doi: 10.3934/dcdss.2021083
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations

1. 

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

2. 

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China

3. 

Department of Mathematics and Informatics, Azerbaijan University 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

4. 

Section of Mathematics, International Telematic University Uninettuno I-00186 Rome, Italy

5. 

Department of Mathematics, College of Education University of Sulaimani, Sulaimani, Kurdistan Region, Iraq

6. 

Department of Applied Mathematics and Statistics, Technical University of Cartagena Hospital de Marina, ES-30203 Cartagena, Spain

7. 

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group

8. 

Department of Mathematics, Faculty of Science, King Abdulaziz University P.O. Box 80203, Jeddah 21589, Saudi Arabia

9. 

Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

* Corresponding author

Received  March 2021 Revised  April 2021 Early access July 2021

We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their $ \varrho $-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its $ \varrho $-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their $ \varrho $-paths.

Citation: Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Y. S. Hamed. Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021083
References:
[1]

F. Atici and P. Eloe, A transform method in discrete fractional calculus, Internat. J. Differ. Equ., 2 (2007), 165-176.   Google Scholar

[2]

Ö. Akgandüller and S. Paşali Atmaca, Discrete normal vector field approximation via time scale calculus, Appl. Math. Nonlinear Sci., 5 (2020), 349-360.  doi: 10.2478/amns.2020.1.00033.  Google Scholar

[3]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ., 2016 (2016). doi: 10.1186/s13662-016-0949-5.  Google Scholar

[4]

T. Abdeljawad, Fractional difference operators with discrete generalized Mittag-Leffler kernels, Chaos Soliton Fract., 126 (2019), 315-324.  doi: 10.1016/j.chaos.2019.06.012.  Google Scholar

[5]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017). doi: 10.1186/s13662-017-1126-1.  Google Scholar

[6]

T. AbdeljawadF. JaradA. Atangana and P. O. Mohammed, On a new type of fractional difference operators on h-step isolated time scales, J. Fract. Calc. & Nonlinear Sys., 1 (2021), 46-74.   Google Scholar

[7]

B. AhmadM. AlghanmiA. AlsaediH. M. Srivastava and S. K. Ntouyas, The Langevin equation in terms of generalized Liouville-Caputo derivatives with nonlocal boundary conditions involving a generalized fractional integral, Mathematics, 7 (2019), 1-10.   Google Scholar

[8]

T. Abdeljawad, On delta and nabla caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013). doi: 10.1155/2013/406910.  Google Scholar

[9]

T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ., 2013 (2013). doi: 10.1186/1687-1847-2013-36.  Google Scholar

[10]

T. Abdeljawad, Different type kernel $h$–fractional differences and their fractional $h$–sums, Chaos Solit. Fract., 116 (2018), 146-56.  doi: 10.1016/j.chaos.2018.09.022.  Google Scholar

[11]

M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. Google Scholar

[12]

M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016. doi: 10.1007/978-3-319-47620-9.  Google Scholar

[13]

L. L. HuangG. C. WuD. Baleanu and H. Y. Wang, Discrete fractional calculus for interval-valued systems, Fuzzy Sets Syst., 404 (2020), 141-158.  doi: 10.1016/j.fss.2020.04.008.  Google Scholar

[14]

C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Berlin, 2015. doi: 10.1007/978-3-319-25562-0.  Google Scholar

[15]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer, Berlin, Germany, 2010. Google Scholar

[16]

L.-L. HuangD. BaleanuZ.-W. Mo and G.-C. Wu, Fractional discrete-time diffusion equation with uncertainty: Applications of fuzzy discrete fractional calculus, Physica A Stat. Mech. Appl., 508 (2018), 166-175.  doi: 10.1016/j.physa.2018.03.092.  Google Scholar

[17]

A. Khan, H. M. Alshehri, T. Abdeljawad and Q. M. Al-Mdallal, Stability analysis of fractional nabla difference COVID-19 model, Results Phys., 22 (2021), 103888. Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[19]

