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An optimal scheme to boost immunity and suppress viruses for HIV by combining a phased immunotherapy with the sustaining antiviral therapy

  • Received: 12 April 2020 Accepted: 21 June 2020 Published: 02 July 2020
  • Despite many approaches to treat HIV virus, the endeavor, due to the inability of therapy to eradicate HIV infection, has been aroused to formulate rational therapeutic strategies to establish sustained immunity to suppress viruses after stopping therapy. In this paper, incorporating the time lag of the expansion of immune cells, we propose an explicit model with continuous antiretroviral therapy (CATT) and an intermittent immunotherapy to describe an interaction of uninfected cells, HIV virus and immune response. Two kinds of bistability and the sensitivities of the amplitude and period of the periodic solution with respect to all of parameters indicate that both ε and b relating to the therapy are scheduled to propose an optimal treatment tactics. Furthermore, taking a patient performed a CATT but with an unsuccessful outcome as a example, we inset a phased immunotherapy into the above CATT and then adjust the therapeutic session as well as the inlaid time to quest the preferable therapeutic regimen. Mathematically, we alter the solution of system from the basin of the attraction of the immune-free equilibrium to the immune control balance when the treatment is ceased, meanwhile minimize the cost function through a period of combined therapy. Due to the particularity of our optimal problem, we contribute a novel optimization approach by meshing a special domain on the antiretroviral and immunotherapy parameters ε and b, to catch an optimal combined treatment scheme. Simulations exhibit that early mediating immunotherapy suppresses the load of virus lower while shortening the combined treatment session does not reduce but magnify the cost function. Our results can provide some insights into the design of optimal therapeutic strategies to boost sustained immunity to quell viruses.

    Citation: Youyi Yang, Yongzhen Pei, Xiyin Liang, Yunfei Lv. An optimal scheme to boost immunity and suppress viruses for HIV by combining a phased immunotherapy with the sustaining antiviral therapy[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4578-4608. doi: 10.3934/mbe.2020253

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  • Despite many approaches to treat HIV virus, the endeavor, due to the inability of therapy to eradicate HIV infection, has been aroused to formulate rational therapeutic strategies to establish sustained immunity to suppress viruses after stopping therapy. In this paper, incorporating the time lag of the expansion of immune cells, we propose an explicit model with continuous antiretroviral therapy (CATT) and an intermittent immunotherapy to describe an interaction of uninfected cells, HIV virus and immune response. Two kinds of bistability and the sensitivities of the amplitude and period of the periodic solution with respect to all of parameters indicate that both ε and b relating to the therapy are scheduled to propose an optimal treatment tactics. Furthermore, taking a patient performed a CATT but with an unsuccessful outcome as a example, we inset a phased immunotherapy into the above CATT and then adjust the therapeutic session as well as the inlaid time to quest the preferable therapeutic regimen. Mathematically, we alter the solution of system from the basin of the attraction of the immune-free equilibrium to the immune control balance when the treatment is ceased, meanwhile minimize the cost function through a period of combined therapy. Due to the particularity of our optimal problem, we contribute a novel optimization approach by meshing a special domain on the antiretroviral and immunotherapy parameters ε and b, to catch an optimal combined treatment scheme. Simulations exhibit that early mediating immunotherapy suppresses the load of virus lower while shortening the combined treatment session does not reduce but magnify the cost function. Our results can provide some insights into the design of optimal therapeutic strategies to boost sustained immunity to quell viruses.


    P.L Čebyšev in the year 1882 has proved the following interesting inequality:

    |1babaf(x)g(x)dx(1babaf(x)dx)(1babag(x)dx)|112(ba)2fg.

    where f,g are absolutely continuous functions defined on [a,b] and f,gL[a,b]. The left hand side of the above equation is denoted by T(f,g) is called Cebysev Functional if the integral exists. The applications of above type of inequalities can be found in the field of coding theory, statistics and other branches of mathematics.

