Abstract
The present work investigates the applicability and effectiveness of the generalized Riemann-Liouville fractional integral operator integral method to obtain new Minkowski, Grüss type and several other associated dynamic variants on an arbitrary time scale, which are communicated as a combination of delta and fractional integrals. These inequalities extend some dynamic variants on time scales, and tie together and expand some integral inequalities. The present method is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractional differential equations applied in mathematical physics.
1. Introduction
Fractional calculus has also been comprehensively utilized in several instances, but the concept has been popularized and implemented in numerous disciplines of science, technology and engineering as a mathematical model (see [, ]). Numerous distinguished generalized fractional integral operators consist of the Hadamard operator, Erdlelyi-Kober operators, the Saigo operator, the Gaussian hypergeometric operator, the Marichev-Saigo-Maeda fractional integral operator, and so on.; out of the ones, the Riemann-Liouville fractional integral operator has been extensively utilized by researchers in theory as well as applications (see [, –]).
Stefan Hilger began the theories of time scales in his doctoral dissertation [] and combined discrete and continuous analysis (see [, ]). From this moment, this hypothesis has received a lot of attention. In the book written by Bohner and Peterson [] on the issues of time scale, a brief summary is given and several time calculations are performed. Over the past decade, many analysts working in specific applications have proved a reasonable number of dynamic inequalities on a time scale (see [–]). Several researchers have created various results relating to fractional calculus on time scales to obtain the corresponding dynamic inequalities (see [–]).
Recently, the idea of the fractional-order derivative has been expounded by Bastos et al. [] via Riemann-Liouville fractional operators on scale versions by considering linear dynamic equations. Another approach on time scales shifts to the inverse Laplace transform []. Following such innovator work, the investigation of fractional calculus on time scales created in a mainstream look into research studies on time scales (see [, –] and references therein). Since the publications in 2015, several researchers made significant contributions to the history of time scales. Sun and Hou [] employed the fractional q-symmetric systems on time scales. Yaslan and Liceli [] obtained the three-point boundary value problem with delta Riemann-Liouville fractional derivative on time scales. Yan et al. [] adopted the Caputo fractional techniques on differential equations on time scales. Zhu and Wu [] employed Caputo nabla fractional derivatives in order to find the existence of solutions for Cauchy problems. As certifiable utilities, we refer to the study of calcium ion channels that are impeded with an infusion of calcium-chelator ethylene glycol tetraacetic acid []. Actually, physical utilization of initial value-fractional problems in diverse time scales proliferates [, , ]. For instance, the continuous time scale 𝕋 = ℝ, the fractional differential equations that oversee the practices of viscoelastic materials with memory and creep tendencies have been investigated in Chidouh et al. [].
Integral Inequalities are an excellent way to investigate many scientific fields of research, including engineering, flow dynamics, biology, chaos, meteorology, vibration analysis, biochemistry, aerodynamics and many more. Since the productions of the above outcome in 1883, several works have been published in the literature of time calculus, with varied evidence, various speculations and improvements [–]. Recently, numerous analysts examined various inequalities, such as Hermite-Hadamard inequalities, Ostrowski inequalities and the expanded version of Hardy-type inequalities (see [–, , ] and the references therein).
Here, we broaden accessible outcomes in the literature [] by presenting increasingly broad ideas of fractional integral inequalities on time scales in the frame of generalized Riemann-Liouville fractional integral. At that point, we study the dynamic variants of corresponding generalized fractional-order on time scales. We obtain the inequalities Grss, Minkowski and several others using the delta integrals in arbitrary time scales. For δ = 1, the integral will become delta integral and for δ = 0, it advances toward turning out to be nabla integral. An astounding audit about the time scale calculus can be found in the paper []. The proposed dynamical integral method is reliable and effective to obtain new solutions. This method has more advantages: it is direct and concise. Thus, the proposed method can be extended to solve many systems of non-linear fractional partial differential equations in mathematical and physical sciences. Also, the new exact analytical solutions can be obtained for the generalized ordinary differential equations to obtain new theorems related to stability and continuous dependence on parameters for dynamic equations on time scales.
