In this article, a class of uncertain conformable fractional-order delay systems under input saturation is considered. By establishing the Lyapunov boundedness theorem for conformable fractional-order delay systems, some sufficient conditions for robust stability and boundedness of the systems are obtained. Examples are given to illustrate the obtained theory.
Citation: Danhua He, Baizeng Bao, Liguang Xu. Robust stability and boundedness of uncertain conformable fractional-order delay systems under input saturation[J]. AIMS Mathematics, 2023, 8(9): 21123-21137. doi: 10.3934/math.20231076
[1] | 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 |
[2] | 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 |
[3] | Gauhar Rahman, Shahid Mubeen, Kottakkaran Sooppy Nisar . On generalized k-fractional derivative operator. AIMS Mathematics, 2020, 5(3): 1936-1945. doi: 10.3934/math.2020129 |
[4] | Saima Naheed, Shahid Mubeen, Gauhar Rahman, M. R. Alharthi, Kottakkaran Sooppy Nisar . Some new inequalities for the generalized Fox-Wright functions. AIMS Mathematics, 2021, 6(6): 5452-5464. doi: 10.3934/math.2021322 |
[5] | Khaled Mehrez, Abdulaziz Alenazi . Bounds for certain function related to the incomplete Fox-Wright function. AIMS Mathematics, 2024, 9(7): 19070-19088. doi: 10.3934/math.2024929 |
[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] | Muajebah Hidan, Mohamed Akel, Hala Abd-Elmageed, Mohamed Abdalla . Solution of fractional kinetic equations involving extended (k,τ)-Gauss hypergeometric matrix functions. AIMS Mathematics, 2022, 7(8): 14474-14491. doi: 10.3934/math.2022798 |
[8] | 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 |
[9] | Ji Hyang Park, Hari Mohan Srivastava, Nak Eun Cho . Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind. AIMS Mathematics, 2021, 6(10): 11380-11402. doi: 10.3934/math.2021660 |
[10] | Kanagaraj Muthuselvan, Baskar Sundaravadivoo, Suliman Alsaeed, Kottakkaran Sooppy Nisar . New interpretation of topological degree method of Hilfer fractional neutral functional integro-differential equation with nonlocal condition. AIMS Mathematics, 2023, 8(7): 17154-17170. doi: 10.3934/math.2023876 |
In this article, a class of uncertain conformable fractional-order delay systems under input saturation is considered. By establishing the Lyapunov boundedness theorem for conformable fractional-order delay systems, some sufficient conditions for robust stability and boundedness of the systems are obtained. Examples are given to illustrate the obtained theory.
Fractional calculus is an essential research area, which is equally useful not only in pure mathematics but also in applied mathematics, physics, biology, engineering, economics and control theory etc. In recent years, study on fractional differential equations is very dynamic and widespread around the world. Some of its applications in diverse fields are discussed in [1,2]. Optical solitons of time-fractional higher-order nonlinear Schr¨odinger equation and soliton molecule solutions of nonlinear Schr¨odinger equations are established by Dai et al. [3,4]. A hybrid analytical solution to examine the fractional model of the nonlinear wave-like equation is explored by Kumar et al. [5]. Numerical solutions with linearization techniques of the fractional Harry Dym equation are established in [6].
Many researchers have discussed fractional calculus operators [7,8]. The composition of Erdélyi-Kober fractional operators is presented in [9]. Mishra et al. discussed the Marichev-Saigo-Maeda fractional calculus operators on the product of Srivastava polynomials and generalized Mittag-Leffler function [10]. Certain Integral operators involving the Gauss hypergeometric functions are elaborated in [11,12]. A brief study of fractional calculus operators on generalized multivariable Mittag-Leffler function is presented by Suthar et al. [13]. A brief systematic history of the generalized fractional calculus operators and their applications is being profoundly analyzed in [14,15]. A concise description of generalized fractional calculus operators together with their applications is available in [16,17,18].
A variety of research publications are continuously in progress regarding the generalization of classical fractional calculus operators. In this continuation, many researchers established generalized fractional operator and their applications[19,20]. Smraiz et al. modified the (k,s) fractional integral operator involving k-Mittag-Leffler function and also discussed the applications of (k,s)-Hilfer-Prabhakar fractional derivative in mathematical physics [21,22]. Can et al. have discussed the global existence for a mild solution of fractional Volterra integro-differential equations [23] and inverse source problem for the time-fractional diffusion equation with Mittag-Leffler kernel [24]. They also explored regularized solution approximation for the fractional pseudo-parabolic problem with a nonlinear source term in [25].
For our study, we start with k versions of Saigo fractional integral and derivative operators involving the k-hypergeometric function in the kernel which were introduced by Gupta and Parihar [26] and are defined as follows:
For w∈R+, ϵ,ϱ,χ∈C with Re(ϵ)>0, k>0, we have
(Iϵ,ϱ,χ0+,kf)(w)=w−ϵ−ϱkkΓk(ϵ)∫w0(w−t)ϵk−1×2F1,k((ϵ+ϱ,k),(−χ,k);(ϵ,k);(1−tw))f(t)dt. | (1.1) |
(Iϵ,ϱ,χ−,kf)(w)=1kΓk(ϵ)∫∞w(t−w)ϵk−1t−ϵ−ϱk×2F1,k((ϵ+ϱ,k),(−χ,k);(ϵ,k);(1−wt))f(t)dt. | (1.2) |
(Dϵ,ϱ,χ0+,kf)(w)=(ddw)n(I−ϵ+n,−ϱ−n,ϵ+χ−n0+,kf)w,n=[Re(ϵ)+1]=(ddw)nwϵ+ϱkkΓk(−ϵ+n)∫w0(w−t)−ϵk+n−1×2F1,k((−ϵ−ϱ,k),(−χ−ϵ+n,k);(−ϵ+n,k);(1−tw))f(t)dt. | (1.3) |
(Dϵ,ϱ,χ−,kf)(w)=(ddw)n(I−ϵ+n,−ϱ−n,ϵ+χ−,kf)w,n=[Re(ϵ)+1]=(ddw)n1kΓk(−ϵ+n)∫∞w(t−w)−ϵ−nk−1tϵ+ϱk×2F1,k((−ϵ−ϱ,k),(−χ−ϵ+n,k);(−ϵ+n,k);(1−wt))f(t)dt. | (1.4) |
Where [Re(ϵ)] is the integer part of Re(ϵ) and 2F1,k((ϵ,k),(ϱ,k);(χ,k);w) is the k-hypergeometric function defined by Mubeen and Habibullah in [19] as:
For w∈C, |w|<1, Re(χ)>Re(ϱ)>0,
2F1,k((ϵ,k),(ϱ,k);(χ,k);w)=∞∑n=0(ϵ)n,k(ϱ)n,kwn(χ)n,kn!. | (1.5) |
The benefit of the generalized k-fractional calculus operators is that they generalize classical Saigo's fractional operators and classical Riemann-Liouville operators. For k→1, (1.1)–(1.4) condense to the Saigo's fractional integral and differential operators [11]. If we take ϱ=−ϵ in (1.1)–(1.4), we have the k- Riemann-Liouville operators as follows:
(Iϵ,ϱ,χ0+,kf)(w)=(Iϵ0+,kf)(w), | (1.6) |
(Iϵ,ϱ,χ−,kf)(w)=(Iϵ−,kf)(w), | (1.7) |
(Dϵ,ϱ,χ0+,kf)(w)=(Dϵ0+,kf)(w), | (1.8) |
(Dϵ,ϱ,χ−,kf)(w)=(Dϵ−,kf)(w) | (1.9) |
and for k→1, Eqs (1.6)–(1.9) reduce to classical Riemann-Liouville fractional operators.
