+

= (−1)𝑛 (𝐷𝑛−1 𝑔 (𝑏) − 𝐷𝑛−1 𝑔 (𝑡))

𝑡

We have the following special case of the theorem on integration by parts. Theorem 16. If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, then

(33)

= (−1)𝑛−1 𝐷𝑛−1 𝑔 (𝑡) 𝑛−1 = 𝐷𝑏− 𝑔 (𝑡) .

The proof is completed. 𝑏

𝑏

𝑎

𝑎

𝛼 𝛼 𝑔) (𝑡) 𝑑𝑡 = ∫ (𝐷𝑎+ 𝑓) (𝑡) 𝑔 (𝑡) 𝑑𝑡 ∫ 𝑓 (𝑡) (𝐷𝑏−

for any 𝑓 ∈

𝛼,1 𝐴𝐶𝑎+

(30)

𝑛

and 𝑔 ∈ 𝐶 satisfying boundary conditions

𝐷𝑖 𝑔 (𝑎) = 𝐷𝑖 𝑔 (𝑏) = 0,

𝑖 = 0, . . . , 𝑛 − 1.

(31)

The next theorem will be used in the last section of the paper. Proof of this theorem is contained in [4] and in [5] its extension to the case of fractional derivatives of higher order is derived (the method of the proof is the same in both papers).


5

Theorem 17. If 0 < 𝛼 < 1, 𝛼 > 1/𝑝, 𝛼 > 1/𝑞, 1 ≤ 𝑝 < ∞, and 1 ≤ 𝑞 < ∞, then

𝛼,𝑝

Now, let us assume that 𝑢 ∈ 𝑊𝑎+ , that is, 𝑢 ∈ 𝐿𝑝 , and there exists a function 𝑔 ∈ 𝐿𝑝 such that

𝑏

𝛼 1−𝛼 𝑔) (𝑡) 𝑑𝑡 = (𝐼𝑎+ 𝑓) (𝑎) 𝑔 (𝑎) ∫ 𝑓 (𝑡) (𝐷𝑏− 1−𝛼 − (𝐼𝑏− 𝑔) (𝑏) 𝑓 (𝑏)

(34)

𝑏

𝛼 𝑓) (𝑡) 𝑔 (𝑡) 𝑑𝑡 + ∫ (𝐷𝑎+ 𝑎

for 𝑓 ∈

𝛼,𝑝 𝐴𝐶𝑎+ ([𝑎, 𝑏], R𝑛 )

𝛼,𝑞

and 𝑔 ∈ 𝐴𝐶𝑏− ([𝑎, 𝑏], R𝑛 ).

4. Fractional Sobolev Spaces Let 0 < 𝛼 ≤ 1 and let 1 ≤ 𝑝 < ∞. By left Sobolev space of 𝛼,𝑝 𝛼,𝑝 order 𝛼 we will mean the set 𝑊𝑎+ = 𝑊𝑎+ ((𝑎, 𝑏), R𝑚 ) given by 𝛼,𝑝 𝛼 = {𝑢 ∈ 𝐿𝑝 ; ∃ 𝑝 ∀ ∞ ∫ 𝑢 (𝑡) 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡 𝑊𝑎+ 𝑎

𝑎

𝑏

1−𝛼 (−𝐷1 𝜑)) (𝑡) 𝑑𝑡 = ∫ 𝑢 (𝑡) (𝐼𝑏− 𝑎

𝑏

1−𝛼 1−𝛼 1−𝛼 𝐼𝑎+ 𝑢) (𝑡) (𝐼𝑏− (−𝐷1 𝜑)) (𝑡) 𝑑𝑡 = ∫ (𝐷𝑎+

A function 𝑔 given above will be called the weak left fractional 𝛼 . derivative of order 𝛼 ∈ (0, 1] of 𝑢; let us denote it by 𝑢𝑎+ Uniqueness of this weak derivative follows from [1, Lemma IV.2 and Propositions IV.18, IV.21]. Let 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2. By the left Sobolev space 𝛼,𝑝 𝛼,𝑝 of order 𝛼 we mean the set 𝑊𝑎+ = 𝑊𝑎+ ([𝑎, 𝑏], R𝑚 ) given by 𝛼,𝑝 𝛼−1,𝑝 𝛼−(𝑛−1) 𝑛−1,𝑝 }. 𝑊𝑎+ = {𝑢 ∈ 𝑊𝑎+ ; 𝑢𝑎+ ∈ 𝑊𝑎+ (1)

(36)

𝐶𝑐∞ ,

= −𝐷 𝜑 = −𝜑 for 𝜑 ∈ we see Remark 18. Since 1 that the weak left fractional derivative 𝑢𝑎+ of 𝑢 coincides with the classical weak derivative 𝑢󸀠 = 𝐷1 𝑢 of 𝑢. Consequently, 𝑛,𝑝 𝑛,𝑝 𝑊𝑎+ = 𝑊𝑛,𝑝 = 𝐴𝐶𝑛,𝑝 = 𝐴𝐶𝑎+

(37)

for 𝑛 ∈ N. 𝛼,𝑝

We have the following characterization of 𝑊𝑎+ . Theorem 19. If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, and 1 ≤ 𝑝 < ∞, then 𝛼,𝑝 𝛼,𝑝 𝑊𝑎+ = 𝐴𝐶𝑎+ ∩ 𝐿𝑝 .

(38)

Proof. Case of 𝛼 = 𝑛 follows from (37) and from the fact that 𝑛,𝑝 𝐴𝐶𝑎+ ⊂ 𝐿𝑝 . So, let us consider the case of 𝑛 − 1 < 𝛼 < 𝑛. We will apply the induction with respect to 𝑛 ∈ N. 𝛼,𝑝 Let 𝑛 = 1. If 𝑢 ∈ 𝐴𝐶𝑎+ ∩ 𝐿𝑝 , then from Theorem 7 𝛼 𝑢 ∈ 𝐿𝑝 . Theorem 16 it follows that 𝑢 has the derivative 𝐷𝑎+ implies that ∫

𝑎

for any 𝜑 ∈

𝛼 𝑢 (𝑡) 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡

𝐶𝑐∞ .

So, 𝑢 ∈

=∫

𝛼,𝑝 𝑊𝑎+

𝑏

𝑎

(41)

𝑏

𝑎

𝑏

𝑎

𝛼,𝑝

= ∫ 𝑔 (𝑡) 𝜑 (𝑡) 𝑑𝑡} .

1

𝑎

for any 𝜑 ∈ 𝐶𝑐∞ . To show that 𝑢 ∈ 𝐴𝐶𝑎+ ∩ 𝐿𝑝 it is sufficient to 𝛼,𝑝 check (cf. Theorem 7 and definition of 𝐴𝐶𝑎+ ) that 𝑢 possesses the left Riemann-Liouville derivative of order 𝛼, belonging to 1−𝛼 𝐿𝑝 , that is, that 𝐼𝑎+ 𝑢 is absolutely continuous on [𝑎, 𝑏] and its classical derivative of the first order (existing a.e. on [𝑎, 𝑏]) belongs to 𝐿𝑝 . 𝛼 (𝐶) and If 𝜑 ∈ 𝐶𝑐∞ , then (cf. Lemma 14) 𝜑 ∈ 𝐼𝑏− 𝛼 1−𝛼 1 𝐷𝑏− 𝜑 = 𝐼𝑏− (−𝐷 𝜑). From the differential form of the classical fractional theorem on the integration by parts (cf. [3, formula (2.64)]) it follows that

(35)

𝑏

1 𝜑 𝐷𝑏−

𝑏

𝛼 𝜑 (𝑡) 𝑑𝑡 ∫ 𝑢 (𝑡) 𝐷𝑏−

𝑏

𝑔∈𝐿 𝜑∈𝐶𝑐

𝑏

𝛼 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝑔 (𝑡) 𝜑 (𝑡) 𝑑𝑡 ∫ 𝑢 (𝑡) 𝐷𝑏−

𝑎

𝛼 (𝐷𝑎+ 𝑢) (𝑡) 𝜑 (𝑡) 𝑑𝑡

(39)

(42)

𝑏

1−𝛼 𝑢) (𝑡) (−𝐷1 𝜑) (𝑡) 𝑑𝑡 = ∫ (𝐼𝑎+ 𝑎

𝑏

1−𝛼 𝑢) (𝑡) (𝐷1 𝜑) (𝑡) 𝑑𝑡. = − ∫ (𝐼𝑎+ 𝑎

This means (cf. (41) and (42)) that 𝑏

𝑏

𝑎

𝑎

1−𝛼 𝑢) (𝑡) (𝐷1 𝜑) (𝑡) 𝑑𝑡 = − ∫ 𝑔 (𝑡) 𝜑 (𝑡) 𝑑𝑡 ∫ (𝐼𝑎+

(43)

1−𝛼 1−𝛼 for any 𝜑 ∈ 𝐶𝑐∞ . So, 𝐼𝑎+ 𝑢 ∈ 𝑊1,𝑝 . Consequently, 𝐼𝑎+ 𝑢 is absolutely continuous and its classical derivative is equal a.e. on [𝑎, 𝑏] to 𝑔 ∈ 𝐿𝑝 . Now, let 𝑛 ∈ N and assume that 𝛼,𝑝 𝛼,𝑝 = 𝐴𝐶𝑎+ ∩ 𝐿𝑝 𝑊𝑎+

(44)

for any 𝛼 ∈ (𝑛 − 1, 𝑛). We will show that this equality holds true for any 𝛼 ∈ (𝑛, 𝑛 + 1). 𝛼,𝑝 𝛼−1,𝑝 Indeed, let 𝛼 ∈ (𝑛, 𝑛 + 1). If 𝑢 ∈ 𝑊𝑎+ , then 𝑢 ∈ 𝑊𝑎+ = 𝛼−1,𝑝 𝑛,𝑝 𝛼−𝑛 𝐴𝐶𝑎+ ∩ 𝐿𝑝 and 𝑢𝑎+ ∈ 𝑊𝑎+ . So, 𝑛−1

𝑐𝑖 (𝑡 − 𝑎)𝛼−1−𝑛+𝑖 Γ − 𝑖=0 (𝛼 𝑛 + 𝑖)

𝑢 (𝑡) = ∑

𝛼−1 + 𝐼𝑎+ 𝜑 (𝑡) ,

(45)

𝑡 ∈ [𝑎, 𝑏] a.e.,

where 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 ∈ R𝑚 and 𝜑 ∈ 𝐿𝑝 . If 𝑛 = 1, from the above formula it follows (cf. (40)) that

with

𝛼 𝛼 = 𝑔 = 𝐷𝑎+ 𝑢 ∈ 𝐿𝑝 . 𝑢𝑎+

𝑎

(40)

𝛼−1 𝛼−1 1,𝑝 𝑢 = 𝑢𝑎+ ∈ 𝑊𝑎+ = 𝑊1,𝑝 . 𝜑 = 𝐷𝑎+

(46)

6


If 𝑛 ≥ 2, from Theorem 13 (b) and (40) it follows that

where 𝑛

𝑐𝑖 (𝑡 − 𝑎)𝛼−(𝑛+1)+𝑖 Γ − 𝑛 + 𝑖) (𝛼 𝑖=0

𝛼−𝑛 𝛼−𝑛 (𝛼−1)−(𝑛−1) = 𝐷𝑎+ 𝑢 = 𝐷𝑎+ 𝑢 𝑢𝑎+

=

𝑛−2

𝑖 𝑐1+𝑖 ∑ 𝐼𝑎+ 𝑖=0

+

𝑢 (𝑡) = ∑ (47)

𝑛−1 𝐼𝑎+ 𝜑.