C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.  Google Scholar

[20]

Z.-Y. Liu, T.-C. Xia and J.-B. Wang, Image encryption technique based on new two-dimensional fractional-order discrete chaotic map and Menezes-Vanstone elliptic curve cryptosystem, Chin. Phys. B, 27 (2018), 030502. Google Scholar

[21]

Q. Lu and Y. Zhu, Comparison theorems and distributions of solutions to uncertain fractional difference equations, J. Comput. Appl., 376 (2020), 112884. doi: 10.1016/j.cam.2020.112884.  Google Scholar

[22]

Q. Lu, Y. Zhu and Z. Lu, Uncertain fractional forward difference equations for Riemann-Liouville type, Adv. Differ. Equ., 2019 (2019). doi: 10.1186/s13662-019-2093-5.  Google Scholar

[23]

P. O. Mohammed, A generalized uncertain fractional forward difference equations of Riemann-Liouville type, J. Math. Res., 11 (2019), 43-50.   Google Scholar

[24]

P. O. Mohammed, F. K. Hamasalh and T. Abdeljawad, Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels, Adv. Differ. Equ., 2021 (2021). doi: 10.1186/s13662-021-03372-2.  Google Scholar

[25]

P. O. MohammedT. AbdeljawadF. Jarad and Y.-M. Chu, Existence and uniqueness of uncertain fractional backward difference equations of Riemann-Liouville type, Math. Probl. Eng., 2020 (2020), 1-8.  doi: 10.1155/2020/6598682.  Google Scholar

[26]

P. O. Mohammed and T. Abdeljawad, Discrete generalized fractional operators defined using h-discrete Mittag-Leffler kernels and applications to AB fractional difference systems, Math. Meth. Appl. Sci., (2020), 1–26. doi: 10.1002/mma.7083.  Google Scholar

[27]

J. ShiM. Han and N. Zhang, Uncertainty principles for discrete signals associated with the fractional Fourier and linear canonical transforms, SIViP, 10 (2016), 1519-1525.   Google Scholar

[28]

H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60 (2020), 73-116.  doi: 10.5666/KMJ.2020.60.1.73.  Google Scholar

[29]

H. M. Srivastava and P. O. Mohammed, A correlation between solutions of uncertain fractional forward difference equations and their paths, Front. Phys., 8 (2020). Google Scholar

[30]

H. M. Srivastava, P. O. Mohammed, C. Ryoo and Y. S. Hamed, Existence and uniqueness of a class of uncertain Liouville-Caputo fractional difference equations, J. King Saud Univ. Sci., 33 (2021), 101497. doi: 10.1016/j.jksus.2021.101497.  Google Scholar

[31]

Z. WangB. Shiri and D. Baleanu, Discrete fractional watermark technique, Front. Inform. Technol. Electron. Eng., 21 (2020), 880-883.   Google Scholar

[32]

G. WuD. Baleanu and Y. Bai, Discrete fractional masks and their applications to image enhancement, De Gruyter, Berlin, 8 (2019), 261-270.   Google Scholar

[33]

B. Zhang and P. Shang, Uncertainty of financial time series based on discrete fractional cumulative residual entropy, Chaos, 29 (2019). doi: 10.1063/1.5091545.  Google Scholar

show all references

References:
[1]

F. Atici and P. Eloe, A transform method in discrete fractional calculus, Internat. J. Differ. Equ., 2 (2007), 165-176.   Google Scholar

[2]

Ö. Akgandüller and S. Paşali Atmaca, Discrete normal vector field approximation via time scale calculus, Appl. Math. Nonlinear Sci., 5 (2020), 349-360.  doi: 10.2478/amns.2020.1.00033.  Google Scholar

[3]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ., 2016 (2016). doi: 10.1186/s13662-016-0949-5.  Google Scholar

[4]