Now, we will state the lemmas presented in [26] which will be helpful to prove our main results.
Lemma 1.1. Let ϵ,ρ,χ,λ∈C, k∈R+(0,∞), Re(λ)>max[0,Re(ϱ−χ)]. Then
(Iϵ,ϱ,χ0+,ktλk−1)(w)=∞∑n=0knΓk(λ)Γk(λ−ϱ+χ)Γk(λ−ϱ)Γk(λ+ϵ+χ)wλ−ϱk−1. | (1.10) |
Lemma 1.2. Let ϵ,ρ,χ,λ∈C, k∈R+(0,∞), Re(λ)>max[Re(−ϱ),Re(−χ)]. Then
(Iϵ,ϱ,χ−,kt−λk)(w)=∞∑n=0knΓk(λ+ϱ)Γk(λ+χ)Γk(λ)Γk(λ+ϵ+ϱ+χ)w−λ−ϱk. | (1.11) |
Lemma 1.3. Let ϵ,ρ,χ,λ∈C, k∈R+(0,∞), n=Re[ϵ]+1 such that Re(λ)>max[0,Re(−ϵ−ϱ−χ)]. Then
(Dϵ,ϱ,χ0+,ktλk−1)(w)=∞∑n=0Γk(λ)Γk(λ+ϱ+χ+ϵ)Γk(λ+χ)Γk(λ+ϱ+n−nk)wλ+ϱ+nk−n−1. | (1.12) |
Lemma 1.4. Let ϵ,ρ,χ,λ∈C, n=Re[ϵ]+1, k∈R+(0,∞) such that Re(λ)>max[Re(−ϵ−χ),Re(ϱ−nk+n)]. Then
(Dϵ,ϱ,χ−,kt−λk)(w)=∞∑n=0Γk(λ−ϱ−n+nk)Γk(λ+ϵ+χ)Γk(λ)Γk(λ−ϱ+χ)w−λ−ϱ+nk−n. | (1.13) |
Gehlot and Prajapati in [27] defined the k-Wright function as follows:
For k∈R+, w,ai,bj∈C, Ai,Bj∈R(Ai,Bj)≠0 where i=1,2,..u;j=1,2,..v and (ai+Ain),(bj+Bjn)∈C∖kZ−,
uψkv[(a1,A1),..(au,Au);(b1,B1),..(bv,Bv);z]=∞∑n=0Γk(a1,nA1)..Γk(au,nAu)znΓk(b1,nB1)..Γk(bv,nBv)n!, | (1.14) |
with convergence condition
1+u∑j=1Bjk−v∑i=1Aik>0, | (1.15) |
for reasonably bounded values of |z|.
The Lommel-Wright k-function is defined as follows:
J℘,mℵ,ℏ,k(z)=(z2)ℵ+2ℏk∞∑n=0(−1)n(z2)2n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘), | (1.16) |
where z∈C|(−∞,0],℘>0,m∈N,k∈R,ℵ,ℏ∈C and Γk(Z) is the k-gamma function introduced by Diaz and Pariguan [28] given by
Γk(z)=limn→∞n!kn(nk)ωk−1(ω)n,k, |
with k-Pochhammer symbol (ω)n,k given by
(ω)n,k=ω(ω+k)(ω+2k)....(ω+(n−1)k),x∈C,k∈R,n∈N+. |
The classical Eulers Gamma function and Gamma k-function are related with following relation
Γk(ω)=kωk−1Γ(ωk). |
The Lommel-Wright k-function can also be expressed in the form of k-Wright function as:
J℘,mℵ,ℏ,k(z)=(z2)ℵ+2ℏk1ψkm+1[(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘);−z24k]. | (1.17) |
For m=1 in (1.16), we define the generalized Bessel-Maitland k-function as:
J℘ℵ,ℏ,k(z)=(z2)ℵ+2ℏk∞∑n=0(−1)n(z2)2nΓk(ℏ+k+nk)Γk(ℵ+ℏ+k+n℘). | (1.18) |
It is observed that for k=1, generalized Lommel-Wright k-function reduces to generalized Lommel-Wright function as given in [29] and for m=k=1, we get the Bessel-Maitland function presented in [29]. It also capitulates connection with the classical Bessel function Jℵ(z) mentioned in [30] for m=℘=k=1 and ℏ=0.
As various kinds of generalized fractional calculus operators involving different special functions are in consistent development. The papers on certain generalized fractional operators and integral transform [31,32,33] serve as inspiration for our presented work. This work backs up the prior results and contributes to the field by making broad generalizations.
The layout of the paper is as follows: In section 2, we established the formulas for generalized Saigo fractional integrals involving generalized Lommel-Wright function and some of its cases are also discussed as corollaries. Section 3 is devoted to developing the generalized Saigo fractional differentiation formulas involving generalized Lommel-Wright function along with its special consequences. In Section 4, extended Beta transform is applied to the generalized Lommel-Wright function. The last section contains concluding remarks.
In this section, we develop the formulas for Saigo k-fractional integrals (1.1) and (1.2) associated with Lommel-Wright k-function. These results are expressed in terms of k-Wright function.
Theorem 2.1. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Reλ>max[0,Re(ϱ−χ)] and Re(λ+χ−ϱ)>0. If condition given by (1.15) is satisfied and Iϵ,ϱ,χ0+,k is the left sided integral operator of the generalized k-fractional integration considering k-hypergeometric function. Then the subsequent formula
(Iϵ,ϱ,χ0+,ktλk−1J℘,mℵ,ℏ,k(tσk))(w)=wσ(ℵ+2ℏk)+λ−ϱk−1(12)(ℵ+2ℏk)×3ψkm+3[(σ(ℵ+2ℏk)+λ,2σ),(σ(ℵ+2ℏk)+λ−ϱ+χ,2σ),(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ−ϱ,2σ),(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ,2σ);−w2σk4] | (2.1) |
holds.
Proof. Using Eq (1.16) in the left hand side of Eq (2.1), we get
=[Iϵ,ϱ,χ0+,ktλk−1∞∑n=0(−1)n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)(tσk2)ℵ+2ℏk+2n](w) | (2.2) |
=∞∑n=0(−1)n12ℵ+2ℏk+2n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)[Iϵ,ϱ,χ0+,k(tσ(ℵ+2ℏk)+λ+2σnk−1)](w). | (2.3) |
Applying Lemma 1.1, we obtain
=wσ(ℵ+2ℏk)+λ−ϱk−1(12)ℵ+2ℏk∞∑n=01(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)×Γk(σ(ℵ+2ℏk)+λ+2σn)Γk(σ(ℵ+2ℏk)+λ−ϱ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ−ϱ+2σn)Γk(σ(ℵ+2ℏk)+λ+ϱ+ϵ+χ+2σn)(−kw2σk4)n. | (2.4) |
Multiplying and dividing by Γ(n+1) and using Γ(n+1)=k−nΓk(nk+k), we get
=wσ(ℵ+2ℏk)+λ−ϱk−1(12)ℵ+2ℏk∞∑n=0k−nΓk(k+nk)(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)n!×Γk(σ(ℵ+2ℏk)+λ+2σn)Γk(σ(ℵ+2ℏk)+λ−ϱ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ−ϱ+2σn)Γk(σ(ℵ+2ℏk)+λ+ϱ+ϵ+χ+2σn)(−kw2σk4)n. | (2.5) |
Using Eq (1.14) in (2.5), we have the desired formula.
