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Research article

Generalized integral inequalities for ABK-fractional integral operators

  • In this paper, we employ new version of the Atangana-Baleanu integral operator namely ABK-fractional integrals to obtain two general integral identities complying second-order derivatives for a given function. Thus allowing us to derive new generalized Hermite-Hadamard type inequalities via ABK- fractional integrals. Moreover we give several new versions of Mid-point and Trapezoid type inequalities by employing Hölder, Young and Jensen inequalities.

    Citation: Saad Ihsan Butt, Erhan Set, Saba Yousaf, Thabet Abdeljawad, Wasfi Shatanawi. Generalized integral inequalities for ABK-fractional integral operators[J]. AIMS Mathematics, 2021, 6(9): 10164-10191. doi: 10.3934/math.2021589

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  • In this paper, we employ new version of the Atangana-Baleanu integral operator namely ABK-fractional integrals to obtain two general integral identities complying second-order derivatives for a given function. Thus allowing us to derive new generalized Hermite-Hadamard type inequalities via ABK- fractional integrals. Moreover we give several new versions of Mid-point and Trapezoid type inequalities by employing Hölder, Young and Jensen inequalities.



    Time-fractional differential equations arise in the mathematical modeling of a variety of real-world phenomena in many areas of sciences and engineering, such as elasticity, heat transfer, circuits systems, continuum mechanics, fluid mechanics, wave theory, etc. For more details, we refer the reader to [4,6,7,8,14,15,17,24] and the references therein. Consequently, the study of time-fractional differential equations attracted much attention of many researchers (see e.g. [1,5,9,10,19,22,23] and the references therein).

    Multi-time differential equations arise, for example, in analyzing frequency and amplitude modulation in oscillators, see Narayan and Roychowdhury [18]. Some methods for solving Multi-time differential equations can be found in [20,21].

    The study of blowing-up solutions to time-fractional differential equations was initiated by Kirane and his collaborators, see e.g. [3,11,12,13]. In particular, Kirane et al. [11] considered the two-times fractional differential equation

    {CDα0|tu(t,s)+CDβ0|s|u|m(t,s)=|u|p(t,s),t,s>0,u(0,s)=u0(s),u(t,0)=u1(t),t,s>0, (1.1)

    where p,m>1, 0<α,β<1, CDα0|t is the Caputo fractional derivative of order α with respect to the first time-variable t, and CDβ0|s is the Caputo fractional derivative of order β with respect to the second time-variable s. Namely, the authors provided sufficient conditions for which any solution to (1.1) blows-up in a finite time. In the same reference, the authors extended their study to the case of systems.

    In this paper, we investigate the nonexistence of global solutions to two-times-fractional differential inequalities of the form

    {HCDαa|tu(t,s)+CDβa|s|u|m(t,s)(sa)γ(lnta)σ|u|p(t,s),t,s>a,u(a,s)=u0(s),u(t,a)=u1(t),t,s>a, (1.2)

    where p>1, m1, γ,σR, a>0, 0<α,β<1, HCDαa|t is the Hadamard-Caputo fractional derivative of order α with respect to the first time-variable t, and CDβa|s is the Caputo fractional derivative of order β with respect to the second time-variable s. Using the test function method (see e.g. [16]) and a judicious choice of a test function, we establish sufficient conditions ensuring the nonexistence of global solutions to (1.2). Our obtained conditions depend on the parameters α,β,p,m,γ,σ, and the initial values.

    Our motivation for considering problems of type (1.2) is to study the combination effect of the two fractional derivatives of different nature HCDαa|t and CDβa|s on the nonexistence of global solutions to (1.2). As far as we know, the study of nonexistence of global solutions for time fractional differential equations (or inequalities) involving both Hadamard-Caputo and Caputo fractional derivatives, was never considered in the literature.

    The rest of the paper is organized as follows: In Section 2, we recall some concepts from fractional calculus and provide some useful lemmas. In Section 3, we state our main results and provide some examples. Section 4 is devoted to the proofs of our main results.

