Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

Hu, Xing; Li, Yongkun

doi:https://doi.org/10.1155/2022/7149356

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 7149356 | https://doi.org/10.1155/2022/7149356

Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

Xing Hu¹and Yongkun Li¹

Academic Editor: Dumitru Motreanu

Received04 Jul 2021

Accepted20 Aug 2021

Published22 Jan 2022

Abstract

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.

1. Introduction

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 [10] 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.

2. Preliminaries

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 [18]). For , we define the forward jump operator by while the backward jump operator is defined by

Remark 2 (see [18]). (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 [18]. 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 [18]). 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 [19]). 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 [20]). is an increasing continuous function on . If is the extension of to the real interval given by then

Definition 7 (see [19]) (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 [19]) (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 [19]). Let , we have .

Proposition 10 (see [19]). For any function that is integrable on , the Riemann-Liouville -fractional integral satisfies for and .

Proposition 11 (see [19]). For any function that is integrable on , one has .

Corollary 12 (see [19]). For , we have and , where denotes the identity operator.

Theorem 13 (see [19]). Let and , then iff

Theorem 14 (see [19]). Let and satisfy the condition in Theorem 13. Then,

Theorem 15 (see [2]). A function is absolutely continuous on iff is -differentiable -a.e. on and

Theorem 16 (see [21]). 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 [22]). 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 [2]) (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 [23]). 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 [1]). Let . Then, the following holds iff there exists a constant such that

Definition 21 (see [1]). 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 [1]). 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 [24]). Fractional integration operators are bounded in , i.e., the following estimate holds.

Proposition 24 (see [1]). 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 [25]) (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 [25]). 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 [25]). 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 [26]) (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.

In the following, let . Inspired by Theorems 15–18, we give the following definition.

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 [1]. 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 .

Proof. The conclusion follows from Theorem 23, Proposition 24, and Proposition 6. The proof is complete.

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 where Consequently, where .
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 [27].

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 where 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 [1], one obtains so and Let , then , , , and hence, The proof is complete.

Proposition 45. Let and . For all , if or , then if and , then

Proof. In view of Remark 36 and Theorem 14, in order to prove inequalities (73) and (74), we only need to prove that for or , and that for and .
Note that , the inequality (75) follows from Lemma 39 directly.
We are now in a position to prove (76). For , choose such that . For all , since is an increasing monotone function, by using Proposition 6, we find that . Taking into account Proposition 24, we have The proof is complete.

Remark 46. (i)According to (73), we can consider with respect to the norm in the following analysis.(ii)It follows from (73) and (74) that is continuously immersed into with the natural norm

Proposition 47. Let and . Assume that and the sequence converges weakly to in . Then, in , i.e., , as .

Proof. If , then by (74) and (78), the injection of into , with its natural norm , is continuous, i.e., in , then in .
Since in , it follows that in . In fact, for any , if in , then in , and thus, . Therefore, , which means that . Hence, if in , then for any , we have , and thus, , i.e., in .
By the Banach-Steinhaus theorem, is bounded in and, hence, in . Now, we prove that the sequence is equicontinuous. Let and with , for all , by using Proposition 24, Proposition 6, and Theorem 27, and noting , we have Therefore, the sequence is equicontinuous since, for , , by applying (79) and (78), we have where and is a constant. By the Arzelà-Ascoli theorem on time scales (Lemma 28), is relatively compact in . By the uniqueness of the weak limit in , every uniformly convergent subsequence of converges uniformly on to . The proof is complete.

Remark 48. It follows from Proposition 47 that is compactly immersed into with the natural norm .

Theorem 49. Let , , , , satisfies the following: (i)For each , is -measurable in (ii)For -almost every , is continuously differentiable in If there exist , , and , , such that, for -a.e. and every , one has Then, the functional defined by is continuously differentiable on , and , one has

Proof. It suffices to prove that has, at every point , a directional derivative given by (84) and that the mapping is continuous. The rest proof is similar to the proof of Theorem 1.4 in [28]. We will omit it here. The proof is complete.

