Home Discrete fractional boundary value problems and inequalities
Article
Licensed
Unlicensed Requires Authentication

Discrete fractional boundary value problems and inequalities

  • Martin Bohner EMAIL logo and Nick Fewster-Young
Published/Copyright: November 22, 2021

Abstract

In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral representation, and, second, the establishment of solutions satisfying certain a priori bounds provided the key inequality holds. In order to establish the latter result, several novel discrete fractional inequalities are developed, each of them interesting in itself and of potential future use in different contexts. We illustrate the usefulness of our existence results by presenting two examples.

References

[1] R. P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001).10.1017/CBO9780511543005Search in Google Scholar

[2] F. M. Atιcι and S. Šengül, Modeling with fractional difference equations. J. Math. Anal. Appl. 369, No 1 (2010), 1–9.10.1016/j.jmaa.2010.02.009Search in Google Scholar

[3] F. M. Atιcι and P. W. Eloe, Two-point boundary value problems for finite fractional difference equations. J. Difference Equ. Appl. 17, No 4 (2011), 445–456.10.1080/10236190903029241Search in Google Scholar

[4] F. M. Atιcι and P. W. Eloe, Gronwall's inequality on discrete fractional calculus. Comput. Math. Appl. 64, No 10 (2012), 3193–3200.10.1016/j.camwa.2011.11.029Search in Google Scholar

[5] M. Bohner and I. M. Stamova, An impulsive delay discrete stochastic neural network fractional-order model and applications in finance. Filomat 32, No 18 (2018), 6339–6352.10.2298/FIL1818339BSearch in Google Scholar

[6] A. Cabada and N. Dimitrov, Nontrivial solutions of non-autonomous Dirichlet fractional discrete problems. Fract. Calc. Appl. Anal. 23, No 4 (2020), 980–995; 10.1515/fca-2020-0051; https://www.degruyter.com/journal/key/fca/23/4/html.Search in Google Scholar

[7] G. E. Chatzarakis, G. M. Selvam, R. Janagaraj, and G. N. Miliaras, Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term. Math. Slovaca 70, No 5 (2020), 1165–1182.10.1515/ms-2017-0422Search in Google Scholar

[8] C. Chen, M. Bohner, and B. Jia, Method of upper and lower solutions for nonlinear Caputo fractional difference equations and its applications. Fract. Calc. Appl. Anal. 22, No 5 (2019), 1307–1320; 10.1515/fca-2019-0069; https://www.degruyter.com/journal/key/fca/22/5/html.Search in Google Scholar

[9] C. Chen, M. Bohner, and B. Jia, Ulam–Hyers stability of Caputo fractional difference equations. Math. Methods Appl. Sci. 42, No 18 (2019), 7461–7470.10.1002/mma.5869Search in Google Scholar

[10] C. Chen, M. Bohner, and B. Jia, Existence and uniqueness of solutions for nonlinear Caputo fractional difference equations. Turkish J. Math. 44, No 3 (2020), 857–869.10.3906/mat-1904-29Search in Google Scholar

[11] C. Chen, R. Mert, B. Jia, L. Erbe, and A. Peterson, Gronwall's inequality for a nabla fractional difference system with a retarded argument and an application. J. Difference Equ. Appl. 25, No 6 (2019), 855–868.10.1080/10236198.2019.1581180Search in Google Scholar

[12] R. Dahal and C. S. Goodrich, A uniformly sharp convexity result for discrete fractional sequential differences. Rocky Mountain J. Math. 49, No 8 (2019), 2571–2586.10.1216/RMJ-2019-49-8-2571Search in Google Scholar

[13] K. Diethelm, The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin (2010).10.1007/978-3-642-14574-2Search in Google Scholar

[14] K. Diethelm and N. J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265, No 2 (2002), 229–248.10.1007/978-3-642-14574-2Search in Google Scholar

[15] K. Diethelm, N. J. Ford, A. D. Freed, and Y. F. Luchko, Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Engrg. 194, Nos 6-8 (2005), 743–773.10.1016/j.cma.2004.06.006Search in Google Scholar

[16] P. W. Eloe, C. M. Kublik, and J. T. Neugebauer, Comparison of Green's functions for a family of boundary value problems for fractional difference equations. J. Difference Equ. Appl. 25, No 6 (2019), 776–787.10.1080/10236198.2018.1531129Search in Google Scholar

[17] N. Fewster-Young and C. C. Tisdell, The existence of solutions to second-order singular boundary value problems. Nonlinear Anal. 75, No 13 (2012), 4798–4806.10.1016/j.na.2012.03.029Search in Google Scholar