Theorem 2.2. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Re(λ+ϵ)>max[−Re(ϱ)−Re(χ)] and Re(ϱ)≠Re(χ). If condition (1.15) is satisfied and Iϵ,ϱ,χ−,k is the right sided integral operator of the generalized k-fractional integration considering k-hypergeometric function. Then the subsequent formula
(Iϵ,ϱ,χ−,kt−ϵ−λkJ℘,mℵ,ℏ,k(t−σk))(w)=w−σ(ℵ+2ℏk)+ϵ+λ−ϱk(12)(ℵ+2ℏk)×3ψkm+3[(σ(ℵ+2ℏk)+ϵ+λ+ϱ,2σ),(σ(ℵ+2ℏk)+ϵ+λ+χ,2σ),(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+ϵ+λ,2σ),(σ(ℵ+2ℏk)+2ϵ+λ+ϱ+χ,2σ);−w−2σk4] | (2.6) |
holds.
Proof. The proof of Theorem 2.2 runs parallel to Theorem 2.1.
The findings in (2.1) and (2.6) are very general in nature and can result in a large number of individual cases. Allocating some acceptable values to the parameters involved, we have the following corollaries.
Using m=1, the results (2.1) and (2.6) take the form.
Corollary 2.3. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, k∈R+, such that Re(ϵ)>0, Reλ>max[0,Re(ϱ−χ)] and Re(λ+χ−ϱ)>0. Then the subsequent formula
(Iϵ,ϱ,χ0+,ktλk−1J℘ℵ,ℏ,k(tσk))(w)=wσ(ℵ+2ℏk)+λ−ϱk−1(12)(ℵ+2ℏk)×3ψk4[(σ(ℵ+2ℏk)+λ,2σ),(σ(ℵ+2ℏk)+λ−ϱ+χ,2σ),(k,k);(ℏ+k,k),(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ−ϱ,2σ),(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ,2σ);−w2σk4] | (2.7) |
is true.
Corollary 2.4. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Re(λ+ϵ)>max[−Re(ϱ)−Re(χ)] and Re(ϱ)≠Re(χ). Then the subsequent formula
(Iϵ,ϱ,χ−,kt−ϵ−λkJ℘ℵ,ℏ,k(t−σk))(w)=w−σ(ℵ+2ℏk)+ϵ+λ−ϱk(12)(ℵ+2ℏk)×3ψk4[(σ(ℵ+2ℏk)+ϵ+λ+ϱ,2σ),(σ(ℵ+2ℏk)+ϵ+λ+χ,2σ),(k,k);(ℏ+k,k),(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+ϵ+λ,2σ),(σ(ℵ+2ℏk)+2ϵ+λ+ϱ+χ,2σ);−w−2σk4] | (2.8) |
is true.
Letting k=1, we have the generalized Lommel-Wright function and the corresponding formulas are presented in subsequent corollaries.
Corollary 2.5. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Reλ>max[0,Re(ϱ−χ)] and Re(λ+χ−ϱ)>0. Then the subsequent formula
(Iϵ,ϱ,χ0+tλ−1J℘,mℵ,ℏ(tσ))(w)=wσ(ℵ+2ℏ)+λ−ϱ−1(12)(ℵ+2ℏ)×3ψm+3[(σ(ℵ+2ℏ)+λ,2σ),(σ(ℵ+2ℏ)+λ−ϱ+χ,2σ),(1,1);(ℏ+1,1)⏟m−times,(ℵ+ℏ+1,℘),(σ(ℵ+2ℏ)+λ−ϱ,2σ),(σ(ℵ+2ℏ)+λ+ϵ+ϱ+χ,2σ);−w2σ4] | (2.9) |
holds.
Corollary 2.6. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Re(λ+ϵ)>max[−Re(ϱ)−Re(χ)] and Re(ϱ)≠Re(χ). Then the subsequent formula
(Iϵ,ϱ,χ−t−ϵ−λJ℘,mℵ,ℏ(t−σ))(w)=w−σ(ℵ+2ℏ)+ϵ+λ+ϱ(12)(ℵ+2ℏ)×3ψm+3[(σ(ℵ+2ℏ)+ϵ+λ+ϱ,2σ),(σ(ℵ+2ℏ)+ϵ+λ+χ,2σ),(1,1);(ℏ+1,1)⏟m−times,(ℵ+ℏ+1,℘),(σ(ℵ+2ℏ)+ϵ+λ,2σ),(σ(ℵ+2ℏ)+2ϵλ+ϱ+χ,2σ);−w−2σ4] | (2.10) |
holds.
For m=k=℘=1 and ℏ=0, the corresponding corollaries are as given below.
Corollary 2.7. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Reλ>max[0,Re(ϱ−χ)] and Re(λ+χ−ϱ)>0. Then the subsequent formula holds
(Iϵ,ϱ,χ0+tλ−1J1,1ℵ(tσ))(w)=wσ(ℵ+)+λ−ϱ−1(12)ℵ×2ψ3[(σℵ+λ,2σ),(σℵ+λ−ϱ+χ,2σ);(ℵ+1,1),(σℵ+λ−ϱ,2σ),(σℵ+λ+ϵ+ϱ+χ,2σ);−w2σ4]. | (2.11) |
Corollary 2.8. Let ϵ,ϱ,χ,λ,σ∈C, m∈N, such that Re(ϵ)>0, Re(λ+ϵ)>max[−Re(ϱ)−Re(χ)] and Re(ϱ)≠Re(χ). Then the subsequent formula holds
(Iϵ,ϱ,χ−t−ϵ−λJ1,1ℵ(t−σ))(w)=w−σℵ+ϵ+λ+ϱ(12)ℵ×2ψ3[(σℵ+ϵ+λ+ϱ,2σ),(σℵ+ϵ+λ+χ,2σ);(ℵ+1,1),(σℵ+ϵ+λ,2σ),(σℵ+2ϵλ+ϱ+χ,2σ);−w−2σ4]. | (2.12) |
In this part, we will present formulas for differentiation using Saigo k-fractional differential operators given by (1.3) and (1.4) involving generalized Lomme-Wright k-function. These formulae are presented in terms of k-Wright function.
Theorem 3.1. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Re(λ)>max[0,Re(−ϵ−ϱ−χ)] and Re(λ+χ+ϱ)>0. If condition (1.15) holds and Dϵ,ϱ,χ0+,k is the left sided operator of the generalized k-fractional differentiation considering k-hypergeometric function. Then the following formula
(Dϵ,ϱ,χ0+,ktλk−1J℘,mℵ,ℏ,k(tσk))(w)=wσ(ℵ+2ℏk)+λ−ϱk−1(12)(ℵ+2ℏk)×3ψkm+3[(σ(ℵ+2ℏk)+λ,2σ),(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ,2σ),(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ+χ,2σ),(σ(ℵ+2ℏk)+λ,2σ−k+1);−w(2σ+1k)−14k] | (3.1) |
holds true.