+

𝛼−1,𝑝

𝑛−1 𝑛−1 𝑛−1 𝑛,𝑝 𝜑 = 𝐷𝑎+ 𝐼𝑎+ 𝜑 = 𝐷𝑛−1 𝐼𝑎+ 𝜑 ∈ 𝐷𝑛−1 (𝑊𝑎+ )

𝑖=0

1 𝜑 = 𝑐𝑛 + 𝐼𝑎+ 𝜆.

𝑛−1

𝛼,𝑝

𝑛+1−𝛼 𝑛+1,𝑝 𝐼𝑎+ 𝑢 ∈ 𝐴𝐶𝑎+ .

(50)

Indeed, we have (below, we use the following elementary formula Γ (𝛿) (𝑡 − 𝑎)]+𝛿−1 , Γ (] + 𝛿)

(51)

𝑡 ∈ [𝑎, 𝑏] a.e., for ] > 0, 𝛿 > 0)

𝑛−1

𝑐𝑖 Γ (𝛼 − 𝑛 + 𝑖) (𝑡 − 𝑎)𝑖 Γ − 𝑛 + 𝑖) Γ + 𝑖) (𝛼 (1 𝑖=0

𝑐𝑖 𝑛 𝑛+1 + 𝐼𝑎+ 𝑐𝑛 (𝑡) + 𝐼𝑎+ 𝜆 (𝑡) Γ + 𝑖) (1 𝑖=0

=∑ 𝑛

(52)

have to check that

𝑛,𝑝 𝑊𝑎+ .

𝛼,𝑝

∈ 𝑊𝑎+ we (53)

Theorem 13(b) implies the equality 𝛼−1 1 𝑢 = 𝑐𝑛 + 𝐼𝑎+ 𝜑, 𝐷𝑎+

𝑏

𝑏

𝑎

𝑎

𝛼−(𝑛−𝑖) ∫ 𝑢 (𝑡) 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝑔𝑖 (𝑡) 𝜑 (𝑡) 𝑑𝑡,

𝑐 𝑛+1 = ∑ 𝑖 (𝑡 − 𝑎)𝑖 + 𝐼𝑎+ 𝜆 (𝑡) 𝑖! 𝑖=0

∈

Theorem 20. If 𝛼 ∈ (0, 1], then the weak left fractional 𝛼,𝑝 𝛼 of a function 𝑢 ∈ 𝑊𝑎+ coincides with its left derivative 𝑢𝑎+ 𝛼 𝑢 a.e. on [𝑎, 𝑏]. Riemann-Liouville fractional derivative 𝐷𝑎+

Theorem 22. Let 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 1 ≤ 𝑝 < ∞, and 𝛼,𝑝 𝑢 ∈ 𝐿𝑝 . Then, 𝑢 ∈ 𝑊𝑎+ if and only if there exist functions 𝑝 𝑔1 , . . . , 𝑔𝑛 ∈ 𝐿 such that

𝑛−1

𝑢∈

From the first part of the above proof (case of 𝑛 = 1) and from the uniqueness of the weak fractional derivative the following theorem follows (cf. also [4, fractional fundamental lemma]).

Now, we will prove an extension of Theorem 1.

𝑛 𝐼𝑎+ 𝜑 (𝑡)

𝛼−𝑛 𝑢𝑎+

𝛼−𝑛 and equality (existence of the weak fractional derivative 𝑢𝑎+ 𝛼−𝑛,𝑝 𝛼−𝑛 𝛼−𝑛 𝑢𝑎+ = 𝐷𝑎+ 𝑢 follow from the relation 𝑢 ∈ 𝑊𝑎+ and (40)).

5. Weak Fractional Derivatives

=∑

𝛼−1,𝑝 , 𝑊𝑎+

𝑛,𝑝 = 𝑊𝑎+

𝛼,𝑝

𝑐𝑖 𝑛+1−𝛼 𝐼𝑎+ =∑ ((⋅ − 𝑎)𝛼−1−𝑛+𝑖 ) (𝑡) Γ − 𝑛 + 𝑖) (𝛼 𝑖=0

𝑛+1,𝑝 𝑛+1−𝛼 𝑢 ∈ 𝐴𝐶𝑎+ . 𝑡 ∈ [𝑎, 𝑏] a.e. Thus, 𝐼𝑎+ 𝛼,𝑝 Conversely, let 𝑢 ∈ 𝐴𝐶𝑎+ ∩ 𝐿𝑝 . To show that 𝑢

(56)

Remark 21. If 𝑛−1 < 𝛼 ≤ 𝑛 and (𝑛−𝛼)𝑝 < 1, then 𝐴𝐶𝑎+ ⊂ 𝐿𝑝 𝛼,𝑝 𝛼,𝑝 𝛼,𝑝 and, consequently, 𝑊𝑎+ = 𝐴𝐶𝑎+ ∩𝐿𝑝 = 𝐴𝐶𝑎+ . If (𝑛−𝛼)𝑝 ≥ 1, 𝛼,𝑝 𝛼,𝑝 𝑝 then 𝑊𝑎+ = 𝐴𝐶𝑎+ ∩ 𝐿 is the set of all functions belonging 𝛼,𝑝 𝑛−𝛼 𝑓)(𝑎) = 0. to 𝐴𝐶𝑎+ that satisfy the condition (𝐼𝑎+

𝑛−1

𝑛+1−𝛼 𝛼−1 + 𝐼𝑎+ 𝐼𝑎+ 𝜑 (𝑡)

for

𝑐1+𝑖 𝑛 𝑛,𝑝 𝜑 ∈ 𝐴𝐶𝑎+ (𝑡 − 𝑎)𝑖 + 𝐼𝑎+ 𝑖! 𝑖=0

=∑

(49)

To show that 𝑢 ∈ 𝐴𝐶𝑎+ ∩ 𝐿𝑝 it is sufficient to check that

+

.

𝑛−1

So, in both cases (𝑛 = 1, 𝑛 ≥ 2) there exist 𝑐𝑛 ∈ R𝑚 and 𝜆 ∈ 𝐿𝑝 such that

𝑛+1−𝛼 𝐼𝑎+ 𝑢 (𝑡)

𝛼−1,𝑝

∩ 𝐿𝑝 = 𝑊𝑎+

𝛼−𝑛 𝛼−𝑛 𝑖 𝑛 𝑢𝑎+ = 𝐷𝑎+ 𝑢 = ∑ 𝐼𝑎+ 𝑐1+𝑖 + 𝐼𝑎+ 𝜑

(48)

= 𝐷𝑛−1 (𝑊𝑛,𝑝 ) = 𝑊1,𝑝 .

(55)

𝑡 ∈ [𝑎, 𝑏] a.e.,

with 𝜑 ∈ 𝐿𝑝 . So (cf. Theorem 12), 𝑢 ∈ 𝐴𝐶𝑎+ From Theorem 13 (b) it also follows that

𝑛,𝑝

𝑛−1 This means that 𝐼𝑎+ 𝜑 ∈ 𝑊𝑎+ and, consequently,

] ((⋅ − 𝑎)𝛿−1 ) (𝑡) = 𝐼𝑎+

𝛼 𝐼𝑎+ 𝜑 (𝑡) ,

(54)

𝜑 ∈ 𝐶𝑐∞ , (57)

for any 𝑖 ∈ {1, . . . , 𝑛}. In such a case there exist the left 𝛼−(𝑛−1) 𝛼−(𝑛−2) 𝛼 Riemann-Liouville derivatives 𝐷𝑎+ 𝑢, 𝐷𝑎+ 𝑢, . . . , 𝐷𝑎+ 𝑢 of 𝑢 and 𝛼−(𝑛−1) 𝑢, 𝑔1 = 𝐷𝑎+

𝛼−(𝑛−2) 𝑔2 = 𝐷𝑎+ 𝑢, . . . ,

𝛼 𝑔𝑛 = 𝐷𝑎+ 𝑢. (58)

Proof. Case of 𝛼 = 𝑛 follows from Theorem 1. So, let us consider the case of 𝑛−1 < 𝛼 < 𝑛. We will apply the induction with respect to 𝑛 ∈ N.


7 𝛼,𝑝

When 𝑛 = 1, it is sufficient to use the definition of 𝑊𝑎+ with 𝛼 ∈ (0, 1) and Theorem 20. So, let 𝑛 ∈ N and assume that the theorem is true for any 𝛼 ∈ (𝑛−1, 𝑛). We will show that it is true for any 𝛼 ∈ (𝑛, 𝑛+1). 𝛼,𝑝 Indeed, if 𝑢 ∈ 𝑊𝑎+ with a fixed 𝛼 ∈ (𝑛, 𝑛 + 1), then 𝛼−1,𝑝 𝑢 ∈ 𝑊𝑎+ , 𝛼−𝑛 𝑢 𝐷𝑎+

=

𝛼−𝑛 𝑢𝑎+

∈

Moreover, condition (64) for 𝑖 = 1, Lemma 14, and the integral form of the classical fractional theorem on the integration by parts (cf. [3, formula (2.20)]) imply that 𝑏

𝑏

𝑎

𝑎

𝛼−𝑛 𝜑 (𝑡) 𝑑𝑡 ∫ 𝑔1 (𝑡) 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝑢 (𝑡) 𝐷𝑏−

(59)

𝑛,𝑝 𝑊𝑎+ .

𝑏

𝑛−𝛼+1 1 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝑢 (𝑡) 𝐼𝑏−

(60)

𝑎

Relation (59) implies (by the induction assumption) the existence of functions 𝑔1 , . . . , 𝑔𝑛 ∈ 𝐿𝑝 such that (of course, 𝛼 − 1 ∈ (𝑛 − 1, 𝑛)) 𝑏

𝑏

𝑎

𝑎

𝛼−((𝑛+1)−𝑖) (𝛼−1)−(𝑛−𝑖) 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝑢 (𝑡) 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡 ∫ 𝑢 (𝑡) 𝐷𝑏− 𝑏

= ∫ 𝑔𝑖 (𝑡) 𝜑 (𝑡) 𝑑𝑡 𝑎

(61)

𝑔1 =

=

𝑏

𝑎

(62)

𝑏

(𝑛+1)−𝛼 𝑖 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝑢 (𝑡) 𝐼𝑏− 𝑎

𝑏

(𝑛+1)−𝛼 𝑖 𝑢 (𝑡) 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝐼𝑎+ 𝑎

𝛼−𝑛 𝑢 and from (60) it follows that From the existence of 𝐷𝑎+ 1−(𝛼−𝑛) 𝑛+1,𝑝 𝐼𝑎+ 𝑢 ∈ 𝑊 . In particular, there exists derivative 𝛼 𝛼 𝑢 ∈ 𝐿𝑝 . If we put 𝑔𝑛+1 = 𝐷𝑎+ 𝑢 we see that 𝑔𝑛+1 = 𝐷𝑎+ 𝛼−((𝑛+1)−(𝑛+1)) 𝐷𝑎+ 𝑢 and

𝑎

𝑛−𝛼+1 for any 𝜑 ∈ This means that 𝐼𝑎+ 𝑢 ∈ 𝑊1,𝑝 . So, if we fix 𝑖 ∈ {2, . . . , 𝑛+1} and use once again Lemma 14 and the integral form of the classical fractional theorem on the integration by parts, we obtain

𝛼−((𝑛+1)−𝑖) 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝑢 (𝑡) 𝐷𝑏−

(𝛼−1)−(𝑛−𝑛) 𝛼−((𝑛+1)−𝑛) 𝑢 = 𝐷𝑎+ 𝑢. 𝑔𝑛 = 𝐷𝑎+

∫

𝑎

𝐶𝑐∞ .

𝑎

𝛼−((𝑛+1)−1) 𝐷𝑎+ 𝑢,

.. .

