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
[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 y∈C/(−∞,0]; α,β,γ∈C, ℜ(α)≥0, ℜ(β)≥−1, ℜ(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 ℜ(ρ)>0 are defined by
Iρa+h(u)=1Γ(ρ)∫ya(y−u)ρ−1h(u)du,a<y | (1.5) |
Iρbh(u)=1Γ(ρ)∫by(u−y)ρ−1h(u)du,y<b. | (1.6) |
Riemann-Liouville fractional differentials operators (RLDO) [12,19] for ℜ(ρ)>0; n=[ℜ(n)−1]
Dρa+h(u)=(d/dy)nIn−ρa+h(y) | (1.7) |
Dρbh(u)=(−d/dy)nIn−ρbh(y). | (1.8) |
Srivastava and Singh [9] defined the following fractional integral operator for α1,β1,r∈C, ℜ(α1)>0, ℜ(β1)≥−1 by
h(y)=def∫y0(y−t)β1Jα1β1(r(y−t)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,γ∈C, ℜ(α1)>max{0,ℜ(k)−1}; min{ℜ(β1),ℜ(k)}>0
(Er;γ,ka+;α1,β1h)(y)=∫ya(y−t)β1−1Eγ,kα1,β1(r(y−t)α1)h(t)dt. | (1.10) |
Prabhakar fractional integral operators for γ,β1∈C, ℜ(α1)>0 are defined in [21] by
E∗(α1,β1;γ;r)h(y)=∘h(y)=∫ya(y−t)β1−1Eγα1,β1(r(y−t)α1)h(t)dt,a<y, | (1.11) |
E∗(α1,β1;γ;r)h(y)=∫by(t−y)β1−1Eγα1,β1(r(t−y)α1)h(t)dt,y<b. | (1.12) |
Tilahun et al. [22] derived the generalized FIO for ℜ(β1)>0, ℜ(α1)>0 and r,γ∈C as
(Ir;α1,γa+,β1,kh)(y)=∫ya(y−t)β1Jγ,kα1,β1(r(y−t)α1;p)h(t)dt,a<y | (1.13) |
and
(Ir;α1,γa+,β1,kh)(y)=∫by(y−t)β1Jγ,kα1,β1(r(y−t)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0.
Ir;γ,ka+;(αj,βj)mh(y)=m∏j=1∫ya(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)h(u)du,a<y | (1.15) |
Ir;γ,kb;(αj,βj)mh(y)=m∏j=1∫by(u−y)βjJ(αj,βj)mγ,k(r(u−y)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
∫dbdy∫ybh(y,z)dz=∫dbdz∫dzh(y,z)dy. | (1.17) |
Kilbas [24] analyzed the generalized Wright function for ξi,ζj∈R, (i=1,2⋯r),(j=1,2⋯s) and bi,cj∈C as
rψs(y)=∞∑n=0∏ri=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 ℜ(y)>0, ℜ(z)>0 and also in gamma form appearance of beta function is defined as follows
B(y,z)=∫10uy−1(1−u)z−1du=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+n−1),for n≥11,for n=0, γ≠0 | (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=β1−1 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,β1−1)mh(y)=Iβ1a+h(y) | (1.23) |
I0;γ,kb;(α1,β1−1)mh(y)=Iβ1bh(y). | (1.24) |
2). Setting j=m=1, β1=β1−1 in Eq (1.15), we have a fractional integral defined in Eq (1.10) as
Ir;γ,ka+;(α1,β1−1)mh(y)=(Er;γ,ka+;α1,β1h)(y). | (1.25) |
3). Setting j=m=1, k=1, β1=β1−1, in Eqs (1.15) and (1.16), we get the FIO defined in Eqs (1.11) and (1.12) respectively
Ir;γ,1a+;(α1,β1−1)mh(y)=E∗(α1,β1;γ;r)h(y)=∘h(y) | (1.26) |
Ir;γ,1b;(α1,β1−1)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(y−t)β1Jα1β1(r(y−t)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and ℜ(ρ)>0 as