    In last few decades many researchers have obtained various extensions and generalizations of above inequalities using various techniques see [1,2]. Study of inequalities have attracted the attention of researchers from various fields due to its wide applications in various fields [3,4].

    During last few years the subject of Fractional Calculus has been developed rapidly due to the applications in various fields of science and engineering. Various new definitions of fractional derivatives and integrals have been obtained by various researchers depending on the applications such as Riemann liouville, Caputo, Saigo, Hilfer, Hadmard, Katugampola and others See [5,6,7,8]. Many results on study of mathematical inequalities using various new fractional definitions such as Conformable and generalized fractional integral were obtained in [9,10]. Recently in [11,12,13,14,15] the authors have obtained the results on Cebysev inequalities using various fractional integral and derivatives definitions.

    In [7] authors have given definations of fractional derivative and integrals of a functions with respect to another functions. Recently in [16,17] authors have studied the ψ Caputo and ψ Hilfer fractional derivative of a function with respect to another functions and its applications. The ψ fractional and integral definations are more generalized and it reduces to Riemann Liouville, Hadmard and Erdelyi-Kober fractional definitions for different values of ψ.

    Motivated from the above mentioned literature the aim of this paper is to obtain ψ Caputo fractional Čebyšev inequalities involving functions of two and three variables.

    Now in this section we give some basic definitions and properties which are useful in our subsequent discussions. In [7,8] the authors have defined the fractional integrals and fractional derivative of a function with respect to another function as follows.

    Definition 2.1 [7,16]. Let I=[a,b] be an interval, α>0, f is an integrable function defined on I and ψC1(I) an increasing function such that ψ(x)0 for all xI then fractional derivative and integral of f is given by

    Iα,ψa+f(x)=1Γ(α)xaψ(t)(ψ(x)ψ(t))α1f(t)dt

    and

    Dα,ψa+f(x)=(1ψ(x)ddx)nInα,ψa+f(x)=1Γ(nα)(1ψ(x)ddx)nxaψ(t)(ψ(x)ψ(t))nα1f(t)dt,

    respectively. Similarly right fractional integral and right fractional derivative are given by

    Iα,ψbf(x)=1Γ(α)xaψ(t)(ψ(t)ψ(x))α1f(t)dt

    and

    Dα,ψbf(x)=(1ψ(x)ddx)nInα,ψbf(x)=1Γ(nα)(1ψ(x)ddx)nxaψ(t)(ψ(t)ψ(x))nα1f(t)dt.

    In [16] Almedia has considered a Caputo type fractional derivative with respect to another function.

    Definition 2.2 [16] Let α>0, nN, I is the interval a<b, f,ψCn(I) two functions such that ψ is increasing and ψ(x)0 for all xI. The left ψ-Caputo fractional derivative of f of order α is given by

    CDα,ψa+f(x)=Inα,ψa+(1ψ(x)ddx)nf(x),

    and the right ψ-Caputo fractional derivative of f is given by

    CDα,ψbf(x)=Inα,ψb(1ψ(x)ddx)nf(x).

    For given αN

    CDα,ψa+f(x)=1Γ(nα)xaψ(t)(ψ(x)ψ(t))nα1f[n]ψ(t)dt

    and

    CDα,ψbf(x)=1Γ(nα)xaψ(t)(ψ(t)ψ(x))nα1(1)nf[n]ψ(t)dt.

    In particular when α(0,1) then

    CDα,ψa+f(x)=1Γ(1α)xa(ψ(x)ψ(t))αf(t)dt

    and

    CDα,ψbf(x)=1Γ(1α)xa(ψ(t)ψ(x))αf(t)dt.