    Let a,TR be such that 0<a<T. The left-sided and right-sided Riemann-Liouville fractional integrals of order θ>0 of a function ϑL1([a,T]), are defined respectively by (see [10])

    (Iθaϑ)(t)=1Γ(θ)ta(tτ)θ1ϑ(τ)dτ

    and

    (IθTϑ)(t)=1Γ(θ)Tt(τt)θ1ϑ(τ)dτ,

    for almost everywhere t[a,T], where Γ is the Gamma function.

    Notice that, if ϑC([a,T]), then Iθaϑ,IθTϑC([a,T]) with

    (Iθaϑ)(a)=(IθTϑ)(T)=0. (2.1)

    The Caputo fractional derivative of order θ(0,1) of a function ϑAC([a,)), is defined by (see [10])

    CDθaϑ(t)=(I1θaϑ)(t)=1Γ(1θ)ta(tτ)θϑ(τ)dτ,

    for almost everywhere ta.

    Lemma 2.1. [see [10]]Let κ>0, p,q1, and 1p+1q1+κ (p1, q1, in the case 1p+1q=1+κ). Let ϑLp([a,T] and wLq([a,T]). Then

    Ta(Iκaϑ)(t)w(t)dt=Taϑ(t)(IκTw)(t)dt.

    The left-sided and right-sided Hadamard fractional integrals of order θ>0 of a function ϑL1([a,T]), are defined respectively by (see [10])

    (Jθaϑ)(t)=1Γ(θ)ta(lntτ)θ1ϑ(τ)1τdτ

    and

    (JθTϑ)(t)=1Γ(θ)Tt(lnτt)θ1ϑ(τ)1τdτ,

    for almost everywhere t[a,T].

    Notice that, if ϑC([a,T]), then Jθaϑ,JθTϑC([a,T]) with

    (Jθaϑ)(a)=(JθTϑ)(T)=0. (2.2)

    The Hadamard-Caputo fractional derivative of order θ(0,1) of a function ϑAC([a,)), is defined by (see [2])

    HCDθaϑ(t)=(J1θaδϑ)(t)=1Γ(1θ)ta(lntτ)θδϑ(τ)1τdτ,

    for almost everywhere ta, where

    δϑ(t)=tϑ(t).

    We have the following integration by parts rule.

    Lemma 2.2. Let κ>0, p,q1, and 1p+1q1+κ (p1, q1, in the case 1p+1q=1+κ). If ϑexpLp([lna,lnT]) and wexpLq([lna,lnT]), then

    Ta(Jκaϑ)(t)w(t)1tdt=Taϑ(t)(JκTw)(t)1tdt.

    Proof. Using the change of variable x=lnτ, we obtain

    (Jκaϑ)(t)=1Γ(κ)ta(lntτ)κ1ϑ(τ)1τdτ=1Γ(κ)lntlna(lntx)κ1(ϑexp)(x)dx,

    that is,

    (Jκaϑ)(t)=(Iκlnaϑexp)(lnt). (2.3)

    Similarly, we have

    (JκTw)(t)=(IκlnTwexp)(lnt). (2.4)

    By (2.3), we obtain

    Ta(Jκaϑ)(t)w(t)1tdt=Ta(Iκlnaϑexp)(lnt)w(t)1tdt.

    Using the change of variable x=lnt, we get

    Ta(Jκaϑ)(t)w(t)1tdt=lnTlna(Iκlnaϑexp)(x)(wexp)(x)dx.

    Since ϑexpLp([lna,lnT]) and wexpLq([lna,lnT]), by Lemma 2.1, we deduce that

    Ta(Jκaϑ)(t)w(t)1tdt=lnTlna(ϑexp)(x)(IκlnTwexp)(x)dx.

    Using again the change of variable x=lnt, there holds

    Ta(Jκaϑ)(t)w(t)1tdt=Taϑ(t)(IκlnTwexp)(lnt)1tdt.

