Research article

Estimation of generalized fractional integral operators with nonsingular function as a kernel

  • Received: 23 November 2020 Accepted: 01 February 2021 Published: 22 February 2021
  • MSC : 11S80, 26A33, 33C10, 33C20

  • Bessel function has a significant role in fractional calculus having immense applications in physical and theoretical approach. Present work aims to introduce fractional integral operators in which generalized multi-index Bessel function as a kernel, and develop some important special cases which are connected with fractional operators in fractional calculus. Here, we construct important links to familiar findings from some individual occurrence with our key outcomes.

    Citation: Iqra Nayab, Shahid Mubeen, Rana Safdar Ali, Gauhar Rahman, Abdel-Haleem Abdel-Aty, Emad E. Mahmoud, Kottakkaran Sooppy Nisar. Estimation of generalized fractional integral operators with nonsingular function as a kernel[J]. AIMS Mathematics, 2021, 6(5): 4492-4506. doi: 10.3934/math.2021266

    Related Papers:

    [1] Shahid Mubeen, Rana Safdar Ali, Iqra Nayab, Gauhar Rahman, Thabet Abdeljawad, Kottakkaran Sooppy Nisar . Integral transforms of an extended generalized multi-index Bessel function. AIMS Mathematics, 2020, 5(6): 7531-7547. doi: 10.3934/math.2020482
    [2] Shahid Mubeen, Rana Safdar Ali, Iqra Nayab, Gauhar Rahman, Kottakkaran Sooppy Nisar, Dumitru Baleanu . Some generalized fractional integral inequalities with nonsingular function as a kernel. AIMS Mathematics, 2021, 6(4): 3352-3377. doi: 10.3934/math.2021201
    [3] Saima Naheed, Shahid Mubeen, Thabet Abdeljawad . Fractional calculus of generalized Lommel-Wright function and its extended Beta transform. AIMS Mathematics, 2021, 6(8): 8276-8293. doi: 10.3934/math.2021479
    [4] Rana Safdar Ali, Saba Batool, Shahid Mubeen, Asad Ali, Gauhar Rahman, Muhammad Samraiz, Kottakkaran Sooppy Nisar, Roshan Noor Mohamed . On generalized fractional integral operator associated with generalized Bessel-Maitland function. AIMS Mathematics, 2022, 7(2): 3027-3046. doi: 10.3934/math.2022167
    [5] D. L. Suthar, D. Baleanu, S. D. Purohit, F. Uçar . Certain k-fractional calculus operators and image formulas of k-Struve function. AIMS Mathematics, 2020, 5(3): 1706-1719. doi: 10.3934/math.2020115
    [6] Sabila Ali, Shahid Mubeen, Rana Safdar Ali, Gauhar Rahman, Ahmed Morsy, Kottakkaran Sooppy Nisar, Sunil Dutt Purohit, M. Zakarya . Dynamical significance of generalized fractional integral inequalities via convexity. AIMS Mathematics, 2021, 6(9): 9705-9730. doi: 10.3934/math.2021565
    [7] Sobia Rafeeq, Sabir Hussain, Jongsuk Ro . On fractional Bullen-type inequalities with applications. AIMS Mathematics, 2024, 9(9): 24590-24609. doi: 10.3934/math.20241198
    [8] Gauhar Rahman, Shahid Mubeen, Kottakkaran Sooppy Nisar . On generalized $\mathtt{k}$-fractional derivative operator. AIMS Mathematics, 2020, 5(3): 1936-1945. doi: 10.3934/math.2020129
    [9] D. L. Suthar, A. M. Khan, A. Alaria, S. D. Purohit, J. Singh . Extended Bessel-Maitland function and its properties pertaining to integral transforms and fractional calculus. AIMS Mathematics, 2020, 5(2): 1400-1410. doi: 10.3934/math.2020096
    [10] Georgia Irina Oros, Gheorghe Oros, Daniela Andrada Bardac-Vlada . Certain geometric properties of the fractional integral of the Bessel function of the first kind. AIMS Mathematics, 2024, 9(3): 7095-7110. doi: 10.3934/math.2024346
  • Bessel function has a significant role in fractional calculus having immense applications in physical and theoretical approach. Present work aims to introduce fractional integral operators in which generalized multi-index Bessel function as a kernel, and develop some important special cases which are connected with fractional operators in fractional calculus. Here, we construct important links to familiar findings from some individual occurrence with our key outcomes.



