Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales
By the concept of fractional derivative of Riemann-Liouville on time scales, we first introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Then, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and some imbeddings. Finally, as an application, by constructing an appropriate variational setting, using fibering mapping and Nehari manifolds, the existence of weak solutions for a class of fractional boundary value problems on time scales is studied, and a result of the existence of weak solutions for this problem is obtained.
The Sobolev space theory was developed by the Soviet mathematician S.L. Sobolev in the 1930s. It was created for the needs of studying modern theories of differential equations and studying many problems in the fields related to mathematical analysis. It has become a basic content in mathematics. In order to study the existence of solutions of differential and difference equations under a unified framework, papers [1–3] study some Sobolev space theories on time scales.
In the past few decades, fractional calculus and fractional differential equations have attracted widespread attention in the field of differential equations, as well as in applied mathematics and science. In addition to true mathematical interest and curiosity, this trend is also driven by interesting scientific and engineering applications that have produced fractional differential equation models to better describe (time) memory effects and (space) nonlocal phenomena [4–9]. It is the rise of these applications that give new vitality to the field of fractional calculus and fractional differential equations and call for further research in this field.
In order to unify the discrete analysis and continuous analysis, Hilger  proposed the time scale theory and established its related basic theory [11, 12]. So far, the study of time scale theory has attracted worldwide attention. It has been widely used in engineering, physics, economics, population dynamics, cybernetics, and other fields [13–17].
As far as we know, no one has studied the fractional Sobolev space and its properties on time scales through the Riemann-Liouville derivative. In order to fill this gap, the main purpose of this article is to establish the fractional Sobolev space on time scales via the Riemann-Liouville derivative and to study its basic properties. Then, as an application of our new theory, we study the solvability of a class of fractional boundary value problems on time scales.
In this section, we briefly collect some basic known notations, definitions, and results that will be used later.
A time scale is an arbitrary nonempty closed subset of the real set with the topology and ordering inherited from . Throughout this paper, we denote by a time scale. We will use the following notations: , , , , .
Definition 1 (see ). For , we define the forward jump operator by while the backward jump operator is defined by
Remark 2 (see ). (1)In Definition 1, we put (i.e., if has a maximum ) and (i.e., if has a minimum ), where denotes the empty set.(2)If , we say that is right-scattered, while if , we say that is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated.(3)If and , we say that is right-dense, while if and , we say that is left-dense. Points that are right-dense and left-dense at the same time are called dense(4)The graininess function is defined by .(5)The derivative makes use of the set , which is derived from the time scale as follows: if has a left-scattered maximum , then ; otherwise, .
Definition 3 . Assume that is a function and let . Then, we define to be the number (provided it exists) with the property that given any , there is a neighborhood of (i.e., for some ) such that for all . We call the delta (or Hilger) derivative of at . Moreover, we say that is delta (or Hilger) differentiable (or in short: differentiable) on provided exists for all . The function is then called the (delta) derivative of on .
Definition 4 (see ). A function is called -continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of -continuous functions will be denoted by The set of functions that are differentiable and whose derivative is -continuous is denoted by
Definition 5 (see ). Let denote a closed bounded interval in . A function is called a delta antiderivative of function provided is continuous on , delta differentiable at , and for all . Then, we define the -integral of from to by
Proposition 6 (see ). is an increasing continuous function on . If is the extension of to the real interval given by then
Definition 7 (see ) (fractional integral on time scales). Suppose is an integrable function on . Let . The left fractional integral of order of is defined by The right fractional integral of order of is defined by where is the gamma function.
Definition 8 (see ) (Riemann-Liouville fractional derivative on time scales). Let , , and . The left Riemann-Liouville fractional derivative of order of is defined by The right Riemann-Liouville fractional derivative of order of is defined by
Proposition 9 (see ). Let , we have .
Proposition 10 (see ). For any function that is integrable on , the Riemann-Liouville -fractional integral satisfies for and .