Proof. By means of Eq (1.16) we can write the left hand side of Eq (3.1) as follows:
=[Dϵ,ϱ,χ0+,ktλk−1∞∑n=0(−1)n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)(tσk2)ℵ+2ℏk+2n](w), | (3.2) |
=∞∑n=0(−1)n12ℵ+2ℏk+2n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)[Dϵ,ϱ,χ0+,k(tσ(ℵ+2ℏk)+λ+2σnk−1)](w). | (3.3) |
Using Lemma 1.3 in Eq (3.3), we obtain
=wσ(ℵ+2ℏk)+λ+ϱk−1(12)ℵ+2ℏk∞∑n=01(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)×Γk(σ(ℵ+2ℏk)+λ+2σn)Γk(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ+2σn−kn+n).(−w2σ+1k−14)n. | (3.4) |
Multiplying and dividing by Γ(n+1) and using Γ(n+1)=k−nΓk(nk+k), we get
=wσ(ℵ+2ℏk)+λ+ϱk−1(12)ℵ+2ℏk∞∑n=0k−nΓk(nk+k)(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)n!×Γk(σ(ℵ+2ℏk)+λ+2σn)Γk(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ+2σn−kn+n)(−w2σ+1k−14)n. | (3.5) |
By means of Definition (1.14) in (3.5), we obtain the formula (3.1).
Theorem 3.2. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Re(ϱ)>max[Re(ϵ+ϱ)+n−Re(χ)] and Re(ϵ+ϱ−χ)+n≠0, where n=[Re(ϵ)+1] If condition (1.15) is satisfied and (Dϵ,ϱ,χ−,k is the right sided operator of the generalized k-fractional differentiation considering k-hypergeometric function. Then the subsequent formula
(Dϵ,ϱ,χ−,kt−ϵ−λkJ℘,mℵ,ℏ,k(t−σk))(w)=w−σ(ℵ+2ℏk)+ϵ−λ+ϱk(12)(ℵ+2ℏk)×3ψkm+3[(σ(ℵ+2ℏk)+λ−ϵ−ϱ,2σ+k−1),(σ(ℵ+2ℏk)+λ+χ,2σ),(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ−ϵ,2σ),(σ(ℵ+2ℏk)+λ−ϵ−ϱ+χ,2σ);−w−2σ+1k−14k] | (3.6) |
holds.
Proof. The proof of Theorem 3.2 is similiar to Theorem 3.1.
Now, we discuss some special cases.
For m=1 the results (3.1) and (3.6) are established in the form of following corollaries.
Corollary 3.3. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, k∈R+, such that Re(ϵ)>0, Re(λ)>max[0,Re(−ϵ−ϱ−χ)] and Re(λ+χ+ϱ)>0. If condition in (1.15) holds then the subsequent formula
(Dϵ,ϱ,χ0+,ktλk−1J℘ℵ,ℏ,k(tσk))(w)=wσ(ℵ+2ℏk)+λ−ϱk−1(12)(ℵ+2ℏk)×3ψk4[(σ(ℵ+2ℏk)+λ,2σ),(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ,2σ),(k,k);(ℏ+k,k),(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ+χ,2σ),(σ(ℵ+2ℏk)+λ,2σ−k+1);−w(2σ+1k)−14k] | (3.7) |
is true.
Corollary 3.4. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, k∈R+, such that Re(ϵ)>0, Re(ϱ)>max[Re(ϵ+ϱ)+n−Re(χ)] and Re(ϵ+ϱ−χ)+n≠0, where n=[Re(ϵ)+1]. If condition in (1.15) is satisfied then the subsequent formula holds
(Dϵ,ϱ,χ−,kt−ϵ−λkJ℘,mℵ,ℏ,k(t−σk))(w)=w−σ(ℵ+2ℏk)+ϵ−λ+ϱk(12)(ℵ+2ℏk)×3ψk4[(σ(ℵ+2ℏk)+λ−ϵ−ϱ,2σ+k−1),(σ(ℵ+2ℏk)+λ+χ,2σ),(k,k);(ℏ+k,k),(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ−ϵ,2σ),(σ(ℵ+2ℏk)+λ−ϵ−ϱ+χ,2σ);−w−2σ+1k−14k]. | (3.8) |
For k=1, in Eqs. (3.1) and (3.6), the obtained corollaries are given below.
Corollary 3.5. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Re(λ)>max[0,Re(−ϵ−ϱ−χ)] and Re(λ+χ+ϱ)>0. If condition in (1.15) holds then the subsequent formula is true
(Dϵ,ϱ,χ0+tλ−1J℘,mℵ,ℏ(tσ))(w)=wσ(ℵ+2ℏ)+λ−ϱ−1(12)(ℵ+2ℏ)×3ψm+3[(σ(ℵ+2ℏ)+λ,2σ),(σ(ℵ+2ℏ)+λ+ϵ+ϱ+χ,2σ),(1,1);(ℏ+1,1)⏟m−times,(ℵ+ℏ+1,℘),(σ(ℵ+2ℏ)+λ+χ,2σ),(σ(ℵ+2ℏ)+λ,2σ);−w2σ4]. | (3.9) |
Corollary 3.6. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Re(ϱ)>max[Re(ϵ+ϱ)+n−Re(χ)] and Re(ϵ+ϱ−χ)+n≠0, where n=[Re(ϵ)+1]. If condition in (1.15) is satisfied then the subsequent formula holds
(Dϵ,ϱ,χ−t−ϵ−λJ℘,mℵ,ℏ(t−σ))(w)=w−σ(ℵ+2ℏ)+ϵ−λ+ϱ(12)(ℵ+2ℏ)×3ψm+3[(σ(ℵ+2ℏ)+λ−ϵ−ϱ,2σ),(σ(ℵ+2ℏ)+λ+χ,2σ),(1,1);(ℏ+1,1)⏟m−times,(ℵ+ℏ+1,℘),(σ(ℵ+2ℏ)+λ−ϵ,2σ),(σ(ℵ+2ℏ)+λ−ϵ−ϱ+χ,2σ);−w−2σ4]. | (3.10) |
For m=k=℘=1 and ℏ=0, the subsequent corollaries are as follows:
Corollary 3.7. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Re(λ)>max[0,Re(−ϵ−ϱ−χ)] and Re(λ+χ+ϱ)>0. If condition in (1.15) holds then the subsequent formula is true
(Dϵ,ϱ,χ0+tλ−1J1,1ℵ(tσ))(w)=wσℵ)+λ−ϱ−1(12)(ℵ)×2ψ3[(σℵ)+λ,2σ),(σℵ+λ+ϵ+ϱ+χ,2σ);(ℵ+1,1),(σℵ+λ+χ,2σ),(σℵ+λ,2σ);−w2σ4]. | (3.11) |
Corollary 3.8. Let ϵ,ϱ,χ,λ,σ∈C such that Re(ϵ)>0, Re(ϱ)>max[Re(ϵ+ϱ)+n−Re(χ)] and Re(ϵ+ϱ−χ)+n≠0, where n=[Re(ϵ)+1] If condition in (1.15) is satisfied then the subsequent formula holds
(Dϵ,ϱ,χ−t−ϵ−λJ1,1ℵ(t−σ))(w)=w−σℵ+ϵ−λ+ϱ(12)ℵ×2ψ3[(σℵ+λ−ϵ−ϱ,2σ),(σℵ+λ+χ,2σ),(1,1);(ℵ+1,1),(σℵ+λ−ϵ,2σ),(σℵ+λ−ϵ−ϱ+χ,2σ);−w−2σ4]. | (3.12) |
In this part, we will discuss some theorems on integral transforms of generalized Lommel-Wright k-function connecting with the results established in previous sections.