    The Bessel function has immense applications in the field of engineering, physics, and applied mathematics. Baricz [1], Generalized Bessel functions of the first kind in (2010), which discussed the geometric properties, functional inequalities of generalized Bessel function and also the inequalities involving circular and hyperbolic functions to Bessel function and modified Bessel functions. Tumakov [2] investigated the numerical algorithms for fast computations of the Bessel functions of an integer order with the required accuracy. Choi and Agarwal[3], Abramowitz and Stegun [4], Heymans and Podlubny [5], Watson [6] and Purohit et al. [7] studied the following Bessel function (Bf) defined by

    Jα(y)=n=0(1)n(y/2)α+2nn!Γ(α+n+1). (1.1)

    Edward Maitland Wright [8] introduced the generalized form of Bessel function with the name of Bessel-Maitland function (B-M1)

    Jαβ(y)=n=0(y)nn!Γ(αn+β+1). (1.2)

    The properties of generalised Bessel function can be found in the work of Srivastava and Singh [9]. Suthar et al. [10,11] discussed the various properties of Bessel-Maitland function. Ali et al. [12] established some fractional operators with the generalized Bessel-Maitland function.

    Waseem et al. [13] established the generalized Bessel-Maitland function (B-M11) and discuss the numerous integral formulas for yC/(,0]; α,β,γC, (α)0, (β)1, (γ)0, k(0,1)N defined by

    Jα,γβ,k(y)=n=0(γ)kn(y)nn!Γ(αn+β+1). (1.3)

    Suthar et al. [14] studied the following generalized multi-index Bessel function (Gm-Bf) defined by

    J(αj,βj)mγ,k(y)=n=0(γ)kn(y)nn!mj=1Γ(αjn+βj+1). (1.4)

    Recently, fractional integrals are widely applied in different branches of mathematics, physics, engineering due to their wide applications (see e.g., [5,15,16,17,18]).

    Riemann-Liouville fractional integral operators for (ρ)>0 are defined by

    Iρa+h(u)=1Γ(ρ)ya(yu)ρ1h(u)du,a<y (1.5)
    Iρbh(u)=1Γ(ρ)by(uy)ρ1h(u)du,y<b. (1.6)

    Riemann-Liouville fractional differentials operators (RLDO) [12,19] for (ρ)>0; n=[(n)1]

    Dρa+h(u)=(d/dy)nInρa+h(y) (1.7)
    Dρbh(u)=(d/dy)nInρbh(y). (1.8)

    Srivastava and Singh [9] defined the following fractional integral operator for α1,β1,rC, (α1)>0, (β1)1 by

    h(y)=defy0(yt)β1Jα1β1(r(yt)α1)h(u)du. (1.9)

    Srivastava and Tomovski [20] established the fractional integral operator (FIO) having Mittag-Leffler function as a kernel, discuss its boundedness and convergence of integral and also derive the product of FIO with Riemann-Liouville fractional integral operator defined for r,γC, (α1)>max{0,(k)1}; min{(β1),(k)}>0

    (Er;γ,ka+;α1,β1h)(y)=ya(yt)β11Eγ,kα1,β1(r(yt)α1)h(t)dt. (1.10)

    Prabhakar fractional integral operators for γ,β1C, (α1)>0 are defined in [21] by

    E(α1,β1;γ;r)h(y)=h(y)=ya(yt)β11Eγα1,β1(r(yt)α1)h(t)dt,a<y, (1.11)
    E(α1,β1;γ;r)h(y)=by(ty)β11Eγα1,β1(r(ty)α1)h(t)dt,y<b. (1.12)

    Tilahun et al. [22] derived the generalized FIO for (β1)>0, (α1)>0 and r,γC as

    (Ir;α1,γa+,β1,kh)(y)=ya(yt)β1Jγ,kα1,β1(r(yt)α1;p)h(t)dt,a<y (1.13)

    and

    (Ir;α1,γa+,β1,kh)(y)=by(yt)β1Jγ,kα1,β1(r(yt)α1;p)h(t)dt, (1.14)

    where y<b.