[18] M. Garić-Demirović, S. Moranjkić, M. Nurkanović, and Z. Nurkanović, Stability, Neimark–Sacker bifurcation, and approximation of the invariant curve of certain homogeneous second-order fractional difference equation. Discrete Dyn. Nat. Soc. 2020 (2020), Art. ID 6254013, 12 pages.10.1155/2020/6254013Search in Google Scholar

[19] C. S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem. Int. J. Difference Equ. 5, No 2 (2010), 195–216.Search in Google Scholar

[20] C. S. Goodrich, Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 217, No 9 (2011), 4740–4753.10.1016/j.amc.2010.11.029Search in Google Scholar

[21] C. S. Goodrich, Some new existence results for fractional difference equations. Int. J. Dyn. Syst. Differ. Equ. 3, Nos 1-2 (2011), 145–162.10.1504/IJDSDE.2011.038499Search in 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. 20, No 1 (2021), 339–358.10.3934/cpaa.2020269Search in Google Scholar

[23] C. S. Goodrich and A. C. Peterson, Discrete Fractional Calculus. Springer, Cham (2015).10.1007/978-3-319-25562-0Search in Google Scholar

[24] P. Hartman, On boundary value problems for systems of ordinary, nonlinear, second order differential equations. Trans. Amer. Math. Soc. 96 (1960), 493–509.10.1090/S0002-9947-1960-0124553-5Search in Google Scholar

[25] J. Henderson, Existence of local solutions for fractional difference equations with Dirichlet boundary conditions. J. Difference Equ. Appl. 25, No 6 (2019), 751–756.10.1080/10236198.2018.1505882Search in Google Scholar

[26] J. Henderson and J. T. Neugebauer, Smallest eigenvalues for a fractional difference equation with right focal boundary conditions. J. Difference Equ. Appl. 23, No 8 (2017), 1317–1323.10.1080/10236198.2017.1321641Search in Google Scholar

[27] M. Holm, Sum and difference compositions in discrete fractional calculus. Cubo 13, No 3 (2011), 153–184.10.4067/S0719-06462011000300009Search in Google Scholar

[28] M. N. Islam and J. T. Neugebauer, Initial value problems for fractional differential equations of Riemann-Liouville type. Adv. Dyn. Syst. Appl. 15, No 2 (2020), 113–124.Search in Google Scholar

[29] J. M. Jonnalagadda, Discrete fractional Lyapunov-type inequalities in nabla sense. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 27, No 6 (2020), 397–419.Search in Google Scholar

[30] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006).Search in Google Scholar

[31] V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications. Commun. Appl. Anal. 11, Nos 3-4 (2007), 395–402.Search in Google Scholar

[32] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations. Nonlinear Anal. 69, No 8 (2008), 2677–2682.10.1016/j.na.2007.08.042Search in Google Scholar

[33] D. B. Pachpatte, A. S. Bagwan, and A. D. Khandagale, Existence of solutions to discrete boundary value problem of fractional difference equations. Malaya J. Mat. 8, No 3 (2020), 832–837.10.26637/MJM0803/0017Search in Google Scholar

[34] I. Podlubny, Fractional Differential Equations. Academic Press, Inc., San Diego, CA (1999).Search in Google Scholar

[35] A. Pratap, R. Raja, J. Cao, C. Huang, M. Niezabitowski, and O. Bagdasar, Stability of discrete-time fractional-order time-delayed neural networks in complex field. Math. Methods Appl. Sci. 44, No 1 (2021), 419–440.10.1002/mma.6745Search in Google Scholar

[36] H. G. Schuster (editor), Reviews of Nonlinear Dynamics and Complexity. Vol. 1. Wiley-VCH Verlag Berlin GmbH, Weinheim (2008).10.1002/9783527626359Search in Google Scholar

[37] A. G. M. Selvam, J. Alzabut, R. Dhineshbabu, S. Rashid, and M. Rehman, Discrete fractional order two-point boundary value problem with some relevant physical applications. J. Inequal. Appl. 2020 (2020), Paper No 221, 19 pages.10.1186/s13660-020-02485-8Search in Google Scholar

Received: 2021-04-24
Revised: 2021-10-19
Published Online: 2021-11-22
Published in Print: 2021-12-20

© 2021 Diogenes Co., Sofia

Downloaded on 9.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2021-0077/html?recommended=sidebar
Scroll to top button