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

Wang, Yanning; Zhou, Jianwen; Li, Yongkun

doi:https://doi.org/10.1155/2016/9636491

Advances in Mathematical Physics

On this page

Abstract Introduction Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2016 | Article ID 9636491 | 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

Yanning Wang,¹Jianwen Zhou,¹and Yongkun Li¹

Academic Editor: Pietro d’Avenia

Received23 Jun 2016

Revised26 Aug 2016

Accepted08 Sept 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 [5–10] 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 [15–18] 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 [23–26]. 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 .

Absolutely continuous functions have the following properties.

Combining Definitions 30, 36, Theorems 18 and 20, we have the following theorems.

Theorem 39. Assume function is absolutely continuous on , then is conformable fractional differentiable of order -a.e. on and the following equality is valid:

Theorem 40. A function is absolutely continuous on , then is absolutely continuous on and the following equality is valid:

3. Conformable Fractional Sobolev’s Spaces on Time Scales and Their Properties

In this section, we develop the fractional Sobolev’s spaces on time scales via conformable fractional calculus and their important properties.

For , we set the spacewith the normWe can obtain the following theorem.

Theorem 41. Assume is a real number. Then the space is a Banach space together with the norm . Especially, when , is a Hilbert space together with the inner product given for every bywhere denotes the inner product in .

Proof. Let be a Cauchy sequence, . Hence, one has It follows from (56) thatfor any . Therefore, () are the Cauchy sequences of . By Theorem 24, there exist such that We let . On one hand, we can getThus, . On the other hand, according to (58), we haveFrom (60), in ; therefore, the space is a Banach space together with the norm .
Evidently, is a Hilbert space together with the inner product given for every byThe proof is complete.

Now, we introduce the conformable fractional Sobolev’s spaces on . For the sake of convenience, in the sequel, we will let and

Definition 42. Let be such that and . One says that if and only if , and there exists such that and For , we denote

Remark 43. Theorems 39 and 40 imply that for every with ,

Next, we will point out that both sets are, as class of functions, equivalent. For the sake of this purpose, we prove the following theorems firstly.

Theorem 44. Suppose . Then, a necessary and sufficient condition for the validity of the equality is the existence of a constant such that

Proof. By Definition 30, implies .
On one hand, If , for any , Combining Theorem 35 with the definition of , Definitions 30 and 32, we haveOn the other hand, , let , then and That is Because , by Theorem 29, there exists such that Analogically, there exists such that Therefore, The proof is finished.

Now, we show the characterization of functions in in terms of functions in .

Theorem 45. Assume that for some with , and that (63) holds for . Then, there exists a unique function such that

Proof. Let denote the canonical basis of . We can take in (63), which guarantees So, it is easy to get thatWe define function asEquations (76) and (77) imply that and by Theorem 40, we assert thatfor every . It follows from Theorem 44 and (78) that there exists such that As a result of the fundamental theorem of conformable fractional calculus we can choose the function defined as Then, the function is the unique function in for which (74) is valid. The proof is complete.

If we identify with its absolutely continuous representative for which (74) holds, then the set can be endowed with the structure of Banach space.

Theorem 46. Suppose and . Then the set is a Banach space together with the norm defined asMoreover, the set is a Hilbert space together with the inner product

Proof. Let be a Cauchy sequence in According to the definition of , we have and there exist such that and Therefore, by Theorem 45, there exists such thatEquations (83) and (84) imply that for all . On account of is a Cauchy sequence in , by (81), we can getIt follows from Theorems 33 and 35 and (86) and (87) that For another, from Theorem 41, (87), and (88), there exist such thatCombining (85) with (89), one hasfor all . By (90), we claim that . Furthermore, it follows from Theorems 33 and 35 and (89) that Hence, by Remark 43, (89), (91), and Theorem 45, there exists such that So, is a Banach space.
Clearly, the set is a Hilbert space together with the inner product The proof is finished.

Next, we will show some important properties of the Banach space .

Theorem 47. The space is a reflexive and uniformly convex Banach space when .

Proof. By Clarkson inequality, we have where Thus, , for any and , if , by (94) and (95), one has if , by (96) and (97), we have Combining (98) with (99), is a uniformly convex Banach space. By functional analysis, we know that a uniformly convex Banach space is reflexive. The proof is complete.