    Definition 1.1. (FIO)Fractional integral operator with generalized multi-index Bessel function (Gm-Bf) kernel for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0.

    Ir;γ,ka+;(αj,βj)mh(y)=mj=1ya(yu)βjJ(αj,βj)mγ,k(r(yu)αj)h(u)du,a<y (1.15)
    Ir;γ,kb;(αj,βj)mh(y)=mj=1by(uy)βjJ(αj,βj)mγ,k(r(uy)αj)h(u)du,y<b. (1.16)

    Dirichlet formula (Fubini's theorem) Samko et al. in [23] and Kelelaw et al. in [22] is defined as

    dbdyybh(y,z)dz=dbdzdzh(y,z)dy. (1.17)

    Kilbas [24] analyzed the generalized Wright function for ξi,ζjR, (i=1,2r),(j=1,2s) and bi,cjC as

    rψs(y)=n=0ri=1Γ(bi+ξin)sj=1Γ(cj+ζjn)ynn!=rψs[(bi,ξi)1,r(cj,ζj)1,s|y]. (1.18)

    The integral representation of beta function [25,26] for (y)>0, (z)>0 and also in gamma form appearance of beta function is defined as follows

    B(y,z)=10uy1(1u)z1du=Γ(y)Γ(z)Γ(y+z). (1.19)

    Pochhammer symbol and its properties can be found [25,26,27] as

    (γ)n={γ(γ+1)(γ+2)(γ+n1),for n11,for n=0γ0 (1.20)
    =Γ(γ+n)Γ(γ) and (γ)kn=Γ(γ+kn)Γ(γ) (k>0). (1.21)

    The space of Lebesgue measurable for complex and real valued functions defined by Kelelaw et al. [22] as follows

    L(a,y)={h:||h||1:=ya|h(u)|du<}. (1.22)

    The following some conditions of fractional integral operators can be obtained by setting the integrals according to requirements:

    1). Setting r=0, j=1=m and β1=β11 in (FIO) defined in Eqs (1.15) and (1.16), we get the Riemann-Liouville fractional integral operator defined in [28] as

    I0;γ,ka+;(α1,β11)mh(y)=Iβ1a+h(y) (1.23)
    I0;γ,kb;(α1,β11)mh(y)=Iβ1bh(y). (1.24)

    2). Setting j=m=1, β1=β11 in Eq (1.15), we have a fractional integral defined in Eq (1.10) as

    Ir;γ,ka+;(α1,β11)mh(y)=(Er;γ,ka+;α1,β1h)(y). (1.25)

    3). Setting j=m=1, k=1, β1=β11, in Eqs (1.15) and (1.16), we get the FIO defined in Eqs (1.11) and (1.12) respectively

    Ir;γ,1a+;(α1,β11)mh(y)=E(α1,β1;γ;r)h(y)=h(y) (1.26)
    Ir;γ,1b;(α1,β11)mh(y)=E(α1,β1;γ;r)h(y). (1.27)

    4). Setting j=m=1, k=0 and limits from [0,y] in Eq (1.15), we get a fractional integral defined in Eq (1.9) as

    Ir;γ,0a+;(α1,β1)mh(y)=y0(yt)β1Jα1β1(r(yt)α1)h(u)du=h(y). (1.28)

    5). Setting j=m=1 in Eq (1.15) then, we get the generalized fractional integral operator defined in Eq (1.13) as

    Ir;γ,ka+;(α1,β1)h(y)=(Ir;α1,γa+,β1,kh)(y). (1.29)

    Lemma 1.1. Consider the Riemann-Liouville fractional integral operator with multi-index power function for αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0 and (ρ)>0 as

    mj=1Iρa+[(ua)βj+αjn](y)=mj=1[Γ(βj+αjn+1)Γ(ρ+βj+αjn+1)(ya)ρ+βj+αjn]. (1.30)

    Remark 1.1. Setting j=m=1 in lemma 1.1 then we obtain the result that defined the Mathai Haubold [29] and Kelelaw et al. [22] as

    Iρa+[(ua)β1+α1n](y)=(ya)ρ+β1+α1nΓ(β1+α1n+1)Γ(ρ+β1+α1b+1). (1.31)

    The preliminary results for generalized multi-index Bessel function which used to proceed the new results is given in this section. We calculate the nth-differential and also develop some results with the coordination of Riemann-Liouville fractional operator and (Gm-Bf).