    In [18] the author has defined the ψ fractional partial integral with respect to another functions as

    Definition 2.3 Let θ=(a,b) and α=(α1,α2) where 0α1,α21. Also put I=[a,k]×[b,m] where a,b and k,m are positive constants. Also let ψ(.) be an increasing positive monotone function on (a,k]×(b,m] having continuous derivative ψ(.) on (a,k]×(b,m]. Then the fractional partial integral is

    Iα;ψθu(x,y)=1Γ(α1)Γ(α2)xaybψ(s)ψ(t)(ψ(x)ψ(s))α11(ψ(y)ψ(t))α21f(s,t)dtds.

    The Caputo fractional partial derivative is defined as follows

    Definition 2.4 Let θ=(a,b) and α=(α1,α2) where 0α1,α21. Also put I=[a,k]×[b,m] where a,b and a,b are positive constants. Also let ψ(.) be an increasing function on (a,k]×(b,m] and ψ(.)0 on (a,k]×(b,m]. The ψ Caputo fractional partial derivative of functions of two variables of order α is given by

    CDα;ψθu(x,y)=I2α;ψθ(1ψ(s)ψ(t)2αyx)u(x,y).

    We use the following notation:

    CDα;ψθu(x,y)=2αψuψyαψxα(x,y).

    We define the norm for a function of two variables as follows

    CDα;ψθf=sup|CDα;ψθf(x,y)|.

    Similarly as in Definition (2.3) and (2.4) we define the ψ fractional partial integral with respect to another functions and ψ Caputo fractional partial derivative of functions of three variables as follows:

    Definition 2.5 Let Θ=(a,b,c) and α=(α1,α2,α3) where 0α1,α2,α31. Also put I=[a,k]×[b,m]×[c,n] where a,b,c and k,m,n are positive constants. Also let ψ(.) be an increasing positive monotone function on (a,k]×(b,m]×[c,n] having continuous derivative ψ(.) on (a,k]×(b,m]×(c,n].

    Then the fractional partial integral is

    Iα;ψΘu(x,y,z)=1Γ(α1)Γ(α2)xaybzcψ(s)ψ(t)ψ(r)×(ψ(x)ψ(s))α11(ψ(y)ψ(t))α21(ψ(z)ψ(r))α31f(s,t,r)drdtds.

    Definition 2.6 Let θ=(a,b,c) and α=(α1,α2,α3) where 0α1,α2,α31. Also put I=[a,k]×[b,m]×[c,n] where a,b,c and k,m,n are positive constants. Also let ψ(.) be an increasing function on (a,k]×(b,m]×(c,n] and ψ(.)0 on (a,k]×(b,m]×(c,n]. The ψ Caputo fractional partial derivative of functions of three variables of order α is given by

    CDα;ψΘu(x,y,z)=I3α;ψΘ(1ψ(s)ψ(t)ψ(r)3zyx)u(x,y,z).

    We use the following notation:

    CDα;ψΘu(x,y,z)=3αψuψzαψyαxα(x,y,z).

    We define the norm for a function of three variables as follows

    CDα;ψΘf=sup|CDα;ψΘf(x,y,z)|.

    Now we give the ψ Caputo fractional Čebyšev inequality involving functions of two variables as follows:

    Theorem 3.1 Let f,g:[a,l]×[b,m]R be a continuous function on [a,l]×[b,m] and 2αfψyαψxα, 2αgψyαψxα exists continuous and bounded on [a,l]×[b,m] and α=(α1,α2). Then

    |lamb[f(x,y)g(x,y)12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]dydx]|18(ψ(l)ψ(a))(ψ(m)ψ(b))lamb[|g(x,y)|Dα;ψθf+g(x,y)Dα;ψθg]dydx, (3.1)

    where

    G(f(x,y))=12[f(a,y)+f(x,m)+f(x,b)+f(l,y)]14[f(a,b)+f(a,m)+f(l,b)+f(l,m)]

    and

    H(2αfψyαψxα(x,y))=1Γ(α1)Γ(α2)××[xaybψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdtxamyψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdtlxybψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt+lxmyψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt].

