On a boundary value problem for fractional Hahn integro-difference equations with four-point fractional integral boundary conditions

Varaporn Wattanakejorn; Sotiris K. Ntouyas; Thanin Sitthiwirattham; Varaporn Wattanakejorn; Sotiris K. Ntouyas; Thanin Sitthiwirattham

doi:10.3934/math.2022040

AIMS Mathematics

2022, Volume 7, Issue 1: 632-650. doi: 10.3934/math.2022040

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Research article

On a boundary value problem for fractional Hahn integro-difference equations with four-point fractional integral boundary conditions

1.
Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand
2.
Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
3.
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 01 August 2021 Accepted: 07 October 2021 Published: 14 October 2021
MSC : 39A10, 39A13, 39A70

In this paper, we study a boundary value problem consisting of Hahn integro-difference equation supplemented with four-point fractional Hahn integral boundary conditions. The novelty of this problem lies in the fact that it contains two fractional Hahn difference operators and three fractional Hahn integrals with different quantum numbers and orders. Firstly, we convert the given nonlinear problem into a fixed point problem, by considering a linear variant of the problem at hand. Once the fixed point operator is available, we make use the classical Banach's and Schauder's fixed point theorems to establish existence and uniqueness results. An example is also constructed to illustrate the main results. Several properties of fractional Hahn integral that will be used in our study are also discussed.
- fractional Hahn integral,
- fractional Hahn difference,
- boundary value problems,
- existence
Citation: Varaporn Wattanakejorn, Sotiris K. Ntouyas, Thanin Sitthiwirattham. On a boundary value problem for fractional Hahn integro-difference equations with four-point fractional integral boundary conditions[J]. AIMS Mathematics, 2022, 7(1): 632-650. doi: 10.3934/math.2022040

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Abstract

In this paper, we study a boundary value problem consisting of Hahn integro-difference equation supplemented with four-point fractional Hahn integral boundary conditions. The novelty of this problem lies in the fact that it contains two fractional Hahn difference operators and three fractional Hahn integrals with different quantum numbers and orders. Firstly, we convert the given nonlinear problem into a fixed point problem, by considering a linear variant of the problem at hand. Once the fixed point operator is available, we make use the classical Banach's and Schauder's fixed point theorems to establish existence and uniqueness results. An example is also constructed to illustrate the main results. Several properties of fractional Hahn integral that will be used in our study are also discussed.