The k-beta function presented in [34] as:
For r,s>0
Bk(r,s)=1k∫10trk−1(1−t)sk−1dt. | (4.1) |
It can also be written as
Bk(l(t);r,s)=1k∫10trk−1(1−t)sk−1l(t)dt. | (4.2) |
The relation between k-beta function and the classical one is
Bk(r,s)=1kB(rk,sk)=Γk(r)Γk(s)Γk(r+s). | (4.3) |
Theorem 4.1. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Reλ>max[0,Re(ϱ−χ)]. Then the following formula is true
Bk((Iϵ,ϱ,χ0+,ktλk−1J℘,mℵ,ℏ,k(zt)σk))(w);r,s)=wσ(ℵ+2ℏk)+λ−ϱk−1(12)(ℵ+2ℏk)Γk(s)×4ψkm+4[(σ(ℵ+2ℏk)+λ,2σ),(σ(ℵ+2ℏk)+λ−ϱ+χ,2σ),(r+σ(ℵ+2ℏk),2σ),(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ−ϱ,2σ),(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ,2σ),(r+s+σ(ℵ+2ℏk,2σ);−w2σk4]. | (4.4) |
Proof. Using Eqs (1.16) and (4.2) in the left hand side of Eq (4.4), we can write
=1k∫10zrk−1(1−z)sk−1[Iϵ,ϱ,χ0+,ktλk−1∞∑n=0(−1)n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)×((zt)σk2)ℵ+2ℏk+2n](w)dz, | (4.5) |
which implies
=∞∑n=0(−1)n12ℵ+2ℏk+2n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)[Iϵ,ϱ,χ0+,k(tσ(ℵ+2ℏk)+λ+2σnk−1)](w)×∫10zr+σ(ℵ+2ℏk)+2σnk−1(1−z)sk−1dz. | (4.6) |
Applying Lemma 1.1 and using Eq (4.3), we obtain
=wσ(ℵ+2ℏk)+λ−ϱk−1(12)ℵ+2ℏk∞∑n=01(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)×Γk(σ(ℵ+2ℏk)+λ+2σn)Γk(σ(ℵ+2ℏk)+λ−ϱ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ−ϱ+2σn)Γk(σ(ℵ+2ℏk)+λ+ϱ+ϵ+χ+2σn)×Γk(r+σ(ℵ+2ℏk)+2σn)Γk(s)Γk(r+s+σ(ℵ+2ℏk)+2σn)(−kw2σk4)n. | (4.7) |
Multiplying and dividing by Γ(n+1) and using Γ(n+1)=k−nΓk(nk+k), we get
=wσ(ℵ+2ℏk)+λ−ϱk−1(12)ℵ+2ℏk∞∑n=0k−nΓk(k+nk)(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)n!×Γk(σ(ℵ+2ℏk)+λ+2σn)Γk(σ(ℵ+2ℏk)+λ−ϱ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ−ϱ+2σn)Γk(σ(ℵ+2ℏk)+λ+ϱ+ϵ+χ+2σn)×Γk(r+σ(ℵ+2ℏk)+2σn)Γk(s)Γk(r+s+σ(ℵ+2ℏk)+2σn)(−w2σk4)n. | (4.8) |
By combining Eqs (1.14) and (4.8), we get our required result (4.4).
Theorem 4.2. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Re(λ+ϵ)>max[−Re(ϱ)−Re(χ)]. Then the following formula is true
Bk((Iϵ,ϱ,χ−,kt−ϵ−λkJ℘,mℵ,ℏ,k(zt)−σk)(w);r,s)=w−σ(ℵ+2ℏk)+ϵ+λ−ϱk(12)(ℵ+2ℏk)Γk(s)×4ψkm+4[(σ(ℵ+2ℏk)+ϵ+λ+ϱ,2σ),(σ(ℵ+2ℏk)+ϵ+λ+χ,2σ),(r+σ(ℵ+2ℏk,2σ),(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+ϵ+λ,2σ),(σ(ℵ+2ℏk)+2ϵ+λ+ϱ+χ,2σ),(r+s+σ(ℵ+2ℏk,2σ);−w−2σk4]. | (4.9) |
Proof. The proof is similar to Theorem 4.1.
Theorem 4.3. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Re(λ)>max[0,Re(−ϵ−ϱ−χ)]. Then the subsequent formula is true
Bk((Dϵ,ϱ,χ0+,ktλk−1J℘,mℵ,ℏ,k(zt)σk)(w);r,s)=wσ(ℵ+2ℏk)+λ−ϱk−1(12)(ℵ+2ℏk)Γk(s)×4ψkm+4[(σ(ℵ+2ℏk)+λ,2σ),(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ,2σ),(r+σ(ℵ+2ℏk,2σ),(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ+χ,2σ),(σ(ℵ+2ℏk)+λ,2σ−k+1),(r+s+σ(ℵ+2ℏk,2σ);−w(2σ+1k)−14k]. | (4.10) |
Proof. By means of Eqs (1.16) and (4.2) in the left hand side of Eq (4.10), we have
=1k∫10zrk−1(1−z)sk−1[Dϵ,ϱ,χ0+,ktλk−1∞∑n=0(−1)n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)×(tσk2)ℵ+2ℏk+2n](w)dz. | (4.11) |
On simplification, we obtain
=∞∑n=0(−1)n12ℵ+2ℏk+2n(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)[Dϵ,ϱ,χ0+,k(tσ(ℵ+2ℏk)+λ+2σnk−1)](w)×∫10zr+σ(ℵ+2ℏk)+2σnk−1(1−z)sk−1dz. | (4.12) |
Using Lemma 1.3 and relation (4.3), we get
=wσ(ℵ+2ℏk)+λ+ϱk−1(12)ℵ+2ℏk∞∑n=01(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)×Γk(σ(ℵ+2ℏk)+λ+2σn)Γk(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ+2σn−kn+n)×Γk(r+σ(ℵ+2ℏk)+2σn)Γk(s)Γk(r+s+σ(ℵ+2ℏk)+2σn)(−w2σ+1k−14)n. | (4.13) |
Multiplying and dividing by Γ(n+1) and using Γ(n+1)=k−nΓk(nk+k), we get
=wσ(ℵ+2ℏk)+λ+ϱk−1(12)ℵ+2ℏk∞∑n=0k−nΓk(nk+k)(Γk(ℏ+k+nk))mΓk(ℵ+ℏ+k+n℘)n!×Γk(σ(ℵ+2ℏk)+λ+2σn)Γk(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ+χ+2σn)Γk(σ(ℵ+2ℏk)+λ+2σn−kn+n)×Γk(r+σ(ℵ+2ℏk)+2σn)Γk(s)Γk(r+s+σ(ℵ+2ℏk)+2σn)(−w2σ+1k−14)n. | (4.14) |
By means of definition (1.14), the proof is done.