m∏j=1Iρa+[(u−a)βj+αjn](y)=m∏j=1[Γ(βj+αjn+1)Γ(ρ+βj+αjn+1)(y−a)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+[(u−a)β1+α1n](y)=(y−a)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, n∈N as
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)]=(y−u)βj−nJ(αj,βj−n)mγ,k(r(y−u)αj). | (2.1) |
Proof. Let the nth-differential of generalized multi-index Bessel function with power function as
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)], | (2.2) |
using the behavior of (1.4), we take as
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)]=(d/dy)n[(y−u)βj∞∑n=0(γ)kn(−r(y−u)αj)nn!∏mj=1Γ(αjn+βj+1)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)(d/dy)n[(y−u)βj+αjn]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)(d/dy)n[(y−u)(β1+β2+⋯+βm)+(α1n+α2n+⋯+αmn)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)(d/dy)n[(y−u)(β1+α1n)+(β2+α2n)+⋯+(βm+αmn)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)×(d/dy)n[(y−u)(β1+α1n)(y−u)(β2+α2n)⋯(y−u)(βm+αmn)]. | (2.3) |
Using the identity result for simplification of (2.3), we get
(d/dy)nyθ=Γ(θ+1)Γ(θ−n+1)yθ−n,θ≥n | (2.4) |
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)×Γ(β1+α1n+1)Γ(β2+α2n+1)⋯Γ(βm+αmn+1)(y−u)αjn+βj−nΓ(β1+α1n−n+1)Γ(β2+α2n−n+1)⋯Γ(βm+αmn−n+1)=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)∏mj=1Γ(αjn+βj+1)∏mj=1Γ(αjn+βj−n+1)(y−u)αjn+βj−n=(y−u)βj−n∞∑n=0(γ)kn(−r(y−u)αj)nn!∏mj=1Γ(αjn+βj−n+1)=(y−u)βj−nJ(αj,βj−n)mγ,k(r(y−u)αj). | (2.5) |
Corollary 2.1. Suppose that αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, n∈N then theorem 2.1 can be expressed as
(d/dy)n[(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)]=1Γ(γ)(y−u)βj−n2ψm+1{(γ,k)(βj+1,αj)(βj−n+1,αj)(βj+1,αj)|mj=1|r(y−u)αj}. | (2.6) |
Corollary 2.2. Suppose that r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, n∈N and setting that βj=βj−1, (r(y−u)αj)=(−r(y−u)αj) in theorem 2.1 we see that
(d/dy)n[(y−u)βj−1J(αj,βj−1)mγ,k(r(y−u)αj)]=(y−u)βj−1−nE(αj,βj−n)mγ,k(r(y−u)αj), | (2.7) |
where E(αj,βj−n)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and y>a, a∈ℜ+(0,∞), ℜ(ρ)>0 as
m∏j=1Iρa+[(u−a)βjJ(αj,βj)mγ,k(r(u−a)αj)](y)=m∏j=1(y−a)βj+ρJ(αj,βj+ρ)mγ,k(r(y−a)αj). | (2.8) |
Proof. Let (RLIO) with (Gm-Bf) is defined in Eq (1.4), we have
m∏j=1Iρa+[(u−a)βjJ(αj,βj)mγ,k(r(u−a)αj)](y)=m∏j=1Iρa+[(u−a)βj∞∑n=0(γ)kn(−r(u−a)αj)nn!∏mj=1Γ(αjn+βj+1)]=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)m∏j=1Iρa+[(u−a)βj+αjn]. | (2.9) |
By using Lemma 1.1 in Eq (2.9) then we attain the equation as