    Theorem 2.1. Consider the nth-differential of generalized multi-index Bessel function with power function for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0, nN as

    (d/dy)n[(yu)βjJ(αj,βj)mγ,k(r(yu)αj)]=(yu)βjnJ(αj,βjn)mγ,k(r(yu)αj). (2.1)

    Proof. Let the nth-differential of generalized multi-index Bessel function with power function as

    (d/dy)n[(yu)βjJ(αj,βj)mγ,k(r(yu)αj)], (2.2)

    using the behavior of (1.4), we take as

    (d/dy)n[(yu)βjJ(αj,βj)mγ,k(r(yu)αj)]=(d/dy)n[(yu)βjn=0(γ)kn(r(yu)αj)nn!mj=1Γ(αjn+βj+1)]=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)(d/dy)n[(yu)βj+αjn]=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)(d/dy)n[(yu)(β1+β2++βm)+(α1n+α2n++αmn)]=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)(d/dy)n[(yu)(β1+α1n)+(β2+α2n)++(βm+αmn)]=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)×(d/dy)n[(yu)(β1+α1n)(yu)(β2+α2n)(yu)(βm+αmn)]. (2.3)

    Using the identity result for simplification of (2.3), we get

    (d/dy)nyθ=Γ(θ+1)Γ(θn+1)yθn,θn (2.4)
    (d/dy)n[(yu)βjJ(αj,βj)mγ,k(r(yu)αj)]=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)×Γ(β1+α1n+1)Γ(β2+α2n+1)Γ(βm+αmn+1)(yu)αjn+βjnΓ(β1+α1nn+1)Γ(β2+α2nn+1)Γ(βm+αmnn+1)=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)mj=1Γ(αjn+βj+1)mj=1Γ(αjn+βjn+1)(yu)αjn+βjn=(yu)βjnn=0(γ)kn(r(yu)αj)nn!mj=1Γ(αjn+βjn+1)=(yu)βjnJ(αj,βjn)mγ,k(r(yu)αj). (2.5)

    Corollary 2.1. Suppose that αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0, nN then theorem 2.1 can be expressed as

    (d/dy)n[(yu)βjJ(αj,βj)mγ,k(r(yu)αj)]=1Γ(γ)(yu)βjn2ψm+1{(γ,k)(βj+1,αj)(βjn+1,αj)(βj+1,αj)|mj=1|r(yu)αj}. (2.6)

    Corollary 2.2. Suppose that r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0, nN and setting that βj=βj1, (r(yu)αj)=(r(yu)αj) in theorem 2.1 we see that

    (d/dy)n[(yu)βj1J(αj,βj1)mγ,k(r(yu)αj)]=(yu)βj1nE(αj,βjn)mγ,k(r(yu)αj), (2.7)

    where E(αj,βjn)mγ,k(.) is generalized multi-index Mittag-Leffler function.

    Theorem 2.2. Consider the Riemann-Liouville fractional integral operator defined in Eq (1.5) with generalized multi-index Bessel function for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0 and y>a, a+(0,), (ρ)>0 as

    mj=1Iρa+[(ua)βjJ(αj,βj)mγ,k(r(ua)αj)](y)=mj=1(ya)βj+ρJ(αj,βj+ρ)mγ,k(r(ya)αj). (2.8)

    Proof. Let (RLIO) with (Gm-Bf) is defined in Eq (1.4), we have

    mj=1Iρa+[(ua)βjJ(αj,βj)mγ,k(r(ua)αj)](y)=mj=1Iρa+[(ua)βjn=0(γ)kn(r(ua)αj)nn!mj=1Γ(αjn+βj+1)]=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)mj=1Iρa+[(ua)βj+αjn]. (2.9)

    By using Lemma 1.1 in Eq (2.9) then we attain the equation as

    mj=1Iρa+[(ua)βjJ(αj,βj)mγ,k(r(ua)αj)](y)=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)mj=1Γ(αjn+βj+1)(ya)αjn+ρ+βjΓ(αjn+βj+ρ+1)=mj=1(ya)ρ+βjn=0(γ)kn(r(ya)αj)nn!mj=1Γ(αjn+βj+ρ+1)=mj=1(ya)βj+ρJ(αj,βj+ρ)mγ,k(r(ya)αj). (2.10)

