On the first-order autonomous interval-valued difference equations under gH-difference

Document Type : Research Paper

Authors

1 Beijing Key Lab on Mathematical Characterization, Analysis, and Applications of Complex Information, School of Mathematics and Statistics, Beijing Institute of Technology, 100081, Beijing, China

2 CITMAga, 15782 Santiago de Compostela, Spain

3 Departamento de Estat\'istica, An\'alise Matem\'atica e Optimizaci\'on, Facultade de Matem\'aticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain

Abstract

The theory of interval-valued difference equations under $gH$-difference is an interesting topic, since it can be applied to study numerical solutions to interval-valued or fuzzy-valued differential equations. In this paper, we estimate the number of solutions to a class of first-order interval-valued difference equations under $gH$-difference, which reveals the complexity of the stability analysis in this area, as well as the difficulty for prediction and control problems. Then, based on the relative positions of initial values and equilibrium points, we provide sufficient conditions for the existence of convergent solutions. We also provide examples to illustrate the validity of our results.

Keywords

References

[1] K. A. Chrysafis, B. K. Papadopoulos, G. Papaschinopoulos, On the fuzzy difference equations of finance, Fuzzy Sets and Systems, 159 (2008), 3259-3270.
[2] T. M. Costa, Y. Chalco-Cano, W. A. Lodwick, G. N. Silva, A new approach to linear interval differential equations as a first step toward solving fuzzy differential equations, Fuzzy Sets and Systems, 347 (2018), 129-141.
[3] P. Cull, M. Flahive, R. Robson, Difference equations, Springer, 2005.
[4] E. Y. Deeba, A. De Korvin, Analysis by fuzzy difference equations of a model of CO2 level in the blood, Applied Mathematics Letters, 12 (1999), 33-40.
[5] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientific, Singapore, 1994.
[6] L. L. Huang, G. C. Wu, D. Baleanu, H. Y. Wang, Discrete fractional calculus for interval-valued systems, Fuzzy Sets and Systems, 404 (2021), 141-158.
[7] M. Hukuhara, Integration des applications mesurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj, 10 (1967), 205-223.
 [8] A. Khastan, New solutions for first order linear fuzzy difference equations, Journal of Computational and Applied Mathematics, 312 (2017), 156-166.
[9] A. Khastan, Fuzzy logistic difference equation, Iranian Journal of Fuzzy Systems, 15 (2018), 55-66.
[10] A. Khastan, Z. Alijani, On the new solutions to the fuzzy difference equation xn+1 = A + B/xn, Fuzzy Sets and
Systems, 358 (2019), 64-83.
[11] H. Kneser, Eine direkte Ableitung des Zornschen Lemmas aus dem Auswahlaxiom, Mathematische Zeitschrift, 53
(1950), 110-113.
[12] J. Lewin, A simple proof of Zorn’s lemma, The American Mathematical Monthly, 98 (1991), 353-354.
[13] G. Papaschinopoulos, B. K. Papadopoulos, On the fuzzy difference equation xn+1 = A + xn/xn−m, Fuzzy Sets and
Systems, 129 (2002), 73-81.
[14] G. Papaschinopoulos, B. K. Papadopoulos, On the fuzzy difference equation xn+1 = A + B/xn, Soft Computing, 6
(2002), 456-461.
[15] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential
equations, Nonlinear Analysis, 71 (2009), 1311-1328.
[16] T. Sun, G. Su, C. Han, F. Zeng, B. Qin, Eventual periodicity of a system of max-type fuzzy difference equations of
higher order, Fuzzy Sets and Systems, 443(A) (2022), 286-303.
[17] T. Sun, G. Su, B. Qin, On the fuzzy difference equation xn = F(xn−1, xn−k), Fuzzy Sets and Systems, 387 (2020),
81-88.
[18] H. Wang, R. Rodr´ıguez-L´opez, On the existence of solutions to boundary value problems for interval-valued differential equations under gH-differentiability, Information Sciences, 553 (2021), 225-246.
[19] H. Wang, R. Rodr´ıguez-L´opez, On the existence of solutions to interval-valued differential equations with length
constraints, Iranian Journal of Fuzzy Systems, 18 (2021), 1-13.
[20] H. Wang, R. Rodr´ıguez-L´opez, Boundary value problems for interval-valued differential equations on unbounded
domains, Fuzzy Sets and Systems, 436 (2022), 102-127.
[21] W. A. Weldon, D. Dubois, Interval linear systems as a necessary step in fuzzy linear systems, Fuzzy Sets and
Systems, 281 (2015), 227-251.
[22] H. Zarei, A. Khastan, R. Rodr´ıguez-L´opez, Suboptimal control of linear fuzzy systems, Fuzzy Sets and Systems,
(2022), in Press, https://doi.org/10.1016/j.fss.2022.05.006.
[23] Q. Zhang, W. Zhang, F. Lin, D. Li, On dynamic behavior of second-order exponential-type fuzzy difference equation,
Fuzzy Sets and Systems, 419 (2021), 169-187.
[24] Q. Zhang, W. Zhang, J. Liu, Y. Shao, On a fuzzy logistic difference equation, WSEAS Transactions on Mathematics,
13 (2014), 282-290.
Iranian Journal of Fuzzy Systems
Volume 20, Issue 2
March and April 2023
Pages 21-32

Files

Share

How to cite

Statistics