    Then, by (2.4), the desired result follows.

    By elementary calculations, we obtain the following properties.

    Lemma 2.3. For sufficiently large λ, let

    ϕ1(t)=(lnTa)λ(lnTt)λ,atT. (2.5)

    Let κ(0,1). Then

    (JκTϕ1)(t)=Γ(λ+1)Γ(κ+λ+1)(lnTa)λ(lnTt)κ+λ, (2.6)
    (JκTϕ1)(t)=Γ(λ+1)Γ(κ+λ)(lnTa)λ(lnTt)κ+λ11t. (2.7)

    Lemma 2.4. For sufficiently large λ, let

    ϕ2(s)=(Ta)λ(Ts)λ,asT. (2.8)

    Let κ(0,1). Then

    (IκTϕ2)(s)=Γ(λ+1)Γ(κ+λ+1)(Ta)λ(Ts)κ+λ, (2.9)
    (IκTϕ2)(s)=Γ(λ+1)Γ(κ+λ)(Ta)λ(Ts)κ+λ1. (2.10)

    First, let us define global solutions to (1.2). To do this, we need to introduce the functional space

    Xa:={uC([a,)×[a,)):u(,s)AC([a,)),|u|m(t,)AC([a,))}.

    We say that u is a global solution to (1.2), if uXa and u satisfies the fractional differential inequality

    HCDαa|tu(t,s)+CDβa|s|u|m(t,s)(sa)γ(lnta)σ|u|p(t,s)

    for almost everywhere t,sa, as well as the initial conditions

    u(a,s)=u0(s),u(t,a)=u1(t),t,s>a.

    Now, we state our main results.

    Theorem 3.1. Let u0L1([a,)), u1Lm([a,),1tdt), and u10.Let

    0<β<1m1,γ>max{m11mβ,m(σ+1)1}β. (3.1)

    If

    mmax{γ+1,σ+1}<p<1+γβ, (3.2)

    then, for all α(0,1), (1.2) admits no global solution.

    Remark 3.1. Notice that by (3.1), the set of exponents p satisfying (3.2) is nonempty.

    Theorem 3.2. Let u0L1([a,)), u1Lm([a,),1tdt), and u10. Let

    0<β<1m1,11m<α<1,σ>(m1)(1α)1mβα. (3.3)

    If

    βmax{m11mβ,m(σ+1)1}<γ<(σ+α)β1α (3.4)

    and

    p=1+γβ, (3.5)

    then (1.2) admits no global solution.

    Remark 3.2. Notice that by (3.3), the set of real numbers γ satisfying (3.4) is nonempty.

    We illustrate our obtained results by the following examples.

    Example 3.1. Consider the fractional differential inequality

    {HCDαa|tu(t,s)+CD14a|su2(t,s)(sa)(lnta)1|u|p(t,s),t,s>a,u(a,s)=(1+s2)1,u(t,a)=exp(t),t,s>a, (3.6)

    where a>0 and 0<α<1. Observe that (3.6) is a special case of (1.2) with

    β=14,m=2,σ=1,γ=1,u0(s)=(1+s2)1,u1(t)=exp(t).

    Moreover, we have

    0<β=14<12=1m<1,max{m11mβ,m(σ+1)1}β=max{2,0}4=12<γ=1,

    and u0L1([a,)), u1Lm([a,),1tdt). Hence, condition (3.1) is satisfied. Then, by Theorem 3.1, we deduce that, if

    mmax{γ+1,σ+1}<p<1+γβ,

    that is,

    4<p<5,

    then (3.6) admits no global solution.

    Example 3.2. Consider the fractional differential inequality

    {HCD34a|tu(t,s)+CD12a|s|u|(t,s)(sa)14(lnta)12|u|32(t,s),t,s>a,u(a,s)=(1+s2)1,u(t,a)=exp(t),t,s>a, (3.7)

    where a>0. Then (3.7) is a special case of (1.2) with

    α=34,β=12,m=1,σ=12,γ=14,p=32,u0(s)=(1+s2)1,u1(t)=exp(t).