The present work investigates the applicability and effectiveness of the several dynamic variants that are presented, which are based primarily on the generalized Riemann-Liouville fractional integral operators. We will show that the Grüss and Minkowski type, that we participated in are very specific to the current work. From an application point of view, the results ultimately relate to the study of Young's inequality, arithmetic, and geometry inequality. Our computed outcomes can be very useful as a starting point of comparison when some approximate methods are applied to this non-linear space-time fractional equation. Furthermore, there are likewise some occurrences that can be derived from our outcomes.
2. Preliminaries
A non-empty closed subsets ℝ of 𝕋 is known as the time scale. The well-known examples of time scales theory are the set of real numbers ℝ and the integers Throughout the paper, we refer 𝕋 as time scale and a time-scaled interval is ϒ𝕋 = [υ1, υ2]𝕋. We need the concept of jump operators. The forward jump operator is denoted by the symbol ♢ and the backward jump operator is denoted by ϑ, are said through the formulas:
We accumulate as:
If ♢ (t) > t, then the term t is allude to be right-scattered and ω is allude to be left-scattered ϱ(ω) < ω. The elements that are most likely all the while appropriate-scattered and scattered are known as isolated. The term t is said to be right dense, if ♢(t) = t, and ω is said to be left dense, if ϱ(ω) = ω. In addition, the focuses t, ω are known to be dense if they are most likely right-dense and left-dense.
The mappings μ, ν:𝕋 → [0, +∞) defined by
are called the forward and backward graininess functions, respectively.
Definition 2.1. [, ] “Let ℏ:𝕋 → ℝ be a real-valued function. Then ℏ is said to be -continuous on ℝ if its left limit at any left dense point of 𝕋 is finite and it is continuous on every right dense point of 𝕋. All -continuous functions are denoted by ”
Definition 2.2. “A function is called a delta antiderivative of ℏ:𝕋 → ℝ if for all t ∈ 𝕋k. Then, one defines the delta integral by ”
Theorem 2.1. []. If ℏ ∈ ℂRDand t ∈ 𝕋k, then
[
].
Letυ
1, υ
2, υ
3∈ 𝕋, β ∈ ℝ
andℏ, ω ∈ ℂ
RD,
then.
From Theorem 2.2 (ix), for ℏ2(ρ) = |ℏ1(ρ)| on [υ1, υ2], we have
Proposition 2.1. [] Consider a time scale 𝕋 and ℏ is an increasing continuous function on ϒ𝕋. An extension of ℏ on ϒ𝕋isgiven as
then
Next we demonstrate the idea of fractional integral on time scale, which is mainly due to [].
Definition 2.3. [] “For 0 < δ < 1, let ϒ𝕋 ⊂ 𝕋 is a time scale and be an integrable function on ϒ𝕋. Then the (left) fractional integral of order δ of is defined by
where Γ is the gamma function.”
Again, we demonstrate the concept of generalized Riemann-Liouville fractional integral operator which is proposed by [].
Definition 2.4. [] “For 0 < δ < 1, let 𝕋 is a time scale and [υ1, υ2] is an interval of 𝕋. Suppose be an integrable function on [υ1, υ2] and Φ is monotone having a delta derivative ΦΔ with ΦΔ ≠ 0 for any η ∈ [υ1, υ2]. Let 0 < δ < 1, then the (left) generalized fractional integral of order δ of with respect to Φ is defined by
Remark 2.1. If 𝕋 = ℝ, then Definitions 2.3 and 2.4 reduces to the well-known Riemann-Liouville and generalized Riemann-Liouville fractional integral, respectively (see []).