    Proof. From the given hypotheses for (x,y)[a,l]×[b,m] we have

    1Γ(α1)Γ(α2)xaybψ(t)ψ(s)×(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt=1Γ(α1)xaψ(s)(ψ(x)ψ(t))α11[αfψsα(s,t)|yc]=1Γ(α1)xaψ(s)(ψ(y)ψ(t))α11[αfψsα(t,y)αfψsα(t,b)]=f(t,y)|xaf(t,b)|xa=f(x,y)f(a,y)f(x,b)+f(a,b). (3.2)

    Similarly we have

    1Γ(α1)Γ(α2)xamyψ(t)ψ(s)×(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt=f(x,y)f(a,m)+f(x,m)+f(a,y), (3.3)
    1Γ(α1)Γ(α2)lxybψ(t)ψ(s)×(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt=f(x,y)f(l,b)+f(x,b)+f(l,y), (3.4)
    1Γ(α1)Γ(α2)lxmyψ(t)ψ(s)×(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(s,t)dsdt=f(x,y)+f(l,b)f(x,b)f(l,y). (3.5)

    Adding the above identities we have

    4f(x,y)2[f(a,y)+f(x,m)+f(x,b)+f(l,y)]+[f(a,b)+f(a,m)+f(l,b)+f(l,m)]=1Γ(α1)Γ(α2)[xaybψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdtxadyψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdtlxybψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt+lxmyψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt]. (3.6)

    From (3.6) we have

    f(x,y)G(f(x,y))=14H(2αfψyαψxα(x,y)), (3.7)

    for (x,y)[a,l]×[b,m]. Similarly we have

    g(x,y)G(g(x,y))=14H(2αgψyαψxα(x,y)), (3.8)

    for (x,y)[a,l]×[b,m].

    Multiplying (3.7) by g(x,y), (3.8) by f(x,y) adding them and Integrating over (x,y)[a,l]×[b,m] we get

    lamb[2f(x,y)g(x,y)g(x,y)G(f(x,y))f(x,y)G(g(x,y))]dydx=18lamb[H(2αfψyαψxα(x,y))g(x,y)+14f(x,y)H(2αgψyαψxα(x,y))]. (3.9)

    From the properties of modulus we have

    |H(2αfψyαψxα(x,y))|1Γ(α1)Γ(α2)lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αfψsαψtα(t,s)|dsdt(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2cDα;ψθf, (3.10)
    |H(2αgψyαψxα(x,y))|1Γ(α1)Γ(α2)lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αgψsαψtα(t,s)|dsdt(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2cDα;ψθg. (3.11)

    From (3.9), (3.10) and (3.11) we have

    |lamb[f(x,y)g(x,y)12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]]dydx|18lamb[|H(2αfψyαψxα(x,y))||g(x,y)|+|H(2αgψyαψxα(x,y))||f(x,y)|]18lamb{|g(x,y)|[1Γ(α1)Γ(α2)×[lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αfψsαψtα(t,s)|dsdt]+|f(x,y)|×[lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αgψsαψtα(t,s)|dsdt]}dydx18(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2×lamb[|g(x,y)|cDα;ψθf+|f(x,y)|cDα;ψθg]dydx, (3.12)

    which is required inequality.

    Theorem 3.2 Let f,g,G(f(x,y)),G(g(f(x,y)),2αfψyαψxα,2αgψyαψxα be as in Theorem 3.1 then

    |lamb{f(x,y)g(x,y)[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)G(f(x,y))G(g(x,y))]}dydx116{(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2}2cDα;ψθfcDα;ψθg, (3.13)

    for (x,y)[a,l]×[b,m].