Theorem 48. There exists such that the inequalityholds for all , where .
Moreover, if , then

Proof. Going to the components of , we can suppose that . If , from Theorem 45, is absolutely continuous on . On the basis of Theorem 15, there exists such thatHence, for , using Theorems 15 and 26 and (102), we obtain where . If , by (103), (101) holds. In the general case, for , from Theorem 33 and Hölder’s inequality, we have where By (104), (100) holds. The proof is complete.

Remark 49. It follows from Theorem 48 that is continuously immersed into with the norm .

Theorem 50. If the sequence converges weakly to in , then converges strongly in to .

Proof. Owing to in , is bounded in and, hence, in . It follows from Remark 49 that in . For , there exists such that In view of this, the sequence is equicontinuous. According to Ascoli-Arzela theorem, is relatively in . By the uniqueness of the weak limit in , every uniformly convergent subsequence of converges to . Consequently, converges strongly in to . The proof is complete.

Remark 51. It follows from Theorem 50 that is compact immersed into with the norm .

Theorem 52. Let be -measurable in for each and continuously conformal fractional differentiable of order in for -almost every . If there exist and such that for -almost and every , one has where , then the functional defined by is continuously differentiable on and

Proof. It suffices to prove that has at every point a directional derivative given by (108) and that the mapping is continuous.
As a matter of fact, on one hand, it follows from (106) that is everywhere finite on . We denote, for and fixed in , In the light of (106), we obtain where By the definition of function , . Since , one has On the other hand, (106) implies thathence . Therefore, from Theorem 48, (114), and (115), there exist positive constants such thatand has a directional derivative at and given by (108).
Furthermore, (106) implies that the mapping from into defined by is continuous, so that is continuous from into . The proof is finished.

4. An Application of the Space

In this section, as an application of the Sobolev’s space . We present a recent approach via variational methods and critical point theory to obtain the existence and multiplicity of solutions for the -Laplacian conformable fractional differential equations boundary value problem on time scale where denotes the conformable fractional derivative of of order at , is the forward jump operator, and satisfies the following assumption:(A) is -measurable in for every and continuously differentiable in for -a.e. and there exist such that for all and -a.e. , where denotes the gradient of in .By making a variational structure on , we can reduce the problem of finding solutions of (118) to the one of seeking the critical points of a corresponding functional.

Problem (118) covers the -Laplacian system (for when ) and the second-order Hamiltonian system on time scale (for when ) as well as the second-order Hamiltonian system (for when ) and the second-order discrete Hamiltonian system (for when )

Consider the functional defined by We will show the following theorems.

Theorem 53. The functional is continuously differentiable on and for all .

Proof. We choose for all and in Theorem 52. It is easy to get that the functional is continuously differentiable on andfor all by condition and Theorem 52. The proof is complete.

Theorem 54. If is a critical of in , that is, , then is a weak solution of problem (118).

Proof. Since , it follows from Theorem 53 that for all . That is for all . Using condition (A) and Definition 42, we obtain that . It follows from Theorem 45 and (74) that there exists a unique function such thatEquation (129) imply that We identify with its absolutely continuous representative
for which (129) holds. Then, is solution of problem (118). The proof is complete.

In order to prove the existence and multiplicity of solutions for (118), we need the following definitions and theorems.

Definition 55 (see [24], P₈₁). Let be a real Banach space, and . is said to be satisfying -condition on if the existence of a sequence such that and as , implies that is a critical value of .

Definition 56 (see [24], P₈₁). Let be a real Banach space and . is said to be satisfying PS condition on if any sequence for which is bounded and as , possesses a convergent subsequence in .

Remark 57. It is clear that the PS condition implies the -condition for each .

Lemma 58 (Theorem , [24]). Let be a Banach space and let . Assume that splits into a direct sum of closed subspace with where . LetThen, if satisfies the -condition, is a critical value of .

As shown in [29], a deformation lemma can be proved with the weaker condition (C) replacing the usual PS condition, and it turns out that the Saddle Point Theorem (Lemma 58) holds under (C) condition.

For the sake of convenience, in the sequel, we denote .

For , let and . Set , then and we have the following theorem.

Theorem 59. In Banach space , for , if and only if

Proof. Firstly, we prove the necessity. Indeed, from Theorems 15 and 48, we getThis inequality impliesThat is
Secondly, we prove the sufficiency. From Theorem 15 and Hölder inequality we getTherefore, we haveThat is . This completes the proof.