4. An Application

As an application of the concepts we introduced and the results obtained in Section 3, in this section, we will use critical point theory to study the solvability of a class of boundary value problems on time scales. More precisely, our goal is to study the following fractional nonlinear Dirichlet problem on time scale : where and are the right and the left Riemann-Liouville fractional derivative operators of order defined on , respectively, is the gradient of at and is homogeneous of degree , is a positive parameter, , and .

We make the following assumptions:

is homogeneous of degree , that is, for all , ;

for all .

By , , we have for some constant .

Our main results are as follows.

Theorem 50. Let , and suppose that satisfies the conditions and . Then, there exists such that for all , (85) has at least two nontrivial solutions.

There have been many results using critical point theory to study boundary value problems of fractional differential equations [29–35] and dynamic equations on time scales [36–40], but the results of using critical point theory to study boundary value problems of fractional dynamic equations on time scales are still rare [3]. This section will explain that critical point theory is an effective way to deal with the existence of solutions of (85) on time scales.

We will use the famous Nehari manifold and fibering map theory to prove our main results.

We say that is a solution to the problem (85), if satisfies the following equality:

As a result, associated to the problem (85), we define the functional where

We need to show that the following lemma holds.

Lemma 51. (i)The functional is well defined on (ii)The functional is of class , and for all , we havewhere

Proof. (i)From (33) in Proposition 45, (87), (88), and the equivalent norm, we obtainwhich implies that is well defined on . (ii)LetThen, we can easily show that for all and for -a.e. , Hence, in view of the Lagrange mean value theorem, (87) and (42), there exists a real number such that and On the other hand, in view of Hölder inequality on time scales, we see that for or . Because is bounded, then, from the above inequalities, we conclude that the expression (97) is in .
As a result, in view of the dominated convergence theorem on time scales, one gets That is to say, is Gteaux differentiable.
In what follows, we prove that the Gteaux derivative of is continuous.
First, we verify that is continuous.
Taking into account (88), we have which combining with in and Lebesgue’s dominated convergence theorem on time scales leads to Namely, Let such that in (). Using the Hölder inequality on time scales and (73) in Proposition 45, we can obtain So, is continuous.
Next, we will prove . For any given , by the Hölder inequality on time scales, we have That is, is bounded. It is obvious that is linear. Hence, for any , .
Define , , , where . Now, it is time for us to demonstrate is continuous in the following two cases: (1)If , then, for , using Hölder inequality on time scales, we can deduce that(2)If , then, for , we haveConsequently, when , is continuous.
Now, for , we will show that In fact, for , by Hölder inequality on time scales, we have Combining with the continuity of , we see that .
In conclusion, (ii) is proven. The proof is complete.

We deduce from Lemma 51 and (87) that

It is easy to see that the energy functional is not bounded below on the space , but it is bounded below on a suitable subset of . In order to study problem (85), we define the constraint set

Note that contains every nonzero solution of (85), and if and only if

In order to get the existence of solutions, we decompose into three parts: corresponding to local minima, local maxima, and points of inflection are -measurable sets defined as follows:

Next, we give some important attributes of , and . Let be such that and put

Then, we have the following crucial result.

Lemma 52. If , then .

Proof. We proceed by contradiction to show that for all . If there exists , then, in view of (111), we get By Proposition 45 and (114), we have which implies that Moreover, combining with Proposition 45, (88), and (115), one has Hence, It follows from (47) that and that Combining with (120) and (120), we gain that Namely, , which leads to a contradiction. This completes the proof of Lemma 52.

Lemma 53. If , then is coercive and bounded below on .

Proof. Let , then, by (88) and Proposition 45, we gain Hence, in view of (111), one gets Since , is coercive and bounded below on . The proof of Lemma 53 is now completed.

Now as we know, the Nehari manifold is closely related to the behavior of the functions defined as

Such maps are called fibering maps. For , we define

Then, one obtains

Then, it is obvious to see that iff , and in particular, iff .

Before using fiber mapping to study the behavior of Nehari manifolds, let us introduce some symbols.

We will study the fibering map according to the signs of and . To this end, let us define by setting

Hence, for , one gets which implies that iff is a solution of the following equation:

Furthermore, obviously, and

Lemma 54. If , then has no critical point.

Proof. In this case, and for all , which yields that is strictly increasing and hence has no critical point. The proof is complete.

