# 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 and Continuous Dynamical Systems - S, 2022, 15 (2) : 427-440. 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. [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. [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. [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. [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. [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. [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. [8] T. Abdeljawad, On delta and nabla caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013). doi: 10.1155/2013/406910. [9] T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ., 2013 (2013). doi: 10.1186/1687-1847-2013-36. [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. [11] M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. [12] M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016. doi: 10.1007/978-3-319-47620-9. [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. [14] C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Berlin, 2015. doi: 10.1007/978-3-319-25562-0. [15] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer, Berlin, Germany, 2010. [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. [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. [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. [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. [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. [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. [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. [23] P. O. Mohammed, A generalized uncertain fractional forward difference equations of Riemann-Liouville type, J. Math. Res., 11 (2019), 43-50. [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. [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. [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. [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. [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. [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). [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. [31] Z. Wang, B. Shiri and D. Baleanu, Discrete fractional watermark technique, Front. Inform. Technol. Electron. Eng., 21 (2020), 880-883. [32] G. Wu, D. Baleanu and Y. Bai, Discrete fractional masks and their applications to image enhancement, De Gruyter, Berlin, 8 (2019), 261-270. [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.

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. [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. [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. [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. [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. [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. [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. [8] T. Abdeljawad, On delta and nabla caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013). doi: 10.1155/2013/406910. [9] T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ., 2013 (2013). doi: 10.1186/1687-1847-2013-36. [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. [11] M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. [12] M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016. doi: 10.1007/978-3-319-47620-9. [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. [14] C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Berlin, 2015. doi: 10.1007/978-3-319-25562-0. [15] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer, Berlin, Germany, 2010. [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. [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. [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. [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. [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. [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. [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. [23] P. O. Mohammed, A generalized uncertain fractional forward difference equations of Riemann-Liouville type, J. Math. Res., 11 (2019), 43-50. [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. [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. [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. [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. [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. [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). [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. [31] Z. Wang, B. Shiri and D. Baleanu, Discrete fractional watermark technique, Front. Inform. Technol. Electron. Eng., 21 (2020), 880-883. [32] G. Wu, D. Baleanu and Y. Bai, Discrete fractional masks and their applications to image enhancement, De Gruyter, Berlin, 8 (2019), 261-270. [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.
 [1] Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete and 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 and 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 and 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] Najeeb Abdulaleem. Optimality and duality for $E$-differentiable multiobjective programming problems involving $E$-type Ⅰ functions. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022004 [6] 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 and Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031 [7] Shihan Di, Dong Ma, Peibiao Zhao. $\alpha$-robust portfolio optimization problem under the distribution uncertainty. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022054 [8] Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 [9] 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 and Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 [10] Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $p$-Laplacian difference equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254 [11] Mathew Gluck. Classification of solutions to a system of $n^{\rm th}$ order equations on $\mathbb R^n$. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 [12] Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $1$-$d$ coupled wave equations. Mathematical Control and Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015 [13] 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 and Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134 [14] 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 [15] Davide Addona, Giorgio Menegatti, Michele Miranda jr.. $BV$ functions on open domains: the Wiener case and a Fomin differentiable case. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2679-2711. doi: 10.3934/cpaa.2020117 [16] 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 [17] 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 [18] 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 [19] Harun Karsli, Purshottam Narain Agrawal. Rate of convergence of Stancu type modified $q$-Gamma operators for functions with derivatives of bounded variation. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022002 [20] Yuan Li. Extremal solution and Liouville theorem for anisotropic elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4063-4082. doi: 10.3934/cpaa.2021144

2020 Impact Factor: 2.425