    Corollary 2.3. Consider the right-sided Riemann-Liouville fractional integral operator with generalized multi-index Bessel function for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0 and y>a, a+(0,), (ρ)>0 as

    mj=1Iρb[(bu)βjJ(αj,βj)mγ,k(r(bu)αj)](y)=mj=1(by)βj+ρJ(αj,βj+ρ)mγ,k(r(by)αj). (2.11)

    Corollary 2.4. Consider the Riemann-Liouville fractional differential operator defined in Eq (1.7) with generalized multi-index Bessel function for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0 and y>a, a+(0,), (ρ)>0 as

    mj=1Dρa+[(ua)βjJ(αj,βj)mγ,k(r(ua)αj)](y)=mj=1(ya)βjρJ(αj,βjρ)mγ,k(r(ya)αj). (2.12)

    Corollary 2.5. Consider the right-sided Riemann-Liouville fractional differential operator with generalized multi-index Bessel function for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0 and y>a, a+(0,), (ρ)>0 as

    mj=1Dρb[(bu)βjJ(αj,βj)mγ,k(r(bu)αj)](y)=mj=1(by)βjρJ(αj,βjρ)mγ,k(r(by)αj). (2.13)

    In this section, we discuss some properties of the generalized fractional integrals with non singular function as a kernel.

    Theorem 3.1. Fractional integral operator having generalized multi-index Bessel function (Gm-Bf) kernel for ϑ,ρ,σ,r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0, when h(u)=(ua)ϑ+ρσ1, then

    [Ir;γ,ka+;(αj,βj)m(ua)ϑ+ρσ1](y)=mj=1(ya)ϑ+ρσ+βjΓ(ϑ+ρσ)J(αj,βj+ϑ+ρσ)mγ,k(r(ya)αj). (3.1)

    Proof. (FIO) defined in Eq (1.15) with Eq (1.4), we obtain as

    [Ir;γ,ka+;(αj,βj)m(ua)ϑ+ρσ1](y)=mj=1ya(yu)βjJ(αj,βj)mγ,k(r(yu)αj)(ua)ϑ+ρσ1du=n=0(γ)kn(r)nn!mj=1Γ(αjn+βj+1)mj=1ya(ua)ϑ+ρσ1(yu)βj+αjndu=n=0(γ)kn(r)nn!1mj=1Γ(αjn+βj+1)mj=1ya(ua)ϑ+ρσ1(yu)βj+αjndu (3.2)

    Substituting u=yz(ua) and using the definition of beta function and the following relation in Eq (3.2), we get

    Iλa+[(ya)u1](y)=Γ(u)Γ(λ+u)(ya)λ+u1.
    [Ir;γ,ka+;(αj,βj)m(ua)ϑ+ρσ](y)=mj=1n=0(γ)kn(r)nn!Γ(ϑ+ρσ)mj=1Γ(ϑ+ρσ+αjn+βj+1)(ya)ϑ+ρσ+βj+αjn=mj=1(ya)ϑ+ρσ+βjΓ(ϑ+ρσ)n=0(γ)kn(r(ya)αj)nn!mj=1Γ(ϑ+ρσ+αjn+βj+1)=mj=1(ya)ϑ+ρσ+βjΓ(ϑ+ρσ)J(αj,βj+ϑ+ρσ)mγ,k(r(ya)αj). (3.3)

    Corollary 3.1. Fractional integral operator having generalized multi-index Bessel function (Gm-Bf) kernel for ϑ,ρ,σ,r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0, when h(u)=(au)ϑ+ρσ, then

    [Ir;γ,kb;(αj,βj)m(bu)ϑ+ρσ](y)=mj=1(by)ϑ+ρσ+βj+1Γ(ϑ+ρσ+1)J(αj,βj+ϑ+ρσ+1)mγ,k(r(by)αj). (3.4)

    Theorem 3.2. Consider the composition of Riemann-Liouville fractional integral operator with (FIO) for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0, (ρ)>0 then