    On the other hand, we have

    0<β=12<1=1m,11m=0<α=34<1,σ=12>34=(m1)(1α)1mβα,

    which shows that condition (3.3) is satisfied. Moreover, we have

    βmax{m11mβ,m(σ+1)1}=14<γ=14<12=(σ+α)β1α,p=32=1+γβ,

    which shows that conditions (3.4) and (3.5) are satisfied. Then, by Theorem 3.2, we deduce that (3.7) admits no global solution.

    In this section, C denotes a positive constant independent on T, whose value may change from line to line.

    Proof of Theorem 3.1. Suppose that uXa is a global solution to (1.2). For sufficiently large T and λ, let

    φ(t,s)=ϕ1(t)ϕ2(s),at,sT,

    where ϕ1 and ϕ2 are defined respectively by (2.5) and (2.8). Multiplying the inequality in (1.2) by 1tφ and integrating over ΩT:=(a,T)×(a,T), we obtain

    ΩT(sa)γ(lnta)σ|u|pφ(t,s)1tdtdsΩTHCDαa|tuφ(t,s)1tdtds+ΩTCDβa|s|u|mφ(t,s)1tdtds. (4.1)

    On the other hand, using Lemma 2.2, integrating by parts, using the initial conditions, and taking in consideration (2.2), we obtain

    TaHCDαa|tuφ(t,s)1tdt=Ta(J1αa|ttut)(t,s)φ(t,s)1tdt=Taut(t,s)(J1αT|tφ)(t,s)dt=[u(t,s)(J1αT|tφ)(t,s)]Tt=aTau(t,s)(J1αT|tφ)t(t,s)dt=u0(s)(J1αT|tφ)(a,s)Tau(t,s)(J1αT|tφ)t(t,s)dt.

    Integrating over (a,T), we get

    ΩTHCDαa|tuφ(t,s)1tdtds=Tau0(s)(J1αT|tφ)(a,s)dsΩTu(t,s)(J1αT|tφ)t(t,s)dtds. (4.2)

    Similarly, using Lemma 2.1, integrating by parts, using the initial conditions, and taking in consideration (2.1), we obtain

    TaCDβa|s|u|mφ(t,s)ds=Ta(I1βa|s|u|ms(t,s))φ(t,s)ds=Ta|u|ms(t,s)(I1βT|sφ)(t,s)ds=[|u|m(t,s)(I1βT|sφ)(t,s)]Ts=aTa|u|m(t,s)(I1βT|sφ)s(t,s)ds=|u1(t)|m(I1βT|sφ)(t,a)Ta|u|m(t,s)(I1βT|sφ)s(t,s)ds.

    Integrating over (a,T), there holds

    ΩTCDβa|s|u|mφ(t,s)1tdtds=Ta|u1(t)|m(I1βT|sφ)(t,a)1tdtΩT|u|m(t,s)(I1βT|sφ)s(t,s)1tdtds. (4.3)

    It follows from (4.1)–(4.3) that

    ΩT(sa)γ(lnta)σ|u|pφ(t,s)1tdtds+Tau0(s)(J1αT|tφ)(a,s)ds+Ta|u1(t)|m(I1βT|sφ)(t,a)1tdtΩT|u||(J1αT|tφ)t|dtds+ΩT|u|m|(I1βT|sφ)s|1tdtds. (4.4)

    On the other hand, by Young's inequality, we have

    ΩT|u||(J1αT|tφ)t|dtds12ΩT(sa)γ(lnta)σ|u|pφ(t,s)1tdtds+CΩTt1p1(sa)γp1(lnta)σp1φ1p1(t,s)|(J1αT|tφ)t|pp1dtds. (4.5)