𝛼−((𝑛+1)−(𝑛+1)) 𝑢 (𝑡) 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡

𝑏

𝑛−𝛼+1 𝑢 (𝑡) 𝐷1 𝜑 (𝑡) 𝑑𝑡 = − ∫ 𝐼𝑎+

∫ 𝑔𝑖 (𝑡) 𝜑 (𝑡) 𝑑𝑡

(𝛼−1)−(𝑛−2) 𝛼−((𝑛+1)−2) 𝑢 = 𝐷𝑎+ 𝑢, 𝑔2 = 𝐷𝑎+

𝑏

𝑎

(66) 𝑛−𝛼+1 1 𝐼𝑎+ 𝑢 (𝑡) 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡

𝑏

for any 𝜑 ∈ 𝐶𝑐∞ and 𝑖 ∈ {1, . . . , 𝑛}. Moreover, (𝛼−1)−(𝑛−1) 𝐷𝑎+ 𝑢

=∫

𝑏

=∫

𝑏

𝑎

=∫

𝑏

𝑎

𝛼−((𝑛+1)−𝑖) 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝑔𝑖 (𝑡) 𝜑 (𝑡) 𝑑𝑡, ∫ 𝑢 (𝑡) 𝐷𝑏−

(63)

𝑏

𝛼−𝑛 𝑢 (𝑡) 𝐷𝑖−1 𝜑 (𝑡) 𝑑𝑡. = (−1)𝑖−1 ∫ 𝐷𝑎+ 𝑎

Thus, from Theorem 1 it follows that

𝜑 ∈ 𝐶𝑐∞ , (64)

for any 𝑖 ∈ {1, . . . , 𝑛 + 1}. Since 𝛼 − ((𝑛 + 1) − 𝑖) = 𝛼 − 1 − (𝑛 − 𝑖), the above condition for 𝑖 ∈ {1, . . . , 𝑛} and the induction assumption mean that 𝛼−1,𝑝 . 𝑢 ∈ 𝑊𝑎+

𝑏

(𝑛+1)−𝛼 𝑢 (𝑡) 𝐷1 𝐷𝑖−1 𝜑 (𝑡) 𝑑𝑡 = (−1)𝑖 ∫ 𝐼𝑎+

𝑎

𝛼 𝐷𝑎+ 𝑢 (𝑡) 𝜑 (𝑡) 𝑑𝑡

for any 𝜑 ∈ 𝐶𝑐∞ (the second equality follows from Theorem 16). Now, let us assume that there exist functions 𝑔1 , . . . , 𝑔𝑛+1 ∈ 𝐿𝑝 such that

𝑎

𝑎

𝑏

𝑎

𝑎

= (−1) ∫

(67) (𝑛+1)−𝛼 𝐼𝑎+ 𝑢 (𝑡) 𝐷𝑖 𝜑 (𝑡) 𝑑𝑡

1−(𝛼−𝑛) 𝑢 (𝑡) 𝐷𝑖−1 𝜑 (𝑡) 𝑑𝑡 = (−1)𝑖−1 ∫ 𝐷1 𝐼𝑎+

= ∫ 𝑔𝑛+1 (𝑡) 𝜑 (𝑡) 𝑑𝑡

𝑏

𝑏

𝑎

𝛼 𝑢 (𝑡) 𝐷𝑏− 𝜑 (𝑡) 𝑑𝑡

𝑏

𝑏

𝑖

(65)

𝛼−𝑛 𝛼−𝑛 = 𝐷𝑎+ 𝑢 ∈ 𝑊𝑛,𝑝 . 𝑢𝑎+ 𝛼−1,𝑝

Using (65) and (68) we assert that 𝑢 ∈ 𝑊𝑎+

(68) .

Functions 𝑔1 , . . . , 𝑔𝑛 will be called the weak left fractional 𝛼,𝑝 derivatives of 𝑢 ∈ 𝑊𝑎+ of orders 𝛼 − (𝑛 − 1), . . . , 𝛼, respectively. Their uniqueness follows from [1, Lemma IV.2 and Propositions IV.18,IV.21]. From the above theorem it follows that they coincide with the appropriate RiemannLiouville derivatives. We have the following counterpart of Remark 2. 𝛼,𝑝

Theorem 23. If 𝑢 ∈ 𝑊𝑎+ , 𝑛 − 1 < 𝛼 ≤ 𝑛, and 𝑛 ∈ N, then 𝛼−(𝑛−1) 𝑔𝑖 = 𝐷𝑖−1 (𝐷𝑎+ 𝑢) ,

𝑖 = 1, . . . , 𝑛.

(69)

8

Journal of Function Spaces and Applications 𝛼,𝑝

𝑛−𝛼 Proof. If 𝑢 ∈ 𝑊𝑎+ , then 𝐼𝑎+ 𝑢 ∈ 𝑊𝑛,𝑝 and from the definition of the Riemann-Liouville derivative and Theorem 22 it follows that

where 𝐾 = (𝑏 − 𝑎)𝛼 /Γ(𝛼 + 1) (cf. [3, formula (2.72)]). Thus (of 1−𝛼 𝛼 𝑢(𝑎) and 𝜑 = 𝐷𝑎+ 𝑢), course, 𝑐 = 𝐼𝑎+ 𝑝 󵄩 󵄩𝑝 ‖𝑢‖𝐿𝑝 ≤ 𝐿 𝛼,0 (|𝑐|𝑝 + 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝐿𝑝 )

𝛼−𝑛+1 1−(𝛼−𝑛+1) 𝑛−𝛼 𝑢 = 𝐷1 𝐼𝑎+ 𝑢 = 𝐷1 (𝐼𝑎+ 𝑢) , 𝑔1 = 𝐷𝑎+

𝑔2 =

𝛼−𝑛+2 𝐷𝑎+ 𝑢

=𝐷

1

=

2−(𝛼−𝑛+2) 𝐷2 𝐼𝑎+ 𝑢

𝑛−𝛼 (𝐷1 𝐼𝑎+ 𝑢)

=𝐷

1

=

𝑔𝑛 =

=

𝑝

𝑎+

(70)

where (𝑏 − 𝑎)1−(1−𝛼)𝑝 + 𝐾𝑝 ) . Γ(𝛼)𝑝 (1 − (1 − 𝛼) 𝑝)

(76)

𝑝 𝑝 𝑝 󵄩 𝛼 󵄩󵄩𝑝 𝑢󵄩󵄩𝐿𝑝 ≤ 𝐿 𝛼,1 ‖𝑢‖𝑎,𝑊𝛼,𝑝 , ‖𝑢‖𝑊𝛼,𝑝 = ‖𝑢‖𝐿𝑝 + 󵄩󵄩󵄩𝐷𝑎+ 𝑎+ 𝑎+

(77)

𝐿 𝛼,0 = 2𝑝−1 (

𝑛−(𝛼−𝑛+𝑛) 𝐷𝑛 𝐼𝑎+ 𝑢

𝑛−𝛼 𝛼−𝑛+1 = 𝐷𝑛−1 (𝐷1 𝐼𝑎+ 𝑢) = 𝐷𝑛−1 (𝐷𝑎+ 𝑢) .

Consequently,

The proof is completed. 𝛼,𝑝

6. Norms in 𝑊𝑎+

Let us fix 𝛼 ∈ (𝑛 − 1, 𝑛] where 𝑛 ∈ N and consider in the space 𝛼,𝑝 𝑊𝑎+ a norm ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 given by

where 𝐿 𝛼,1 = 𝐿 𝛼,0 + 1. Now, we will prove that there exists a constant 𝑀𝛼,1 > 0 such that 𝑝

𝑝

‖𝑢‖𝑎,𝑊𝛼,𝑝 ≤ 𝑀𝛼,1 ‖𝑢‖𝑊𝛼,𝑝 , 𝑎+

𝑛−1 󵄩 𝛼−(𝑛−1)+𝑖 󵄩󵄩𝑝 𝑝 𝑝 𝑢󵄩󵄩󵄩𝐿𝑝 , ‖𝑢‖𝑊𝛼,𝑝 = ‖𝑢‖𝐿𝑝 + ∑ 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑎+

𝑖=0

𝛼,𝑝 𝑢 ∈ 𝑊𝑎+

(71)

(here ‖ ⋅ ‖𝐿𝑝 denotes the classical norm in 𝐿𝑝 ). We have the following theorem.

𝑛−1

󵄨 𝑛−𝛼 󵄨𝑝 󵄩 𝛼 󵄩󵄩𝑝 = ∑ 󵄨󵄨󵄨󵄨𝐷𝑖 𝐼𝑎+ 𝑢(𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝 ,

𝑢∈

𝑖=0

𝛼,𝑝 𝑊𝑎+ .

𝑐 1 𝛼 𝑢 (𝑡) = + 𝐼𝑎+ 𝜑 (𝑡) Γ (𝛼) (𝑡 − 𝑎)1−𝛼

≤ 2𝑝−1 (

𝑡

𝑖

𝑡0

(80)

for any 𝑡 ∈ [𝑎, 𝑏]. Consequently, 𝑡 󵄨󵄨 1−𝛼 𝑖 󵄨󵄨 󵄨󵄨(𝐼 𝑢) (𝑡)󵄨󵄨 ≤ 1 󵄩󵄩󵄩󵄩𝐼1−𝛼 𝑢󵄩󵄩󵄩󵄩 1 + ∫ 󵄨󵄨󵄨𝐷𝛼 𝑢 (𝑠)󵄨󵄨󵄨 𝑑𝑠 󵄨 󵄨󵄨 𝑏 − 𝑎 󵄩 𝑎+ 󵄩𝐿 󵄨󵄨 𝑎+ 󵄨 𝑎+ 𝑡0

≤

1 󵄩󵄩 1−𝛼 󵄩󵄩 󵄩󵄩𝐼 𝑢󵄩󵄩 + 󵄩󵄩󵄩𝐷𝛼 𝑢󵄩󵄩󵄩 1 𝑏 − 𝑎 󵄩 𝑎+ 󵄩𝐿1 󵄩 𝑎+ 󵄩𝐿

≤

1 (𝑏 − 𝑎)1−𝛼 󵄩 𝛼 󵄩󵄩 𝑢󵄩󵄩𝐿1 ‖𝑢‖𝐿1 + 󵄩󵄩󵄩𝐷𝑎+ 𝑏 − 𝑎 Γ (2 − 𝛼)

(81)

for 𝑡 ∈ [𝑎, 𝑏]. In particular, 1−𝛼 󵄨󵄨 1−𝛼 𝑖 󵄨󵄨 󵄨󵄨(𝐼 𝑢) (𝑎)󵄨󵄨 ≤ 1 (𝑏 − 𝑎) ‖𝑢‖𝐿1 + 󵄩󵄩󵄩𝐷𝛼 𝑢󵄩󵄩󵄩 1 . 󵄨󵄨 𝑎+ 󵄨󵄨 𝑏 − 𝑎 Γ (2 − 𝛼) 󵄩 𝑎+ 󵄩𝐿

𝑏

1 |𝑐|𝑝 󵄩 󵄩𝑝 ≤ 2𝑝−1 ( (𝑏 − 𝑎)(𝛼−1)𝑝+1 +𝐾𝑝 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝐿𝑝 ) , Γ(𝛼)𝑝 (𝛼 − 1)𝑝 + 1 (74)

𝑖

1−𝛼 1−𝛼 1−𝛼 (𝐼𝑎+ 𝑢) (𝑡) = (𝐼𝑎+ 𝑢) (𝑡0 ) + ∫ 𝐷1 (𝐼𝑎+ 𝑢) (𝑠) 𝑑𝑠

(73)

|𝑐| 󵄩 𝛼 󵄩󵄩𝑝 (𝛼−1)𝑝 𝑑𝑡 + 󵄩󵄩󵄩𝐼𝑎+ 𝜑󵄩󵄩𝐿𝑝 ) 𝑝 ∫ (𝑡 − 𝑎) Γ(𝛼) 𝑎

(79)

𝑖

with 𝑐 ∈ R𝑛 and 𝜑 ∈ 𝐿𝑝 , we have

𝑝

𝑏 1 1−𝛼 𝑖 𝑢) (𝑠) 𝑑𝑠. ∫ (𝐼𝑎+ 𝑏−𝑎 𝑎

1−𝛼 From the absolute continuity of (𝐼𝑎+ 𝑢) it follows that

(72)

󵄨󵄨𝑝 󵄨󵄨 1 󵄨󵄨 󵄨 𝑐 𝛼 󵄨󵄨 𝑑𝑡 = ∫ 󵄨󵄨󵄨󵄨 + 𝐼 𝜑(𝑡) 𝑎+ 1−𝛼 󵄨󵄨 𝑎 󵄨󵄨 Γ(𝛼) (𝑡 − 𝑎) 󵄨

(78)

𝛼,𝑝

𝑖

𝑏

𝛼,𝑝 𝑢 ∈ 𝑊𝑎+ .