Proposition 11 (see ). For any function that is integrable on , one has .
Corollary 12 (see ). For , we have and , where denotes the identity operator.
Theorem 13 (see ). Let and , then iff
Theorem 15 (see ). A function is absolutely continuous on iff is -differentiable -a.e. on and
Theorem 16 (see ). A function is absolutely continuous on iff the following conditions are satisfied: (i) is -differentiable -a.e. on and (ii)The equality holds for every .
Theorem 17 (see ). A function is absolutely continuous iff there exist a constant and a function such that In this case, we have and , a.e.
Theorem 18 (see ) (integral representation). Let and . Then, has a left-sided Riemann-Liouville derivative of order iff there exist constant and a function such that In this case, we have and , a.e.
Theorem 19 (see ). Let , , and , where and in the case when . Moreover, let then the following integration by part formulas hold. (a)If and , then (b)If and , then
Lemma 20 (see ). Let . Then, the following holds iff there exists a constant such that
Definition 21 (see ). Let be such that and . Say that belongs to iff and there exists such that and with where is the set of all continuous functions on such that they are -differential on and their -derivatives are -continuous on .
Theorem 22 (see ). Let be such that . Then, the set is a Banach space together with the norm defined for every as Moreover, is a Hilbert space together with the inner product given for every by
Theorem 23 (see ). Fractional integration operators are bounded in , i.e., the following estimate holds.
Proposition 24 (see ). Suppose and . Let be such that . Then, if and , then and This expression is called Hölder’s inequality and Cauchy-Schwarz’s inequality whenever .
Theorem 25 (see ) (the first mean value theorem). Let and be bounded and integrable functions on , and let be nonnegative (or nonpositive) on . Let us set Then, there exists a real number satisfying the inequalities such that
Corollary 26 (see ). Let be an integrable function on and let and be the infimum and supremum, respectively, of on . Then, there exists a number between and such that
Theorem 27 (see ). Let be a function defined on and let with . If is -integrable from to and from to , then is -integrable from to and
Lemma 28 (see ) (a time scale version of the Arzelà-Ascoli theorem). Let be a subset of satisfying the following conditions: (i) is bounded(ii)For any given , there exists such that , implies for all Then, is relatively compact.
3. Fractional Sobolev Spaces on Time Scales and Their Properties
In this section, we present and prove some lemmas, propositions, and theorems, which are of utmost significance for our main results.
Definition 29. Let . By , we denote the set of all functions that have the representation
with and .
Then, we have the following result.
Theorem 30. Let and . Then, function has the left Riemann-Liouville derivative of order on the interval iff ; that is, has the representation (27). In such a case,
Proof. Let us assume that has a left-sided Riemann-Liouville derivative . This means that is (identified to) an absolutely continuous function. From the integral representation of Theorems 15 and 17, there exist a constant and a function such that
with and ,
By Proposition 10 and applying to (29), we obtain The result follows from the -differentiability of (30).
Conversely, let us assume that (27) holds true. From Proposition 10 and applying to (27), we obtain and then, has an absolutely continuous representation. Further, has a left-sided Riemann-Liouville derivative . This completes the proof.
Remark 31. (i)By , we denote the set of all functions possessing representation (27) with and (ii)It is easy to see that Theorem 30 implies that for any , has the left Riemann-Liouville derivative iff ; that is, has the representation (27) with
Definition 32. Let and let . By the left Sobolev space of order , we will mean the set given by
Remark 33. A function given in Definition 32 will be called the weak left fractional derivative of order of ; let us denote it by . The uniqueness of this weak derivative follows from 1.
We have the following characterization of .
Theorem 34. If and , then
Proof. On the one hand, if , then from Theorem 30, it follows that has derivative . Theorem 19 implies that
for any . So, with .
On the other hand, if , then , and there exists a function such that for any . To show that , it suffices to check (Theorem 30 and definition of ) that possesses the left Riemann-Liouville derivative of order , which belongs to ; that is, is absolutely continuous on and its delta derivative of order (existing -a.e. on ) belongs to .