    Similarly, since p>m, we have

    ΩT|u|m|(I1βT|sφ)s|1tdtds12ΩT(sa)γ(lnta)σ|u|pφ(t,s)1tdtds+CΩT1t(sa)γmpm(lnta)σmpmφmpm(t,s)|(I1βT|sφ)s|ppmdtds. (4.6)

    Hence, combining (4.4)–(4.6), we deduce that

    Tau0(s)(J1αT|tφ)(a,s)ds+Ta|u1(t)|m(I1βT|sφ)(t,a)1tdtC(K1+K2), (4.7)

    where

    K1=ΩTt1p1(sa)γp1(lnta)σp1φ1p1(t,s)|(J1αT|tφ)t|pp1dtds

    and

    K2=ΩT1t(sa)γmpm(lnta)σmpmφmpm(t,s)|(I1βT|sφ)s|ppmdtds.

    By the definition of the function φ, we have

    (J1αT|tφ)(a,s)=ϕ2(s)(J1αT|tϕ1)(a).

    Thus, using (2.6), we obtain

    (J1αT|tφ)(a,s)=Cϕ2(s)(lnTa)1α.

    Integrating over (a,T), we get

    Tau0(s)(J1αT|tφ)(a,s)ds=C(lnTa)1αTau0(s)(Ta)λ(Ts)λds. (4.8)

    Similarly, by the definition of the function φ, we have

    (I1βT|sφ)(t,a)=ϕ1(t)(I1βT|sϕ2)(a).

    Thus, using (2.9), we obtain

    (I1βT|sφ)(t,a)=Cϕ1(t)(Ta)1β.

    Integrating over (a,T), we get

    Ta|u1(t)|m(I1βT|sφ)(t,a)1tdt=C(Ta)1βTa|u1(t)|m(lnTa)λ(lnTt)λ1tdt. (4.9)

    Combining (4.8) with (4.9), there holds

    Tau0(s)(J1αT|tφ)(a,s)ds+Ta|u1(t)|m(I1βT|sφ)(t,a)1tdt=C(lnTa)1αTau0(s)(Ta)λ(Ts)λds+C(Ta)1βTa|u1(t)|m(lnTa)λ(lnTt)λ1tdt.

    Since u0L1([a,)), u1Lm([a,),1tdt), and u10, by the dominated convergence theorem, we deduce that for sufficiently large T,

    Tau0(s)(J1αT|tφ)(a,s)ds+Ta|u1(t)|m(I1βT|sφ)(t,a)1tdtC(Ta)1βa|u1(t)|m1tdt. (4.10)

    Now, we shall estimate the terms Ki, i=1,2. By the definition of the function φ, the term K1 can be written as

    K1=(Ta(sa)γp1ϕ2(s)ds)(Tat1p1(lnta)σp1ϕ1p11(t)|(J1αT|tϕ1)(t)|pp1dt). (4.11)

    Next, by (2.8), we obtain

    Ta(sa)γp1ϕ2(s)ds=(Ta)λTa(sa)γp1(Ts)λdsTa(sa)γp1ds.

    On the other hand, by (3.1) and (3.2), it is clear that γ<p1. Thus, we deduce that

    Ta(sa)γp1ϕ2(s)dsC(Ta)1γp1. (4.12)

    By (2.5) and (2.7), we have

    Tat1p1(lnta)σp1ϕ1p11(t)|(J1αT|tϕ1)(t)|pp1dt=(lnTa)λTa(lnTt)λαpp1(lnta)σp11tdt(lnTa)αpp1Ta(lnta)σp11tdt.

