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Original Research ARTICLE

Front. Phys., 03 June 2020 | https://doi.org/10.3389/fphy.2020.00165

A New Dynamic Scheme via Fractional Operators on Time Scale

  • 1Department of Mathematics, Government College University, Faisalabad, Pakistan
  • 2Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
  • 3Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia
  • 4Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey
  • 5Institute of Space Sciences, Magurele, Romania
  • 6Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
  • 7Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Pakistan

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 [1, 2]). 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 [1, 38]).

Stefan Hilger began the theories of time scales in his doctoral dissertation [9] and combined discrete and continuous analysis (see [10, 11]). From this moment, this hypothesis has received a lot of attention. In the book written by Bohner and Peterson [12] 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 [1315]). Several researchers have created various results relating to fractional calculus on time scales to obtain the corresponding dynamic inequalities (see [1620]).

Recently, the idea of the fractional-order derivative has been expounded by Bastos et al. [16] via Riemann-Liouville fractional operators on scale versions by considering linear dynamic equations. Another approach on time scales shifts to the inverse Laplace transform [18]. Following such innovator work, the investigation of fractional calculus on time scales created in a mainstream look into research studies on time scales (see [18, 2129] and references therein). Since the publications in 2015, several researchers made significant contributions to the history of time scales. Sun and Hou [30] employed the fractional q-symmetric systems on time scales. Yaslan and Liceli [29] obtained the three-point boundary value problem with delta Riemann-Liouville fractional derivative on time scales. Yan et al. [31] adopted the Caputo fractional techniques on differential equations on time scales. Zhu and Wu [32] 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 [33]. Actually, physical utilization of initial value-fractional problems in diverse time scales proliferates [10, 34, 35]. 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. [36].

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 [3751]. Recently, numerous analysts examined various inequalities, such as Hermite-Hadamard inequalities, Ostrowski inequalities and the expanded version of Hardy-type inequalities (see [1315, 24, 52] and the references therein).

Here, we broaden accessible outcomes in the literature [53] 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 Gru¨ss, 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 [54]. 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 Z. 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:

(t)=inf {λ𝕋:ρ>t}𝕋,ϑ(ω)=sup {ρ𝕋:ρ<ω}𝕋.

We accumulate as:

inf :=sup 𝕋,sup :=inf 𝕋.

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

μ(t):=(t)t,
ν(t):=tϑ(t)

are called the forward and backward graininess functions, respectively.

Definition 2.1. [12, 55] “Let ℏ:𝕋 → ℝ be a real-valued function. Then ℏ is said to be RD-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 RD-continuous functions are denoted by RD.

Definition 2.2. “A function F:𝕋 is called a delta antiderivative of ℏ:𝕋 → ℝ if FΔ(t)=(t), for all t ∈ 𝕋k. Then, one defines the delta integral by υ1t(s)Δs=F(t)-F(υ1).

Theorem 2.1. [55]. If ℏ ∈ ℂRD and t ∈ 𝕋k, then

t(t)(s)Δs=μ(t)(t).

Theorem 2.2. [55]. Let υ1, υ2, υ3 ∈ 𝕋, β ∈ ℝ and ℏ, ω ∈ ℂRD, then

(i). υ1υ2(1(ρ)+2(ρ))Δρ=υ1υ21(ρ)Δρ+υ1υ22(ρ)Δρ;

(ii). υ1υ2β(ρ)Δρ=βυ1υ2(ρ)Δρ;

(iii). υ1υ2(ρ)Δρ=-υ2υ1(ρ)Δρ;

(iv). υ1υ2(ρ)Δρ=υ1ς3(ρ)Δρ+ς3υ2(ρ)Δρ;

(v). υ1υ21(ρ)2ΔΔρ  =  (12)(υ2) - (12)(υ1) - υ1υ21Δ(ρ)2(ρ)Δ(ρ);

(vi). υ1υ21(ρ)2ΔΔρ  =  (12)(υ2) - (12)(υ1) - υ1υ21Δ(ρ)2(ρ)Δ(ρ);

(vii). υ1υ2(ρ)Δ(ρ)=0;

(viii). If (ρ)0 for all ρ,thenυ1υ2(ρ)Δ(ρ)0;

(ix). If|1(ρ)|  2(ρ)on[υ1,υ2],then|υ1υ21(ρ)Δρ|  υ1υ22(ρ)Δ(ρ).