    {Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)={Ir;γ,ka+;(αj,βj+ρ)m}(y)={Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y). (3.5)

    Proof. Let the left side of Eq (3.5), we seen as

    {Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=1Γ(ρ)ya(yu)ρ1[Ir;γ,ka+;(αj,βj)mh](u)du=1Γ(ρ)ya(yu)ρ1[mj=1ua(ut)βjJ(αj,βj)mγ,k(r(ut)αj)h(t)dt]du. (3.6)

    By interchanging the order of integrations and using the Eq (1.17) in Eq (3.6), we attain as

    {Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=ya[1Γ(ρ)mj=1yt(yu)ρ1(ut)βjJ(αj,βj)mγ,k(r(ut)αj)du]×h(t)dt. (3.7)

    Setting ut=η, we have

    {Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=ya[mj=11Γ(ρ)yt0(ytη)ρ1ηβjJ(αj,βj)mγ,k(r(η)αj)dη]×h(t)dt=yamj=1Iρ0+[ηβjJ(αj,βj)mγ,k(r(η)αj)](yt)×h(t)dt. (3.8)

    applying theorem 2.2, we see

    {Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=mj=1ya[(yt)βj+ρJ(αj,βj+ρ)mγ,k(r(yt)αj)]h(t)dt={Ir;γ,ka+;(αj,βj+ρ)m}(y). (3.9)

    We start the right side to determine the second part of (3.5) with (FIO) defined in Eq (1.15) as

    {Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y)=mj=1ya(yu)βjJ(αj,βj)mγ,k(r(yu)αj)[Iρa+h](u)du=mj=1ya(yu)βjJ(αj,βj)mγ,k(r(yu)αj)[1Γ(ρ)ua(ut)ρ1h(t)dt]du. (3.10)

    Interchanging the order of integration and using Eq (1.17), we get

    {Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y)=mj=1ya[1Γ(ρ)yt(yu)βjJ(αj,βj)mγ,k(r(yu)αj)(ut)ρ1du]h(t)dt. (3.11)

    Setting yu=x and using theorem (2.2) then

    {Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y)=mj=1ya[1Γ(ρ)0yt(x)βjJ(αj,βj)mγ,k(r(x)αj)(yxt)ρ1(dx)]h(t)dt=mj=1ya[1Γ(ρ)yt0(x)βjJ(αj,βj)mγ,k(r(x)αj)(ytx)ρ1dx]h(t)dt=mj=1yaIρ0+[(x)βjJ(αj,βj)mγ,k(r(x)αj)](yt)×h(t)dt=mj=1ya(yt)βj+ρJ(αj,βj+ρ)mγ,k(r(yt)αj)h(t)dt={Ir;γ,ka+;(αj,βj+ρ)m}(y). (3.12)

    Thus, we obtain the desired results by combining Eqs (3.9) and (3.12).

    CCorollary 3.2. Composition of right-sided (FIO) with right-sided Riemann-Liouville fractional integral for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0, (ρ)>0 then

    {Iρb[Ir;γ,kb;(αj,βj)mh]}(y)={Ir;γ,kb;(αj,βj+ρ)m}(y)={Ir;γ,kb;(αj,βj)m[Iρbh]}(y). (3.13)

    Corollary 3.3. Consider the composition of Riemann-Liouville fractional differential operator with (FIO) for r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0, (ρ)>0 then

    {Dρa+[Ir;γ,ka+;(αj,βj)mh]}(y)={Ir;γ,ka+;(αj,βjρ)m}(y)={Ir;γ,ka+;(αj,βj)m[Dρa+h]}(y). (3.14)

    Theorem 3.3. If r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0 then

    ||Ir;γ,ka+;(αj,βj)mh||1A||h||1, (3.15)

    where

    A=mj=1n=0|(ya)(βj)+1||(γ)kn||(r(ya)(αj))n|n!|mj=1Γ(αjn+βj+1)|(αj)n+(βj)+1. (3.16)