m∏j=1Iρa+[(u−a)βjJ(αj,βj)mγ,k(r(u−a)αj)](y)=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)m∏j=1Γ(αjn+βj+1)(y−a)αjn+ρ+βjΓ(αjn+βj+ρ+1)=m∏j=1(y−a)ρ+βj∞∑n=0(γ)kn(−r(y−a)αj)nn!∏mj=1Γ(αjn+βj+ρ+1)=m∏j=1(y−a)βj+ρJ(αj,βj+ρ)mγ,k(r(y−a)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and y>a, a∈ℜ+(0,∞), ℜ(ρ)>0 as
m∏j=1Iρb[(b−u)βjJ(αj,βj)mγ,k(r(b−u)αj)](y)=m∏j=1(b−y)βj+ρJ(αj,βj+ρ)mγ,k(r(b−y)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and y>a, a∈ℜ+(0,∞), ℜ(ρ)>0 as
m∏j=1Dρa+[(u−a)βjJ(αj,βj)mγ,k(r(u−a)αj)](y)=m∏j=1(y−a)βj−ρJ(αj,βj−ρ)mγ,k(r(y−a)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 and y>a, a∈ℜ+(0,∞), ℜ(ρ)>0 as
m∏j=1Dρb[(b−u)βjJ(αj,βj)mγ,k(r(b−u)αj)](y)=m∏j=1(b−y)βj−ρJ(αj,βj−ρ)mγ,k(r(b−y)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, when h(u)=(u−a)ϑ+ρσ−1, then
[Ir;γ,ka+;(αj,βj)m(u−a)ϑ+ρσ−1](y)=m∏j=1(y−a)ϑ+ρσ+βjΓ(ϑ+ρσ)J(αj,βj+ϑ+ρσ)mγ,k(r(y−a)αj). | (3.1) |
Proof. (FIO) defined in Eq (1.15) with Eq (1.4), we obtain as
[Ir;γ,ka+;(αj,βj)m(u−a)ϑ+ρσ−1](y)=m∏j=1∫ya(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)(u−a)ϑ+ρσ−1du=∞∑n=0(γ)kn(−r)nn!∏mj=1Γ(αjn+βj+1)m∏j=1∫ya(u−a)ϑ+ρσ−1(y−u)βj+αjndu=∞∑n=0(γ)kn(−r)nn!1∏mj=1Γ(αjn+βj+1)m∏j=1∫ya(u−a)ϑ+ρσ−1(y−u)βj+αjndu | (3.2) |
Substituting u=y−z(u−a) and using the definition of beta function and the following relation in Eq (3.2), we get
Iλa+[(y−a)u−1](y)=Γ(u)Γ(λ+u)(y−a)λ+u−1. |
[Ir;γ,ka+;(αj,βj)m(u−a)ϑ+ρσ](y)=m∏j=1∞∑n=0(γ)kn(−r)nn!Γ(ϑ+ρσ)∏mj=1Γ(ϑ+ρσ+αjn+βj+1)(y−a)ϑ+ρσ+βj+αjn=m∏j=1(y−a)ϑ+ρσ+βjΓ(ϑ+ρσ)∞∑n=0(γ)kn(−r(y−a)αj)nn!∏mj=1Γ(ϑ+ρσ+αjn+βj+1)=m∏j=1(y−a)ϑ+ρσ+βjΓ(ϑ+ρσ)J(αj,βj+ϑ+ρσ)mγ,k(r(y−a)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, when h(u)=(a−u)ϑ+ρσ, then
[Ir;γ,kb;(αj,βj)m(b−u)ϑ+ρσ](y)=m∏j=1(b−y)ϑ+ρσ+βj+1Γ(ϑ+ρσ+1)J(αj,βj+ϑ+ρσ+1)mγ,k(r(b−y)αj). | (3.4) |
Theorem 3.2. Consider the composition of Riemann-Liouville fractional integral operator with (FIO) for r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, ℜ(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(y−u)ρ−1[Ir;γ,ka+;(αj,βj)mh](u)du=1Γ(ρ)∫ya(y−u)ρ−1[m∏j=1∫ua(u−t)βjJ(αj,βj)mγ,k(r(u−t)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)m∏j=1∫yt(y−u)ρ−1(u−t)βjJ(αj,βj)mγ,k(r(u−t)αj)du]×h(t)dt. | (3.7) |
Setting u−t=η, we have
{Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=∫ya[m∏j=11Γ(ρ)∫y−t0(y−t−η)ρ−1ηβjJ(αj,βj)mγ,k(r(η)αj)dη]×h(t)dt=∫yam∏j=1Iρ0+[ηβjJ(αj,βj)mγ,k(r(η)αj)](y−t)×h(t)dt. | (3.8) |
applying theorem 2.2, we see