3. Minkowski Type Inequalities for Generalized Riemann-Liouville Fractional Integral on Time Scale
This section is inaugurated to establishing generalizations of some reverse Minkowski inequality by introducing the generalized Riemann-Liouville fractional integral on time scale.
Theorem 3.1.Let δ, γ > 1, and 𝕋 is a time scale. Supposebe two positive functions on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔwith ΦΔ ≠ 0 such that for allIfthen
Proof: Since we find that
and
Taking product on both sides of (5) which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have
which implies that
It follows that
Accordingly, therefore we have
Taking product (9) by we arrive at
Taking product on both sides of (11) which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have
Hence, we can write
Conducting product between (8) and (12), we can draw the desired conclusion easily.
□
Corollary 3.1. Letting 𝕋 = ℝ, then under the assumption of Theorem 3.1, we have the following inequality in generalized Riemann-Liouville fractional integral:
Theorem 3.2.Let δ, γ > 1, and 𝕋 is a time scale. Supposebe two positive functions on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔwith ΦΔ ≠ 0 such that for allIfthen
where
Proof: Replacing and by and in Theorem 3.1, we acquire the desired result. This completes the proof.
□
4. Grüss Type Inequalities via Generalized Riemann-Liouville Fractional Integral on Time Scale
Our coming result is the generalization of Grüss type inequality via generalized Reimann-Liouville fractional integral operator on time scale.
Theorem 4.1. Let δ, γ > 1, and 𝕋 is a time scale. Suppose there is a positive functionon [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔwith ΦΔ ≠ 0 such that for all η > 0. Assume that the subsequent.
(I)There exist two integrable functions φ1, φ2on [0, ∞)𝕋such that
Then, for η > 0, δ, γ > 1, one has
Proof: From (I), for all θ ≥ 0, λ ≥ 0, we have
Therefore,
Taking product on both sides of (17) which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have
arrives at
Taking product on both sides of (19) which is positive because λ ∈ (0, η), η > 0, we integrate the resulting identity with respect to λ from 0 to η we have
Hence, we conclude the desired inequality. This completes the proof.
□
Special cases of Theorem 4.1, we attain the subsequent results.
Corollary 4.1. Letting Φ(η) = η, then Theorem 4.1 will lead to the Riemann-Liouville fractional integral on time scales:
Remark 4.1. If 𝕋 = ℝ, then Theorem 4.1 will lead to Theorem 2.11 in [] and corollary 4.1 will lead to Corollary 3 in []. Also, if we choose 𝕋 = ℝ along with Φ(η) = η, then Theorem 4.1 will lead to Theorem 2 in [].
Theorem 4.2.Let δ, γ > 1, and 𝕋 is a time scale. Suppose there are two positive functionson [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔwith ΦΔ ≠ 0 such that for all η > 0. Suppose that (I) holds and moreover one assumes the following. (II) There exist ω1 and ω2integrable functions on [0, ∞)𝕋such that
Then, for η > 0, δ, γ > 1, the following inequalities hold:
Proof: To prove (A1), from (I) and (II), we have for x ∈ [0, ∞)𝕋 that
Therefore,
Taking product on both sides of (24) which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have
Then we have
Again, multiplying both sides of (26) by which is positive because λ ∈ (0, η), η > 0, we integrate the resulting identity with respect to λ from 0 to η we have
This follows that
we acquire the desired inequality (A1).
To prove (B1) − (D1), we utilizes the subsequent variants:
□
Special case of Theorem 4.2, we have the subsequent corollaries.
Corollary 4.2. Letting Φ(η) = η, then Theorem 4.2 will lead to a new result for Riemann-Liouville fractional integral on time scales:
Remark 4.2. If 𝕋 = ℝ, then Theorem 4.2 will lead to Theorem 2.15 in [] and corollary 4.2 will lead to Corollary 2.16 in []. Also, If we choose 𝕋 = ℝ along with Φ(η) = η, then Theorem 4.2 will lead to Theorem 5 in [].