In fact, let , then and . From Theorem 19, it follows that In view of (34) and (35), we get for any . So, . Consequently, is absolutely continuous and its delta derivative is equal -a.e. on to . The proof is complete.
From the proof of Theorem 34 and the uniqueness of the weak fractional derivative, the following theorem follows.
Theorem 35. If and , then the weak left fractional derivative of a function coincides with its left Riemann-Liouville fractional derivative -a.e. on .
Remark 36. (1)If and , then and, consequently, (2)If and , then is the set of all functions belonging to that satisfy the condition
By using the definition of with and Theorem 35, one can easily prove the following result.
Theorem 37. Let , and . Then, iff there exists a function such that
In such a case, there exists the left Riemann-Liouville derivative of and .
Remark 38. Function will be called the weak left fractional derivative of of order . Its uniqueness follows from . From the above theorem, it follows that it coincides with an appropriate Riemann-Liouville derivative.
Let us fix and consider in the space a norm given by
(Here denotes the delta norm in (Theorem 22)).
Lemma 39. Let and , then where . That is to say, the fractional integration operator is bounded in .
Theorem 40. If , then the norm is equivalent to the norm given by
Proof. (1)Assume that . On the one hand, in view of Remarks 31 and 36, for , we can write it aswith and . Since is an increasing monotone function, by using Proposition 6, we can write that . And taking into account Lemma 39, we have
where comes from Lemma 39. Noting that , , one can obtain
On the other hand, we will prove that there exists a constant such that Indeed, let and consider coordinate functions of with . Lemma 39, Theorem 25, and Corollary 26 imply that there exist constants such that Hence, for a fixed , if for all , then we can take constants such that Therefore, we have From the absolute continuity (Theorem 16) of , it follows that for any . Consequently, combining with Proposition 9 and Lemma 39, we see that for . In particular, So, where and . Thus, and, consequently, where .
If for belongs to some subset of , from the above argument process, one can easily see that there exists a constant such that (32) holds. (2)When , then (Remark 36) is the set of all functions that belong to that satisfy the condition . Hence, in the same way as in the case of (putting ), we obtain the inequalityThe inequality, is obvious (it is sufficient to put and use the fact that ).
The proof is complete.
Now, we are in a position to prove some basic properties of the space .
Theorem 41. The space is complete with respect to each of the norms and for any , .
Proof. In view of Theorem 40, we only need to show that with the norm is complete. Let be a Cauchy sequence with respect to this norm. So, the sequences and are Cauchy sequences in and , respectively.
Let and be the limits of the above two sequences in and , respectively. Then, the function belongs to and it is the limit of in with respect to . The proof is complete.
The proof method of the following two theorems is inspired by the method used in the proof of Proposition 8.1 (b) and (c) in .
Theorem 42. The space is reflexive with respect to the norm for any and .
Proof. Let us consider with the norm and define a mapping
It is obvious that
which means that the operator is an isometric isomorphic mapping and the space is isometric isomorphic to the space , which is a closed subset of as is closed.
Since is reflexive, the Cartesian product space is also a reflexive space with respect to the norm , where .
Thus, is reflexive with respect to the norm . The proof is complete.
Theorem 43. The space is separable with respect to the norm for any and .
Proof. Let us consider with the norm and the mapping defined in the proof of Theorem 42. Obviously, is separable as a subset of separable space . Since is the isometry, is also separable with respect to the norm . The proof is complete.
Theorem 44. Let , , then and where with .
Proof. We will divide the proof into the following three major cases. (i)When , we can take , the conclusion is evident(ii)When , we can take , the conclusion is obvious(iii)Let In this case, if there exist such that , then In view of , we have Hence, we obtain that Therefore, when satisfy the following conditions that is to say, by Proposition 2.6 in from , one obtains so and Let