Theorem 60. Assume that (A) and the following conditions are satisfied:()there exist such that for all and -a.e. .()there exists such that for all and -a.e. .()there exists a subset of with such that for -a.e. .Then problem (118) has at least one solution.

Proof. We will use the Saddle Point Theorem (Lemma 58) to prove Theorem 60. It suffices to prove that(i) satisfies (C)-condition,(ii) as ,(iii) as Firstly, we prove (i).
Let be (C)-sequence of , that is is bounded and as Then there exists a positive constant such that for all . In accordance with , (), (124), and (125), we havefor all . This inequality gives that there exists a constant such that for all . From condition (A), we have Combining (143) with (144), one has where . Moreover, by (124), (141), and (145), one hasfor all . This shows that there exists constant such that for all . It follows from Theorem 48 and (147) that there exists constant such that for all .
Now, we will assert the sequence is bounded. If not, we may assume, without loss of generality, that as Set , then is bounded in . By Remark 51, there is a subsequence of (for simplicity denoted again by ) such thatCombining with (148), is bounded in , hence, So as for all . By (), we havewhich contradicts (145).
Consequently, by Theorem 59, is bounded in . Again, from Remark 51, there is a subsequence of (for simplicity denoted again by ) such that According to (152), is bounded in , then from condition (A), there exists positive constant such thatCombining with (152), we getBe aware ofone hasMoreover, it is easy to derive from (152) thatDefine Then one has as . By Hölder inequality, we getThis combined with (159) yields . It is easy to derive from (151) and Theorem 47 that in
Secondly, we prove (ii).
For any , combining (124) with () and (), we haveas . This shows (ii) holds.
Thirdly, we prove (iii).
For and a.e. , we set Make use of (), when , holds. In addition, using (162), satisfiesSo, when ,Besides, depending on condition (A) and (163), for and a.e. , we assertwhere . This implies thatfor all and a.e. . Moreover, by virtue of Theorem 48 and (124), we obtainfor all . Since , (iii) holds.
It follows from Lemma 58 and Theorem 54 that Theorem 60 holds.

Example 61. Let . Consider the -Laplacian conformable fractional differential equation boundary value problem on time scale where .
In virtue of , , all conditions of Theorem 60 hold with . It is easy to derive from Theorem 60 that problem (169) has at least one solution. Moreover, is not the solution of problem (169). Thus, problem (169) has at least one nontrivial solution.

According to Theorem 60, we have the following corollary.

Corollary 62. Assume that , , and the following condition are satisfied.() uniformly for -a.e. .Then problem (118) has at least one solution.

Theorem 63. Suppose that and the following condition are satisfied:() as .() Assume that is -subconvex with -a.e. , that is, for all and -a.e. .Then problem (118) has at least one solution.

Proof. It follows directly from () that Analogous to the proof of Theorem 60, we can obtain Next, we will verify satisfies (C)-condition. Let be a (C)-sequence; that is, is bounded and as Using the same method as that of (145), (147), and (148) in the proof of (i) in Theorem 60, there exist constants and such thatfor all . By means of (), we obtainfor all , this conclusion shows that is bounded. Combining with (174), we are sure is bounded. In a way similar to the proof of (i) in Theorem 60 we can prove that has a convergent subsequence, thus, satisfies condition (C).
It follows from Lemma 58 and Theorem 54 that Theorem 63 holds.

Example 64. Let . Consider the -Laplacian differential equation boundary value problemwhere .
In virtue of , and = (), all conditions of Theorem 63 hold . It follows directly from Theorem 63 that problem (176) has at least one solution. Furthermore, is not the solution of problem (176). Therefore, problem (176) has at least one nontrivial solution.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grants 11361072 and 11561072 and the Natural Sciences Foundation of Yunnan Province under Grant 2016FB011.