From Theorem 2.2 (ix), for ℏ2(ρ) = |ℏ1(ρ)| on [υ1, υ2], we have

|υ1υ2(ρ)Δρ|υ1υ2|(ρ)|Δ(ρ).

Proposition 2.1. [56] Consider a time scale 𝕋 andis an increasing continuous function on ϒ𝕋. An extension ofon ϒ𝕋 is F given as

(θ):={(θ),  ifθ𝕋(η),  ifθ(η,σ(η))𝕋, 

then

υ1υ2(η)Δυ1υ2()d.

Next we demonstrate the idea of fractional integral on time scale, which is mainly due to [16].

Definition 2.3. [16] “For 0 < δ < 1, let ϒ𝕋 ⊂ 𝕋 is a time scale and F be an integrable function on ϒ𝕋. Then the (left) fractional integral of order δ of F is defined by

Jηδυ1𝕋(η)=1Γ(δ)υ1η(ηθ)δ1(θ)Δθ,    (1)

where Γ is the gamma function.”

Again, we demonstrate the concept of generalized Riemann-Liouville fractional integral operator which is proposed by [24].

Definition 2.4. [24] “For 0 < δ < 1, let 𝕋 is a time scale and [υ1, υ2] is an interval of 𝕋. Suppose F 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 F with respect to Φ is defined by

Jηδυ1;Φ𝕋(η)=1Γ(δ)υ1η(Φ(η)Φ(θ))δ1ΦΔ(θ)(θ)Δθ.    (2)

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 [7]).

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. Suppose F,G be two positive functions on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔ with ΦΔ ≠ 0 such that for all η>0,0+;Φ𝕋JηδF(η)<,0+;Φ𝕋JηδG(η)<. If 0<𝔪F(θ)G(θ)M,θ[0,η], then

[0+;Φ𝕋Jηδ(η)]1α[0+;Φ𝕋JηδG(η)]1β             (𝔪)1αβ[0+;Φ𝕋Jηδ((θ))1α(G(η))1β].    (3)

Proof: Since F(θ)G(θ)M,θ[0,η],η>0, we find that

(G(θ))1α1β((θ))1β    (4)

and

((θ))1α(G(θ))1β1β(θ).    (5)

Taking product on both sides of (5) (Φ(η)Φ(θ))δ1ΦΔ(θ)Γ(δ) which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have

1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)((θ))1α(G(θ))1βΔθ1βΓ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)(θ)Δθ,    (6)

which implies that

0+;Φ𝕋Jηδ((θ))1α(G(η))1β1β0+;Φ𝕋Jηδ(η).    (7)

It follows that

(0+;Φ𝕋Jηδ((θ))1α(G(η))1β)1α1αβ(0+;Φ𝕋Jηδ(η))1α.    (8)

Accordingly, 𝔪G(θ)F(θ),θ(0,η),η>0, therefore we have

((θ))1α𝔪1α(G(θ))1α.    (9)

Taking product (9) by (G(θ))1β, we arrive at

(G(θ))1β((θ))1α𝔪1αG(θ).    (10)

Taking product on both sides of (11) (Φ(η)-Φ(θ))δ-1ΦΔ(θ)Γ(δ), which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have

1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)(G(θ))1β((θ))1αΔθ𝔪1α1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)G(θ)Δθ.    (11)

Hence, we can write

(0+;Φ𝕋Jηδ((θ))1α(G(η))1β)1β𝔪1αβ(0+;Φ𝕋JηδG(η))1β.    (12)

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:

[ΦJηδ(η)]1α[ΦJηδG(η)]1β(𝔪)1αβ[ΦJηδ((θ))1α(G(η))1β].

