Advances in Mathematical Physics

Advances in Mathematical Physics / 2016 / Article

Research Article | Open Access

Volume 2016 |Article ID 9636491 | https://doi.org/10.1155/2016/9636491

Yanning Wang, Jianwen Zhou, Yongkun Li, "Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales", Advances in Mathematical Physics, vol. 2016, Article ID 9636491, 21 pages, 2016. https://doi.org/10.1155/2016/9636491

Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

Academic Editor: Pietro d’Avenia
Received23 Jun 2016
Revised26 Aug 2016
Accepted08 Sep 2016
Published23 Nov 2016

Abstract

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a -Laplacian conformable fractional differential equation boundary value problem on time scale , , , where denotes the conformable fractional derivative of of order at , is the forward jump operator, , and . By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.

1. Introduction

A time scale is an arbitrary nonempty closed subset of the real numbers, which has the topology inherited from the real numbers with the standard topology. In order to unify and generalize continuous and discrete analysis, the calculus on time scales was initiated by Hilger in 1990 (see [1, 2]). In view of the fact that time scale calculus can be used to model dynamic processes whose time domains are more complicated than the set of integers or real numbers, it plays an important role in various equations and systems arising in economy, biology, ecology, astronomy, and so on (see [1, 3, 4]). During the last decade, there has been a great deal of interest in the study of dynamic equations on time scales and the research in this area is rapidly growing, see [510] and the references therein. Nevertheless, these studies are all about integer order dynamic equations on time scales. The existence and multiplicity of solutions for fractional dynamic equations on time scales has received considerably less attention (see [11, 12]).

It is well known that the fractional calculus refers to differentiation and integration of an arbitrary (noninteger) order. The theory is owed to mathematicians such as Leibniz, Liouville, Riemann, Letnikov, and Grunwald. In this day and age, fractional calculus is one of the most intensively developing areas of mathematical analysis, including several definitions of fractional calculus like Riemann-Liouville fractional calculus, Caputo fractional calculus, Grunwald-Letnikov fractional calculus, Hadamard fractional calculus, Riesz fractional calculus, Weyl fractional calculus, Kolwankar-Gangal fractional calculus and so on. Especially, in [13], the authors introduce the conformable fractional calculus on . In order to unify and generalize the Hilger calculus and the conformable fractional calculus on , the authors introduce the conformable fractional calculus on time scales and study its properties in [14]. The generation of fractional differential equations is born with the birth of the fractional calculus. Fractional differential equations have gained importance due to their numerous applications in many fields of science and engineering including fluid flow, electrical networks, probability and statistics, viscoelasticity, chemical physics and signal processing, and so on, see [1518] and references therein.

On one hand, there have been many approaches to study solutions of boundary value problems for the fractional differential equations such as lower and upper solution method, monotone iterative method ([19]), fixed-point theorems ([20]), Leray-Schauder theory ([21]), critical point theory ([22]) and so on. But until now, as far as I am concerned, no researchers have applied the critical point theory to study conformable fractional differential equations on time scales. Since it is often very difficult to establish a suitable space and variational functional for conformable fractional differential equations on time scales.

On the other hand, Sobolev spaces are regarded as one of most fundamental tools, especially in the use of variational methods to solve boundary value problems in ordinary and partial differential equations and difference equations, see [2326]. Sobolev’s spaces on a closed interval of are well known in [24]. For the sake of the study for differential equations on time scales, the authors defined the Sobolev spaces on time scales and studied some of their important properties in [25]. In [26], Jiao and Zhou developed a Caputo fractional derivative space and some of their properties which can be used to study Caputo fractional differential equation boundary value problems via critical point theory.

In view of the above reasons, our main purpose of this paper is to construct fractional Sobolev spaces on time scales via conformable fractional calculus and investigate some of their important properties. As an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a -Laplacian conformable fractional differential equations boundary value problem on time scale .

This paper is organized as follows. In Section 2, we introduce the definition of conformable fractional calculus and their important properties. In Section 3, we construct fractional Sobolev spaces on time scales via conformable fractional calculus and investigate some of their important properties. Section 4, as an application of the fractional Sobolev’s spaces on time scales, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for -Laplacian conformable fractional differential equations boundary value problem on time scale . By making a variational structure on the fractional Sobolev’s spaces on time scales, we can reduce the problem of finding solutions of a -Laplacian conformable fractional differential equation boundary value problem on time scale to the one of seeking the critical points of a corresponding functional.