Indeed, let 𝑢 ∈ 𝑊𝑎+ and consider a coordinate function 1−𝛼 𝑖 1−𝛼 (𝐼𝑎+ 𝑢) of 𝐼𝑎+ 𝑢 with a fixed 𝑖 ∈ {1, . . . , 𝑚}. The mean value theorem implies the existence of 𝑡0 ∈ (𝑎, 𝑏) such that 𝑖

Proof. We will use the induction with respect to 𝑛 ∈ N. 𝛼,𝑝 Assume that 𝑛 = 1 and (1 − 𝛼)𝑝 < 1. Then, for 𝑢 ∈ 𝑊𝑎+ given by

𝑝 ‖𝑢‖𝐿𝑝

𝑎+

1−𝛼 (𝐼𝑎+ 𝑢) (𝑡0 ) =

Theorem 24. If 𝑛 ∈ N and 𝛼 ∈ (𝑛 − 1, 𝑛], then the norm ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 is equivalent to a norm ‖𝑢‖𝑎,𝑊𝑎+𝛼,𝑝 given by 𝑝 ‖𝑢‖𝑎,𝑊𝛼,𝑝 𝑎+

(75)

= 𝐿 𝛼,0 ‖𝑢‖𝑎,𝑊𝛼,𝑝 ,

𝛼−𝑛+1 (𝐷𝑎+ 𝑢) ,

.. . 𝛼−𝑛+𝑛 𝐷𝑎+ 𝑢

󵄨𝑝 󵄩 𝛼 󵄩󵄩𝑝 󵄨 1−𝛼 = 𝐿 𝛼,0 (󵄨󵄨󵄨󵄨𝐼𝑎+ 𝑢 (𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝 )

𝑛−𝛼 𝐷2 𝐼𝑎+ 𝑢

(82)

So, 󵄨 󵄨󵄨 1−𝛼 𝛼 󵄩 󵄨󵄨𝐼𝑎+ 𝑢 (𝑎)󵄨󵄨󵄨 ≤ 𝑚𝑀𝛼,0 (‖𝑢‖𝐿1 + 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿1 ) 󵄨 󵄨 󵄩 𝛼 󵄩󵄩 ≤ 𝑚𝑀𝛼,0 (𝑏 − 𝑎)(𝑝−1)/𝑝 (‖𝑢‖𝐿𝑝 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝 ) ,

(83)


9 1,𝑝

where 𝑀𝛼,0 = (𝑏 − 𝑎)−𝛼 /Γ(2 − 𝛼) + 1. Thus,

𝛼−1 𝑛−(𝛼−1) 𝑛+1−𝛼 𝑢 = 𝐷𝑛 𝐼𝑎+ 𝑢 = 𝐷𝑛 𝐼𝑎+ 𝑢 ∈ 𝑊𝑎+ (because of Since 𝐷𝑎+ 𝑛+1,𝑝 𝑛+1−𝛼 𝐼𝑎+ 𝑢 ∈ 𝑊𝑎+ ), therefore (cf. case of 𝑛 = 1)

󵄨𝑝 󵄨󵄨 1−𝛼 𝑝 󵄨󵄨𝐼𝑎+ 𝑢 (𝑎)󵄨󵄨󵄨 ≤ 𝑚𝑝 𝑀𝛼,0 (𝑏 − 𝑎)𝑝−1 2𝑝−1 󵄨 󵄨 𝑝 󵄩 𝛼 󵄩󵄩𝑝 × (‖𝑢‖𝐿𝑝 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝 )

(84)

𝑝 𝑝 󵄩󵄩 𝛼−1 󵄩󵄩𝑝 󵄩 󵄩 𝑛−(𝛼−1) 󵄩 𝑛−(𝛼−1) 󵄩 󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩 𝑝 = 󵄩󵄩󵄩𝐷𝑛 𝐼𝑎+ 𝑢󵄩󵄩󵄩󵄩𝐿𝑝 ≤ 󵄩󵄩󵄩󵄩𝐷𝑛 𝐼𝑎+ 𝑢󵄩󵄩󵄩󵄩𝑊1,𝑝 󵄩 󵄩𝐿 󵄩 𝑎+

󵄩 󵄩𝑝 𝑛−(𝛼−1) 󵄩 ≤ 𝐿 1,1 󵄩󵄩󵄩󵄩𝐷𝑛 𝐼𝑎+ 𝑢󵄩󵄩󵄩𝑎,𝑊1,𝑝 𝑎+

and, consequently,

󵄨𝑝 󵄩 󵄨 𝑛−(𝛼−1) 𝑛−(𝛼−1) 󵄩 = 𝐿 1,1 (󵄨󵄨󵄨󵄨𝐷𝑛 𝐼𝑎+ 𝑢 (𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩𝐷1 𝐷𝑛 𝐼𝑎+ 𝑢󵄩󵄩󵄩󵄩𝐿𝑝 ) 󵄨𝑝 󵄩 󵄨 𝑛−(𝛼−1) 𝑛−(𝛼−1) 󵄩 𝑢 (𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩𝐷𝑛+1 𝐼𝑎+ 𝑢󵄩󵄩󵄩󵄩𝐿𝑝 ) , = 𝐿 1,1 (󵄨󵄨󵄨󵄨𝐷𝑛 𝐼𝑎+ (88)

󵄨𝑝 󵄩 𝛼 󵄩󵄩𝑝 󵄨 1−𝛼 𝑝 𝑢 (𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝 ‖𝑢‖𝑎,𝑊𝛼,𝑝 = 󵄨󵄨󵄨󵄨𝐼𝑎+ 𝑎+ 𝑝

≤ (𝑚𝑝 𝑀𝛼,0 (𝑏 − 𝑎)𝑝−1 2𝑝−1 + 1)

(85)

𝑝 󵄩 𝛼 󵄩󵄩𝑝 𝑢󵄩󵄩𝐿𝑝 ) × (‖𝑢‖𝐿𝑝 + 󵄩󵄩󵄩𝐷𝑎+

=

where 𝐿 1,1 is such that

𝑝 𝑀𝛼,1 ‖𝑢‖𝑊𝛼,𝑝 , 𝑎+

‖V‖

𝑝

1,𝑝

𝑊𝑎+

≤ 𝐿 1,1 ‖V‖

𝑝

1,𝑝

𝑎,𝑊𝑎+

(89)

𝑝

where 𝑀𝛼,1 = 𝑚𝑝 𝑀𝛼,0 (𝑏 − 𝑎)𝑝−1 2𝑝−1 + 1. 𝛼,𝑝 𝛼,𝑝 When (1 − 𝛼)𝑝 ≥ 1, then (cf. Remark 21) 𝑊𝑎+ = 𝐴𝐶𝑎+ ∩ 𝛼,𝑝 𝐿𝑝 is the set of all functions 𝑢 belonging to 𝐴𝐶𝑎+ that satisfy 1−𝛼 the condition (𝐼𝑎+ 𝑢)(𝑎) = 0. Consequently, in the same way as in the case of (1 − 𝛼)𝑝 < 1 (putting 𝑐 = 0) we obtain the 𝑝 𝑝 inequality ‖𝑢‖𝑊𝛼,𝑝 ≤ 𝐿 𝛼,1 ‖𝑢‖𝑎,𝑊𝛼,𝑝 with some 𝐿 𝛼,1 > 0. The 𝑝

𝑎+

1,𝑝

for V ∈ 𝑊𝑎+ . Thus, 𝑛−1 󵄨 𝑛−(𝛼−1) 󵄨𝑝 𝑝 𝑢(𝑎)󵄨󵄨󵄨󵄨 ‖𝑢‖𝑊𝛼,𝑝 ≤ 𝐿 𝛼−1,𝑛 ∑ 󵄨󵄨󵄨󵄨𝐷𝑖 𝐼𝑎+ 𝑎+

+ 𝐿 𝛼−1,𝑛 𝐿 1,1

𝑝

inequality ‖𝑢‖𝑎,𝑊𝛼,𝑝 ≤ 𝑀𝛼,1 ‖𝑢‖𝑊𝛼,𝑝 with some 𝑀𝛼,1 > 0 is 𝑎+ 𝑎+ obvious (it is sufficient to put 𝑀𝛼,1 = 1 and use the fact that 1−𝛼 (𝐼𝑎+ 𝑢)(𝑎) = 0). Now, let us assume that the assertion holds true for some 𝑛 ∈ N. We will prove that it is true for 𝑛 + 1. Let 𝛼 ∈ (𝑛, 𝑛 + 1]. We know that there exist constants 𝐿 𝛼,𝑛 and 𝑀𝛼,𝑛 > 0 such that (

1 𝑝 𝑝 𝑝 ) ‖𝑢‖ 𝛼−1,𝑝 ≤ ‖𝑢‖ 𝛼−1,𝑝 ≤ 𝐿 𝛼,𝑛 ‖𝑢‖ 𝛼−1,𝑝 𝑎,𝑊𝑎+ 𝑊𝑎+ 𝑎,𝑊𝑎+ 𝑀𝛼,𝑛 𝛼−1,𝑝

𝛼,𝑝

𝛼−1,𝑝

for any 𝑢 ∈ 𝑊𝑎+ . If 𝑢 ∈ 𝑊𝑎+ , then 𝑢 ∈ 𝑊𝑎+ therefore (from the induction assumption)

󵄨 󵄨𝑝 󵄩 𝑛−(𝛼−1) 𝑛−(𝛼−1) 󵄩 × (󵄨󵄨󵄨󵄨𝐷𝑛 𝐼𝑎+ 𝑢 (𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩𝐷𝑛+1 𝐼𝑎+ 𝑢󵄩󵄩󵄩󵄩𝐿𝑝 ) 󵄩 𝛼 󵄩󵄩𝑝 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝

𝑖=0 𝑝

= 𝐿 𝛼,𝑛+1 ‖𝑢‖𝑎,𝑊𝛼,𝑝 ,

(86)

and

𝑎+

where 𝐿 𝛼,𝑛+1 = 𝐿 𝛼−1,𝑛 + 𝐿 𝛼−1,𝑛 𝐿 1,1 + 1 (we used the equality 𝑛−(𝛼−1) 𝑛+1−𝛼 𝛼 𝐷𝑛+1 𝐼𝑎+ 𝑢 = 𝐷𝑛+1 𝐼𝑎+ 𝑢 = 𝐷𝑎+ 𝑢). 𝛼−1,𝑝 On the other hand, since 𝑢 ∈ 𝑊𝑎+ , therefore (from the induction assumption) 𝑛−1

󵄨 𝑛−(𝛼−1) 󵄨𝑝 󵄩 𝛼−1 󵄩󵄩𝑝 𝑢(𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 ∑ 󵄨󵄨󵄨󵄨𝐷𝑖 𝐼𝑎+

𝑖=0

𝑖=0

𝑛−1 󵄩 (𝛼−1)−(𝑛−1)+𝑖 󵄩󵄩𝑝 𝑝 = ‖𝑢‖𝐿𝑝 + ∑ 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 𝑖=0