5. Some Other Bounds via Generalized Riemann-Liouville Fractional Integral on Time Scale
Theorem 5.1.Let δ, γ > 1, and 𝕋 is a time scale. Suppose there are two positive functionson [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔ with ΦΔ ≠ 0 such that for all η > 0, α, β > 1 satisfyingThen, for η > 0, one has
Proof: Taking into account the Young's inequality []:
setting and we have
Taking product on both sides of (29) which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have
we get
Again, multiplying both sides of (31) by which is positive because λ ∈ (0, η), η > 0, we integrate the resulting identity with respect to λ from 0 to η we have
consequently, we get
which implies (A3). The remaining variants can be proved by adopting the same technique as we did in (A3).
Repeating the foregoing argument, we obtain (B3) − (D3).
□
Theorem 5.2.Let δ, γ > 1, and 𝕋 is a time scale. Supposebe two positive functions on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔwith ΦΔ ≠ 0 such that for all η > 0, and α, β > 0 satisfying α + β = 1. Then, for η > 0, one has
Proof: Taking into account the weighted AM − GM inequality
by setting and we have
Multiplying both sides of (36) by which is positive because θ, λ ∈ (0, η), η > 0 and integrating the resulting identity from 0 to η we have
we conclude that
which implies (A4). The rest of inequalities can be shown in similar way by the following choice of parameters in AM − GM inequality.
□
Example 5.1. Let δ, γ > 1, and 𝕋 is a time scale. Suppose be two positive functions on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔ with ΦΔ ≠ 0 such that for all η > 0, and α, β > 0 satisfying Let
Then, for η > 0, δ, γ > 1, one has the following inequalities:
Proof: From Equation (39) and the inequality
then we can write as,
Multiplying both sides of (41) by which is positive because θ ∈ (0, η), η > 0 and integrating the resulting identity from 0 to η, we have
implies that
on the other hand, it follows from and
that
then from equation (43) and (45), we obtain,
Which implies (1). By some transformation of (1), similarly, we obtain (2) and (3).
□
6. Conclusion
The succinct view of this paper to establish numerous inequalities on an arbitrary time scale for generalized Riemann-Liouville fractional integrals. For the suitable selection of Φ on time scale, one can discover numerous novel and existing outcomes as specific cases. This shows the idea of generalized Riemann-Liouville fractional integral is wide and unifying one, yet additionally, improve few consequences in the study on the time scale hypothesis. Numerous variants are explored, when 𝕋 = ℝ. Finally, we introduced various dynamic variants by employing generalized Riemann-Liouville fractional integral as an example. Our consequences have potential applications in calcium ion channels, fractional calculus of variations on time scales, involving fractional fundamentalism in mechanics and physics, quantization, control theory, and description of conservative, nonconservative, and constrained systems. The performance of the fractional dynamical integral method is reliable and effective to obtain new solutions. This method has more advantages: it is direct and concise. Thus, the proposed method can be extended to solve many systems of nonlinear fractional partial differential equations in mathematical and physical sciences. Also, the new exact analytical solutions, can be obtained for the generalized ordinary differential equations to obtain new theorems related to stability and continuous dependence on parameters for dynamic equations on time scales. Our computed outcomes can be very useful as a starting point of comparison when some approximate methods are applied to this nonlinear space-time fractional equation.
Statements
Author contributions
SR and MA: Conceptualization; SR, MA, and KN: Writing original draft preparation; DB and GR: Formal Analysis; SR and DB: Methodology; KN, DB, and GR: Writing review and Editing.
Acknowledgments
Authors were grateful to the referees for their valuable suggestions and comments.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
References
1.
AlmeidaR. A Caputo fractional derivative of a function with respect to another function. Commun Nonlin Sci Numer Simulat. (2017) 44:460–81. 10.1016/j.cnsns.2016.09.006
2.