References

[1]	I. Podlubny, Fractional differential equations, SanDiego: Academic Press, 1999.
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[3]	J. Sabatier, O. P. Agrawal, J. A. Machado, Advances in fractional calculus: Theoretical developments and applications in physics and engineering, Dordrecht: Springer, 2007.
[4]	D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus models and numerical methods, Singapore: World Scientific, 2012.
[5]	S. Rezapour, S. K. Ntouyas, M. Q. Iqbal, A. Hussain, S. Etemad, J. Tariboon, An analytical survey on the solutions of the generalized double-order $\varphi$-integrodifferential equation, J. Funct. Space., 2021 (2021). doi: 10.1155/2021/6667757. doi: 10.1155/2021/6667757
[6]	S. Sitho, S. Etemad, B. Tellab, S. Rezapour, S. K. Ntouyas, J. Tariboon, Approximate solutions of an extended multi-order boundary value problem by implementing two numerical algorithms, Symmetry, 13 (2021), 1–26. doi: 10.3390/sym13081341. doi: 10.3390/sym13081341
[7]	S. Heidarkhani, G. Caristi, A. Salari, Nontrivial solutions for impulsive elastic beam equations of Kirchhoff-type, J. Nonlinear Funct. Anal., 2020 (2020), 1–16. doi: 10.23952/jnfa.2020.4. doi: 10.23952/jnfa.2020.4
[8]	M. Kamenski, G. Petrosyan, C. F. Wen, An existence result for a periodic boundary value problem of fractional semilinear differential equations in Banach spaces, J. Nonlinear Var. Anal., 5 (2021), 155–177. doi: 10.23952/jnva.5.2021.1.10. doi: 10.23952/jnva.5.2021.1.10
[9]	W. Yukunthorn, B. Ahmad, S. K. Ntouyas, J. Tariboon, On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Anal.-Hybri., 19 (2016), 77–92. doi: 10.1016/j.nahs.2015.08.001. doi: 10.1016/j.nahs.2015.08.001
[10]	R. P. Feynman, A. R. Hibbs, Quantum mechanics and path integrals, New York: McGraw-Hill, 1965.
[11]	B. Ahmad, S. Ntouyas, J. Tariboon, Quantum calculus: New concepts, impulsive IVPs and BVPs, inequalities, Singapore: World Scientific, 2016.
[12]	M. H. Annaby, Z. S. Mansour, $q$-fractional calculus and equations, Berlin: Springer, 2012. doi: 10.1007/978-3-642-30898-7.
[13]	V. Kac, P. Cheung, Quantum calculus, New York: Springer, 2002. doi: 10.1007/978-1-4613-0071-7.
[14]	D. L. Jagerman, Difference equations with applications to queues, New York: CRC Press, 2000. doi: 10.1201/9780203909737.
[15]	T. Sitthiwirattham, A. Zeb, S. Chasreechai, Z. Eskandari, M. Tilioua, S. Djilali, Analysis of a discrete mathematical COVID-19 model, Results Phys., 28 (2021), 104668. doi: 10.1016/j.rinp.2021.104668. doi: 10.1016/j.rinp.2021.104668
[16]	K. A. Aldowah, A. B. Malinowska, D. F. M. Torres, The power quantum calculus and variational problems, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms, 19 (2012), 93–116.
[17]	A. M. C. Birto da Cruz, N. Martins, D. F. M. Torres, Symmetric differentiation on time scales, Appl. Math. Lett., 26 (2013), 264–269. doi: 10.1016/j.aml.2012.09.005. doi: 10.1016/j.aml.2012.09.005
[18]	A. M. C. Birto da Cruz, Symmetric quantum calculus, Ph. D. thesis, Universidade de Aveiro, 2012.
[19]	G. C. Wu, D. Baleanu, New applications of the variational iteration method-from differential equations to q-fractional difference equations, Adv. Differ. Equ., 2013 (2013), 1–16. doi: 10.1186/1687-1847-2013-21. doi: 10.1186/1687-1847-2013-21
[20]	J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 2013 (2013), 1–19. doi: 10.1186/1687-1847-2013-282. doi: 10.1186/1687-1847-2013-282
[21]	R. Álvarez-Nodarse, On characterization of classical polynomials, J. Comput. Appl. Math., 196 (2006), 320–337. doi: 10.1016/j.cam.2005.06.046. doi: 10.1016/j.cam.2005.06.046
[22]	R. P. Agarwal, D. Baleanu, V. Hedayati, S. Rezapour, Two fractional derivative inclusion problems via integral boundary condition, Appl. Math. Comput., 257 (2015), 205–212. doi: 10.1016/j.amc.2014.10.082. doi: 10.1016/j.amc.2014.10.082
[23]	R. P. Agarwal, D. Baleanu, S. Rezapour, S. Salehi, The existence of solution for some fractional finite difference equations via sum boundary conditions, Adv. Differ. Equ., 2014 (2014), 1–16. doi: 10.1186/1687-1847-2014-282. doi: 10.1186/1687-1847-2014-282
[24]	N. Nyamoradi, D. Baleanu, R. P. Agarwal, Existence and uniqueness of positivesolutions to fractional boundary valueproblems with nonlinear boundaryconditions, Adv. Differ. Equ., 2013 (2013), 1–11. doi: 10.1186/1687-1847-2013-266. doi: 10.1186/1687-1847-2013-266
[25]	W. Hahn, Über orthogonalpolynome, die q-differenzenlgleichungen genügen, Math. Nachr., 2 (1949), 4–34. doi: 10.1002/mana.19490020103. doi: 10.1002/mana.19490020103
[26]	R. S. Costas-Santos, F. Marcellán, Second structure relation for $q$-semiclassical polynomials of the Hahn Tableau, J. Math. Anal. Appl., 329 (2007), 206–228. doi: 10.1016/j.jmaa.2006.06.036. doi: 10.1016/j.jmaa.2006.06.036