    Proof. Multiplying left hand side and right hand side of (3.7) and (3.8) we have

    f(x,y)g(x,y)[f(x,y)G(g(x,y))+g(x,y)G(f(x,y))]=116H(2αfψyαψxα(x,y))H(2αgψyαψxα(x,y)). (3.14)

    Integrating (3.14) over [a,l]×[b,m] and from the properties of modulus we get

    |lamb{f(x,y)g(x,y)[G(g(x,y))f(x,y)+G(f(x,y))g(x,y)]G(f(x,y))G(g(x,y))}dydx|116lamb|H(2αfψyαψxα(x,y))||H(2αgψyαψxα(x,y))|dydx. (3.15)

    Now using (3.13),(3.14) in (3.19) we get required inequality (3.13).

    Now in our result we give the ψ Caputo fractional Čebyšev inequality involving functions of three variables. We use some notations as follows:

    A(p(u,v,w))=18[p(a,b,c)+p(k,m,n)]14[p(u,b,c)+p(u,m,n)+p(u,m,c)+p(u,b,n)]14[p(a,v,c)+p(k,v,n)+p(a,v,n)+p(k,v,c)]14[p(a,b,w)+p(k,m,w)+p(k,b,w)+p(a,m,w)]+12[p(a,v,w)+p(k,v,w)]+12[p(u,b,w)+p(u,m,w)]+12[p(u,v,c)+p(u,v,n)] (4.1)

    and

    B(3αpψwαψvαψuα(u,v,w))=1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)uavbncψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)uamvwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)kuvbwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(u)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)uamrnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)kumvwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)kuvbnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)kumvnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr. (4.2)

    Now we give our next result as

    Theorem 4.1 Let f,g:[a,k]×[b,m]×[c,n]R be a continuous function on [a,l]×[b,m] and 3αfψtαψsαψrα, 3αgψtαψsαψrα exists and continuous and bounded on [a,k]×[b,m]×[c,n]. Then

    kambnc[f(u,v,w)g(u,v,w)12[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))]]dwdvdu116(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3×kambnc[|g(u,v,w)|cDα;ψΘf+|f(u,v,w)|cDα;ψΘg]dwdvdu, (4.3)

    where A,B are as given in (4.1),(4.2).

    Proof. From the hypotheses we have for u,v,w[a,k]×[b,m]×[c,n]

    1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr=1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,t)|wcdsdr=1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,w)dsdr1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,c)dsdr=1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,s,w)|vbdr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,s,c)|vbdr=1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,v,w)dr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,b,w)dr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,v,c)dr+1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,b,c)dr=f(r,v,w)|uaf(r,b,w)|uaf(r,v,c)|ua+f(r,b,c)|ua=f(u,v,w)f(a,v,w)f(u,b,w)+f(a,b,w)f(u,v,c)+f(a,v,c)+f(u,b,c)+f(a,b,c).

    Thus we have

    f(u,v,w)=f(a,v,w)+f(u,b,w)f(a,b,w)+f(u,v,c)f(a,v,c)f(u,b,c)f(a,b,c)1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, (4.4)

    Similarly we have

    f(u,v,w)=f(u,v,n)+f(a,v,w)+f(u,b,w)+f(a,b,n)f(a,b,w)f(a,v,n)f(v,b,n)1Γ(α1)Γ(α2)Γ(α3)uavbnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, (4.5)
    f(u,v,w)=f(u,m,w)+f(u,v,c)+f(a,m,c)+f(a,v,w)f(u,m,c)f(a,m,w)f(a,v,c)1Γ(α1)Γ(α2)Γ(α3)uamvwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, (4.6)
    f(u,v,w)=f(k,s,t)+f(k,b,c)+f(u,v,c)+f(u,b,w)f(k,v,c)f(k,b,w)f(u,b,c)1Γ(α1)Γ(α2)Γ(α3)kuvbwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, (4.7)
    f(u,v,w)=f(u,m,w)+f(u,v,n)+f(a,m,n)+f(a,v,w)f(u,m,n)f(a,m,w)f(a,v,n)+1Γ(α1)Γ(α2)Γ(α3)uamvnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, (4.8)
    f(u,v,w)=f(r,m,t)+f(u,v,c)+f(k,s,t)+f(k,m,c)f(k,m,w)f(k,v,c)f(u,m,c)+1Γ(α1)Γ(α2)Γ(α3)kumvwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, (4.9)
    f(u,v,w)=f(k,v,w)+f(k,b,n)+f(u,v,n)+f(u,b,t)f(k,v,n)f(k,b,w)f(u,b,n)+1Γ(α1)Γ(α2)Γ(α3)kuvbnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr (4.10)

