# American Institute of Mathematical Sciences

February  2022, 15(2): 427-440. doi: 10.3934/dcdss.2021083

## 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 Published  February 2022 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, 2022, 15 (2) : 427-440. doi: 10.3934/dcdss.2021083
##### References:
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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. Mohammed, T. Abdeljawad, F. 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. Shi, M. 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. Wang, B. Shiri and D. Baleanu, Discrete fractional watermark technique, Front. Inform. Technol. Electron. Eng., 21 (2020), 880-883.   Google Scholar [32] G. Wu, D. 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

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##### 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. Abdeljawad, F. Jarad, A. 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. Ahmad, M. Alghanmi, A. Alsaedi, H. 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. Huang, G. C. Wu, D. 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. Huang, D. Baleanu, Z.-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. Mohammed, T. Abdeljawad, F. 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. Shi, M. 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. Wang, B. Shiri and D. Baleanu, Discrete fractional watermark technique, Front. Inform. Technol. Electron. Eng., 21 (2020), 880-883.   Google Scholar [32] G. Wu, D. 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
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