    Proof. Let Vn be the nth-terms of (3.16); we have

    |Vn+1Vn|=mj=1|(γ)kn+k|(γ)kn|n!(n+1)!|mj=1Γ(αjn+βj+1)mj=1Γ(αjn+αj+βj+1)|×(αj)n+(βj)+1(αj)n+(αj)+(βj)+1|(1)n+1(1)n||r(ya)(αj)|mj=1(kn)k|r(ya)(αj)|(n+1)|mj=1|(αj)|n(αj)|,asn (3.17)

    hence, |Vn+1Vn|0 as n, and k<(αj) which means that right hand side of (3.16) is convergent and finite under the given condition. The condition of boundedness of the integral operator (Ir;γ,ka+;(αj,βj)mh)(y) is discussed in the space of Lebesgue measure L(a,y) of a continuous function, where y>a. Consider the Lebesgue measurable space (1.22) and FIO (1.15), we have

    ||Ir;γ,ka+;(αj,βj)mh||1=ya|Ir;γ,ka+;(αj,βj)mh|du=ya|mj=1ua(uτ)βjJ(αj,βj)mγ,k(r(uτ)αj)h(τ)dτ|duya[mj=1yτ(uτ)βj|J(αj,βj)mγ,k(r(uτ)αj)|du]|h(τ)|dτ. (3.18)

    Putting uτ=λ,u=yλ=yτ;u=τλ=0,du=dλ in Eq (3.18), we get

    ||Ir;γ,ka+;(αj,βj)mh||1ya[mj=1yτ0λ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλ]|h(τ)|dτya[mj=1ya0λ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλ]|h(τ)|dτ,

    where

    mj=1ya0λ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλmj=1ya0λ(βj)|n=0(γ)kn|(r)n(λ)αjb|n!mj=1Γ(αjn+βj+1)|dλn=0|(γ)kn||(r)n|n!|mj=1Γ(αjn+βj+1)|mj=1ya0λ(αj)n+(βj)dλmj=1n=0|(ya)(βj)+1||(γ)kn||(r(ya)(αj))n|n!|mj=1Γ(αjn+βj+1)|(αj)n+(βj)+1=A. (3.19)

    Therefore,

    ||Ir;γ,ka+;(αj,βj)mh||1yaA|h(τ)|dτA||h||1||Ir;γ,ka+;(αj,βj)mh||1A||h||1

    Corollary 3.4. If r,αj,βj,γC, (j=1,2m), (αj)>0, (βj)>1, mj=1(α)j>max{0;(k)1}, k>0 then

    ||Ir;γ,kb;(αj,βj)mh||1A||h||1, (3.20)

    where

    A=mj=1(by)(βj)+1n=0|(γ)kn||(r(by)(αj))n|n!|mj=1Γ(αjn+βj+1)|(αj)n+(βj)+1. (3.21)

    The results we discussed in this paper is creating a chain of fractional operators with kernels, having convergence and boundedness, continuity, symmetric properties and composition with Riemann-Liouville operators [9,12,20,21,22,25,30,31] in fractional calculus. We constructed the fractional operator with generalized multi-index Bessel function as a kernel, and discussed its properties, continuity, and check the behaviour with Riemann-Liouville fractional operators. We analyzed the generalized multi-index Bessel function nth-derivative and integral in the field of fractional calculus.

    Taif University Researchers Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.

    The authors declare that they have no competing interests.