Theorem 4.4. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Re(λ)>max[Re(−ϵ−χ)−nk+n]. Then the subsequent formula is true
Bk((Dϵ,ϱ,χ−,kt−ϵ−λkJ℘,mℵ,ℏ,k(zt)−σk)(w);r,s)=w−σ(ℵ+2ℏk)+ϵ−λ+ϱk(12)(ℵ+2ℏk)Γk(s)×4ψkm+4[(σ(ℵ+2ℏk)+λ−ϵ−ϱ,2σ+k−1),(σ(ℵ+2ℏk)+λ+χ,2σ),(r+σ(ℵ+2ℏk,2σ),(k,k);(ℏ+k,k)⏟m−times,(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ−ϵ,2σ),(σ(ℵ+2ℏk)+λ−ϵ−ϱ+χ,2σ),(r+s+σ(ℵ+2ℏk,2σ);−w−2σ+1k−14k]. | (4.15) |
Proof. The proof of Theorem 4.4 runs parallel to Theorem 4.3.
Now, we will discuss some special cases.
By substituting m=1 in Eqs (4.4), (4.9), (4.10) and (4.15), we establish the following corollaries.
Corollary 4.5. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, k∈R+, such that Re(ϵ)>0, Reλ>max[0,Re(ϱ−χ)]. Then the following formula is true
Bk((Iϵ,ϱ,χ0+,ktλk−1J℘ℵ,ℏ,k(zt)σk))(w);r,s)=wσ(ℵ+2ℏk)+λ−ϱk−1(12)(ℵ+2ℏk)Γk(s)×4ψk5[(σ(ℵ+2ℏk)+λ,2σ),(σ(ℵ+2ℏk)+λ−ϱ+χ,2σ),(r+σ(ℵ+2ℏk,2σ),(k,k);(ℏ+k,k),(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ−ϱ,2σ),(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ,2σ),(r+s+σ(ℵ+2ℏk,2σ);−w2σk4]. | (4.16) |
Corollary 4.6. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, k∈R+, such that Re(ϵ)>0, Re(λ+ϵ)>max[−Re(ϱ)−Re(χ)]. Then prove the following formula
Bk((Iϵ,ϱ,χ−,kt−ϵ−λkJ℘ℵ,ℏ,k(zt)−σk)(w);r,s)=w−σ(ℵ+2ℏk)+ϵ+λ−ϱk(12)(ℵ+2ℏk)Γk(s)×4ψk5[(σ(ℵ+2ℏk)+ϵ+λ+ϱ,2σ),(σ(ℵ+2ℏk)+ϵ+λ+χ,2σ),(r+σ(ℵ+2ℏk,2σ),(k,k);(ℏ+k,k),(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+ϵ+λ,2σ),(σ(ℵ+2ℏk)+2ϵ+λ+ϱ+χ,2σ),(r+s+σ(ℵ+2ℏk,2σ);−w−2σk4]. | (4.17) |
Corollary 4.7. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, k∈R+, such that Re(ϵ)>0, Re(λ)>max[0,Re(−ϵ−ϱ−χ)]. Then the subsequent formula is true
Bk((Dϵ,ϱ,χ0+,ktλk−1J℘ℵ,ℏ,k(zt)σk)(w);r,s)=wσ(ℵ+2ℏk)+λ−ϱk−1(12)(ℵ+2ℏk)Γk(s)×4ψk5[(σ(ℵ+2ℏk)+λ,2σ),(σ(ℵ+2ℏk)+λ+ϵ+ϱ+χ,2σ),(r+σ(ℵ+2ℏk,2σ),(k,k);(ℏ+k,k),(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ+χ,2σ),(σ(ℵ+2ℏk)+λ,2σ−k+1),(r+s+σ(ℵ+2ℏk,2σ);−w(2σ+1k)−14k]. | (4.18) |
Corollary 4.8. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, k∈R+, such that Re(ϵ)>0, Re(λ)>max[Re(−ϵ−χ)−nk+n]. Then the subsequent formula is true
Bk((Dϵ,ϱ,χ−,kt−ϵ−λkJ℘ℵ,ℏ,k(zt)−σk)(w);r,s)=w−σ(ℵ+2ℏk)+ϵ−λ+ϱk(12)(ℵ+2ℏk)Γk(s)×4ψk5[(σ(ℵ+2ℏk)+λ−ϵ−ϱ,2σ+k−1),(σ(ℵ+2ℏk)+λ+χ,2σ),(r+σ(ℵ+2ℏk,2σ),(k,k);(ℏ+k,k),(ℵ+ℏ+k,℘),(σ(ℵ+2ℏk)+λ−ϵ,2σ),(σ(ℵ+2ℏk)+λ−ϵ−ϱ+χ,2σ),(r+s+σ(ℵ+2ℏk,2σ);−w−2σ+1k−14k]. | (4.19) |
For k=1, we establish the following formulas from Eqs (4.4), (4.9), (4.10) and (4.15).
Corollary 4.9. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Reλ>max[0,Re(ϱ−χ)]. Then the following formula is true
B((Iϵ,ϱ,χ0+tλ−1J℘,mℵ,ℏ(zt)σ))(w);r,s)=wσ(ℵ+2ℏ)+λ−ϱ−1(12)ℵ+2ℏΓ(s)×4ψm+4[(σ(ℵ+2ℏ)+λ,2σ),(σ(ℵ+2ℏ)+λ−ϱ+χ,2σ),(r+σ(ℵ+2ℏ),2σ),(1,1);(ℏ+1,1)⏟m−times,(ℵ+ℏ+1,℘),(σ(ℵ+2ℏ)+λ−ϱ,2σ),(σ(ℵ+2ℏ)+λ+ϵ+ϱ+χ,2σ),(r+s+σ(ℵ+2ℏ),2σ);−w2σ4]. | (4.20) |
Corollary 4.10. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Re(λ+ϵ)>max[−Re(ϱ)−Re(χ)]. Then the following formula is true
B((Iϵ,ϱ,χ−t−ϵ−λJ℘,mℵ,ℏ(zt)−σ)(w);r,s)=w−σ(ℵ+2ℏ)+ϵ+λ−ϱ(12)(ℵ+2ℏ)Γ(s)×4ψm+4[(σ(ℵ+2ℏ)+ϵ+λ+ϱ,2σ),(σ(ℵ+2ℏ)+ϵ+λ+χ,2σ),(r+σ(ℵ+2ℏ),2σ),(1,1);(ℏ+1,1)⏟m−times,(ℵ+ℏ+1,℘),(σ(ℵ+2ℏ)+ϵ+λ,2σ),(σ(ℵ+2ℏk)+2ϵ+λ+ϱ+χ,2σ),(r+s+σ(ℵ+2ℏ),2σ);−w−2σ4]. | (4.21) |
Corollary 4.11. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Re(λ)>max[0,Re(−ϵ−ϱ−χ)]. Then the subsequent formula is true
B((Dϵ,ϱ,χ0+tλ−1J℘,mℵ,ℏ(zt)σ)(w);r,s)=wσ(ℵ+2ℏ)+λ−ϱ−1(12)(ℵ+2ℏ)Γ(s)×4ψm+4[(σ(ℵ+2ℏ)+λ,2σ),(σ(ℵ+2ℏ)+λ+ϵ+ϱ+χ,2σ),(r+σ(ℵ+2ℏ),2σ),(1,1);(ℏ+1,1)⏟m−times,(ℵ+ℏ+1,℘),(σ(ℵ+2ℏ)+λ+χ,2σ),(σ(ℵ+2ℏ)+λ,2σ),(r+s+σ(ℵ+2ℏ),2σ);−w2σ4]. | (4.22) |
Corollary 4.12. Let ϵ,ϱ,χ,λ,σ∈C, ℘>0, m∈N, such that Re(ϵ)>0, Re(λ)>max[Re(−ϵ−χ)]. Then the subsequent formula is true
B((Dϵ,ϱ,χ−t−ϵ−λJ℘,mℵ,ℏ(zt)−σ)(w);r,s)=w−σ(ℵ+2ℏ)+ϵ−λ+ϱ(12)(ℵ+2ℏ)Γ(s)×4ψm+4[(σ(ℵ+2ℏ)+λ−ϵ−ϱ,2σ),(σ(ℵ+2ℏ)+λ+χ,2σ),(r+σ(ℵ+2ℏ),2σ),(1,1);(ℏ+1,1)⏟m−times,(ℵ+ℏ+1,℘),(σ(ℵ+2ℏ)+λ−ϵ,2σ),(σ(ℵ+2ℏ)+λ−ϵ−ϱ+χ,2σ),(r+s+σ(ℵ+2ℏ),2σ);−w−2σ4]. | (4.23) |
For m=k=℘=1 and ℏ=0, the subsequent corollaries are as follows.