󵄩 𝛼 󵄩󵄩𝑝 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝 𝑝 󵄩 𝛼 󵄩󵄩𝑝 = ‖𝑢‖ 𝛼−1,𝑝 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝 𝑊𝑎+ 𝑝

≤ 𝐿 𝛼−1,𝑛 ‖𝑢‖

𝛼−1,𝑝

𝑎,𝑊𝑎+

𝑛−1

󵄩 𝛼 󵄩󵄩𝑝 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝

󵄨 𝑛−(𝛼−1) 󵄨𝑝 󵄩 𝛼−1 󵄩󵄩𝑝 = 𝐿 𝛼−1,𝑛 ( ∑ 󵄨󵄨󵄨󵄨𝐷𝑖 𝐼𝑎+ 𝑢 (𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 ) 𝑖=0

󵄩 𝛼 󵄩󵄩𝑝 𝑢󵄩󵄩𝐿𝑝 . + 󵄩󵄩󵄩𝐷𝑎+

(90)

𝑛 󵄨𝑝 󵄩 𝛼 󵄩󵄩𝑝 󵄨 𝑛+1−𝛼 ≤ 𝐿 𝛼,𝑛+1 (∑󵄨󵄨󵄨󵄨𝐷𝑖 𝐼𝑎+ 𝑢 (𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝 )

𝑛 󵄩 𝛼−𝑛+𝑖 󵄩󵄩𝑝 𝑝 𝑝 𝑢󵄩󵄩󵄩𝐿𝑝 ‖𝑢‖𝑊𝛼,𝑝 = ‖𝑢‖𝐿𝑝 + ∑󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑎+

𝑖=0

𝑎+

≤ (87)

𝑝 𝑀𝛼−1,𝑛 (‖𝑢‖𝐿𝑝

𝑛−1

(91)

󵄩 𝛼−1−(𝑛−1)+𝑖 󵄩󵄩𝑝 + ∑ 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 ) . 𝑖=0

So, 𝑛−1

󵄨𝑝 󵄨 𝑛−(𝛼−1) 𝑢 (𝑎)󵄨󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨󵄨𝐷𝑖 𝐼𝑎+ 𝑖=0

𝑛−2 󵄩 𝛼−1−(𝑛−1)+𝑖 󵄩󵄩𝑝 󵄩󵄩 𝛼−1 󵄩󵄩𝑝 𝑝 ≤ 𝑀𝛼−1,𝑛 (‖𝑢‖𝐿𝑝 + ∑ 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 ) . 𝑖=0

(92)

10


𝑛+1−𝛼 𝑛+1−𝛼 Let us recall that 𝐼𝑎+ 𝑢 ∈ 𝑊𝑛+1,𝑝 , so 𝐼𝑎+ 𝑢 ∈ 𝑊𝑛,𝑝 , 𝑛 𝑛+1−𝛼 1,𝑝 𝛼 𝑛+1 𝑛+1−𝛼 1 𝑛 𝑛+1−𝛼 𝐷 𝐼𝑎+ 𝑢 ∈ 𝑊 , and 𝐷𝑎+ 𝑢 = 𝐷 𝐼𝑎+ 𝑢 = 𝐷 𝐷 𝐼𝑎+ 𝑢. Thus,

󵄨󵄨 𝑛 𝑛+1−𝛼 󵄨𝑝 𝑝 𝛼 󵄩 󵄨󵄨𝐷 𝐼𝑎+ 𝑢(𝑎)󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 󵄨 󵄨 𝑝 󵄩 𝑛+1−𝛼 󵄩 = 󵄩󵄩󵄩󵄩𝐷𝑛 𝐼𝑎+ 𝑢󵄩󵄩󵄩󵄩𝑎,𝑊1,𝑝 𝑎+

𝑛−1

(93)

𝑝 󵄩 𝑛+1−𝛼 󵄩 ≤ 𝑀1,1 󵄩󵄩󵄩󵄩𝐷𝑛 (𝐼𝑎+ 𝑢)󵄩󵄩󵄩󵄩𝑊1,𝑝 𝑎+

1,𝑝

𝑎,𝑊𝑎+

≤ 𝑀1,1 ‖V‖

𝑝

1,𝑝

𝑊𝑎+

.

(94)

Consequently,

(96)

𝑡 ∈ [𝑎, 𝑏] a.e.,

𝛼,𝑝

where 𝑀1,1 is such that 𝑝

𝑐𝑖 (𝑡 − 𝑎)𝛼−𝑛+𝑖 𝑖=0 Γ (𝛼 − 𝑛 + 1 + 𝑖)

𝑢 (𝑡) = ∑

𝛼 + 𝐼𝑎+ 𝜑 (𝑡) ,

𝑝 󵄩 󵄩 𝛼 󵄩󵄩𝑝 𝑛−(𝛼−1) 󵄩 = 𝑀1,1 (󵄩󵄩󵄩󵄩𝐷𝑛 (𝐼𝑎+ 𝑢)󵄩󵄩󵄩󵄩𝐿 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢)󵄩󵄩𝐿 ) , 𝑝 𝑝

‖V‖

𝛼,𝑝

is complete. Let (𝑢𝑘 ) ⊂ 𝑊𝑎+ be a Cauchy sequence with 𝑛−𝛼 𝑢𝑘 (𝑎)), 𝑖 = respect to this norm. So, the sequences (𝐷𝑖 𝐼𝑎+ 𝑚 𝛼 𝑢𝑘 ) is the 0, . . . , 𝑛 − 1, are Cauchy sequences in R and (𝐷𝑎+ Cauchy sequence in 𝐿𝑝 . Let 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 ∈ R𝑚 and 𝜑 ∈ 𝐿𝑝 be limits of the above sequences in R𝑚 and 𝐿𝑝 , respectively. Then the function

𝛼,𝑝

belongs to 𝑊𝑎+ and is the limit of (𝑢𝑘 ) in 𝑊𝑎+ with respect to ‖ ⋅ ‖𝑎,𝑊𝑎+𝛼,𝑝 (to assert that 𝑢 ∈ 𝐿𝑝 it is sufficient to consider the cases (𝑛 − 𝛼)𝑝 < 1 and (𝑛 − 𝛼)𝑝 ≥ 1—in the second case 𝑛−𝛼 𝑢𝑘 (𝑎) = 0 for any 𝑘 ∈ N and, consequently, 𝑐0 = 0). 𝐼𝑎+ In the proofs of the next two theorems we use the method presented in [1].

𝑛−1

󵄨 𝑛−(𝛼−1) 󵄨𝑝 𝑝 𝑢(𝑎)󵄨󵄨󵄨󵄨 ‖𝑢‖𝑎,𝑊𝛼,𝑝 = ∑ 󵄨󵄨󵄨󵄨𝐷𝑖 𝐼𝑎+ 𝑎+

𝛼,𝑝

Theorem 26. The space 𝑊𝑎+ is reflexive with respect to each of the norms ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 and ‖ ⋅ ‖𝑎,𝑊𝑎+𝛼,𝑝 , for any 𝛼 > 0 and 1 < 𝑝 < ∞.

𝑖=0

󵄨 󵄨𝑝 󵄩 𝛼 󵄩󵄩𝑝 𝑛−(𝛼−1) + 󵄨󵄨󵄨󵄨𝐷𝑛 𝐼𝑎+ 𝑢(𝑎)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿𝑝

𝛼,𝑝

𝑛−2 󵄩 𝛼−1−(𝑛−1)+𝑖 󵄩󵄩𝑝 𝑝 ≤ 𝑀𝛼−1,𝑛 (‖𝑢‖𝐿𝑝 + ∑ 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝

Proof. Let us consider 𝑊𝑎+ with the norm ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 and define a mapping

𝑖=0

𝛼,𝑝 𝛼−(𝑛−1) 𝛼−1 𝛼 ∋ 𝑢 󳨃󳨀→ (𝑢, 𝐷𝑎+ 𝑢, . . . , 𝐷𝑎+ 𝑢, 𝐷𝑎+ 𝑢) 𝜆 : 𝑊𝑎+

󵄩 𝛼−1 󵄩󵄩𝑝 +󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 ) 󵄩 𝛼−1 󵄩󵄩𝑝 󵄩󵄩 𝛼 󵄩󵄩𝑝 𝑢󵄩󵄩󵄩𝐿 + 󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿 ) + 𝑀1,1 (󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑝

𝑝

∈ 𝐿𝑝 × 𝐿𝑝 × ⋅ ⋅ ⋅ × 𝐿𝑝 × 𝐿𝑝 . (95)

𝑛−1 󵄩 𝛼−1−(𝑛−1)+𝑖 󵄩󵄩𝑝 𝑝 ≤ 𝑀𝛼,𝑛+1 (‖𝑢‖𝐿𝑝 + ∑ 󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 𝑖=0

󵄩 𝛼 󵄩󵄩𝑝 +󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩𝐿 ) 𝑝

𝑛 󵄩 𝛼−𝑛+𝑖 󵄩󵄩𝑝 𝑝 = 𝑀𝛼,𝑛+1 (‖𝑢‖𝐿𝑝 + ∑󵄩󵄩󵄩󵄩𝐷𝑎+ 𝑢󵄩󵄩󵄩𝐿𝑝 ) 𝑖=0

(97)

𝛼,𝑝

Since 𝜆 is the isometry, 𝜆(𝑊𝑎+ ) is the closed linear subspace of the reflexive space 𝐿𝑝 × ⋅ ⋅ ⋅ × 𝐿𝑝 . So (cf. [8, Corollary 1 in 𝛼,𝑝 Part V.73]), it is reflexive and, consequently, 𝑊𝑎+ is reflexive with respect to ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 . 𝛼,𝑝 From the equivalence of the norms it follows that 𝑊𝑎+ 𝛼,𝑝 with the norm ‖ ⋅ ‖𝑎,𝑊𝑎+ is also reflexive (it is sufficient 𝛼,𝑝 to consider the identity mapping 𝑖 : (𝑊𝑎+ , ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 ) → 𝛼,𝑝 (𝑊𝑎+ , ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 ) being the linear homeomorphism and use [8, Remark in Part V.7.3]). 𝛼,𝑝

Theorem 27. The space 𝑊𝑎+ is separable with respect to each of the norms ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 and ‖ ⋅ ‖𝑎,𝑊𝑎+𝛼,𝑝 , for any 𝛼 > 0 and 1 ≤ 𝑝 < ∞.

𝑝

= 𝑀𝛼,𝑛+1 ‖𝑢‖𝑊𝛼,𝑝 , 𝑎+

where 𝑀𝛼,𝑛+1 = 𝑀𝛼−1,𝑛 + 𝑀1,1 .

𝛼,𝑝

𝛼,𝑝

7. Basic Properties of 𝑊𝑎+

Now, we are in a position to prove some basic properties of the introduced spaces. 𝛼,𝑝

Theorem 25. The space 𝑊𝑎+ is complete with respect to each of the norms ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 and ‖ ⋅ ‖𝑎,𝑊𝑎+𝛼,𝑝 , for any 𝛼 > 0 and 1 ≤ 𝑝 < ∞. Proof. Let 𝑛 ∈ N be such that 𝛼 ∈ (𝑛 − 1, 𝑛]. Of course, 𝛼,𝑝 it is sufficient to show that 𝑊𝑎+ with the norm ‖ ⋅ ‖𝑎,𝑊𝑎+𝛼,𝑝

Proof. Let us consider 𝑊𝑎+ with the norm ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 and mapping 𝜆 defined in the proof of Theorem 26. Of course, 𝛼,𝑝 𝜆(𝑊𝑎+ ) is separable as a subset of separable space 𝐿𝑝 ×⋅ ⋅ ⋅×𝐿𝑝 . 𝛼,𝑝 Since 𝜆 is the isometry, 𝑊𝑎+ is also separable with respect to the norm ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 . Equivalence of the norms ‖ ⋅ ‖𝑎,𝑊𝑎+𝛼,𝑝 𝛼,𝑝 and ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 implies separability of 𝑊𝑎+ with respect to ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 .