{Iρa+[Ir;γ,ka+;(αj,βj)mh]}(y)=m∏j=1∫ya[(y−t)βj+ρJ(αj,βj+ρ)mγ,k(r(y−t)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)=m∏j=1∫ya(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)[Iρa+h](u)du=m∏j=1∫ya(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)[1Γ(ρ)∫ua(u−t)ρ−1h(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)=m∏j=1∫ya[1Γ(ρ)∫yt(y−u)βjJ(αj,βj)mγ,k(r(y−u)αj)(u−t)ρ−1du]h(t)dt. | (3.11) |
Setting y−u=x and using theorem (2.2) then
{Ir;γ,ka+;(αj,βj)m[Iρa+h]}(y)=m∏j=1∫ya[1Γ(ρ)∫0y−t(x)βjJ(αj,βj)mγ,k(r(x)αj)(y−x−t)ρ−1(−dx)]h(t)dt=m∏j=1∫ya[1Γ(ρ)∫y−t0(x)βjJ(αj,βj)mγ,k(r(x)αj)(y−t−x)ρ−1dx]h(t)dt=m∏j=1∫yaIρ0+[(x)βjJ(αj,βj)mγ,k(r(x)αj)](y−t)×h(t)dt=m∏j=1∫ya(y−t)βj+ρJ(αj,βj+ρ)mγ,k(r(y−t)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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, ℜ(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,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0, ℜ(ρ)>0 then
{Dρa+[Ir;γ,ka+;(αj,βj)mh]}(y)={Ir;γ,ka+;(αj,βj−ρ)m}(y)={Ir;γ,ka+;(αj,βj)m[Dρa+h]}(y). | (3.14) |
Theorem 3.3. If r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 then
||Ir;γ,ka+;(αj,βj)mh||1≤A||h||1, | (3.15) |
where
A=m∏j=1∞∑n=0|(y−a)ℜ(βj)+1||(γ)kn||(−r(y−a)ℜ(αj))n|n!|∏mj=1Γ(αjn+βj+1)|ℜ(αj)n+ℜ(βj)+1. | (3.16) |
Proof. Let Vn be the nth-terms of (3.16); we have
|Vn+1Vn|=m∏j=1|(γ)kn+k|(γ)kn|n!(n+1)!|∏mj=1Γ(αjn+βj+1)∏mj=1Γ(αjn+αj+βj+1)|×ℜ(αj)n+ℜ(βj)+1ℜ(αj)n+ℜ(αj)+ℜ(βj)+1|(−1)n+1(−1)n||r(y−a)ℜ(αj)|≈m∏j=1(kn)k|−r(y−a)ℜ(αj)|(n+1)|∏mj=1|(αj)|n(αj)|,asn→∞ | (3.17) |
hence, |Vn+1Vn|→0 as n→∞, and k<ℜ(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|m∏j=1∫ua(u−τ)βjJ(αj,βj)mγ,k(r(u−τ)αj)h(τ)dτ|du≤∫ya[m∏j=1∫yτ(u−τ)βj|J(αj,βj)mγ,k(r(u−τ)αj)|du]|h(τ)|dτ. | (3.18) |
Putting u−τ=λ,u=y⇒λ=y−τ;u=τ⇒λ=0,du=dλ in Eq (3.18), we get
||Ir;γ,ka+;(αj,βj)mh||1≤∫ya[m∏j=1∫y−τ0λℜ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλ]|h(τ)|dτ≤∫ya[m∏j=1∫y−a0λℜ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλ]|h(τ)|dτ, |
where
m∏j=1∫y−a0λℜ(βj)|J(αj,βj)mγ,k(r(λ)αj)|dλ≤m∏j=1∫y−a0λℜ(βj)|∞∑n=0(γ)kn|(−r)n(λ)αjb|n!∏mj=1Γ(αjn+βj+1)|dλ≤∞∑n=0|(γ)kn||(−r)n|n!|∏mj=1Γ(αjn+βj+1)|m∏j=1∫y−a0λℜ(αj)n+ℜ(βj)dλ≤m∏j=1∞∑n=0|(y−a)ℜ(βj)+1||(γ)kn||(−r(y−a)ℜ(αj))n|n!|∏mj=1Γ(αjn+βj+1)|ℜ(αj)n+ℜ(βj)+1=A. | (3.19) |
Therefore,
||Ir;γ,ka+;(αj,βj)mh||1≤∫yaA|h(τ)|dτ≤A||h||1⇒||Ir;γ,ka+;(αj,βj)mh||1≤A||h||1 |
Corollary 3.4. If r,αj,βj,γ∈C, (j=1,2⋯m), ℜ(αj)>0, ℜ(βj)>−1, ∑mj=1ℜ(α)j>max{0;ℜ(k)−1}, k>0 then
||Ir;γ,kb;(αj,βj)mh||1≤A||h||1, | (3.20) |
where
A=m∏j=1(b−y)ℜ(βj)+1∞∑n=0|(γ)kn||(−r(b−y)ℜ(αj))n|n!|∏mj=1Γ(αjn+βj+1)|ℜ(αj)n+ℜ(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.
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