Corollary 4.13. Let ϵ,ϱ,χ,λ,σ∈C, such that Re(ϵ)>0, Reλ>max[0,Re(ϱ−χ)]. Then the following formula is true
B((Iϵ,ϱ,χ0+tλ−1J1,1ℵ(zt)σ))(w);r,s)=wσℵ+λ−ϱ−1(12)ℵΓ(s)×3ψ4[(σℵ+λ,2σ),(σℵ+λ−ϱ+χ,2σ),(r+σℵ,2σ);(ℵ+1,1),(σℵ+λ−ϱ,2σ),(σℵ+λ+ϵ+ϱ+χ,2σ),(r+s+σℵ,2σ);−w2σ4]. | (4.24) |
Corollary 4.14. Let ϵ,ϱ,χ,λ,σ∈C, such that Re(ϵ)>0, Re(λ+ϵ)>max[−Re(ϱ)−Re(χ)]. Then the following formula is true
B((Iϵ,ϱ,χ−t−ϵ−λJ1,1ℵ(zt)−σ)(w);r,s)=w−σℵ+ϵ+λ−ϱ(12)ℵΓ(s)×3ψ4[(σℵ+ϵ+λ+ϱ,2σ),(σℵ+ϵ+λ+χ,2σ),(r+σℵ,2σ);(ℵ+1,1),(σℵ+ϵ+λ,2σ),(σℵ+2ϵ+λ+ϱ+χ,2σ),(r+s+σℵ,2σ);−w−2σ4]. | (4.25) |
Corollary 4.15. Let ϵ,ϱ,χ,λ,σ∈C, such that Re(ϵ)>0, Re(λ)>max[0,Re(−ϵ−ϱ−χ)]. Then the subsequent formula is true
B((Dϵ,ϱ,χ0+tλ−1J1,1ℵ(zt)σ)(w);r,s)=wσℵ+λ−ϱ−1(12)ℵΓ(s)×3ψ4[(σℵ+λ,2σ),(σℵ+λ+ϵ+ϱ+χ,2σ),(r+σℵ,2σ);(ℵ+1,1),(σℵ+λ+χ,2σ),(σℵ+λ,2σ),(r+s+σℵ,2σ);−w2σ4]. | (4.26) |
Corollary 4.16. Let ϵ,ϱ,χ,λ,σ∈C, such that Re(ϵ)>0, Re(λ)>max[Re(−ϵ−χ)]. Then the subsequent formula is true
B((Dϵ,ϱ,χ−t−ϵ−λJ1,1ℵ(zt)−σ)(w);r,s)=w−σℵ+ϵ−λ+ϱ(12)ℵΓ(s)×3ψ4[(σℵ+λ−ϵ−ϱ,2σ),(σℵ+λ+χ,2σ),(r+σℵ+,2σ);(ℵ+1,1),(σℵ+λ−ϵ,2σ),(σℵ+λ−ϵ−ϱ+χ,2σ),(r+s+σℵ,2σ);−w−2σ4]. | (4.27) |
In this article, we established the relations of fractional integration and differentiation associated with the generalized Lommel-Wright function. We conclude that many other interesting image formulas can be derived as the specific cases of our main results. Like the generalized Lommel-Wright function certain other special functions can also be discussed in the same perspective. Briefly, the recent study confirms the earlier results and plays a significant role by making generalizations. Furthermore, for the choice ϱ=−ϵ in our main results and corollaries, we obtain the results for k-Riemann-Liouville fractional operators. We also deduce the results for Saigo's fractional operators by substituting k=1 and for Riemann-Liouville fractional operators, we need to opt k=1 and ϱ=−ϵ in our main results.
The authors declares that there is no conflict of interests regarding the publication of this paper.
[1] |
R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
![]() |
[2] |
F. F. Du, J. G. Lu, Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities, Appl. Math. Comput., 375 (2020), 125079. https://doi.org/10.1016/j.amc.2020.125079 doi: 10.1016/j.amc.2020.125079
![]() |
[3] |
F. F. Du, J. G. Lu, New criteria on finite-time stability of fractional-order Hopfield neural networks with time delays, IEEE Trans. Neural Netw. Learn. Syst., 32 (2021), 3858–3866. https://doi.org/10.1109/TNNLS.2020.3016038 doi: 10.1109/TNNLS.2020.3016038
![]() |
[4] |
Y. C. Ding, H. Liu, A new fixed-time stability criterion for fractional-order systems, AIMS Math., 7 (2022), 6173–6181. https://doi.org/10.3934/math.2022343 doi: 10.3934/math.2022343
![]() |
[5] |
Y. J. Gu, H. Wang, Y. G. Yu, Synchronization for commensurate Riemann-Liouville fractional-order memristor-based neural networks with unknown parameters, J. Frank. Inst., 357 (2020), 8870–8898. https://doi.org/10.1016/j.jfranklin.2020.06.025 doi: 10.1016/j.jfranklin.2020.06.025
![]() |
[6] |
S. A. Murad, Z. A. Ameen, Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives, AIMS Math., 7 (2022), 6404–6419. https://doi.org/10.3934/math.2022357 doi: 10.3934/math.2022357
![]() |
[7] |
N. Sene, Fractional input stability for electrical circuits described by the Riemann-Liouville and the Caputo fractional derivatives, AIMS Math., 4 (2019), 147–165. https://doi.org/10.3934/Math.2019.1.147 doi: 10.3934/Math.2019.1.147
![]() |
[8] |
E. S. A. Shahri, A. Alfia, J. A. T. Machado, Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation, Appl. Math. Model., 81 (2020), 663–672. https://doi.org/10.1016/j.apm.2020.01.013 doi: 10.1016/j.apm.2020.01.013
![]() |
[9] | F. A. Rihan, Delay differential equations and applications to biology, Singapore: Springer, 2021. https://doi.org/10.1007/978-981-16-0626-7 |
[10] |
Z. S. Aghayan, A. Alfi, J. A. T. Machado, Robust stability of uncertain fractional order systems of neutral type with distributed delays and control input saturation, ISA Trans., 111 (2021), 144–155. https://doi.org/10.1016/j.isatra.2020.11.009 doi: 10.1016/j.isatra.2020.11.009
![]() |
[11] |
L. Chen, Y. W. Wang, W. Yang, J. W. Xiao, Robust consensus of fractional-order multi-agent systems with input saturation and external disturbances, Neurocomputing, 303 (2018), 11–19. https://doi.org/10.1016/j.neucom.2018.04.002 doi: 10.1016/j.neucom.2018.04.002
![]() |
[12] |
D. H. He, L. G. Xu, Exponential stability of impulsive fractional switched systems with time delays, IEEE Trans. Circuits Syst. Ⅱ Exp. Briefs, 68 (2021), 1972–1976. https://doi.org/10.1109/TCSII.2020.3037654 doi: 10.1109/TCSII.2020.3037654
![]() |
[13] |
C. Li, K. Chen, J. G. Lu, R. N. Tang, Stability and stabilization analysis of fractional-order linear systems subject to actuator saturation and disturbance, IFAC, 50 (2017), 9718–9723. https://doi.org/10.1016/j.ifacol.2017.08.2055 doi: 10.1016/j.ifacol.2017.08.2055
![]() |
[14] |