8. Imbeddings We have the following extension of Theorem 12.


11

Theorem 28. (a) If 0 < 𝛽 < 𝛼 ≤ 1, then 𝛼,𝑝 𝛽 𝛽,𝑞 𝐴𝐶𝑎+ ⊂ 𝐼𝑎+ (𝐿𝑞 ) ⊂ 𝐴𝐶𝑎+

(98)

for 1 ≤ 𝑞 ≤ 𝑝 < ∞ and 1 ≤ 𝑞 < 1/(1 − 𝛼 + 𝛽); consequently, 𝛼,𝑝 𝛽,𝑞 𝑊𝑎+ ⊂ 𝑊𝑎+

(99)

for 1 ≤ 𝑞 ≤ 𝑝 < ∞ and 1 ≤ 𝑞 < 1/(1 − 𝛼 + 𝛽). (b) If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, and 0 < 𝛽 < 𝛼 − (𝑛 − 1), then 𝛼,𝑝 𝐴𝐶𝑎+

⊂

𝛽 𝐼𝑎+

𝑞

(𝐿 ) ⊂

𝛽,𝑞 𝐴𝐶𝑎+

(100)

for 1 ≤ 𝑝 < ∞ and 1 ≤ 𝑞 < 1/(𝑛 − 𝛼 + 𝛽); consequently, 𝛼,𝑝 𝛽,𝑞 𝑊𝑎+ ⊂ 𝑊𝑎+

(101)

for 1 ≤ 𝑝 < ∞ and 1 ≤ 𝑞 < (1/(𝑛 − 𝛼 + 𝛽)). (c) If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, 𝛽 = 𝛼 − 𝑖, and 𝑖 ∈ {1, . . . , 𝑛 − 1}, then 𝛼,𝑝 𝛽,𝑞 ⊂ 𝐴𝐶𝑎+ 𝐴𝐶𝑎+

(102)

for 1 ≤ 𝑝 < ∞ and 1 ≤ 𝑞 < ∞; consequently, 𝛼,𝑝 𝛽,𝑞 𝑊𝑎+ ⊂ 𝑊𝑎+

(103)

for 1 ≤ 𝑝 < ∞, (𝑛 − 𝛼)𝑝 < 1, 1 ≤ 𝑞 < ∞, and (𝑛 − 𝛼)𝑞 < 1 or 1 ≤ 𝑝 < ∞, (𝑛 − 𝛼)𝑝 ≥ 1, and 1 ≤ 𝑞 < ∞. (d) If 0 < 𝛼 = 𝛽, then 𝛼,𝑝 𝛽,𝑞 ⊂ 𝐴𝐶𝑎+ 𝐴𝐶𝑎+

(104)

for 1 ≤ 𝑞 ≤ 𝑝 < ∞; consequently, 𝛼,𝑝 𝑊𝑎+

⊂

𝛼,𝑞 𝑊𝑎+

(105)

for 1 ≤ 𝑞 ≤ 𝑝 < ∞. Proof. (a) Let us fix 𝑝 ∈ [1, ∞). Theorems 12 and 13 (a) imply that 𝛼,𝑝 𝛼,1 𝛽 𝛽,𝑞 ⊂ 𝐴𝐶𝑎+ ⊂ 𝐼𝑎+ (𝐿𝑞 ) ⊂ 𝐴𝐶𝑎+ 𝐴𝐶𝑎+

(106)

provided that (𝛼 − 𝛽 − 1)𝑞 > −1 and 1 ≤ 𝑞 ≤ 𝑝 (cf. (20), (21)); that is, 1 ≤ 𝑞 < 1/(1 − 𝛼 + 𝛽) and 1 ≤ 𝑞 ≤ 𝑝. Consequently, for such 𝑝 and 𝑞, 𝛼,𝑝 𝛼,𝑝 𝛽,𝑞 𝛽,𝑞 = 𝐴𝐶𝑎+ ∩ 𝐿𝑝 ⊂ 𝐴𝐶𝑎+ ∩ 𝐿𝑞 = 𝑊𝑎+ . 𝑊𝑎+

(107)

(b) Let us fix 𝑝 ∈ [1, ∞). In this case 𝛼 − 𝛽 > 1 and, 𝛼−𝛽 1 𝛼−𝛽−1 𝐼𝑎+ 𝜑 ∈ 𝐶 for any 𝜑 ∈ 𝐿1 . So (cf. consequently, 𝐼𝑎+ 𝜑 = 𝐼𝑎+ Theorems 12 and 13 (a)) 𝛼,𝑝 𝐴𝐶𝑎+

⊂

𝛼,1 𝐴𝐶𝑎+

⊂

𝛽 𝐼𝑎+

𝑞

(𝐿 ) ⊂

𝛽,𝑞 𝐴𝐶𝑎+

(108)

provided that (𝛼 − 𝛽 − 𝑛)𝑞 > −1 and 1 ≤ 𝑞 < ∞ (cf. (20), (21)); that is, 1 ≤ 𝑞 < 1/(𝑛 − 𝛼 + 𝛽). Consequently, for such 𝑞 and 1 ≤ 𝑝 < ∞, 𝛼,𝑝 𝛼,𝑝 𝛽 = 𝐴𝐶𝑎+ ∩ 𝐿𝑝 ⊂ 𝐼𝑎+ (𝐿𝑞 ) 𝑊𝑎+ 𝛽 𝛽,𝑞 𝛽,𝑞 = 𝐼𝑎+ (𝐿𝑞 ) ∩ 𝐿𝑞 ⊂ 𝐴𝐶𝑎+ ∩ 𝐿𝑞 = 𝑊𝑎+

(109)

𝛽

(we used here the inclusion 𝐼𝑎+ (𝐿𝑞 ) ⊂ 𝐿𝑞 ). (c) Let us fix 𝑝 ∈ [1, ∞). Theorems 12 and 13 (b) imply that 𝛼,𝑝 𝛼,1 𝛽,𝑞 ⊂ 𝐴𝐶𝑎+ ⊂ 𝐴𝐶𝑎+ 𝐴𝐶𝑎+

(110)

for any 𝑞 ∈ [1, ∞) (cf. (22), (24)). 𝛼,𝑝 𝛼,𝑝 If (𝑛 − 𝛼)𝑝 < 1, then (cf. Remark 21) 𝐴𝐶𝑎+ ∩ 𝐿𝑝 = 𝐴𝐶𝑎+ . So, 𝛼,𝑝 𝛼,𝑝 𝛼,𝑝 𝛽,𝑞 𝛽,𝑞 𝑊𝑎+ = 𝐴𝐶𝑎+ ∩ 𝐿𝑝 = 𝐴𝐶𝑎+ ⊂ 𝐴𝐶𝑎+ ⊂ 𝐴𝐶𝑎+ ∩ 𝐿𝑞 𝛽,𝑞 = 𝑊𝑎+

(111)

provided that (cf. Theorem 13 (b)) (𝑛 − 𝛼)𝑞 < 1. If (𝑛 − 𝛼)𝑝 ≥ 𝛼,𝑝 1, then (cf. Remark 21) 𝐴𝐶𝑎+ ∩ 𝐿𝑝 is the set of all functions 𝛼,𝑝 𝑛−𝛼 𝑓)(𝑎) = 0. belonging to 𝐴𝐶𝑎+ that satisfy the condition (𝐼𝑎+ 𝛽,𝑞 𝛼,𝑝 𝑝 Consequently (cf. Theorem 13 (b)), 𝐴𝐶𝑎+ ∩ 𝐿 ⊂ 𝐴𝐶𝑎+ ∩ 𝐿𝑞 , 𝛽,𝑞 𝛼,𝑝 that is, 𝑊𝑎+ ⊂ 𝑊𝑎+ for any 1 ≤ 𝑞 < ∞. (d) These facts are obvious. The first part of the point (c) of the above theorem implies the following. Corollary 29. If 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, 𝛽 = 𝛼 − 𝑖, and 𝑖 ∈ {1, . . . , 𝑛 − 1}, then 𝛼,𝑝 𝛽,𝑞 ⊂ 𝑊𝑎+ 𝑊𝑎+

(112)

for 1 ≤ 𝑞 ≤ 𝑝 < ∞. In the next section we will use the following important result obtained in [9, Lemma 1.1]. Theorem 30. If 𝛼 > 0 and 1 ≤ 𝑝 < ∞, then the operator 𝛼 𝐼𝑎+ : 𝐿𝑝 → 𝐿𝑝 is completely continuous, that is, it maps the bounded sets onto relatively compact ones. Now, we are in a position to prove theorems on compactness of some imbeddings. Theorem 31. The imbedding 𝛼,𝑝 𝛽,𝑞 ⊂ 𝑊𝑎+ 𝑊𝑎+

(113)

for 0 < 𝛽 < 𝛼 ≤ 1, 1 ≤ 𝑞 ≤ 𝑝 < ∞, and 1 ≤ 𝑞 < 1/(1 − 𝛼 + 𝛽), given in Theorem 28 (a), is compact. Proof. Let 0 < 𝛽 < 𝛼 ≤ 1, 1 ≤ 𝑞 ≤ 𝑝 < ∞, 1 ≤ 𝑞 < 1/(1 − 𝛼 + 𝛽), and (𝑢𝑘 ), where 𝑢𝑘 (𝑡) =

𝑐𝑘 1 𝛼 + 𝐼𝑎+ 𝜑𝑘 (𝑡) , Γ (𝛼) (𝑡 − 𝑎)1−𝛼 𝑡 ∈ [𝑎, 𝑏] a.e., 𝑘 ∈ N,

(114)

12

Journal of Function Spaces and Applications 𝛼,𝑝

be a bounded sequence in 𝑊𝑎+ . We will show that it contains 𝛽,𝑞 a subsequence which is convergent in 𝑊𝑎+ . Since (𝑢𝑘 ) is bounded (cf. Theorem 24), the sequences (𝑐𝑘 ) and (𝜑𝑘 ) are bounded in R𝑚 and 𝐿𝑝 , respectively. So, one can choose a subsequence (𝑘𝑗 )𝑗∈N of positive integers such that (𝑐𝑘𝑗 ) is convergent to some 𝑐0 in R𝑚 and (cf. Theorem 30) 𝛼−𝛽

such that (𝐼𝑎+ 𝜑𝑘𝑗 ) is convergent to some 𝜓0 in 𝐿𝑝 . Of course, 𝛼−𝛽

(𝐼𝑎+ 𝜑𝑘𝑗 ) is convergent to 𝜓0 in 𝐿𝑞 . Moreover, the sequence ((𝑐𝑘𝑗 /Γ(𝛼−𝛽))(⋅−𝑎)𝛼−1−𝛽 ) converges in 𝐿𝑞 to (𝑐0 /Γ(𝛼−𝛽))(⋅− 𝛼−1−𝛽

𝑎)

. This means (cf. (20), (21)) that the sequence (𝑢𝑘𝑗 ) 𝛽,𝑞

converges in 𝑊𝑎+ to 𝑢0 given by 𝛽 𝑢0 (𝑡) = 𝐼𝑎+ 𝜓0 (𝑡) ,

𝑡 ∈ [𝑎, 𝑏] a.e.,

(115)

where 𝜓0 (𝑡) = (𝑐0 /Γ(𝛼 − 𝛽))(𝑡 − 𝑎)𝛼−1−𝛽 + 𝜓0 (𝑡). The proof is completed.

𝛾

𝜔0 where 𝛾 = (𝛼 − 𝛽) − 1 > 0. Consequently, (𝐼𝑎+ 𝜑𝑘𝑗 ) is convergent in 𝐿1 to 𝜔0 and

𝛼−𝛽 1 𝛾 1 𝐼𝑎+ 𝜑𝑘𝑗 = 𝐼𝑎+ 𝐼𝑎+ 𝜑𝑘𝑗 󳨀→ 𝐼𝑎+ 𝜔0

in 𝐿𝑞 , because 𝑏󵄨 󵄨󵄨𝑞 󵄨 1 𝛾 1 𝐼𝑎+ 𝜑𝑘𝑗 (𝑡) − 𝐼𝑎+ 𝜔0 (𝑡)󵄨󵄨󵄨 𝑑𝑡 ∫ 󵄨󵄨󵄨𝐼𝑎+ 󵄨 𝑎 󵄨 𝑞 𝑏 𝑡󵄨 󵄨󵄨 󵄨 𝛾 𝜑𝑘𝑗 (𝑠) − 𝜔0 (𝑠)󵄨󵄨󵄨 𝑑𝑠) 𝑑𝑡 ≤ ∫ (∫ 󵄨󵄨󵄨𝐼𝑎+ 󵄨 𝑎 𝑎 󵄨

𝛽

So, the sequence (𝐷𝑎+ 𝑢𝑘𝑗 ), where 𝛽 𝑢𝑘𝑗 (𝑡) = 𝐷𝑎+

𝑘

𝑐0 𝑗 (𝑡 − 𝑎)𝛼−𝑛−𝛽 Γ (𝛼 − 𝑛 + 1 − 𝛽) 𝑘

𝑐1 𝑗 + (𝑡 − 𝑎)𝛼−𝑛−𝛽+1 Γ (𝛼 − 𝑛 + 2 − 𝛽)

Corollary 32. The imbedding 𝛼,𝑝 𝑊𝑎+ ⊂ 𝐿𝑞

(116)

𝑘

+ ⋅⋅⋅ +

for 0 < 𝛼 < 1, 1 ≤ 𝑞 ≤ 𝑝 < ∞, 1 ≤ 𝑞 < 1/(1 − 𝛼) or 𝛼 = 1, and 1 ≤ 𝑞 ≤ 𝑝 < ∞ is compact. Theorem 33. The imbedding

𝑗 𝑐𝑛−1 𝛼−𝛽 𝜑𝑘𝑗 (𝑡) , (𝑡 − 𝑎)𝛼−𝛽−1 + 𝐼𝑎+ Γ (𝛼 − 𝛽) (121)

converges to the function

𝛼,𝑝 𝛽,𝑞 𝑊𝑎+ ⊂ 𝑊𝑎+ ,

(117)

for 𝑛−1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, 0 < 𝛽 < 𝛼−(𝑛−1), 1 ≤ 𝑝 < ∞, and 1 ≤ 𝑞 < 1/(𝑛 − 𝛼 + 𝛽), given in Theorem 28 (b), is compact. 𝛼,𝑝

Proof. Let us consider a bounded sequence (𝑢𝑘 ) in 𝑊𝑎+ where 𝑐0𝑘 (𝑡 − 𝑎)𝛼−𝑛 Γ (𝛼 − 𝑛 + 1) 𝑐1𝑘 + (𝑡 − 𝑎)𝛼−𝑛+1 Γ (𝛼 − 𝑛 + 2) + ⋅⋅⋅ +

(120)

󵄩󵄩 𝛾 󵄩󵄩𝑞 ≤ 󵄩󵄩󵄩𝐼𝑎+ 𝜑𝑘𝑗 − 𝜔0 󵄩󵄩󵄩 1 (𝑏 − 𝑎) . 󵄩 󵄩𝐿

The above theorem implies the following.

𝑢𝑘 (𝑡) =

(119)

𝑘 𝑐𝑛−1

Γ (𝛼)

𝛼 + 𝐼𝑎+ 𝜑𝑘 (𝑡) ,

𝜓0 (𝑡) =

𝑐0 (𝑡 − 𝑎)𝛼−𝑛−𝛽 Γ (𝛼 − 𝑛 + 1 − 𝛽) +

𝑐1 (𝑡 − 𝑎)𝛼−𝑛−𝛽+1 Γ (𝛼 − 𝑛 + 2 − 𝛽)

(122)

𝑐𝑛−1 + ⋅⋅⋅ + (𝑡 − 𝑎)𝛼−𝛽−1 Γ (𝛼 − 𝛽) 1 + 𝐼𝑎+ 𝜔0 (𝑡) 𝛽,𝑞

(118)

𝛽

in 𝐿𝑞 . Thus, the sequence (𝑢𝑘𝑗 ) converges in 𝑊𝑎+ to 𝐼𝑎+ 𝜓0 . The proof is completed. The above theorem implies the following.

(𝑡 − 𝑎)𝛼−1 𝑡 ∈ [𝑎, 𝑏] a.e.,

𝑘 ∈ R𝑚 and 𝜑𝑘 ∈ 𝐿𝑝 . We will show that it with 𝑐0𝑘 , 𝑐1𝑘 , . . . , 𝑐𝑛−1 𝛽,𝑞 contains a subsequence which is convergent in 𝑊𝑎+ . is bounded, the sequences Since (𝑢𝑘 ) 𝑘 ) ∈ R𝑚 and (𝜑𝑘 ) are bounded in R𝑚 (𝑐0𝑘 ), (𝑐1𝑘 ), . . . , (𝑐𝑛−1 and 𝐿𝑝 , respectively. So, one can choose a subsequence 𝑘 𝑘 𝑘𝑗 ) are (𝑘𝑗 )𝑗∈N of positive integers such that (𝑐0 𝑗 ), (𝑐1 𝑗 ), . . . , (𝑐𝑛−1 𝑚 convergent to some 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 in R and (cf. Theorem 30) 𝛾 such that the sequence (𝐼𝑎+ 𝜑𝑘𝑗 ) is convergent in 𝐿𝑝 to some

Corollary 34. The imbedding 𝛼,𝑝 ⊂ 𝐿𝑞 𝑊𝑎+

(123)

for 𝑛 − 1 < 𝛼 < 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, 1 ≤ 𝑝 < ∞, 1 ≤ 𝑞 < 1/(𝑛 − 𝛼) or 𝛼 = 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, 1 ≤ 𝑝 < ∞, and 1 ≤ 𝑞 < ∞ is compact. Theorem 35. The imbedding 𝛼,𝑝 𝛽,𝑞 𝑊𝑎+ ⊂ 𝑊𝑎+ ,

(124)

for 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, 𝛽 = 𝛼 − 𝑖, 𝑖 ∈ {1, . . . , 𝑛 − 1}, and 1 ≤ 𝑞 ≤ 𝑝 < ∞, given in Corollary 29, is compact.


13 𝛼,𝑝

Proof. Let us consider a bounded sequence (𝑢𝑘 ) in 𝑊𝑎+ where

Corollary 36. The imbedding 𝛼,𝑝 ⊂ 𝐿𝑞 𝑊𝑎+

𝑐0𝑘 𝑢𝑘 (𝑡) = (𝑡 − 𝑎)𝛼−𝑛 Γ (𝛼 − 𝑛 + 1) +

𝑐1𝑘 (𝑡 − 𝑎)𝛼−𝑛+1 Γ (𝛼 − 𝑛 + 2)

for 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, 𝑛 ≥ 2, and 1 ≤ 𝑞 ≤ 𝑝 < ∞ is compact. (125)

𝑐𝑘 + ⋅ ⋅ ⋅ + 𝑛−1 (𝑡 − 𝑎)𝛼−1 Γ (𝛼) 𝛼 + 𝐼𝑎+ 𝜑𝑘 (𝑡) ,

𝑡 ∈ [𝑎, 𝑏] a.e.,

𝑘 ∈ R𝑚 and 𝜑𝑘 ∈ 𝐿𝑝 . We will show that it with 𝑐0𝑘 , 𝑐1𝑘 , . . . , 𝑐𝑛−1 𝛽,𝑞 contains a subsequence which is convergent in 𝑊𝑎+ . Since (𝑢𝑘 ) is bounded, the sequences (𝑐0𝑘 ), (𝑐1𝑘 ), . . . , 𝑘 (𝑐𝑛−1 ) ∈ R𝑚 and (𝜑𝑘 ) are bounded in R𝑚 and 𝐿𝑝 , respectively. So, one can choose a subsequence (𝑘𝑗 )𝑗∈N of positive inte𝑘

𝑘

𝛼,𝑝

𝛼,𝑞

Remark 37. It is easy to see that imbeddings 𝑊𝑎+ ⊂ 𝑊𝑎+ for 𝛼 > 0 and 1 ≤ 𝑞 ≤ 𝑝 < ∞, given in Theorem 28 (d), are not compact. Remark 38. From Corollary 32 it follows that the imbedding 𝑊1,𝑝 ⊂ 𝐿𝑝 is compact for any 1 ≤ 𝑝 < ∞. Corollary 34 implies the compactness of the imbedding 𝑊𝑛,1 ⊂ 𝐿𝑞 for any 𝑛 ∈ N, 𝑛 ≥ 2, and 1 ≤ 𝑞 < ∞. The following problem is open: is it possible to strengthen Theorem 31 or Theorem 33 to deduce the compactness of the imbedding 𝑊1,1 ⊂ 𝐿𝑞 for any 1 ≤ 𝑞 < ∞?

𝑘

𝑗 ) are convergent to some gers such that (𝑐0 𝑗 ), (𝑐1 𝑗 ), . . . , (𝑐𝑛−1 𝑚 𝑐0 , 𝑐1 , . . . , 𝑐𝑛−1 in R and (cf. Theorem 30) such that

𝑘

𝑘

1 𝑖−1 𝑗 𝑖 𝑗 (𝐼𝑎+ ) , . . . , (𝐼𝑎+ 𝑐𝑛−𝑖+1 𝑐𝑛−1 ) , (𝐼𝑎+ 𝜑𝑘𝑗 )

(126)

are convergent to some 𝜓1,0 , . . . , 𝜓𝑖−1,0 , 𝜓0 in 𝐿𝑝 and, consequently, in 𝐿𝑞 . So (cf. Theorem 13 (b)), 𝑘

(131)

𝑘

9. Application to Boundary Value Problems In this section, we will demonstrate an application of the obtained results to fractional boundary value problems. Namely, let us fix 𝛼 ∈ (1/2, 1) and consider the following problem: 𝛼 𝛼 𝐷𝑎+ 𝑥 (𝑡) + 𝑥 (𝑡) = 𝑓 (𝑡) , 𝐷𝑏−

𝑘

𝛽 1 𝑖−1 𝑗 𝑖 𝑗 𝑗 𝐷𝑎+ 𝑢𝑘𝑗 = 𝑐𝑛−𝑖 + 𝐼𝑎+ 𝑐𝑛−𝑖+1 + ⋅ ⋅ ⋅ + 𝐼𝑎+ 𝑐𝑛−1 + 𝐼𝑎+ 𝜑𝑘𝑗

(127)

𝛼 𝐷𝑎+ 𝑥 (𝑎) = 0,

󳨀→ 𝜓0 in 𝐿𝑞 , where 𝜓0 ∈ 𝐿𝑞 is given by 𝜓0 (𝑡) = 𝑐𝑛−𝑖 + 𝜓1,0 (𝑡) + ⋅ ⋅ ⋅ + 𝜓𝑖−1,0 (𝑡) + 𝜓0 (𝑡) , 𝑡 ∈ [𝑎, 𝑏] a.e.

(128)

This means (cf. (22), (23), and (24)) that the sequence (𝑢𝑘𝑗 )

𝑡 ∈ [𝑎, 𝑏] a.e.

