RESULTS ON APPROXIMATE SOLUTIONS FOR SINGULAR DIFFERENCE SYSTEMS WITH MAXIMA

Xiang Liu; Christopher S. Goodrich

doi:10.11948/20250344

2027 Volume 17 Issue 1

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Xiang Liu, Christopher S. Goodrich. RESULTS ON APPROXIMATE SOLUTIONS FOR SINGULAR DIFFERENCE SYSTEMS WITH MAXIMA[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 71-86. doi: 10.11948/20250344

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Xiang Liu, Christopher S. Goodrich. RESULTS ON APPROXIMATE SOLUTIONS FOR SINGULAR DIFFERENCE SYSTEMS WITH MAXIMA[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 71-86. doi: 10.11948/20250344

RESULTS ON APPROXIMATE SOLUTIONS FOR SINGULAR DIFFERENCE SYSTEMS WITH MAXIMA

Xiang Liu^1, ,,
Christopher S. Goodrich^2,

1.
School of Mathematical Sciences, Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, 050024 Shijiazhuang, China
2.
School of Mathematics and Statistics, UNSW Sydney, NSW 2052 Sydney, Australia

Author Bio: Email: c.goodrich@unsw.edu.au(C. S. Goodrich)
Corresponding author: Email: xliu@hebtu.edu.cn(X. Liu)
Fund Project: The authors were supported by the Youth Top Talent Project of Hebei Education Department (No. BJK2024125) and the National Nature Science Foundation of China (No. 12201175)

Abstract

Abstract

In this paper, by introducing a new singular fractional difference comparison theorem, the existence of maximal and minimal quasi-solutions are proved for the singular fractional difference system with “maxima” combined with the method of upper and lower solutions and the monotone iterative technique on the monotone and nonmonotone case. Finally, we give an example to show the validity of the established results.
- Singular fractional difference systems /
- maxima /
- extremal solutions /
- iterative technique /
- delay equations
MSC: 39A12, 39A70