Theorem 3.2. Let δ, γ > 1, and 𝕋 is a time scale. Suppose F,G be two positive functions on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔ with ΦΔ ≠ 0 such that for all η>0,0+;Φ𝕋JηδFα(η)<,0+;Φ𝕋JηδGβ(η)<. If 0<𝔪Fα(θ)Gβ(θ)M,θ[0,η], then

[0+;Φ𝕋Jηδα(η)]1α[0+;Φ𝕋JηδGβ(η)]1β              (𝔪)1αβ[0+;Φ𝕋Jηδ((θ)G(η))],    (13)

where α>1,1α+1β=1.

Proof: Replacing F(θ) and G(θ) by Fα(θ) and Gβ(θ),θ[0,η],η>0 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 function F on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔ with ΦΔ ≠ 0 such that for all η > 0. Assume that the subsequent.

(I)There exist two integrable functions φ1, φ2 on [0, ∞)𝕋 such that

φ1(η)(η)φ2(η),  η[0,)𝕋.    (14)

Then, for η > 0, δ, γ > 1, one has

0+;Φ𝕋Jηδφ2(η)0+;Φ𝕋Jηγ(η)+0+;Φ𝕋Jηδ(η)0+;Φ𝕋Jηλφ1(η)0+;Φ𝕋Jηδφ2(η)0+;Φ𝕋Jηλφ1(η)+0+;Φ𝕋Jηδ(η)0+;Φ𝕋Jηλ(η),    (15)

Proof: From (I), for all θ ≥ 0, λ ≥ 0, we have

(φ2(θ)(θ))((λ)φ1(λ))0.    (16)

Therefore,

φ2(θ)(λ)+φ1(λ)(θ)φ1(λ)φ2(θ)+(θ)(λ).    (17)

Taking product on both sides of (17) (Φ(η)-Φ(θ))δ-1ΦΔ(θ)Γ(δ), which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have

(λ)1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)φ2(θ)Δθ  +φ1(λ)1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)(θ)Δθφ1(λ)1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)φ2(θ)Δθ  +(λ)1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)(θ)Δθ,    (18)

arrives at

(λ)0+;Φ𝕋Jηδφ2(η)+φ1(λ)0+;Φ𝕋Jηδ(η)φ1(λ)0+;Φ𝕋Jηδφ2(η)                                           +(λ)0+;Φ𝕋Jηδ(η).    (19)

Taking product on both sides of (19) (Φ(η)-Φ(λ))γ-1ΦΔ(λ)Γ(γ), which is positive because λ ∈ (0, η), η > 0, we integrate the resulting identity with respect to λ from 0 to η we have

0+;Φ𝕋Jηδφ2(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)(λ)Δλ  +0+;Φ𝕋Jηδ(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)φ1(λ)Δλ0+;Φ𝕋Jηδφ2(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)φ1(λ)Δλ  +0+;Φ𝕋Jηδ(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)(λ)Δλ.    (20)

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:

0+𝕋Jηδφ2(η)0+𝕋Jηγ(η)+0+𝕋Jηδ(η)0+𝕋Jηλφ1(η)0+𝕋Jηδφ2(η)0+𝕋Jηλφ1(η)+0+𝕋Jηδ(η)0+𝕋Jηλ(η).

Remark 4.1. If 𝕋 = ℝ, then Theorem 4.1 will lead to Theorem 2.11 in [57] and corollary 4.1 will lead to Corollary 3 in [57]. Also, if we choose 𝕋 = ℝ along with Φ(η) = η, then Theorem 4.1 will lead to Theorem 2 in [58].