Lemma 55. If , then possesses a unique critical point that corresponds to a global maximum point. Moreover, there exists such that and .

Proof. In this case, there exists a unique such that . In addition, for and for . Note that and as . So, for , there exists a unique such that . Consequently, according to (131), we get for , and for . That is, is increasing on and decreasing on . Therefore, has exactly one critical point at , which is a global maximum point. Thus, by (128), . The proof is complete.

Lemma 56. If , then possesses a unique critical point that corresponds to a global minimum point. Moreover, there exists such that and .

Proof. In this case, it is easy to see that and for all , which implies that is strictly increasing. Since , there exists a unique such that . This implies that is decreasing on , increasing on and . Thus, has exactly one critical point corresponding to global minimum point. Hence, . Moreover, since , then we have . The proof is complete.

Lemma 57. If , then there exists such that for , has a positive value and has exactly two critical points that correspond to the local minimum and local maximum. Moreover, there exists such that and .

Proof. Let . As in above, we define Then, Let , we have which is the maximum value point of . Moreover, one has In consideration of Proposition 45, we deduce that which is independent of . We now prove that there exists such that . Taking (16) and Proposition 45 into consideration, one obtains Hence, we have for , where is the constant given in (138) and The same arguments used in the proof of Lemma 55 show that has exactly two critical points which correspond to the local minimum and local maximum. Furthermore, there exists such that and . The proof of Lemma 57 is now completed.

From now on, we define by

Note that if , then all the above related lemmas are true.

Lemma 58. Let be a local minimizer for on subsets or of such that . Then, is a critical of .

Proof. Since is a minimizer for under the constraint Then, applying the theory of Lagrange multipliers, we get the existence of such that Therefore, one has but and so . Hence, , which gives the proof of Lemma 58. The proof is complete.

In the following, we assume that and . Let be the constant given by (56). Then, the proof of Theorem 50 is based on the following two propositions.

Proposition 59. Suppose that assumptions of Theorem 50 are satisfied. Then, for all , achieves its minimum on .

Proof. In view of and Lemma 53, we have which is bounded below on and also on . Therefore, there exists a minimizing sequence such that As is coercive on , is a bounded sequence in up to a subsequence; there is such that weakly in .
Let such that . So, using Lemmas 56 and 57, there is such that and . Therefore, .
Because of , we have which yields that Letting go to infinity in the above equation, we obtain Now, we declare that strongly in . Otherwise, we have Since , it follows from (150) that for sufficiently large . Hence, we must have .
However, , and so, which gives a contradiction. Thus, strongly in ; as a consequence, . In addition, it is easy to check by contradiction that . Therefore, from (149), is a nontrivial solution of (85). The proof is complete.

Proposition 60. Suppose that assumptions of Theorem 50 are satisfied. Then, for all , achieves its minimum on .

Proof. Let . Hence, by the result of Lemma 55, we obtain the existence of such that . Therefore, there is a minimizing sequence such that Furthermore, in view of Lemma 53, we know that is coercive, so is a bounded sequence in up to a subsequence, there is such that weakly in .
Because of , then we have Letting go to infinity in (153), it follows from (152) that Therefore, , and so, has a global maximum at some point . Consequently, .
On the other hand, implies that is a global maximum point for , i.e., Now, as in the proof of Proposition 59, we assert that in . Assuming it is not true, then It follows from 4.23 that which gives a contradiction. Therefore, , and so, .
Using Lemma 52, we have , so is a minimizer for on .
On the other hand, by (22), is a nontrivial solution of (1). Finally, since , and are distinct. That is, the result of Theorem 50 holds true. The proof is complete.

5. Conclusions

In this paper, we introduced a class of fractional Sobolev spaces via the fractional derivative of Riemann-Liouville on time scales and obtain some of their basic properties. As an application, we use critical point theory to study the solvability of a class of fractional boundary value problems on time scales. The results and methods in this paper can also be used to study the solvability of other boundary value problems on time scales. At present, the concept of fractional derivatives in different meanings is constantly being proposed. Therefore, studying the theory and application of fractional Sobolev spaces on time scales in other meanings is our future direction.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

The authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work is supported by the Natural Science Foundation of China (11861072).

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Copyright © 2022 Xing Hu and Yongkun Li. 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.

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