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References

[1]	T. Abdeljawad and B. Abdalla, Monotonicity results for delta and nabla caputo and riemann fractional differences via dual identities, Filomat, 2017, 31, 3671–3683. doi: 10.2298/FIL1712671A CrossRef Google Scholar
[2]	R. Agarwal, S. Hristova and D. O'Regan, Iterative technique for the initial value problem for Caputo fractional differential equations with non-instantaneous impluses, Appl. Math. Comput., 2018, 334, 407–421. Google Scholar
[3]	F. M. Atici, M. Atici, M. Belcher and D. Marshall, A new approach for modeling with discrete fractional equations, Fund. Inform., 2017, 151, 313–324. Google Scholar
[4]	F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., Special Edition I, 2009, 3, 1–12. Google Scholar
[5]	F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 2009, 137(3), 981–989. Google Scholar
[6]	F. M. Atici and P. W. Eloe, Linear systems of fractional nabla difference equations, Rocky Mountain J. Math., 2011, 41(2), 353–370. Google Scholar
[7]	F. M. Atici, N. Nguyen, K. Dadashova, S. E. Pedersen and G. Koch, Pharmacokinetics and pharmacodynamics models of tumor growth and anticancer effects in discrete time, Comput. Math. Biophys., 2020, 8(1), 114–125. doi: 10.1515/cmb-2020-0105 CrossRef Google Scholar
[8]	F. M. Atici and S. Şengül, Modeling with fractional difference equations, J. Math. Anal. Appl., 2010, 369(1), 1–9. Google Scholar
[9]	D. D. Bainov and S. G. Hristova, Monotone-iterative techniques of lakshmikantham for a boundary value problem for systems of differential equations with maxima, J. Math. Anal. Appl., 1995, 190(2), 391–401. doi: 10.1006/jmaa.1995.1083 CrossRef Google Scholar
[10]	Z. Baitiche, C. Derbazi, M. Benchohra and J. J. Nieto, Monotone iterative technique for a new class of nonlinear sequential fractional differential equations with nonlinear boundary conditions under the ψ-Caputo operator, Mathematics, 2022, 10(7), 1–11. Google Scholar
[11]	Z. Baitiche, C. Derbazi, A. Salim and M. Benchohra, Monotone iterative technique for a sequential δ-Caputo fractional differential equations with nonlinear boundary conditions, Stud. Univ. Babes-Bolyai Math., 2024, 69(3), 553–565. doi: 10.24193/subbmath.2024.3.06 CrossRef Google Scholar
[12]	N. Benkhettou, A. Salim, J. E. Lazreg, S. Abbas and M. Benchohra, Lakshmikantham monotone iterative principle for hybrid Atangana-Baleanu-Caputo fractional differential equations, An. Univ. Vest Timis. Ser. Mat. -Inform., 2023, 59(1), 79–91. Google Scholar
[13]	S. L. Campbell, Singular Systems of Differential Equations Ⅰ, Pitman Advanced Publishing Program, London, 1980. Google Scholar
[14]	S. L. Campbell, Singular Systems of Differential Equations Ⅱ, Pitman Advanced Publishing Program, London, 1982. Google Scholar
[15]	R. Dahal and C. S. Goodrich, An application of a nonstandard cone to discrete boundary value problems with unbounded indefinite forcing, J. Difference Equ. Appl., 2019, 25(6), 882–903. doi: 10.1080/10236198.2019.1639684 CrossRef Google Scholar
[16]	F. F. Du, B. G. Jia, L. H. Erbe and A. C. Peterson, Monotonicity and convexity for nabla fractional (q, h)-differences, J. Difference Equ. Appl., 2016, 22, 1224–1243. doi: 10.1080/10236198.2016.1188089 CrossRef Google Scholar
[17]	R. A. C. Ferreira, Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one, J. Difference Equ. Appl., 2013, 19(5), 712–718. doi: 10.1080/10236198.2012.682577 CrossRef Google Scholar
[18]	C. S. Goodrich, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl., 2012, 385(1), 111–124. doi: 10.1016/j.jmaa.2011.06.022 CrossRef Google Scholar
[19]	C. S. Goodrich and C. Lizama, Positivity, monotonicity, and convexity for convolution operators, Discrete Contin. Dyn. Syst., 2020, 40(8), 4961–4983. doi: 10.3934/dcds.2020207 CrossRef Google Scholar
[20]	C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity, Israel J. Math., 2020, 236(2), 533-589. doi: 10.1007/s11856-020-1991-2 CrossRef Google Scholar
[21]	C. S. Goodrich, B. Lyons, A. Scapellato and M. T. Velcsov, Analytical and numerical convexity results for discrete fractional sequential differences with negative lower bound, J. Difference Equ. Appl., 2021, 27(3), 1–25. Google Scholar
[22]	C. S. Goodrich, B. Lyons and M. T. Velcsov, Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound, Commun. Pure Appl. Anal., 2021, 20(1), 339–358. doi: 10.3934/cpaa.2020269 CrossRef Google Scholar
[23]	C. S. Goodrich and M. Muellner, An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators, Appl. Math. Lett., 2019, 98(0), 446–452. Google Scholar