T. Abdeljawad, Fractional difference operators with discrete generalized Mittag-Leffler kernels, Chaos Soliton Fract., 126 (2019), 315-324.  doi: 10.1016/j.chaos.2019.06.012.  Google Scholar

[5]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017). doi: 10.1186/s13662-017-1126-1.  Google Scholar

[6]

T. AbdeljawadF. JaradA. Atangana and P. O. Mohammed, On a new type of fractional difference operators on h-step isolated time scales, J. Fract. Calc. & Nonlinear Sys., 1 (2021), 46-74.   Google Scholar

[7]

B. AhmadM. AlghanmiA. AlsaediH. M. Srivastava and S. K. Ntouyas, The Langevin equation in terms of generalized Liouville-Caputo derivatives with nonlocal boundary conditions involving a generalized fractional integral, Mathematics, 7 (2019), 1-10.   Google Scholar

[8]

T. Abdeljawad, On delta and nabla caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013). doi: 10.1155/2013/406910.  Google Scholar

[9]

T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ., 2013 (2013). doi: 10.1186/1687-1847-2013-36.  Google Scholar

[10]

T. Abdeljawad, Different type kernel $h$–fractional differences and their fractional $h$–sums, Chaos Solit. Fract., 116 (2018), 146-56.  doi: 10.1016/j.chaos.2018.09.022.  Google Scholar

[11]

M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. Google Scholar

[12]

M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016. doi: 10.1007/978-3-319-47620-9.  Google Scholar

[13]

L. L. HuangG. C. WuD. Baleanu and H. Y. Wang, Discrete fractional calculus for interval-valued systems, Fuzzy Sets Syst., 404 (2020), 141-158.  doi: 10.1016/j.fss.2020.04.008.  Google Scholar

[14]

C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Berlin, 2015. doi: 10.1007/978-3-319-25562-0.  Google Scholar

[15]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer, Berlin, Germany, 2010. Google Scholar

[16]

L.-L. HuangD. BaleanuZ.-W. Mo and G.-C. Wu, Fractional discrete-time diffusion equation with uncertainty: Applications of fuzzy discrete fractional calculus, Physica A Stat. Mech. Appl., 508 (2018), 166-175.  doi: 10.1016/j.physa.2018.03.092.  Google Scholar

[17]

A. Khan, H. M. Alshehri, T. Abdeljawad and Q. M. Al-Mdallal, Stability analysis of fractional nabla difference COVID-19 model, Results Phys., 22 (2021), 103888. Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[19]

C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.  Google Scholar

[20]

Z.-Y. Liu, T.-C. Xia and J.-B. Wang, Image encryption technique based on new two-dimensional fractional-order discrete chaotic map and Menezes-Vanstone elliptic curve cryptosystem, Chin. Phys. B, 27 (2018), 030502. Google Scholar

[21]

Q. Lu and Y. Zhu, Comparison theorems and distributions of solutions to uncertain fractional difference equations, J. Comput. Appl., 376 (2020), 112884. doi: 10.1016/j.cam.2020.112884.  Google Scholar

[22]

Q. Lu, Y. Zhu and Z. Lu, Uncertain fractional forward difference equations for Riemann-Liouville type, Adv. Differ. Equ., 2019 (2019). doi: 10.1186/s13662-019-2093-5.  Google Scholar

[23]

P. O. Mohammed, A generalized uncertain fractional forward difference equations of Riemann-Liouville type, J. Math. Res., 11 (2019), 43-50.   Google Scholar

[24]

P. O. Mohammed, F. K. Hamasalh and T. Abdeljawad, Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels, Adv. Differ. Equ., 2021 (2021). doi: 10.1186/s13662-021-03372-2.  Google Scholar

[25]

P. O. MohammedT. AbdeljawadF. Jarad and Y.-M. Chu, Existence and uniqueness of uncertain fractional backward difference equations of Riemann-Liouville type, Math. Probl. Eng., 2020 (2020), 1-8.  doi: 10.1155/2020/6598682.  Google Scholar

[26]