Theorem 4.2. Let δ, γ > 1, and 𝕋 is a time scale. Suppose there are two positive functions F,G on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔ with ΦΔ ≠ 0 such that for all η > 0. Suppose that (I) holds and moreover one assumes the following. (II) There exist ω1 and ω2 integrable functions on [0, ∞)𝕋 such that

ω1(η)G(η)ω2(η)  η[0,)𝕋.    (21)

Then, for η > 0, δ, γ > 1, the following inequalities hold:

(A1)    0+;Φ𝕋Jηδφ2(η)0+;Φ𝕋JηγG(η)+0+;Φ𝕋Jηδ(η)0+;Φ𝕋Jηγω1(η)             0+;Φ𝕋Jηδφ2(η)0+;Φ𝕋Jηγω1(η)             +0+;Φ𝕋Jηδ(η)0+;Φ𝕋JηγG(η),(B1)    0+;Φ𝕋Jηγφ1(η)0+;Φ𝕋JηδG(η)+0+;Φ𝕋Jηγω2(η)0+;Φ𝕋Jηγ(η)             0+;Φ𝕋Jηγφ1(η)0+;Φ𝕋Jηδω2(η)             +0+;Φ𝕋Jηγ(η)0+;Φ𝕋JηδG(η),(C1)    0+;Φ𝕋Jηγω2(η)0+;Φ𝕋Jηδφ2(η)+0+;Φ𝕋Jηδ(η)0+;Φ𝕋JηγG(η)             0+;Φ𝕋Jηδφ2(η)0+;Φ𝕋JηγG(η)             +0+;Φ𝕋Jηδ(η)0+;Φ𝕋Jηγω2(η),(D1)    0+;Φ𝕋Jηδφ1(η)0+;Φ𝕋Jηγω1(η)             +0+;Φ𝕋Jηδ(η)0+;Φ𝕋Jηδ0+;Φ𝕋JηγG(η)             0+;Φ𝕋Jηδφ1(η)0+;Φ𝕋JηγG(η)             0+;Φ𝕋Jηγω1(η)0+;Φ𝕋JηδF(η).    (22)

Proof: To prove (A1), from (I) and (II), we have for x ∈ [0, ∞)𝕋 that

(φ2(θ)(θ))(G(λ)ω1(λ))0.    (23)

Therefore,

φ2(θ)G(λ)+ω1(λ)(θ)ω1(λ)φ2(θ)+G(λ)(θ).    (24)

Taking product on both sides of (24) (Φ(η)-Φ(θ))δ-1ΦΔ(θ)Γ(δ), which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have

G(λ)1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)φ2(θ)Δθ    +ω1(λ)1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)(θ)Δθω1(λ)1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)φ2(θ)Δθ    +G(λ)1Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)(θ)Δθ.    (25)

Then we have

G(λ)0+;Φ𝕋Jηδφ2(η)+ω1(λ)0+;Φ𝕋Jηδ(η)ω1(λ)0+;Φ𝕋Jηδφ2(η)+G(λ)0+;Φ𝕋Jηδ(η).    (26)

Again, multiplying both sides of (26) by (Φ(η)-Φ(λ))γ-1ΦΔ(λ)Γ(γ), which is positive because λ ∈ (0, η), η > 0, we integrate the resulting identity with respect to λ from 0 to η we have

0+;Φ𝕋Jηδφ2(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)G(λ)Δλ  +0+;Φ𝕋Jηδ(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)ω1(λ)Δλ0+;Φ𝕋Jηδφ2(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)ω1(λ)Δλ  +0+;Φ𝕋Jηδ(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)G(λ)Δλ.

This follows that

0+;Φ𝕋Jηδφ2(η)0+;Φ𝕋JηγG(η)+0+;Φ𝕋Jηδ(η)0+;Φ𝕋Jηδ0+;Φ𝕋Jηγω1(η)0+;Φ𝕋Jηδφ2(η)0+;Φ𝕋Jηγω1(η)+0+;Φ𝕋Jηδ(η)0+;Φ𝕋JηγG(η),

we acquire the desired inequality (A1).

To prove (B1) − (D1), we utilizes the subsequent variants:

(B1)     (ω2(θ)G(θ))((λ)φ1(λ))0,(C1)     (φ2(θ)(θ))(G(λ)ω2(λ))0,(D1)     (φ1(θ)(θ))(G(λ)ω1(λ))0.

 

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:

(A2)    0+;Φ𝕋Jηδφ2(η)0+𝕋JηγG(η)+0+𝕋Jηδ(η)0+𝕋Jηγω1(η)            0+𝕋Jηδφ2(η)0+𝕋Jηγω1(η)+0+𝕋Jηδ(η)0+𝕋JηγG(η),(B2)    0+;Φ𝕋Jηγφ1(η)0+𝕋JηδG(η)+0+𝕋Jηγω2(η)0+𝕋Jηγ(η)            0+𝕋Jηγφ1(η)0+𝕋Jηδω2(η)+0+𝕋Jηγ(η)0+𝕋JηδG(η),(C2)    0+;Φ𝕋Jηγω2(η)0+𝕋Jηδφ2(η)+0+𝕋Jηδ(η)0+𝕋JηγG(η)            0+𝕋Jηδφ2(η)0+𝕋JηγG(η)+0+𝕋Jηδ(η)0+𝕋Jηγω2(η),(D2)    0+;Φ𝕋Jηδφ1(η)0+𝕋Jηγω1(η)+0+𝕋Jηδ(η)0+𝕋Jηδ0+𝕋JηγG(η)            0+𝕋Jηδφ1(η)0+𝕋JηγG(η)+0+𝕋Jηγω1(η)0+𝕋Jηδ(η).

Remark 4.2. If 𝕋 = ℝ, then Theorem 4.2 will lead to Theorem 2.15 in [57] and corollary 4.2 will lead to Corollary 2.16 in [57]. Also, If we choose 𝕋 = ℝ along with Φ(η) = η, then Theorem 4.2 will lead to Theorem 5 in [58].

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 functions F,G on [0, ∞)𝕋, and Φ is monotone, delta differentiable ΦΔ with ΦΔ ≠ 0 such that for all η > 0, α, β > 1 satisfying 1α+1β=1. Then, for η > 0, one has

(A3)    1α0+;Φ𝕋Jηδα(η)0+;Φ𝕋JηγGα(η)            +1β0+;Φ𝕋JηδGβ(η)0+;Φ𝕋Jηγβ(η)            0+;Φ𝕋Jηδ(η)G(η)0+;Φ𝕋JηγG(η)(η),(B3)    1α0+;Φ𝕋JηγGβ(η)0+;Φ𝕋Jηδα(η)            +1β0+;Φ𝕋Jηγα(η)0+;Φ𝕋JηδGβ(η)            0+;Φ𝕋JηγGβ1(η)α1(η)0+;Φ𝕋Jηδ(η)G(η),(C3)    1α0+;Φ𝕋JηγG2(η)0+;Φ𝕋Jηδα(η)            +1β0+;Φ𝕋Jηγ2(η)0+;Φ𝕋JηδGβ(η)            0+;Φ𝕋Jηγ2β(η)G2α(η)0+;Φ𝕋Jηδ(η)G(η),(D3)    1α0+;Φ𝕋JηγGβ(η)0+;Φ𝕋Jηδ2(η)            +1β0+;Φ𝕋Jηγα(η)0+;Φ𝕋JηδG2(η)            0+;Φ𝕋Jηγα1(η)Gβ1(η)0+;Φ𝕋Jηδ2α(η)G2β(η).    (27)

Proof: Taking into account the Young's inequality [59]:

1αaα+1βbβab,  a,b0,α,β>0,1α+1β=1,    (28)

setting a=F(θ)G(λ) and b=F(λ)G(θ),θ,λ>0, we have

1α((θ)G(λ))α+1β((λ)G(θ))β((θ)G(λ))((λ)G(θ)).    (29)

Taking product on both sides of (29) (Φ(η)-Φ(θ))δ-1ΦΔ(θ)Γ(δ), which is positive because θ ∈ (0, η), η > 0, we integrate the resulting identity with respect to θ from 0 to η we have

Gα(λ)αΓ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)Γ(δ)α(θ)Δθ  +β(λ)βΓ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)Γ(δ)Gβ(θ)Δθ  G(λ)(λ)Γ(δ)0η(Φ(η)Φ(θ))δ1ΦΔ(θ)Γ(δ)(θ)G(θ)Δθ,    (30)

we get

Gα(λ)α0+;Φ𝕋Jηδα(η)+β(λ)β0+;Φ𝕋JηδGβ(η)G(λ)(λ)0+;Φ𝕋Jηδ(η)G(η).    (31)

Again, multiplying both sides of (31) by (Φ(η)-Φ(λ))γ-1ΦΔ(λ)Γ(γ), which is positive because λ ∈ (0, η), η > 0, we integrate the resulting identity with respect to λ from 0 to η we have

1α0+;Φ𝕋Jηδα(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)Gα(λ)Δλ  +1β0+;Φ𝕋JηδGβ(η)1Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)β(λ)Δλ   0+;Φ𝕋Jηδ(η)G(η)Γ(γ)0η(Φ(η)Φ(λ))γ1ΦΔ(λ)G(λ)(λ)Δλ,    (32)

consequently, we get

1α0+;Φ𝕋Jηδα(η)0+;Φ𝕋JηγGα(η)+1β0+;Φ𝕋JηδGβ(η)0+;Φ𝕋Jηγβ(η)0+;Φ𝕋Jηδ(η)G(η)0+;Φ𝕋JηγG(η)(η),    (33)

which implies (A3). The remaining variants can be proved by adopting the same technique as we did in (A3).

(B3)  a=(θ)(λ),      b=G(θ)G(λ),(λ),G(λ)0,(C3)  a=(θ)G2α(λ),  b=2β(λ)G(θ),(D3)  a=2α(θ)(λ),  b=G2β(θ)G(λ),(λ),G(λ)0.

Repeating the foregoing argument, we obtain (B3) − (D3). 

Theorem 5.2. Let δ, γ > 1, and 𝕋 is a time scale. Suppose F,G 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 α + β = 1. Then, for η > 0, one has

(A4)    p0+;Φ𝕋Jηδ(η)0+;Φ𝕋JηγG(η)            +q0+;Φ𝕋Jηγ(η)0+;Φ𝕋JηδG(β)            0+;Φ𝕋Jηδ(α(η)Gβ(η))0+;Φ𝕋Jηγ(β(η)Gα(η)),(B4)    p0+;Φ𝕋Jηδα1(η)0+;Φ𝕋Jηγ((η)Gβ(η))            +q0+;Φ𝕋JηγGβ1(η)0+;Φ𝕋Jηδ(β(η)G(η))            0+;Φ𝕋JηδGβ(η)0+;Φ𝕋Jηγα(η),(C4)    p0+;Φ𝕋Jηδ(η)0+;Φ𝕋JηγG2α(η)            +q0+;Φ𝕋JηδG(η)0+;Φ𝕋Jηγ2β(η)            0+;Φ𝕋Jηδα(η)G(η)0+;Φ𝕋JηγGβ(η)2(η),(D4)    p0+;Φ𝕋Jηδ2α(η)Gβ(η)0+;Φ𝕋JηγGα1(η)            +q0+;Φ𝕋JηδGβ1(η)0+;Φ𝕋Jηγ2β(η)Gα(η)            0+;Φ𝕋Jηδ2(η)0+;Φ𝕋JηγG2(η).    (34)

Proof: Taking into account the weighted AMGM inequality

αa+βbaαbβ,  a,b0,α,β>0,α+β=1,    (35)

by setting a=F(θ)G(λ) and b=F(λ)G(θ),λ,θ>0, we have

α(θ)G(λ)+β(λ)G(θ)((θ)G(λ))α((λ)G(θ))β.    (36)

Multiplying both sides of (36) by 1Γ(δ)Γ(γ)(Φ(η)Φ(θ))δ1ΦΔ(θ)(Φ(η)Φ(λ))γ1ΦΔ(λ) which is positive because θ, λ ∈ (0, η), η > 0 and integrating the resulting identity from 0 to η we have

αΓ(δ)Γ(γ)0η0η(Φ(η)Φ(θ))δ1(Φ(η)Φ(λ))γ1ΦΔ(θ)ΦΔ(λ)(θ)G(λ)ΔθΔλ  +βΓ(δ)Γ(γ)0η0η(Φ(η)Φ(θ))δ1(Φ(η)Φ(λ))γ1ΦΔ(θ)ΦΔ(λ)(λ)G(θ)ΔθΔλ1Γ(δ)Γ(γ)0η0η(Φ(η)Φ(θ))δ1(Φ(η)Φ(λ))γ1ΦΔ(θ)ΦΔ(λ)  ×((θ)G(λ))α((λ)G(θ))βΔλΔθ,    (37)

we conclude that

p0+;Φ𝕋Jηδ(η)0+;Φ𝕋JηγG(η)+q0+;Φ𝕋Jηγ(η)0+;Φ𝕋JηδG(η)0+;Φ𝕋Jηδ(α(η)Gβ(η))0+;Φ𝕋Jηγ(β(η)Gα(η)),    (38)

which implies (A4). The rest of inequalities can be shown in similar way by the following choice of parameters in AMGM inequality.

(B4)  a=(λ)(θ),            b=G(θ)G(λ),(θ),G(λ)0.(C4)  a=(θ)G2α(λ),    b=2β(λ)G(θ),(D4)  a=2α(θ)G(λ),            b=2β(λ)G(θ),G(θ),G(θ)0.

 

Example 5.1. Let δ, γ > 1, and 𝕋 is a time scale. Suppose F,G 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 1α+1β=1. Let

𝔪=min0θη(θ)G(θ)    and    =max0θη(θ)G(θ).    (39)

Then, for η > 0, δ, γ > 1, one has the following inequalities:

(1)  00+;Φ𝕋Jηδ2(η)0+;Φ𝕋JηδG2(η)        𝔪+4𝔪(0+;Φ𝕋Jηδ(η)G(η))2,(2)  000+;Φ𝕋Jηδ2(η)0+;Φ𝕋JηδG2(η)(0+;Φ𝕋Jηδ(η)G(η))        𝔪2𝔪(0+;Φ𝕋Jηδ(η)G(η)),(3)  00+;Φ𝕋Jηδ2(η)0+;Φ𝕋JηδG2(η)(0+;Φ𝕋Jηδ(η)G(η))2        𝔪4𝔪(0+;Φ𝕋Jηδ(η)G(η))2.

Proof: From Equation (39) and the inequality

((θ)G(θ)𝔪)((θ)G(θ))G2(θ)0,0θη,    (40)

then we can write as,

2(θ)+𝔪G2(θ)(𝔪+)(θ)G(θ).    (41)

Multiplying both sides of (41) by 1Γ(δ)(Φ(η)-Φ(θ))ΦΔ(θ), which is positive because θ ∈ (0, η), η > 0 and integrating the resulting identity from 0 to η, we have

1Γ(δ)0η(Φ(η)Φ(θ))ΦΔ(θ)2(θ)Δθ    +𝔪1Γ(δ)0η(Φ(η)Φ(θ))ΦΔ(θ)G2(θ)Δθ(𝔪+)1Γ(δ)0η(Φ(η)Φ(θ))ΦΔ(θ)(θ)G(θ)Δθ,    (42)

implies that

0+;Φ𝕋Jηδ2(η)+𝔪0+;Φ𝕋JηδG2(η)(𝔪+)0+;Φ𝕋Jηδ(η)G(η),    (43)

on the other hand, it follows from 𝔪M>0 and

(0+;Φ𝕋Jηδ2(η)𝔪0+;Φ𝕋JηδG2(η))20,    (44)

that

20+;Φ𝕋Jηδ2(η)𝔪0+;Φ𝕋JηδG2(η)0+;Φ𝕋Jηδ2(η)                                                                                 +𝔪0+;Φ𝕋JηδG2(η)    (45)

then from equation (43) and (45), we obtain,

4𝔪0+;Φ𝕋Jηδ2(η)0+;Φ𝕋JηδG2(η)(𝔪+)2(0+;Φ𝕋Jηδ(η)G(η)).    (46)

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.

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.

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.

Acknowledgments

Authors were grateful to the referees for their valuable suggestions and comments.

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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.

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

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