𝑥 (𝑏) = 0,

(132)

where 𝑓 ∈ 𝐿2 . By a solution to this problem we mean 𝛼,2 𝛼 𝛼 𝛼 a function 𝑥 ∈ 𝑊𝑎+ such that 𝐷𝑎+ 𝑥 and 𝐷𝑏− 𝐷𝑎+ 𝑥 exist, and satisfying the above equation and boundary conditions. Under the assumption on 𝛼 the boundary conditions make sense ([10, Property 4]). Let 𝑎 : 𝐻 × 𝐻 → R, where

𝛽,𝑞

converges in 𝑊𝑎+ to 𝑢0 given by 𝑢0 (𝑡) =

𝛼,2 ; 𝑥 (𝑏) = 0} , 𝐻 = {𝑥 ∈ 𝑊𝑎+

𝑑0 (𝑡 − 𝑎)𝛽−(𝑛−𝑖) Γ (𝛽 − (𝑛 − 𝑖) + 1) 𝑑1 + (𝑡 − 𝑎)𝛽−(𝑛−𝑖)+1 Γ (𝛽 − (𝑛 − 𝑖) + 2) + ⋅⋅⋅ +

𝑑𝑛−(𝑖+1) Γ (𝛽)

𝛽 + 𝐼𝑎+ 𝜓0 (𝑡) ,

to be in the following bilinear form (129)

(𝑡 − 𝑎)𝛽−1

𝑡 ∈ [𝑎, 𝑏] a.e.,

where 𝑑0 = 𝑐0 ,

𝑑1 = 𝑐1 , . . . ,

𝑑𝑛−(𝑖+1) = 𝑐𝑛−(𝑖+1) .

The proof is completed.

(130)

𝑏

𝑏

𝑎

𝑎

𝛼 𝛼 𝑎 (𝑥, 𝑦) = ∫ 𝐷𝑎+ 𝑥 (𝑡) 𝐷𝑎+ 𝑦 (𝑡) 𝑑𝑡 + ∫ 𝑥 (𝑡) 𝑦 (𝑡) 𝑑𝑡. (134)

It is easy to see that 𝐻 is the closed subspace of the Hilbert 𝛼,2 space (𝑊𝑎+ , ‖ ⋅ ‖𝑊𝑎+𝛼,𝑝 ); it is sufficient to observe that it is closed 𝛼,2 in 𝑊𝑎+ with respect to ‖ ⋅ ‖𝑎,𝑊𝑎+𝛼,𝑝 and to use Theorem 24. Of course, 𝑎 is a scalar product in 𝐻 and the norm generated by 𝑎 is simply the norm ‖ ⋅ ‖𝑊𝑎+𝛼,2 restricted to 𝐻. Clearly, 𝑎 is continuous and coercive; that is, there exists a constant 𝑐 > 0 𝛼,2 ; in fact 𝑎(𝑥, 𝑥) = such that 𝑎(𝑥, 𝑥) ≥ 𝑐‖𝑥‖2𝑊𝛼,2 for 𝑥 ∈ 𝑊𝑎+ 𝑎+

The above theorem implies the following.

(133)

‖𝑥‖2𝑊𝛼,2 for 𝑥 ∈ 𝐻. So, Lax-Milgram theorem [1] or simply 𝑎+

14


𝛼,2 Riesz-Frechet theorem implies that there exists 𝑥 ∈ 𝑊𝑎+ such that 𝑏

𝑎 (𝑥, ℎ) = ∫ 𝑓 (𝑡) ℎ (𝑡) 𝑑𝑡, 𝑎

ℎ ∈ 𝐻,

𝑏

1 = min { ‖ℎ‖2𝑊𝛼,2 − ∫ 𝑓 (𝑡) ℎ (𝑡) 𝑑𝑡} . 𝑎+ ℎ∈𝐻 2 𝑎

(136)

Condition (135) means that 𝑏

𝑎

𝑎

𝛼 𝛼 𝑥 (𝑡) 𝐷𝑎+ ℎ (𝑡) 𝑑𝑡 = ∫ (𝑓 (𝑡) − 𝑥 (𝑡)) ℎ (𝑡) 𝑑𝑡, ∫ 𝐷𝑎+

ℎ ∈ 𝐻. (137)

From a counterpart of Theorem 20 for the weak right frac𝛼 𝛼,2 tional derivative it follows that 𝐷𝑎+ 𝑥 ∈ 𝑊𝑏− and 𝛼 𝛼 𝐷𝑎+ 𝑥 (𝑡) + 𝑥 (𝑡) = 𝑓 (𝑡) , 𝐷𝑏−

𝑡 ∈ [𝑎, 𝑏] a.e.

(138)

Since 𝑥 ∈ 𝐻, 𝑥(𝑏) = 0. Moreover, applying Theorem 17 to the left side of (137) we obtain 𝑏

𝛼 𝛼 𝐷𝑎+ 𝑥 (𝑡) + 𝑥 (𝑡) − 𝑓 (𝑡)) ℎ (𝑡) 𝑑𝑡 ∫ (𝐷𝑏− 𝑎

1−𝛼 𝛼 1−𝛼 𝛼 ℎ (𝑎) 𝐷𝑎+ 𝑥 (𝑎) + 𝐼𝑏− 𝐷𝑎+ 𝑥 (𝑏) ℎ (𝑏) − 𝐼𝑎+

(139)

=0 for ℎ ∈ 𝐻. So, 1−𝛼 𝛼 𝐼𝑎+ ℎ (𝑎) 𝐷𝑎+ 𝑥 (𝑎) = 0,

ℎ ∈ 𝐻.

(140)

𝛼 𝑥(𝑎) = 0. Indeed, it is sufficient to This means that 𝐷𝑎+ consider functions ℎ𝑖 , 𝑖 = 1, . . . , 𝑚, of the form

𝑐 1 ℎ𝑖 (𝑡) = (0, . . . , 0, Γ (𝛼) (𝑡 − 𝑎)1−𝛼 +

1−𝛼 𝑥 (𝑎) = 𝐶, 𝐼𝑎+

(135)

𝑥 (𝑏) = 𝐷,

(145)

where 𝐶, 𝐷 ∈ R𝑚 , is investigated using the Stampacchia theorem (cf. [1]). Nonlinear system of the form

𝑏 1 ‖𝑥‖2𝑊𝛼,2 − ∫ 𝑓 (𝑡) 𝑥 (𝑡) 𝑑𝑡 𝑎+ 2 𝑎

𝑏

𝛼,2 . In [4], system (142) with nonhomogeneous in the space 𝑊𝑎+ boundary conditions

𝑡

1 1 , 0, . . . , 0) , ∫ Γ (𝛼) 𝑎 (𝑡 − 𝜏)1−𝛼

(141)

𝛼 𝛼 𝐷𝑎+ 𝑥 (𝑡) = −𝐹𝑥 (𝑡, 𝑥 (𝑡)) , 𝐷𝑏−

𝑡 ∈ [𝑎, 𝑏] a.e.,

(146)

where 𝐹𝑥 is the gradient (in 𝑥) of a potential 𝐹 = 𝐹(𝑡, 𝑥), with the above boundary conditions can be studied using the direct method of calculus of variations (cf. [11] for the case of (144)). 𝛼,2 , and It is worth noting that if 𝛼 ∈ (1/2, 1), 𝑥 ∈ 𝑊𝑎+ 1−𝛼 𝐼𝑎+ 𝑥(𝑎) = 0, then 𝑥(𝑎) = 0. It follows from the integral 𝛼 𝑓 is H¨older representation of 𝑥 and from the fact that 𝐼𝑎+ 𝛼 continuous on (𝑎, 𝑏] and lim𝑡 → 𝑎+ 𝐼𝑎+ 𝑓(𝑡) = 0 when 𝑓 ∈ 𝐿2 and 𝛼 ∈ (1/2, 1) (cf. [10, Property 4]). So, it is natural to put in such a case 𝑥(𝑎) = 0. It seems that the most accurate functions for investigating the above systems with boundary conditions involving condition 𝑥(𝑎) = 0 (or more general 𝑥(𝑎) = 𝐶), in general case of 𝛼 ∈ (0, 1), are the functions possessing the fractional derivatives in Caputo sense; on such functions one assumes that they are absolutely continuous on [𝑎, 𝑏] and, consequently, condition 𝑥(𝑎) = 0 makes sense. To our best knowledge, fractional Sobolev spaces via Caputo derivatives have not been investigated up to now and are an open problem. Remark 40. From the condition (136) it follows that one can search approximate solutions to (132) using numerical methods, for example, the gradient or projection of gradient methods applied to the functional 𝑏 1 ℎ 󳨃󳨀→ ‖ℎ‖2𝑊𝛼,2 − ∫ 𝑓 (𝑡) ℎ (𝑡) 𝑑𝑡 𝑎+ 2 𝑎

(147)

𝛼,2 defined on 𝐻 and 𝑊𝑎+ , respectively.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

𝑏

where 𝑐 = −(𝑏 − 𝑎)1−𝛼 ∫𝑎 (1/(𝑏 − 𝜏)1−𝛼 )𝑑𝑡, with nonzero 1−𝛼 ℎ𝑖 (𝑎) = 𝑖th coordinate function (of course, ℎ𝑖 ∈ 𝐻 and 𝐼𝑎+ (0, . . . , 0, 𝑐, 0 . . . , 0) ≠ 0). Remark 39. In the same way one can prove the existence of a solution to system 𝛼 𝛼 𝐷𝑎+ 𝑥 (𝑡) + 𝑥 (𝑡) = 𝑓 (𝑡) , 𝐷𝑏−

𝑡 ∈ [𝑎, 𝑏] a.e.

(142)

with boundary conditions 1−𝛼 𝐼𝑎+ 𝑥 (𝑎) = 0,

1−𝛼 𝛼 𝐼𝑏− 𝐷𝑎+ 𝑥 (𝑏) = 0

(143)

𝑥 (𝑏) = 0,

(144)

or 1−𝛼 𝐼𝑎+ 𝑥 (𝑎) = 0,

Acknowledgment The project was financed with funds of National Science Centre, granted on the basis of decision DEC-2011/ 01/B/ST7/03426.

References [1] H. Brezis, Analyse Fonctionnelle, Theorie et Applications, Masson, Paris, France, 1983. [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, Ny, USA, 1975. [3] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives—Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1993.

Journal of Function Spaces and Applications [4] L. Bourdin and D. Idczak, “Fractional fundamental lemma and fractional integration by parts formula—applications to critical points of Bolza functionals and to linear boundary value problems,” submitted. [5] D. Idczak and M. Majewski, “Fractional fundamental lemma of order 𝛼 ∈ (𝑛 − 1/2, 𝑛) with 𝑛 ∈ 𝑁, 𝑛 ≥ 2,” Dynamic Systems and Applications, vol. 21, no. 2-3, pp. 251–268, 2012. [6] D. Idczak and S. Walczak, “A fractional imbedding theorem,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 418– 425, 2012. [7] D. Idczak and S. Walczak, “Compactness of fractional imbeddings,” in Proceedings of the 17th International Conference on Methods & Models in Automation & Robotics (MMAR ’12), pp. 585–588, 2012. [8] L. W. Kantorowitch and G. P. Akilov, Functional Analysis, Science, Moscow, Russia, 1984, (Russian). [9] M. W. Michalski, “Derivatives of noninteger order and their applications,” in Dissertationes Mathematicae, vol. 338, Polish Academy of Sciences, Warsaw, Poland, 1993. [10] L. Bourdin, “Existence of a weak solution for fractional EulerLagrange equations,” Journal of Mathematical Analysis and Applications, vol. 399, no. 1, pp. 239–251, 2013. [11] R. Kamocki and M. Majewski, “On a fractional Dirichlet problem,” in Proceedings of the 17th International Conference on Methods & Models in Automation & Robotics (MMAR ’12), pp. 60–63, 2012.

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