KhalilRHoraniMAYousefASababhehM. A new definition of fractional derivative. J Comp Appl Math. (2014) 264:65–70. 10.1016/j.cam.2014.01.002
3.
AtanganaAGomez-AguilarJF. Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu. Numer Methods Partial Differ Equat. (2018) 34:1502–23. 10.1002/num.22195
4.
ChengJFChuYM. Solution to the linear fractional differential equation using Adomian decomposition method. Math Prob Eng. (2011) 2011:587068. 10.1155/2011/587068
5.
ChengJFChuYM. On the fractional difference equations of order (2,q). Abst Appl Anal. (2011) 2011:497259. 10.1155/2011/497259
6.
ChengJFChuYM. Fractional difference equations with real variable. Abst Appl Anal. (2012) 2012:918529. 10.1155/2012/918529
7.
KilbasAASrivastavaHMTrujilloJJ. Theory and applications of fractional differential equations. North-Holland Math Stud. (2006) 204:540.
8.
KumarDSinghJBaleanuD. On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law. Math Methods Appl Sci. (2019) 43:443–57. 10.1002/mma.5903
9.
HilgerS. Ein Mabkettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten (Ph.D. thesis), Universitot Wurzburg (1988).
10.
HilgerS. Analysis on measure chains a unified approach to continuous and discrete calculus. Results Math. (1990) 18:18–56. 10.1007/BF03323153
11.
HilgerS. Differential and difference calculus unified. Nonlin Anal. (1997) 30:2683–94.
12.
BohnerMPetersonA. Dynamic Equations on Time Scales. Boston, MA: Birkhauser Boston, Inc. (2001).
13.
BohnerEABohnerMAkinF. Pachpatte inequalities on time scales. J Inequal Pure Appl Math. (2005) 6:1–23.
14.
DinuC. Hermite-Hadamard inequality on time scale. J Inequal Appl. (2008) 2008:24. 10.1155/2008/287947
15.
DinuC. Ostrowski type inequalities on time scales. An Univ Craiova Math Comput Sci Ser. (2007) 34:43–58.
16.
BastosNRO. Fractional calculus on time scales (Ph.D. thesis), University of Aveiro (2012).
17.
BastosNROMozyrskaDTorresDFM. Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform. Int J Math Comput. (2011) 11:1–9.
18.
BenkhettouNHassaniSTorresDFM. A conformable fractional calculus on arbitrary time scales. J King Saud Univ Sci. (2016) 28:93–8. 10.1016/j.jksus.2015.05.003
19.
Gomez-AguilarJFYepez-MartinezHEscobar-JimenezRFOlivares-PeregrinoVHReyesJMSosaIO. Series solution for the time-fractional coupled mKdV equation using the homotopy analysis method. Math Prob Eng. (2016) 2016:7047126. 10.1155/2016/7047126
20.
Yepez-MartinezHGomez-AguilarJFSosaIOReyesJMTorres-JimenezJ. The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation. Rev Mex Fis. (2016) 62:310–6.
21.
Gomez-AguilarJFYepez-MartinezHTorres-JimenezJCordova-FragaTEscobar-JimenezRFOlivares-PeregrinoVH. Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. Adv Differ Equat. (2017) 2017:1–18. 10.1186/s13662-017-1120-7
22.
GoswamiASinghJKumarDSushila. An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Phys A Stat Mech Appl. (2019) 524:563–75. 10.1016/j.physa.2019.04.058
23.
KumarDSinghJAl QurashiMBaleanuD. A new fractional SIRS-SI malaria disease model with application of vaccines, anti-malarial drugs, and spraying. Adv Differ Equat. (2019) 2019:278. 10.1186/s13662-019-2199-9
24.
MekhalfiKTorresDFM. Generalized fractional operators on time scales with applications to dynamic equations. Eur Phy J Spec Top. (2019) 226:3489–99. 10.1140/epjst/e2018-00036-0
25.