    and

    f(u,v,w)=f(k,m,n)+f(k,v,w)+f(u,m,w)+f(u,v,n)f(k,m,w)f(k,v,n)f(u,m,n)+1Γ(α1)Γ(α2)Γ(α3)kumvnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr. (4.11)

    Adding the above identities we have

    f(u,v,w)A(f(u,v,w))=18B(3αfψwαψvαψuα(u,v,w)), (4.12)

    for (u,v,w)[a,k]×[b,m]×[c,n].

    Similarly we have

    g(u,v,w)A(g(u,v,w))=18B(3αgψwαψvαψuα(u,v,w)), (4.13)

    for (u,v,w)[a,k]×[b,m]×[c,n].

    Now multiplying (4.12) and (4.13) by g(u,v,w) and f(u,v,w) respectively, adding them and Integrating over [a,k]×[b,m]×[c,n] we have

    kambnc[f(u,v,w)g(u,v,w)12[g(u,v,w)A(f(u,v,w))g(u,v,w)A(f(u,v,w))]]dwdvdu=116kambnc[g(u,v,w)B(3αfψwαψvαψuα(u,v,w))+f(u,v,w)B(3αgψwαψvαψuα(u,v,w))]. (4.14)

    From the properties of modulus we have

    |B(3αfψwαψvαψuα(u,v,w))|kambncψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21×(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3CDα;ψΘf, (4.15)
    |B(3αgψwαψvαψuα(u,v,w))|kambncψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21×(ψ(n)ψ(t))α313αgψtαψsαψrα(r,s,t)dtdsdr(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3CDα;ψΘg. (4.16)

    Now by substituting the values from equation (4.15) and (4.16) in (4.14) we get the required inequality (4.3).

    Theorem 4.2 Let f,g, 3αfψtαψsαψrα and 3αgψtαψsαψrα be as in Theorem 4.1. Then

    |kambnc[f(u,v,w)g(u,v,w)[A(f(u,v,w))g(u,v,w)A(g(u,v,w))f(u,v,w)A(f(u,v,w))A(g(u,v,w))|dwdvdu164{(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3}2CDα;ψΘfCDα;ψΘg, (4.17)

    for (r,s,t)[a,k]×[b,m]×[c,n] and A,B are as given in (4.1),(4.2).

    Proof. Multiplying left hand and right hand side of equation (4.12) and (4.13) we have

    f(u,v,w)g(u,v,w)[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))A(f(u,v,w))A(g(u,v,w))]=164B(3αfψwαψvαψuα(u,v,w))B(3αgψwαψvαψuα(u,v,w)). (4.18)

    Integrating over [a,k]×[b,m]×[c,n] and from the properties of modulus we have

    |kambnc[f(u,v,w)g(u,v,w)[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))A(f(u,v,w))A(g(u,v,w))]]|dwdvdu164kambnc|B(3αfψwαψvαψuα(u,v,w))B(3αfψwαψvαψuα(u,v,w))|dwdvdu. (4.19)

    Using (4.15) and (4.16) in (4.19) we get the required inequality (4.17).

    Remark: If we put different values for ψ(x) as x,lnx,xσthen it reduces to various types of fractional Čebyšev inequalities such as Riemann Liouville fractional, Hadmard Fractional and Erdelyi-Kober fractional inequalities respectively.

    In this paper, we studied Čebyšev like inequalities. We proved some new ψ Caputo fractional Čebyšev type inequalities involving functions of two and three variables.

    All authors declare no conflict of interest in this paper.



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