[24]	C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015. Google Scholar
[25]	H. D. Gou and M. Shi, Monotone iterative technique for multi-term time fractional measure differential equations, Fract. Calc. Appl. Anal., 2024, 27, 1428–1470. doi: 10.1007/s13540-024-00273-5 CrossRef Google Scholar
[26]	Z. M. He, P. G. Wang and W. G. Ge, Periodic boundary value problem for first order impulsive differential equations with supremum, Indian J. Pure Appl. Math., 2003, 34(1), 133–143. Google Scholar
[27]	S. Hristova and A. Golev, Monotone-iterative method for the initial value problem with initial time difference for differential equations with “maxima”, Abstr. Appl. Anal., 2012, 2012, 2096–2105. Google Scholar
[28]	T. Jankowski, Minimal and maximal solutions to systems of differential equations with a singular matrix, ANZIAM J., 2003, 45(02), 223–231. doi: 10.1017/S1446181100013286 CrossRef Google Scholar
[29]	K. Jeet, N. Sukavanam and D. Bahuguna, Monotone iterative technique for nonlocal impulsive finite delay differential equations of fractional order, Differ. Equ. Dyn. Syst., 2022, 30(4), 801–816. doi: 10.1007/s12591-019-00498-4 CrossRef Google Scholar
[30]	B. G. Jia, L. H. Erbe and A. C. Peterson, Convexity for nabla and delta fractional differences, J. Difference Equ. Appl., 2015, 21(4), 360–373. doi: 10.1080/10236198.2015.1011630 CrossRef Google Scholar
[31]	B. G. Jia, L. H. Erbe and A. C. Peterson, Two monotonicity results for nabla and delta fractional differences, Arch. Math., 2015, 104(6), 589–597. doi: 10.1007/s00013-015-0765-2 CrossRef Google Scholar
[32]	B. G. Jia, L. H. Erbe and A. C. Peterson, Monotonicity and convexity for nabla fractional q-differences, Dynam. Systems Appl., 2016, 25, 47–60. Google Scholar
[33]	A. R. A. E. Kamar, G. M. Attia, K. Vajravelu and M. Mosaad, Generalized quasilinearization for singular system of differential equations, Appl. Math. Comput., 2000, 114(1), 69–74. Google Scholar
[34]	G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Technique for Nonlinear Differential Equations, Pitman Advanced Publishing Program, Boston, 1985. Google Scholar
[35]	J. Liang, The reaction-diffusion system without quasi-monotone conditions, Acta Sci. Natur. Univ. Pekinesis, 1987, 1987(3), 1–20. Google Scholar
[36]	X. Liu, C. S. Goodrich and P. G. Wang, Convergence of approximate solutions for singular difference systems with maxima, preprint, Authorea, 2021. Google Scholar
[37]	X. Liu, B. G. Jia and P. G. Wang, Some new results for nonlinear fractional h-difference systems with “maxima”, Rocky Mountain J. math., 2020, 50(3), 1073–1084. Google Scholar
[38]	X. Liu, A. C. Peterson, B. G. Jia and L. H. Erbe, A generalized h-fractional gronwall's inequality and its applications for nonlinear h-fractional difference systems with 'maxima', J. Difference Equ. Appl., 2019, 25, 815–836. doi: 10.1080/10236198.2018.1551382 CrossRef Google Scholar
[39]	I. Masubuchi, Y. Kamitane, A. Ohara and N. Suda, H_∞ control for descriptor systems: A matrix inequalities approach, Automatica, 1997, 33(4), 669–673. doi: 10.1016/S0005-1098(96)00193-8 CrossRef Google Scholar
[40]	H. H. Rosenbrock, Structural properties of linear dynamical systems, Int. J. Control, 1974, 20(2), 191–202. doi: 10.1080/00207177408932729 CrossRef Google Scholar
[41]	J. A. Uvah and A. S. Vatsala, Monotone method for first order singular systems with boundary conditions, Int. J. Stoch. Anal., 2007, 2(4), 217–224. Google Scholar
[42]	P. G. Wang and P. Li, Monotone iterative technique for partial dynamic equations of first order on time scales, Discrete Dyn. Nat. Soc., 2008, 2008, Article ID 265609, 7 pp. Google Scholar
[43]	P. G. Wang, S. H. Tian and Y. H. Wu, Monotone iterative method for first-order functional difference equations with nonlinear boundary value conditions, Appl. Math. Comput., 2008, 203(1), 266-272. Google Scholar
[44]	P. G. Wang and J. Zhang, Monotone iterative technique for initial-value problems of nonlinear singular discrete systems, J. Comput. Appl. Math., 2008, 221(1), 158–164. doi: 10.1016/j.cam.2007.10.002 CrossRef Google Scholar
[45]	Z. R. Wang, B. Shiri and D. Baleanu, Discrete fractional watermark technique, Front. Inform. Technol. Electron. Eng., 2020, 21(6), 880–883. doi: 10.1631/FITEE.2000133 CrossRef Google Scholar
[46]	G. C. Wu, D. Baleanu and Y. R. Bai, Discrete fractional masks and their applications to image enhancement, Applications in Engineering, Life and Social Sciences, Part B, 2019, 8, 261–270. Google Scholar
[47]	E. Zeidler, Nonlinear Functional Analysis and its Applications, V. 1: Fixed-Point Theorems, Springer, New York, 1986. Google Scholar