Morales-DelgadoVFGomez-AguilarJFYepez-MartinezMBaleanuDEscobar-JimenezRFOlivares-PeregrinoVH. Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular. Adv Differ Equat. (2016) 2016:1–16. 10.1186/s13662-016-0891-6
26.
SaadKMKhaderMMGomez-AguilarJFBaleanuD. Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods. Chaos Interdiscipl J Nonlin Sci. (2019) 29:1–13. 10.1063/1.5086771
27.
SinghJKumarDBaleanuD. New aspects of fractional Biswas-Milovic model with Mittag-Leffler law. Math Model Nat Phenom. (2019) 14:303. 10.1051/mmnp/2018068
28.
SinghJKumarDBaleanuDRathoreS. On the local fractional wave equation in fractal strings. Math Method Appl Sci. (2019) 42:1588–95. 10.1002/mma.5458
29.
YaslanILiceliO. Three-point boundary value problems with delta Riemann-Liouville fractional derivative on time scales. Fract. Differ. Calc. (2016) 6:1–16. 10.7153/fdc-06-01
30.
SunMHouC. Fractional q-symmetric calculus on a time scale. Adv Differ Equat. (2017) 2017:166. 10.1186/s13662-017-1219-x
31.
YanRASunRHanZL. Existence of solutions of boundary value problems for Caputo fractional differential equations on time scales. Bull Iranian Math Soc. (2016) 42:247–62.
32.
ZhuJWuL. Fractional Cauchy problem with Caputo nabla derivative on time scales. Abstr Appl Anal. (2015) 2015:486054. 10.1155/2015/486054
33.
GaoSL. Fractional time scale in calcium ion channels model. Int J Biomath. (2013) 6:1350023. 10.1142/S179352451350023X
34.
HilferR. Applications of Fractional Calculus in Physics. River Edge, NJ: World Scientific Publishing Co., Inc. (2000).
35.
MohanJS. Variation of parameters for nabla fractional difference equations. Novi Sad J Math. (2014) 44:149–59.
36.
ChidouhAGuezane-LakoudABebbouchiRBouarichaATorresDFM. Linear and nonlinear fractional Voigt models. In: Babiarz A, Czornik A, Klamka J, Niezabitowski M, editors. Theory and Applications of Non-integer Order Systems, Lecture Notes in Electrical Engineering, Vol. 407. Cham: Springer (2017). p. 157–67.
37.
GrüssG. Uber das Maximum des absoluten Betrages von . Math Zeitschr. (1935) 39:215–26. 10.1007/BF01201355
38.
Adil KhanMBegumSKhurshidYChuYM. Ostrowski type inequalities involving conformable fractional integrals. J Inequal Appl. (2018) 2018:70. 10.1186/s13660-018-1664-4
39.
Adil KhanMChuYMKashuriALikoRAliG. Conformable fractional integrals versions of Hermite-Hadamard inequalities and their generalizations. J Fun Spaces. (2018) 2018:6928130. 10.1155/2018/6928130
40.
Adil KhanMIqbalASulemanMChuYM. Hermite-Hadamard type inequalities for fractional integrals via Green's function. J Inequal Appl. (2018) 2018:161. 10.1186/s13660-018-1751-6
41.
Adil KhanMKhurshidYDuTSChuYM. Generalization of Hermite-Hadamard type inequalities via conformable fractional integrals. J Fun Spaces. (2018) 2018:5357463. 10.1155/2018/5357463
42.
DahmaniZ. New inequalities in fractional integrals. Int J Nonlin Sci. (2010) 9:493–7.
43.
DahmaniZTabharitLTafS. New generalisations of Grüss inequality using Riemann-Liouville fractional integrals. Bull Math Anal Appl. (2010) 2:93–9.
44.
RashidSAbdeljawadTJaradFNoorMA. Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications. Mathematics. (2019) 7:807. 10.3390/math7090807
45.