2. Conformable Fractional Calculus on Time Scales and Their Properties

In this section, we introduce some definitions of conformable fractional derivative and integral and study some of their important properties.

Throughout this paper, we assume . We start by the definitions of conformable fractional derivative.

For convenience, we denote the intervals and in by respectively. Note that if is left-dense and if is left-scattered. We denote , therefore if is left-dense and if is left-scattered.

Let and . We define the neighborhood of as . We begin to introduce a new notation: the conformable fractional derivative of order for functions defined on arbitrary time scales.

Definition 1 (Definition  1, [14]). Let , , and . For , we define to be the number (provided it exists) with the property that given any , there is a neighborhood of , such that We call the conformable fractional derivative of of order at , and we define the conformable fractional derivative at as .

Definition 2 (Definition  23, [14]). Let , and be times delta differentiable at . We define the conformable fractional derivative of of order as .

Definition 3 (Definition  26, [14]). Let be a regulated function. Then the -fractional integral of , is defined by

Definition 4 (Definition  28, [14]). Suppose is a regulated function. Denote the indefinite fractional integral of of order , as follows: . Then, for all , we define the Cauchy fractional integral by

The -measure and -integration are defined the same as those in [27].

Definition 5 (Definition  2.3, [28]). Let . is called -null set if . Say that a property holds -almost everywhere (-a.e.) on , or for -almost all (-a.a.) if there is a -null set such that holds for all .

Definition 6. Assume is a function. Let is a -measurable subset of . is -integrable on if and only if is integrable on , and .

The conformable fractional calculus of has the following important properties. Before the statement of the properties, we denote

Lemma 7 (Theorem  4, [14]). Let . Assume and let . The following properties hold:(i)If is conformal fractional differentiable of order at , then is continuous at .(ii)If is continuous at and is right-scattered, then is conformable fractional differentiable of order at with (iii)If is right-dense, then is conformable fractional differentiable of order at if and only if the limit exists as a finite number. In this case, (iv)If is conformable fractional differentiable of order at , then

Lemma 8 (Theorem  15, [14]). Assume are conformable fractional differentiable of order . Then(i)the sum is conformable fractional differentiable with ;(ii)for any , is conformable fractional differentiable with ;(iii)if and are continuous, then the product is conformable fractional differentiable with ;(iv)if is continuous, then is conformable fractional differentiable with (v)if and are continuous, then is conformable fractional differentiable with

Lemma 9 (Theorem  25, [14]). Let . The following relation holds:

Remark 10. In (10), when , we have .

Lemma 11 (Theorem  30, [14]). Let . Then, for any rd-continuous function , there exists a function such that for all . Function is said to be an -antiderivative of .

Lemma 12 (Theorem  31, [14]). Let and be two rd-continuous functions. Then,(i);(ii);(iii);(iv);(v);(vi)if there exist with for all , then ;(vii)if for all , then .

Lemma 13 (Theorem  33, [14]). Let be a time scale. If for all , then is an increasing function on .

Theorem 14. Let be a continuous function on that is conformal fractional differentiable of order on and satisfies . Then there exist such that .

Proof. Since the function is continuous on the compact set , assumes its minimum and its maximum . Therefore there exist such that and . Since , we may assume that . By Lemma 13, we have . The proof is complete.

Theorem 15 (mean value theorem). Let be a continuous function on which is conformal fractional differentiable of order on . Then there exist such that

Proof. It follows from Lemma 9 that Let . Then, the function is continuous function on which is conformal fractional differentiable of order on and . Combining Lemma 8 and (12), we have Applying Theorem 14 to , there exist such that . That is The proof is complete.

Similar to the [Definition , [26]], we give the following definition of absolutely continuous function.

Definition 16. A function is said to be absolutely continuous on (i.e., ), if for every , there exists such that if is a finite pairwise disjoint family of subintervals of satisfying , then .

Lemma 17 (Theorem , [28]). A function is absolutely continuous on if and only if is delta differentiable -a.e. on and

Theorem 18. Assume function is absolutely continuous on , then is conformable fractional differentiable of order -a.e. on and the following equality is valid:

Proof. According to Lemma 17, is delta differentiable -a.e. on . Then, by Remark 10, is conformable fractional differentiable of order -a.e. on . Therefore, Definition 4 implies thatThe proof is complete.

Lemma 19 (Theorem  2.11, [28]). A function is absolutely continuous on , then is absolutely continuous on and the following equality is valid:

Theorem 20. Assume function is absolutely continuous on , then is absolutely continuous on and the following equality is valid:

Proof. The result is obtained by applying Lemmas 8 and 19 and Theorem 18 to the function and . The proof is complete.

Definition 21 (Definition  2.4, [28]). Let be a -measurable set and let be such that and let be a -measurable function. Say that belongs to provided that eitheror there exists a constant such that

Definition 22. Let be a -measurable set and let be such that and let be a -measurable function. Say that belongs to provided that eitheror there exists a constant such that

Lemma 23 (Theorem  2.5, [28]). Let be such that . Then the set is a Banach space together with the norm defined for asMoreover, is a Hilbert space together with the inner product given for every by

Theorem 24. Let be such that . Then the set is a Banach space together with the norm defined for asMoreover, is a Hilbert space together with the inner product given for every by

Proof. Let be a Cauchy sequence, then we have From Lemma 23 and (28), there exists such that Therefore, we can getThus, the space is a Banach space together with the norm .
Clearly, is a Hilbert space together with the inner product given for every by

Lemma 25 (Proposition  2.6, [28]). Suppose and . Let be such that . Then, if and , then and

Theorem 26. Suppose and . Let be such that . Then, if and , then and

Proof. By Lemma 25, we assertThe proof is complete.

Lemma 27 (Proposition  2.7, [28]). If and , then, the set is dense in .

Theorem 28. Let be such that the following equality is true:then

Proof. For every , the density of in guarantees the existence of such thatand so, by Lemma 12, (35) and (37), we deduce that for every , it is true that Because the sets are compact and disjoint subsets of , Urysohn’s lemma allows constructing a function which belongs to and it verifies so that, by defining , Theorem 15, (38), (39), and (40), we have that As a consequence of the arbitrary choice of , by (41), we achieve (36). The proof is complete.

Theorem 29. Let . Then, a necessary and sufficient condition for the validity of the equalityis the existence of a constant such that

Proof. If a.e. on , for any , from Lemma 12 and the definition of , one hasConversely, take , by defining as the fundamental theorem of conformable fractional calculus establishes that and so equality (42) yields Thereby, Theorem 28 and (46) allow to deduce (43) with . The proof is complete.

Next, we introduce the conformable fractional calculus on time scales for vector-valued functions and study some of their important properties.

Definition 30. Assume is a function, and let . Then one defines (provided it exists). One calls the conformable fractional derivative of of order at . The function is conformal fractional differentiable of order provided exists for all . The function is then called the conformable fractional derivative of of order .

Definition 31. Let be a time scale, , and let be times delta differentiable at . We define the conformable fractional derivative of of order as .

Definition 32. Assume is a function and . Let be a -measurable subset of . Then is -integrable on if and only if are -integrable on , and .

From Definitions 30 and 32, we have the following theorems.

Theorem 33. Let . Assume and let . The following properties hold:(i)If is conformal fractional differentiable of order at , then is continuous at .(ii)If is continuous at and is right-scattered, then is conformable fractional differentiable of order at with (iii)If is right-dense, then is conformable fractional differentiable of order at if and only if the limit exists as a finite number. In this case, (iv)If is conformable fractional differentiable of order at , then

Theorem 34. Assume are conformable fractional differentiable of order . Then,(i)the sum is conformable fractional differentiable with ;(ii)for any , is conformable fractional differentiable with ;(iii)if and are continuous, then the product is conformable fractional differentiable with .

Theorem 35. Let and be two rd-continuous functions. Then,(i);(ii);(iii);(iv);(v);(vi)if there exist with for all , then .

Definition 36 (Definition  2.6, [25]). A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in .

As we know from general theory of Sobolev’s spaces, another important class of functions is just the absolutely continuous functions on time scales. Similar to Definition , [23], we give the following definition of absolutely continuous function.

Definition 37. A function . We say is absolutely continuous on (i.e. ), if for every , there exist such that if is a finite pairwise disjoint family of subintervals of satisfying , then .

Remark 38 (Remark  2.1, [25]). By Definitions 22 and 30, we have that if and only if