P. O. Mohammed and T. Abdeljawad, Discrete generalized fractional operators defined using h-discrete Mittag-Leffler kernels and applications to AB fractional difference systems, Math. Meth. Appl. Sci., (2020), 1–26. doi: 10.1002/mma.7083.  Google Scholar

[27]

J. ShiM. Han and N. Zhang, Uncertainty principles for discrete signals associated with the fractional Fourier and linear canonical transforms, SIViP, 10 (2016), 1519-1525.   Google Scholar

[28]

H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60 (2020), 73-116.  doi: 10.5666/KMJ.2020.60.1.73.  Google Scholar

[29]

H. M. Srivastava and P. O. Mohammed, A correlation between solutions of uncertain fractional forward difference equations and their paths, Front. Phys., 8 (2020). Google Scholar

[30]

H. M. Srivastava, P. O. Mohammed, C. Ryoo and Y. S. Hamed, Existence and uniqueness of a class of uncertain Liouville-Caputo fractional difference equations, J. King Saud Univ. Sci., 33 (2021), 101497. doi: 10.1016/j.jksus.2021.101497.  Google Scholar

[31]

Z. WangB. Shiri and D. Baleanu, Discrete fractional watermark technique, Front. Inform. Technol. Electron. Eng., 21 (2020), 880-883.   Google Scholar

[32]

G. WuD. Baleanu and Y. Bai, Discrete fractional masks and their applications to image enhancement, De Gruyter, Berlin, 8 (2019), 261-270.   Google Scholar

[33]

B. Zhang and P. Shang, Uncertainty of financial time series based on discrete fractional cumulative residual entropy, Chaos, 29 (2019). doi: 10.1063/1.5091545.  Google Scholar

[1]

Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050

[2]

Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171

[3]

Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056

[4]

Jianqin Zhou, Wanquan Liu, Xifeng Wang, Guanglu Zhou. On the $ k $-error linear complexity for $ p^n $-periodic binary sequences via hypercube theory. Mathematical Foundations of Computing, 2019, 2 (4) : 279-297. doi: 10.3934/mfc.2019018

[5]

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130

[6]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[7]

Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031

[8]

Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254

[9]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[10]

Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations. Mathematical Control & Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015

[11]

Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134

[12]

Jiao Du, Longjiang Qu, Chao Li, Xin Liao. Constructing 1-resilient rotation symmetric functions over $ {\mathbb F}_{p} $ with $ {q} $ variables through special orthogonal arrays. Advances in Mathematics of Communications, 2020, 14 (2) : 247-263. doi: 10.3934/amc.2020018

[13]

Davide Addona, Giorgio Menegatti, Michele Miranda jr.. $ BV $ functions on open domains: the Wiener case and a Fomin differentiable case. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2679-2711. doi: 10.3934/cpaa.2020117

[14]

Junchao Zhou, Yunge Xu, Lisha Wang, Nian Li. Nearly optimal codebooks from generalized Boolean bent functions over $ \mathbb{Z}_{4} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020121

[15]

Sugata Gangopadhyay, Constanza Riera, Pantelimon Stănică. Gowers $ U_2 $ norm as a measure of nonlinearity for Boolean functions and their generalizations. Advances in Mathematics of Communications, 2021, 15 (2) : 241-256. doi: 10.3934/amc.2020056

[16]

Chunming Tang, Maozhi Xu, Yanfeng Qi, Mingshuo Zhou. A new class of $ p $-ary regular bent functions. Advances in Mathematics of Communications, 2021, 15 (1) : 55-64. doi: 10.3934/amc.2020042

[17]

Yuan Li. Extremal solution and Liouville theorem for anisotropic elliptic equations. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4063-4082. doi: 10.3934/cpaa.2021144

[18]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033

[19]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

[20]

Ziqing Yuan, Jianshe Yu. Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3285-3303. doi: 10.3934/dcdss.2020281

2020 Impact Factor: 2.425

Article outline

[Back to Top]