References

S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
View at: Publisher Site | Google Scholar | MathSciNet
S. Hilger, “Differential and difference calculus Unified!,” Nonlinear Analysis: Theory, Methods & Applications, vol. 30, no. 5, pp. 2683–2694, 1997.
View at: Publisher Site | Google Scholar
G. Nordlund, “Time-scales in futures research and forecasting,” Futures, vol. 44, no. 4, pp. 408–414, 2012.
View at: Publisher Site | Google Scholar
Q. Sheng, M. Fadag, J. Henderson, and J. M. Davis, “An exploration of combined dynamic derivatives on time scales and their applications,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 7, no. 3, pp. 395–413, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
H. T. Zhang and Y. K. Li, “Almost periodic solutions to dynamic equations on time scales,” Journal of the Egyptian Mathematical Society, vol. 21, no. 1, pp. 3–10, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
X.-H. Deng, Q.-R. Wang, and Z. Zhou, “Oscillation criteria for second order nonlinear delay dynamic equations on time scales,” Applied Mathematics and Computation, vol. 269, pp. 834–840, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
S. H. Hong and Y. Z. Peng, “Almost periodicity of set-valued functions and set dynamic equations on time scales,” Information Sciences, vol. 330, pp. 157–174, 2016.
View at: Publisher Site | Google Scholar
S.-Y. Zhang, Q.-R. Wang, and Q. Kong, “Asymptotics and oscillation of $n$ th-order nonlinear dynamic equations on time scales,” Applied Mathematics and Computation, vol. 275, pp. 324–334, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
J. W. Zhou and Y. K. Li, “Variational approach to a class of p-Laplacian systems on time scales,” Advances in Difference Equations, vol. 2013, article 297, 2013.
View at: Publisher Site | Google Scholar
Y. K. Li and J. W. Zhou, “Existence of solutions for a class of damped vibration problems on time scales,” Advances in Difference Equations, vol. 2010, Article ID 727486, 27 pages, 2010.
View at: Google Scholar | MathSciNet
P. A. Williams, “Fractional calculus on time scales with Taylor's theorem,” Fractional Calculus and Applied Analysis, vol. 15, no. 4, pp. 616–638, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
N. Benkhettou, A. Hammoudi, and D. F. M. Torres, “Existence and uniqueness of solution for a fractional Riemann-Liouville initial value problem on time scales,” Journal of King Saud University—Science, vol. 28, no. 1, pp. 87–92, 2016.
View at: Publisher Site | Google Scholar
R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
N. Benkhettou, S. Hassani, and D. F. M. Torres, “A conformable fractional calculus on arbitrary time scales,” Journal of King Saud University-Science, vol. 28, no. 1, pp. 93–98, 2016.
View at: Publisher Site | Google Scholar
R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000.
View at: Publisher Site | Google Scholar
V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 69, no. 8, pp. 2677–2682, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
G. H. Salgado and L. A. Aguirre, “A hybrid algorithm for Caputo fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 33, pp. 133–140, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
S. Q. Zhang, “Existence of a solution for the fractional differential equation with nonlinear boundary conditions,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1202–1208, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
K. H. Zhao and P. Gong, “Positive solutions of Riemann-Stieltjes integral boundary problems for the nonlinear coupling system involving fractional-order differential,” Advances in Difference Equations, vol. 2014, article 254, 18 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
W. H. Jiang, “The existence of solutions for boundary value problems of fractional differential equations at resonance,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 74, no. 5, pp. 1987–1994, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
F. Jiao and Y. Zhou, “Existence of solutions for a class of fractional boundary value problems via critical point theory,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1181–1199, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
P. H. Rabinowitz, “Minimax method in critical point theory with applications to differential equations,” in Proceedings of the CBMS Regional Conference Series in Mathematics, vol. 65 of American Mathematical Society, Providence, RI, USA, 1986.
View at: Google Scholar
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
View at: Publisher Site | MathSciNet
J. W. Zhou and Y. K. Li, “Sobolev's spaces on time scales and its application to a class of second order Hamiltonian systems on time scales,” Nonlinear Analysis, vol. 73, pp. 1375–1388, 2010.
View at: Google Scholar
F. Jiao and Y. Zhou, “Existence results for fractional boundary value problem via critical point theory,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 22, no. 4, Article ID 1250086, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
G. S. Guseinov, “Integration on time scales,” Journal of Mathematical Analysis and Applications, vol. 285, no. 1, pp. 107–127, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Basic properties of Sobolev's spaces on time scales,” Advances in Difference Equations, vol. 2006, Article ID 38121, 14 pages, 2006.
View at: Google Scholar | MathSciNet
C. K. Zhong, X. L. Fan, and W. Y. Chen, Introduction to Nonlinear Functional Analysis, Lanzhou University Press, Lanzhou, China, 2004 (Chinese).

Copyright

Copyright © 2016 Yanning Wang et al. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1775

Downloads

1474

Citations