Y. H. Lim, K. K. Oh, H. S. Ahn, Stability and stabilization of fractional-order linear systems subject to input saturation, IEEE Trans. Autom. Control, 58 (2013), 1062–1067. https://doi.org/10.1109/TAC.2012.2218064 doi: 10.1109/TAC.2012.2218064
![]() |
[15] |
E. S. A. Shahri, A. Alfi, J. A. T. Machado, Stability analysis of a class of nonlinear fractional-order systems under control input saturation, Int. J. Robust Nonlinear Control, 28 (2018), 2887–2905. https://doi.org/10.1002/rnc.4055 doi: 10.1002/rnc.4055
![]() |
[16] |
L. G. Xu, X. Y. Chu, H. X. Hu, Quasi-synchronization analysis for fractional-order delayed complex dynamical networks, Math. Comput. Simul., 185 (2021), 594–613. https://doi.org/10.1016/j.matcom.2021.01.016 doi: 10.1016/j.matcom.2021.01.016
![]() |
[17] |
D. H. He, L. G. Xu, Ultimate boundedness of nonautonomous dynamical complex networks under impulsive control, IEEE Trans. Circuits Syst. Ⅱ Exp. Briefs, 62 (2015), 997–1001. https://doi.org/10.1109/TCSII.2015.2436191 doi: 10.1109/TCSII.2015.2436191
![]() |
[18] |
Y. C. Liu, Q. D. Zhu, Adaptive neural network asymptotic control design for MIMO nonlinear systems based on event-triggered mechanism, Inform. Sci., 603 (2022), 91–105. https://doi.org/10.1016/j.ins.2022.04.048 doi: 10.1016/j.ins.2022.04.048
![]() |
[19] |
Y. C. Liu, Q. D. Zhu, G. X. Wen, Adaptive tracking control for perturbed strict-feedback nonlinear systems based on optimized backstepping technique, IEEE Trans. Neural Netw. Learn. Syst., 33 (2022), 853–865. https://doi.org/10.1109/TNNLS.2020.3029587 doi: 10.1109/TNNLS.2020.3029587
![]() |
[20] |
L. G. Xu, X. Y. Chu, H. X. Hu, Exponential ultimate boundedness of non-autonomous fractional differential systems with time delay and impulses, Appl. Math. Lett., 99 (2020), 106000. https://doi.org/10.1016/j.aml.2019.106000 doi: 10.1016/j.aml.2019.106000
![]() |
[21] |
L. G. Xu, H. X. Hu, F. J. Qin, Ultimate boundedness of impulsive fractional differential equations, Appl. Math. Lett., 62 (2016), 110–117. https://doi.org/10.1016/j.aml.2016.06.011 doi: 10.1016/j.aml.2016.06.011
![]() |
[22] |
L. G. Xu, J. K. Li, S. S. Ge, Impulsive stabilization of fractional differential systems, ISA Trans., 70 (2017), 125–131. https://doi.org/10.1016/j.isatra.2017.06.009 doi: 10.1016/j.isatra.2017.06.009
![]() |
[23] |
L. G. Xu, W. Liu, H. X. Hu, W. S. Zhou, Exponential ultimate boundedness of fractional-order differential systems via periodically intermittent control, Nonlinear Dyn., 96 (2019), 1665–1675. https://doi.org/10.1007/s11071-019-04877-y doi: 10.1007/s11071-019-04877-y
![]() |
[24] |
S. Haghighatnia, H. T. Shandiz, A. Alfi, Conformable fractional order sliding mode control for a class of fractional order chaotic systems, Int. J. Ind. Electron. Control Optim., 2 (2019), 177–188. https://doi.org/10.22111/ieco.2018.25403.1049 doi: 10.22111/ieco.2018.25403.1049
![]() |
[25] |
Y. F. Qi, X. H. Wang, Asymptotical stability analysis of conformable fractional systems, J. Taibah Univ. Sci., 14 (2020), 44–49. https://doi.org/10.1080/16583655.2019.1701390 doi: 10.1080/16583655.2019.1701390
![]() |
[26] |
A. Souahi, A. B. Makhlouf, M. A. Hammami, Stability analysis of conformable fractional-order nonlinear systems, Indagat. Math., 28 (2017), 1265–1274. https://doi.org/10.1016/j.indag.2017.09.009 doi: 10.1016/j.indag.2017.09.009
![]() |
[27] |
X. Y. Chu, L. G. Xu, H. X. Hu, Exponential quasi-synchronization of conformable fractional-order complex dynamical networks, Chaos Solitons Fract., 140 (2020), 110268. https://doi.org/10.1016/j.chaos.2020.110268 doi: 10.1016/j.chaos.2020.110268
![]() |
[28] |
D. H. He, B. Z. Bao, H. X. Hu, L. G. Xu, Asymptotic boundedness of conformable fractional delay differential systems, IEEE Trans. Circuits Syst. Ⅱ Exp. Briefs, 2023. https://doi.org/10.1109/TCSII.2023.3282232 doi: 10.1109/TCSII.2023.3282232
![]() |
[29] |
P. P. Khargonakar, I. R. Petersen, K. Zhou, Robust stabilization of uncertain linear systems: quadratic stabilizability and H∞ control theory, IEEE Trans. Autom. Control, 35 (1990), 356–361. https://doi.org/10.1109/9.50357 doi: 10.1109/9.50357
![]() |
[30] |
E. S. A. Shahri, S. Balochian, Analysis of fractional-order linear systems with saturation using Lyapunov's second method and convex optimization, Int. J. Autom. Comput., 12 (2015), 440–447. https://doi.org/10.1007/s11633-014-0856-8 doi: 10.1007/s11633-014-0856-8
![]() |
[31] |
E. S. A. Shahri, A. Alfi, J. A. T. Machado, Stabilization of fractional-order systems subject to saturation element using fractional dynamic output feedback sliding mode control, J. Comput. Nonlinear Dyn., 12 (2017), 1–6. https://doi.org/10.1115/1.4035196 doi: 10.1115/1.4035196
![]() |
[32] |
E. S. A. Shahri, A. Alfi, J. A. T. Machado, An extension of estimation of domain of attraction for fractional order linear system subject to saturation control, Appl. Math. Lett., 47 (2015), 26–34. https://doi.org/10.1016/j.aml.2015.02.020 doi: 10.1016/j.aml.2015.02.020
![]() |