RashidSJaradFNoorMAKalsoomHChuYM. Inequalities by means of generalized proportional fractional integral operators with respect to another function. Mathematics. (2020) 7:1225. 10.3390/math7121225
46.
RashidSLatifMAHammouchZChuYM. Fractional integral inequalities for strongly h-preinvex functions for a kth order differentiable functions. Symmetry. (2019) 11:1448. 10.3390/sym11121448
47.
RashidSNoorMANoorKI. Some generalize Riemann-Liouville fractional estimates involving functions having exponentially convexity property. Punjab Univ J Math. (2019) 51:1–15.
48.
RashidSNoorMANoorKIAkdemirAO. Some new generalizations for exponentially s-convex functions and inequalities via fractional operators. Fractal Fract. (2019) 3:24. 10.3390/fractalfract3020024
49.
RashidSNoorMANoorKISafdarF. Integral inequalities for generalized preinvex functions. Punjab Univ J Math. (2019) 51:77–91. 10.1186/s13660-019-2248-7
50.
RashidSNoorMANoorKISafdarFChuYM. Hermite-Hadamard inequalities for the class of convex functions on time scale. Mathematics. (2019) 7:956. 10.3390/math7100956
51.
RashidSSafdarFAkdemirAONoorMANoorKI. Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function. J Inequal Appl. (2019) 2019:299. 10.1109/ICAEM.2019.8853807
52.
LiJFRashidSLiuJBAkdemirAOSafdarF. Inequalities involving conformable approach for exponentially convex functions and their applications. J Fun Spaces. (2020) 2020:6517068. 10.1155/2020/6517068
53.
NwaezeERTorresDFM. Chain rules and inequalities for the BHT fractional calculus on arbitrary timescales. Arab J Math. (2017) 6:13–20. 10.1007/s40065-016-0160-2
54.
ShengQFadagMHendersonJDavisJM. An exploration of combined dynamic derivatives on time scales and their applications. Nonlin Anal Real World Appl. (2006) 7:395–413. 10.1016/j.nonrwa.2005.03.008
55.
AgarwalRBohnerMReganDPetersonA. Dynamic equations on time scales: a survey, J Comput Appl Math. (2002) 141:1–26. 10.1016/S0377-0427(01)00432-0
56.
AhmadkhanluAJahanshahiM. On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales. Bull Iranian Math Soc. (2012) 38:241–52.
57.
KacarEKacarZYildirimH. Integral inequalities for Riemann-Liouville fractional integrals of a function with respect to another function. Iranian J Math Sci Inform. (2018) 13: 1–13. 10.7508/ijmsi.2018.1.001
58.
TariboonJNtouyasSKSudsutadW. Some new Riemann-Liouville fractional integral inequalities. Int J Math Sci. (2014) 2014:869434. 10.1186/s13661-014-0253-9
59.
KreyszigE. Introductory Functional Analysis with Applications.New York, NY: Wiley (1989).
Summary
Keywords
Minkowski' inequlity, gruss inequality, fractional calculus, rimenn-liouville fractional integral operator, generalized riemann-liouville fractional integral operator, time sccale, holder inequality
Citation
Rashid S, Aslam Noor M, Nisar KS, Baleanu D and Rahman G (2020) A New Dynamic Scheme via Fractional Operators on Time Scale. Front. Phys. 8:165. doi: 10.3389/fphy.2020.00165
Received
03 December 2019
Accepted
21 April 2020
Published
03 June 2020
Volume
8 - 2020
Edited by
Cosmas K. Zachos, Argonne National Laboratory (DOE), United States
Reviewed by
Yudhveer Singh, Amity University Jaipur, India; Sushila Rathore, Vivekananda Global University, India
Updates
Copyright
© 2020 Rashid, Aslam Noor, Nisar, Baleanu and Rahman.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Kottakkaran Sooppy Nisar n.sooppy@psau.edu.sa
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics
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