    Notice that by (3.1) and (3.2), we have σ<p1. Thus, we get

    Tat1p1(lnta)σp1ϕ1p11(t)|(J1αT|tϕ1)(t)|pp1dtC(lnTa)1αp+σp1. (4.13)

    Hence, it follows from (4.11)–(4.13) that

    K1C(Ta)1γp1(lnTa)1αp+σp1. (4.14)

    Similarly, we can write the term K2 as

    K2=(Ta1t(lnta)σmpmϕ1(t)dt)(Ta(sa)γmpmϕmpm2(s)|(I1βT|sϕ2)(s)|ppmds). (4.15)

    By (2.5), we have

    Ta1t(lnta)σmpmϕ1(t)dt=(lnTa)λTa(lnta)σmpm(lnTt)λ1tdtTa(lnta)σmpm1tdt.

    Notice that by (3.2), we have σm<pm. Thus, we get

    Ta1t(lnta)σmpmϕ1(t)dtC(lnTa)1σmpm. (4.16)

    On the other hand, by (2.8) and (2.10), we have

    Ta(sa)γmpmϕmpm2(s)|(I1βT|sϕ2)(s)|ppmds=(Ta)λTa(Ts)λβppm(sa)γmpmds(Ta)βppmTa(sa)γmpmds.

    Notice that by (3.2), we have p>m(γ+1). Therefore, we obtain

    Ta(sa)γmpmϕmpm2(s)|(I1βT|sϕ2)(s)|ppmdsC(Ta)1γm+βppm. (4.17)

    Combining (4.16) with (4.17), there holds

    K2C(lnTa)1σmpm(Ta)1γm+βppm. (4.18)

    Hence, it follows from (4.14) and (4.18) that

    K1+K2C[(lnTa)1αp+σp1(Ta)1γp1+(lnTa)1σmpm(Ta)1γm+βppm]. (4.19)

    Thus, by (4.7), (4.10), and (4.19), we deduce that

    a|u1(t)|m1tdtC[(lnTa)1αp+σp1(Ta)βγp1+(lnTa)1σmpm(Ta)βγm+βppm]. (4.20)

    Notice that by (3.1) and (3.2), we have

    βγp1<0,βγm+βppm<0.

    Hence, passing to the limit as T in (4.20), we obtain a contradiction with u10. Consequently, (1.2) admits no global solution. The proof is completed.

    Proof of Theorem 3.2. Suppose that uXa is a global solution to (1.2). Notice that in the proof of Theorem 3.1, to obtain (4.20), we used that

    p>m1,p>σ+1,p>m(σ+1),p>m(γ+1).

    On the other hand, by (3.3)–(3.5), it can be easily seen that the above conditions are satisfied. Thus, (4.20) holds. Hence, taking p=1+γβ in (4.20), we obtain

    a|u1(t)|m1tdtC[(lnTa)1αp+σp1+(lnTa)1σmpm(Ta)βγm+βppm]. (4.21)

    On the other hand, by (3.3)–(3.5), we have

    1αp+σp1<0,βγm+βppm<0.

    Hence, passing to the limit as T in (4.21), we obtain a contradiction with u10. This shows that (1.2) admits no global solution. The proof is completed.

    The two-times fractional differential inequality (1.2) is investigated. Namely, using the test function method and a judicious choice of a test function, sufficient conditions ensuring the nonexistence of global solutions to (1.2) are obtained. Two cases are discussed separately: 1<p<1+γβ (see Theorem 3.1) and p=1+γβ (see Theorem 3.2). In the first case, no assumption is imposed on the fractional order α(0,1) of the Hadamard-Caputo fractional derivative, while in the second case, it is supposed that α>11m. About the initial conditions, in both cases, it is assumed that u0L1([a,)), u1Lm([a,),1tdt), and u10.

    Finally, it would be interesting to extend this study to two-times fractional evolution equations. For instance, the tow-times fractional semi-linear heat equation

    {HCDαa|tu(t,s,x)+CDβa|s|u|m(t,s,x)(sa)γ(lnta)σ|u|p(t,s,x),t,s>a,xRN,u(a,s,x)=u0(s,x),u(t,a,x)=u1(t,x),t,s>a,xRN,

    deserves to be studied.

    The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-02.

    The authors declare that they have no competing interests.



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