[27]	K. H. Kwon, D. W. Lee, S. B. Park, B. H. Yoo, Hahn class orthogonal polynomials, Kyungpook Math. J., 38 (1998), 259–281.
[28]	M. Foupouagnigni, Laguerre-Hahn orthogonal polynomials with respect to the Hahn operator: Fourth-order difference equation for the $r$th associated and the Laguerre-Freud equations recurrence coefficients, Ph. D. thesis, Université Nationale du Bénin, 1998.
[29]	K. A. Aldwoah, Generalized time scales and associated difference equations, Ph. D. thesis, Cairo University, 2009.
[30]	M. H. Annaby, A. E. Hamza, K. A. Aldwoah, Hahn difference operator and associated Jackson-Nörlund integrals, J. Optim. Theory App., 154 (2012), 133–153. doi: 10.1007/s10957-012-9987-7. doi: 10.1007/s10957-012-9987-7
[31]	F. H. Jackson, Basic integration, Q. J. Math., 2 (1951), 1–16. doi: 10.1093/qmath/2.1.1. doi: 10.1093/qmath/2.1.1
[32]	A. B. Malinowska, D. F. M. Torres, The Hahn quantum variational calculus, J. Optim. Theory App., 147 (2010), 419–442. doi: 10.1007/s10957-010-9730-1, doi: 10.1007/s10957-010-9730-1,
[33]	A. B. Malinowska, D. F. M. Torres, Quantum variational calculus, Cham: Springer, 2014. doi: 10.1007/978-3-319-02747-0.
[34]	A. B. Malinowska, N. Martins, Generalized transversality conditions for the Hahn quantum variational calculus, Optimization, 62 (2013), 323–344. doi: 10.1080/02331934.2011.579967. doi: 10.1080/02331934.2011.579967
[35]	A. E. Hamza, S. M. Ahmed, Theory of linear Hahn difference equations, J. Adv. Math., 4 (2013), 441–461.
[36]	A. E. Hamza, S. M. Ahmed, Existence and uniqueness of solutions of Hahn difference equations, Adv. Differ. Equ., 2013 (2013), 1–15. doi: 10.1186/1687-1847-2013-316. doi: 10.1186/1687-1847-2013-316
[37]	A. E. Hamza, S. D. Makharesh, Leibniz' rule and Fubinis theorem associated with Hahn difference operator, J. Adv. Math., 12 (2016), 6335–6345. doi: 10.24297/jam.v12i6.3836
[38]	T. Sitthiwirattham, On a nonlocal boundary value problem for nonlinear second-order Hahn difference equation with two different $q, \omega$-derivatives, Adv. Differ. Equ., 2016 (2016), 1–25. doi: 10.1186/s13662-016-0842-2. doi: 10.1186/s13662-016-0842-2
[39]	U. Sriphanomwan, J. Tariboon, N. Patanarapeelert, S. K. Ntouyas, T. Sitthiwirattham, Nonlocal boundary value problems for second-order nonlinear Hahn integro-difference equations with integral boundary conditions, Adv. Differ. Equ., 2017 (2017), 1–18. doi: 10.1186/s13662-017-1228-9. doi: 10.1186/s13662-017-1228-9
[40]	J. $\check{C}$erm$\acute{a}$k, L. Nechv$\acute{a}$tal, On $(q, h)$-analogue of fractional calculus, J. Nonlinear Math. Phy., 17 (2010), 51–68. doi: 10.1142/S1402925110000593. doi: 10.1142/S1402925110000593
[41]	J. $\check{C}$erm$\acute{a}$k, T. Kisela, L. Nechv$\acute{a}$tal, Discrete Mittag-Leffler functions in linear fractional difference equations, Abstr. Appl. Anal., 2011 (2011), 1–21. doi: 10.1155/2011/565067. doi: 10.1155/2011/565067
[42]	M. R. S. Rahmat, The $(q, h)$-Laplace transform on discrete time scales, Comput. Math. Appl., 62 (2011), 272–281. doi: 10.1016/j.camwa.2011.05.008. doi: 10.1016/j.camwa.2011.05.008
[43]	M. R. S. Rahmat, On some $(q, h)$-analogues of integral inequalities on discrete time scales, Comput. Math. Appl., 62 (2011), 1790–1797. doi: 10.1016/j.camwa.2011.06.022. doi: 10.1016/j.camwa.2011.06.022
[44]	F. F. Du, B. G. Jai, L. Erbe, A. Peterson, Monotonicity and convexity for nabla fractional $(q, h)$-difference, J. Differ. Equ. Appl., 22 (2016), 1224–1243. doi: 10.1080/10236198.2016.1188089. doi: 10.1080/10236198.2016.1188089
[45]	T. Brikshavana, T. Sitthiwirattham, On fractional Hahn calculus, Adv. Differ. Equ., 2017 (2017), 1–15. doi: 10.1186/s13662-017-1412-y. doi: 10.1186/s13662-017-1412-y
[46]	N. Patanarapeelert, T. Sitthiwirattham, Existence results for fractional Hahn difference and fractional Hahn integral boundary value problems, Discrete Dyn. Nat. Soc., 2017 (2017), 1–13. doi: 10.1155/2017/7895186. doi: 10.1155/2017/7895186
[47]	N. Patanarapeelert, T. Brikshavana, T. Sitthiwirattham, On nonlocal Dirichlet boundary value problem for sequential Caputo fractional Hahn integrodifference equations, Bound. Value Probl., 2018 (2018), 1–17. doi: 10.1186/s13661-017-0923-5. doi: 10.1186/s13661-017-0923-5
[48]	N. Patanarapeelert, T. Sitthiwirattham, On nonlocal Robin boundary value problems for Riemann-Liouville fractional Hahn integrodifference equation, Bound. Value Probl., 2018 (2018), 1–16. doi: 10.1186/s13661-018-0969-z. doi: 10.1186/s13661-018-0969-z
[49]	D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, New York: Academic Press, 1988.

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