    [1] A. Baricz, Generalized Bessel functions of the first kind, Springer, 2010.
    [2] D. N. Tumakov, The faster methods for computing Bessel functions of the first kind of an integer order with application to graphic processors, Lobachevskii J. Math., 40 (2019), 1725–1738. doi: 10.1134/S1995080219100287
    [3] J. Choi, P. Agarwal, Certain unified integrals involving a product of Bessel functions of first kind, Honam Math. J., 35 (2013), 667–677. doi: 10.5831/HMJ.2013.35.4.667
    [4] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables (Vol. 55), US Government printing office, 1948.
    [5] N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta, 45 (2006), 765–771. doi: 10.1007/s00397-005-0043-5
    [6] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge university press, 1995.
    [7] S. D. Purohit, D. J. Suthar, S. L. Kalla, Marichev-Saigo-Maeda fractional integration operators of the Bassel functions, Le Matematiche, 67 (2012), 21–32.
    [8] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. Lond. Math. Soc., 1 (1935), 286–293.
    [9] T. N. Srivastava, Y. P. Singh, On Maitland's generalised Bessel Function, Can. Math. Bull., 11 (1968), 739–741. doi: 10.4153/CMB-1968-091-5
    [10] D. L. Suthar, H. Amsalu, Certain integrals associated with the generalized Bessel-Maitland function, Applications and Applied Mathematics, 12 (2017), 1002–1016.
    [11] D. L. Suthar, H. Habenom, Integrals involving generalized Bessel-Maitland function, JOSA, 16 (2016), 357.
    [12] R. S. Ali, S. Mubeen, I. Nayab, S. Araci, G. Rahman, K. S. Nisar, Some fractional operators with the generalized Bessel-Maitland function, Discrete Dyn. Nat. Soc., 2020 (2020), 1378457.
    [13] W. A. Khan, K. S. Nisar, J. Choi, An integral formula of the Mellin transform type involving the extended Wright-Bessel function, FJMS, 102 (2017), 2903–2912. doi: 10.17654/MS102112903
    [14] D. L. Suthar, S. D. Purohit, R. K. Parmar, Generalized fractional calculus of the multiindex Bessel function, Math. Nat. Sci., 1 (2017), 26–32. doi: 10.22436/mns.01.01.03
    [15] M. Z. Sarikaya, H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration, Abstr. Appl. Anal., 2012 (2020), 428983.
    [16] B. Ahmad, J. J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl., 2011 (2011), 36. doi: 10.1186/1687-2770-2011-36
    [17] M. U. Awan, S. Talib, Y. M. Chu, M. A. Noor, K. I. Noor, Some new refinements of Hermite-Hadamard-type inequalities involving-Riemann-Liouville fractional integrals and applications, Math. Probl. Eng., 2020 (2020), 3051920.
    [18] Y. S. Liang, Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions, Fract. Calc. Appl. Anal., 21 (2018), 1651–1658. doi: 10.1515/fca-2018-0087
    [19] R. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Differ. Equ., 2009 (2009), 1–47.
    [20] H. M. Srivastava, ˇZ. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag -Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198–210.
    [21] T. R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, J. Math. Comput. Sci., 22 (1971), 266–281.
    [22] K. Tilahun, H. Tadessee, D. L. Suthar, The extended Bessel-Maitland function and integral operators associated with fractional calculus, J. Math., 2020 (2020), 7582063.
    [23] S. G. Samko, A. A. Kilbas, I. O. Marichev, Fractional integrals and derivatives: theory and applications, Yverdon, Switzerland: Gordon and Breach Science Publishers, 1993.
    [24] A. Kilbas, Fractional calculus of the generalized Wright function. Fract. Calc. Appl. Anal., 8 (2005), 113–126.
    [25] S. Mubeen, R. S. Ali, I. Nayab, G. Rahman, T. Abdeljawad, K. S. Nisar, Integral transforms of an extended generalized multi-index Bessel function, AIMS Mathematics, 5 (2020), 7531–7547. doi: 10.3934/math.2020482
    [26] A. Petojevic, A note about the Pochhammer symbol, Mathematica Moravica, 12-1 (2008), 37–42.
    [27] S. Mubeen, R. S. Ali, Fractional operators with generalized Mittag-Leffler k-function, Adv. Differ. Equ., 2019 (2019), 520. doi: 10.1186/s13662-019-2458-9
    [28] R. S. Ali, S. Mubeen, M. M. Ahmad, A class of fractional integral operators with multi-index Mittag-Leffler k-function and Bessel k-function of first kind, J. Math. Comput. Sci., 22 (2020), 266–281. doi: 10.22436/jmcs.022.03.06
    [29] A. M. Mathai, H. J. Haubold, Special functions for applied scientists, New York: Springer Science+ Business Media, 2008.
    [30] T. O. Salim, A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl., 3 (2012), 1–13. doi: 10.1142/9789814355216_0001
    [31] G. Rahman, D. Baleanu, M. A. Qurashi, S. D. Purohit, S. Mubeen, M. Arshad, The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl., 10 (2017), 4244–4253. doi: 10.22436/jnsa.010.08.19
  • This article has been cited by:

    1. Alina Alb Lupaş, Mugur Acu, Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative, 2023, 56, 2391-4661, 10.1515/dema-2022-0249
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2708) PDF downloads(156) Cited by(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog