Open access peer-reviewed chapter - ONLINE FIRST

Integral Inequalities and Differential Equations via Fractional Calculus

By Zoubir Dahmani and Meriem Mansouria Belhamiti

Submitted: August 28th 2019Reviewed: January 10th 2020Published: February 12th 2020

DOI: 10.5772/intechopen.91140

Abstract

In this chapter, fractional calculus is used to develop some results on integral inequalities and differential equations. We develop some results related to the Hermite-Hadamard inequality. Then, we establish other integral results related to the Minkowski inequality. We continue to present our results by establishing new classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations of order n for some other classical/fractional integral results published recently. As applications on inequalities, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to the classical case, are generalised for any α≥1,β≥1. For the part of differential equations, we present a contribution that allow us to develop a class of fractional chaotic electrical circuit. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equation. Then, by establishing some sufficient conditions, another result for the existence of at least one solution is also discussed.

Keywords

• fractional calculus
• fixed point
• Riemann-Liouville integral
• Caputo derivative
• integral inequality

1. Introduction

During the last few decades, fractional calculus has been extensively developed due to its important applications in many field of research [1, 2, 3, 4]. On the other hand, the integral inequalities are very important in probability theory and in applied sciences. For more details, we refer the reader to [5, 6, 7, 8, 9, 10, 11, 12] and the references therein. Moreover, the study of integral inequalities using fractional integration theory is also of great importance; we refer to [1, 13, 14, 15, 16, 17] for some applications.

Also, boundary value problems of fractional differential equations have occupied an important area in the fractional calculus domain, since these problems appear in several applications of sciences and engineering, like mechanics, chemistry, electricity, chemistry, biology, finance, and control theory. For more details, we refer the reader to [3, 18, 19, 20].

In this chapter, we use the Riemann-Liouville integrals to present some results related to Minkowski and Hermite-Hadamard inequalities [21]. We continue to present our results by establishing several classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations for some other fractional and classical integral results published recently [22]. Then, as applications, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to α=1, are generalized for any α1and β1; see [23].

For the part of differential equations, with my coauthor, we present a contribution that allows us to develop a class of fractional differential equations generalizing the chaotic electrical circuit model. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equations. Then, by establishing some sufficient conditions on the data of the problem, another result for the existence of at least one solution is also discussed. The considered class has some relationship with the good paper in [20].

The chapter is structured as follows: In Section 2, we recall some preliminaries on fractional calculus that will be used in the chapter. Section 3 is devoted to the main results on integral inequalities as well as to some estimates on continuous random variables. The Section 4 deals with the class of differential equations of Langevin type: we study the existence and uniqueness of solutions for the considered class by means of Banach contraction principle, and then using Schaefer fixed point theorem, an existence result is discussed. At the end, the Conclusion follows.

2. Preliminaries on fractional calculus

In this section, we present some definitions and lemmas that will be used in this chapter. For more details, we refer the reader to [2, 13, 15, 24].

Definition 1.1. The Riemann-Liouville fractional integral operator of order α0, for a continuous function f on abis defined as

Jaαft=1Γαattτα1fτ,α>0, a<tb,Ja0ft=ft,E1

where Γα0euuα1du.

Note that for α>0,β>0, we have

JaαJaβft=Jα+βaft,E2

and

JaαJaβft=JaβJaαft.E3

In the rest of this chapter, for short, we note a probability density function by p.d.f. So, let us consider a positive continuous function ωdefined on ab. We recall the ωconcepts:

Definition 1.2. The fractional ωweighted expectation of order α>0, for a random variable Xwith a positive p.d.f.fdefined on ab, is given by

Eα,ωXJaαfb=1Γαabbτα1τωτfτ,α>0,a<tb,E4

Definition 1.3. The fractional ωweighted variance of order α>0for a random variable Xhaving a p.d.f.fon [a, b] is given by

σα,ω2X=Vα,ωX1Γαabbτα1τEX2ωτfτ,α>0.E5

Definition 1.4. The fractional ωweighted moment of orders r>0,α>0for a continuous random variable Xhaving a p.d.f.fdefined on [a, b] is defined by the quantity:

Eα,ωXr1Γαabbτα1τrωτfτ,α>0.E6

We introduce the covariance of fractional order as follows.

Definition 1.5. Let f1and f2be two continuous on ab.We define the fractional ωweighted covariance of order α>0for f1Xf2Xby

Covα,ωf1Xf2X1Γαabbτα1f1τf1μf2τf2μωτfτ,α>0,E7

where μis the classical expectation of X.

It is to note that when ωx=1,xab, then we put

Varα,ωXVarαX,Covα,ωXCovαX,Eα,ωXEαX

Definition 1.6. For a function KCnabRand n1<αn, the Caputo fractional derivative of order αis defined by

DαKt=JnαdndtnKt=1Γnαattsnα1Knsds.

We recall also the following properties.

Lemma 1.7. Let nN, and n1<α<n. The general solution of Dαyt=0,tabis given by

yt=i=0n1citai,E8

where ciR,i=0,1,2,..,n1.

Lemma 1.8. Let nNand n1<α<n. Then

JαDαyt=yt+i=0n1citai,tab,E9

for some ciR,i=0,1,2,..,n1.

3. Some integral inequalities

3.1 On Minkowski and Hermite-Hadamard fractional inequalities

In this subsection, we present some fractional integral results related to Minkowski and Hermite-Hadamard integral inequalities. For more details, we refer the reader to [21].

Theorem 1.9. Let α>0,p1and let f,gbe two positive functions on [0,[,such that for all t>0,Jαfpt<,Jαgpt<.If 0<mfτgτM,τ0t,then we have

Jαfpt1p+Jαgpt1p1+Mm+2m+1M+1Jαf+gpt1p.E10

Proof: We use the hypothesis fτgτ<M,τ0t,t>0.We can write

M+1pΓα0ttτα1fpτMpΓα0ttτα1f+gpτ.E11

Hence, we have

JαfptMpM+1pJαf+gpt.E12

Thus, it yields that

Jαfpt1pMM+1Jαf+gpt1p.E13

In the same manner, we have

1+1mgτ1mfτ+gτ.E14

And then,

Jαgpt1p1m+1Jαf+gpt1p.E15

Combining (13) and (15), we achieve the proof.

Remark 1.10. Applying the above theorem for α=1,we obtain Theorem 1.2 of [25] on 0t.

With the same arguments as before, we present the following theorem.

Theorem 1.11. Let α>0,p1and let f,gbe two positive functions on [0,[,such that for all t>0,Jαfpt<,Jαgpt<.If 0<mfτgτM,τ0t,then we have

Jαfpt1p+Jαgpt1p1+Mm+2m+1M+1Jαf+gpt1p.E16

Remark 1.12. Taking α=1in this second theorem, we obtain Theorem 2.2 in [26] on 0t.

Using the notions of concave and Lpfunctions, we present to the reader the following result.

Theorem 1.13. Suppose that α>0,p>1,q>1and let f,gbe two positive functions on [0,[.If fp,gqare two concave functions on [0,[,then we have

2pqf0+ftpg0+gtqJαtα12Jαtα1fptJαtα1gqt.E17

The proof of this theorem is based on the following auxiliary result.

Lemma 1.14. Let hbe a concave function on ab.Then for any xab,we have

ha+hbhb+ax+hx2ha+b2.E18

3.2 A family of fractional integral inequalities

We present to the reader some integral results for a family of functions [22]. These results generalize some integral inequalities of [27]. We have

Theorem 1.15. Suppose that fii=1,nare npositive, continuous, and decreasing functions on ab.Then, the following inequality

JαipnfiγifpβtJαi=1nfiγitJαtaδipnfiγifpβtJαtaδi=1nfiγitE19

holds for any a<tb,α>0,δ>0,βγp>0,where pis a fixed integer in 12n.

Proof: It is clear that

ρaδτaδfpβγpτfpβγpρ0,E20

for any fixed p1nand for any βγp>0,δ>0,τ,ρat;a<tb.

Taking

Kpτρtτα1Γαi=1nfiγiτρaδτaδfpβγpτfpβγpρ,E21

we observe that

Kpτρ0.E22

Also, we have

0atKpτρ= ρaδJαipnfiγifpβt+fpβγpρJαtaδi=1nfiγitJαtaδipnfiγifpβtρaδfβγpρJαi=1nfiγit.E23

Hence, we get

Jαtaδi=1nfiγitJαipnfiγifpβtJαi=1nfiγitJαtaδipnfiγifpβt.E24

The proof is thus achieved.

Remark 1.16. Applying Theorem 1.15 for α=1,t=b,n=1,we obtain Theorem 3 in [27].

Using other sufficient conditions, we prove the following generalization.

Theorem 1.17. Suppose that fii=1,nare positive, continuous, and decreasing functions on ab.Then for any fixed pin 12nand for any a<tb,α>0,ω>0,δ>0,βγp>0,we have

JαipnfiγifpβtJωtaδi=1nfiγit+JωipnfiγifpβtJαtaδi=1nfiγitJαtaδipnfiγifpβtJωi=1nfiγit+JωtaδipnfiγifpβtJαi=1nfiγit1.E25

Proof: Multiplying both sides of (23) by tρω1Γωi=1nfiγiρ,ω>0,then integrating the resulting inequality with respect to ρover at,a<tband using Fubini’s theorem, we obtain the desired inequality.

Remark 1.18.

1. Applying Theorem 1.17 for α=ω,we obtain Theorem 1.15.

2. Applying Theorem 1.17 for α=ω=1,t=b,n=1,we obtain Theorem 3 of [27].

Introducing a positive increasing function gto the family fii=1,n, we establish the following theorem.

Theorem 1.19. Let fii=1,nand gbe positive continuous functions on ab,such that gis increasing and fii=1,nare decreasing on ab.Then, the following inequality

JαipnfiγifpβtJαgδti=1nfiγitJαgδtipnfiγifpβtJαi=1nfiγit1E26

holds for any a<tb,α>0,δ>0,βγp>0,where pis a fixed integer in 12n.

Remark 1.20. Applying Theorem 1.19 for α=1,t=b,n=1,we obtain Theorem 4 of [27].

3.3 Some estimations on random variables

3.3.1 Bounds for fractional moments of beta distribution

In what follows, we present some fractional results on the beta distribution [23]. So let us prove the following αversion.

Theorem 1.21. Let X,Y,U, and Vbe four random variables, such that XBpq,YBmn,UBpn, and VBmq. If pmqn0, then

EαXrEαYrEαUrEαVrBpnBmqBpqBmn,α1.

For the proof of this result, we can apply a weighted version of the fractional Chebyshev inequality as is mentioned in [1].

Remark 1.22. The above theorem generalizes Theorem 3.1 of [7].

We propose also the following αβversion that generalizes the above result. We have

Theorem 1.23. Let X,Y,U, and Vbe four random variables, such that XBpq,YBmn,UBpn, and VBmq. If pmqn0, then

EαXrEβYr+EβXrEαYrEαUrEβVr+EβUrEαVrBpnBmqBpqBmn,α,β1.

Remark 1.24. If α=β=1, then the above theorem reduces to Theorem 3.1 of [7].

3.3.2 Identities and lower bounds

In the following theorem, the fractional covariance of Xand gXis expressed with the derivative of gX. It can be considered as a generalization of a covariance identity established by the authors of [28]. So, we prove the result:

Theorem 1.25. Let Xbe a random variable having a p.d.fdefined on ab;μ=EX. Then, we have

CovαXgX=1Γαabgxdxaxbtα1μtftdt,α1.E27

We can prove this result by the application of the covariance definition in the case where ωx=1.

The following theorem establishes a lower bound for VarαgXof any function gC1ab. We have

Theorem 1.26. Let Xbe a random variable having a p.d.fdefined on ab, such that μ=EX. Then, we have

VarαgX1VarX,α1Γαabgxdxaxbtα1μtftdt2,E28

for any gC1ab.

To prove this result, we use fractional Cauchy-Schwarz inequality established in [29].

Remark 1.27. Let us consider ΩCabthat satisfies axbtα1μtftdt=bxα1σ2Ωxfx. Then, we present the following result.

Theorem 1.28. Let Xbe a random variable having a p.d.f.defined on ab,such that μ=EX,σ2=VarXand ΩCab;axbtα1μtftdt=bxα1σ2Ωxfx. Then, we have

VarαgXσ4XVarαXEα2gXΩX.E29

Proof: We have

Covα2XgX=1Γαabgxdxbxα1σ2Ωxfxdx2.E30

On the other hand, we can see that

1Γαabgxdxbxα1σ2Ωxfxdx2=σ4Eα2gXΩXE31

Thanks to the fractional version of Cauchy Schwarz inequality [29], and using the fact that

Covα2XgXVarαXVarαgX,E32

we obtain

σ4Eα2gXΩXVarαXVarαgX.E33

This ends the proof.

Remark 1.29. Thanks to (30) and (31), we obtain the following fractional covariance identity

σ2EαgXΩX=CovαXgX.

It generalizes the good standard identity obtained in [28] that corresponds to α=1and it is given by

σ2EgXΩX=CovXgX.

We end this section by proving the following fractional integral identity between covariance and expectation in the fractional case.

Theorem 1.30. Let Xbe a continuous random variable with a p.d.f.having a support an interval ab, EX=μ. Then, for any α1, the following general covariance identity holds

CovαhXgX=EαgXZX,E34

where gC1ab, with EZXgX<,hxis a given function and Zxfxbxα1Γα=axEhXhtbtα1Γαftdt.

Proof: We have

CovαhXgX=1Γαabbxα1hxhμgxgμfxdxE35

and

EαgXZX=1Γαabbxα1gxZxfxdx.E36

The definition of ZXimplies that

EαgXZX=1Γαaμgμgtbtα1hμhtftdtE37
+1Γαμbgtgμbtα1hthμftdt.

Hence, we obtain

EαZXgX=CovαgXhX.E38

Remark 1.31. Taking α=1, in the above theorem, we obtain Theorem 2.2 of [10].

4. A class of differential equations of fractional order

Inspired by the work in [4, 20], in what follows we will be concerned with a more general class of Langevin equations of fractional order. The considered class will contain a nonlinearity that depends on a fractional derivative of order δ.So, let us consider the following problem:

cDαD2+λ2ut=ftut,cDδut,t01,λR+0<α1,0δ<α,E39

associated with the conditions

u0=0,u0=0,u1=βuη,η01,E40

where cDαdenotes the Caputo fractional derivative of fractional order α, D2is the two-order classical derivative, f:01×R×RRis a given function, and βR, such that βsinληsinλ.

4.1 Integral representation

We recall the following result [20]:

Lemma 1.32. Let θbe a continuous function on 01. The unique solution of the problem

cDαD2+λ2ut=θt,t01,λR+n1<αn,nN,E41

is given by

ut=1λ0tsinλts0ssτα1Γαθτ+i=1n1cisids+cncosλt+cn+1sinλt,E42

where ciR,i=1n+1.

Thanks to the above lemma, we can state that

The class of Langevin equations (39) and (40) has the following integral representation:

ut=1λ0tsinλts0ssτα1Γαf(τuτDδτ)ds+sinλtΔβ0ηsinληs0ssτα1Γαf(τuτDδτ)ds01sinλ1s0ssτα1Γαf(τuτDδτ)ds,E43

where

Δλsinλβsinλη.E44

4.2 Existence and uniqueness of solutions

Using the above integral representation (43), we can prove the following existence and uniqueness theorem.

Theorem 1.33. Assume that the following hypotheses are valid:

(H1): The function f:01×R×RRis continuous, and there exist two constants Λ1,Λ2>0,such that for all t01and ui,viR,i=1,2,

ftu1u2ftv1v1Λ1u1v1+Λ2u2v2.E45

(H2): Suppose that Λ1Φ+ϒ,

where

ΦΔ1+λ+β1ληα+1Γα+2λΔ1,ΨΔ1sαα+1+λ2+β1λ2ηα+1Γα+2λΔ1,ϒΨΓ2δ,
ΛmaxΛ1Λ2,Δ1=Δ,β1=β.

Then problem (39) and (40) has a unique solution on 01.

E2:Jaα+βft=JaαJaβft

Proof: We introduce the space

E=uuC01DδuC01,

endowed with the norm uEu+Dδu.

Then, E.Eis a Banach space.

Also, we consider the operator T:EEdefined by

Tut1λ0tsinλtsJ0αfsusDδusds+sinλtΔβ0ηsinληsJ0αf(susDδs)ds01sinλ1sJ0αf(susDδus)dsE46

We shall prove that the above operator is contractive over the space E.

Let u1,u2E. Then, for each t01, we have

Tu1tTu2t1λ0tsinλtsJ0αfsu1sDδu1sfsu2sDδu2sds+sinλtΔβ0ηsinληsJ0αf(su1sDδu1s)f(su2sDδu2s)ds+01sinλ1sJ0αf(su1sDδu1s)f(su2sDδu2s)dsA

By (H1), we have

AΛΓα+21λ+1Δ+βΔηα+1u1u2+Dδu1Dδu2.

Hence, it yields that

Tu1Tu2ΛΦu1u2E.E47

With the same arguments as before, we can write.

Tu1tTu2t1λsinλtsJ0αfsu1sDδu1sfsu2sDδu2sds+λcosλtΔβ0ηsinληsJ0αf(su1sDδu1s)f(su2sDδu2s)ds+01sinλ1sJ0αf(su1sDδu1s)f(su2sDδu2s)dsB.

Again, by (H1), we obtain

BΛΔ1sαα+1+λ2+β1λ2ηα+1Γα+2λΔ1u1u2+Dδu1Dδu2.

Consequently, we get

Tu1Tu2ΛΨu1u2E.

This implies that

DδTu1DδTu2Λϒu1u2E.E48

Using (47) and (48), we can state that

Tu1Tu2EΛΦ+ϒu1u2E.

Thanks to (H2), we can say that the operator Tis contractive.

Hence, by Banach fixed point theorem, the operator has a unique fixed point which corresponds to the unique solution of our Langevin problem.

4.3 Existence of solutions

We prove the following theorem.

Theorem 1.34. Assume that the following conditions are satisfied:

(H3): The function f:01×R×RRis jointly continuous.

(H4): There exists a positive constant M;ftuvMfor any t01,u,vR.

Then the problem (39), (40) has at least one solution on 01.

Proof: We use Schaefer fixed point theorem to prove this result. So we proceed into three steps.

Step 1: We prove that Tis continuous and bounded.

Since the function fis continuous by (H3), then the operator is also continuous; this proof is trivial and hence it is omitted.

Let ΩEbe a bounded set. We need to prove that TΩis a bounded set.

Let uΩ.Then, for any t01,we have

Tut1λ+1Δ01J0αfsusDδusds+βΔ0ηJ0αfsusDδsdsC

Using (H4), we get

TuΦM.E49

In the same manner, we find that

DδTuϒM.E50

From (49) and (50), we have

TuEΦ+ϒM.

The operator is thus bounded.

Step 2: Equicontinuity.

Let uE.Then, for each t1,t201, we have

Tut2Tut11λ0t2sinλt2sJ0αf(susDδus)ds0t1sinλt1sJ0αf(susDδus)ds+sinλt2sinλt1Δβ0ηsinληsJ0αf(susDδus)ds+01sinλ1sJ0αf(susDδus)dsΘΔsinλt2sinλt1+1λt1t2sinλt2sJ0αfsusDδusds+1λ0t2(sinλt2ssinλt1sJ0αfsusDδusds,E51

where

Θβ0ηsinληsJ0αfsusDδusds+01sinλ1sJ0αfsusDδusds.

Analogously, we can obtain

Tut2Tut1λΘΔcosλt2cosλt1+1λsinλt2ssinλt1sJ0αfsusDδus.

Consequently, we can write

DδTut2DδTut1J1δTut2Tut1E52

As t1t2, the right-hand sides of (51) and (52) tend to zero.

Therefore,

Tut2Tut1E0.

The operator Tis thus equicontinuous.

As a consequence of Step 1 and Step 2 and thanks to Arzela-Ascoli theorem, we conclude that Tis completely continuous.

Step 3: We prove that ΣuEu=λTu0<λ<1is a bounded set.

Let uΣ. Then, for each t01, the following two inequalities are valid:

ut=λTutTutMΦ

and

Dδut=λDδTutDδTutMϒ.

Therefore,

uEMϒ+Φ.

Thanks to steps 1, 2, and 3 and by Schaefer fixed point theorem, the operator Thas at least one fixed point. This ends the proof of the above theorem.

5. Conclusions

In this chapter, the fractional calculus has been applied for some classes of integral inequalities. In fact, using Riemann-Liouville integral, some Minkowski and Hermite-Hadamard-type inequalities have been established. Several other fractional integral results involving a family of positive functions have been also generated. The obtained results generalizes some classical integral inequalities in the literature. In this chapter, we have also presented some applications on continuous random variables; new identities have been established, and some estimates have been discussed.

The existence and the uniqueness of solutions for nonlocal boundary value problem including the Langevin equations with two fractional parameters have been studied. We have used Caputo approach together with Banach contraction principle to prove the existence and uniqueness result. Then, by application of Schaefer fixed point theorem, another existence result has been also proved. Our approach is simple to apply for a variety of real-world problems.

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Zoubir Dahmani and Meriem Mansouria Belhamiti (February 12th 2020). Integral Inequalities and Differential Equations via Fractional Calculus [Online First], IntechOpen, DOI: 10.5772/intechopen.91140. Available from:

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