Multiplicity result to a system of over-determined Fredholm fractional integro-differential equations on time scales

Xing Hu; Yongkun Li; Xing Hu; Yongkun Li

doi:10.3934/math.2022149

AIMS Mathematics

2022, Volume 7, Issue 2: 2646-2665. doi: 10.3934/math.2022149

Previous Article Next Article

Research article

Multiplicity result to a system of over-determined Fredholm fractional integro-differential equations on time scales

Xing Hu ,
Yongkun Li ^,

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received: 26 September 2021 Accepted: 09 November 2021 Published: 18 November 2021
MSC : 34A08, 26A33, 34B15, 34N05

In present paper, several conditions ensuring existence of three distinct solutions of a system of over-determined Fredholm fractional integro-differential equations on time scales are derived. Variational methods are utilized in the proofs.
- Riemann-Liouville derivatives,
- fractional boundary value problem,
- time scales,
- variational methods,
- critical points,
- multiplicity
Citation: Xing Hu, Yongkun Li. Multiplicity result to a system of over-determined Fredholm fractional integro-differential equations on time scales[J]. AIMS Mathematics, 2022, 7(2): 2646-2665. doi: 10.3934/math.2022149

Related Papers:

Abstract

In present paper, several conditions ensuring existence of three distinct solutions of a system of over-determined Fredholm fractional integro-differential equations on time scales are derived. Variational methods are utilized in the proofs.

References

[1]	S. Rafeeq, H. Kalsoom, S. Hussain, S. Rashid, Y. M. Chu, Delay dynamic double integral inequalities on time scales with applications, Adv. Differ. Equ., 2020 (2020), 40. doi: 10.1186/s13662-020-2516-3. doi: 10.1186/s13662-020-2516-3
[2]	Y. Li, S. Shen, Pseudo almost periodic synchronization of Clifford-valued fuzzy cellular neural networks with time-varying delays on time scales, Adv. Differ. Equ., 2020 (2020), 593. doi: 10.1186/s13662-020-03041-w. doi: 10.1186/s13662-020-03041-w
[3]	Y. Li, S. Shen, Compact almost automorphic function on time scales and its application, Qual. Theory Dyn. Syst., 20 (2021), 86. doi: 10.1007/s12346-021-00522-5. doi: 10.1007/s12346-021-00522-5
[4]	C. Park, Y. M. Chu, M. S. Saleem, S. Mukhtar, N. Rehman, Hermite-Hadamard-type inequalities for $\eta _h$-convex functions via $\Psi$-Riemann-Liouville fractional integrals, Adv. Differ. Equ., 2020 (2020), 602. doi: 10.1186/s13662-020-03068-z. doi: 10.1186/s13662-020-03068-z
[5]	M. U. Awan, S. Talib, Y. M. Chu, M. A. Noor, K. I. Nooret, Some new refinements of Hermite-Hadamard-type inequalities involving Riemann-Liouville fractional integrals and applications, Math. Probl. Eng., 2020 (2020), 3051920. doi: 10.1155/2020/3051920. doi: 10.1155/2020/3051920
[6]	X. Qiang, A. Mahboob, Y. M. Chu, Numerical approximation of fractional-order Volterra integrodifferential equation, J. Funct. Space., 2020 (2020), 8875792. doi: 10.1155/2020/8875792. doi: 10.1155/2020/8875792
[7]	M. K. Wang, S. Rashid, Y. Karaca, D. Baleanu, Y. M. Chu, New multi-functional approach for kth-order differentiability governed by fractional calculus via approximately generalized $(\Psi, \hbar)$-convex functions in Hilbert space, Fractals, 29 (2021), 2140019. doi: 10.1142/S0218348X21400193. doi: 10.1142/S0218348X21400193
[8]	S. B. Chen, S. Soradi-Zeid, M. Alipour, Y. M. Chu, J. F. G$\acute{o}$mez-Aguilar, H. Jahanshahi, Optimal control of nonlinear time-delay fractional differential equations with Dickson polynomials, Fractals, 29 (2021), 2150079. doi: 10.1142/S0218348X21500791. doi: 10.1142/S0218348X21500791
[9]	Y. Li, Y. Wang, B. Li, Existence and finite-time stability of a unique almost periodic positive solution for fractional-order Lasota-Wazewska red blood cell models, Int. J. Biomath., 13 (2020), 2050013. doi: 10.1142/S1793524520500138. doi: 10.1142/S1793524520500138
[10]	K. S. Miller, B. Ross, An Introduction to the fractional calculus and fractional difference equations, New York: Wiley, 1993.
[11]	K. Oldham, J. Spanier, The fractional calculus: Theory and applications of differentiation and integration to arbitrary order, New York: Dover Publications, 2002.
[12]	I. Podlubny, Fractional differential equations, New York: Academic Press, 1999.
[13]	C. Goodrich, A. C. Peterson, Discrete fractional calculus, Cham: Springer, 2015.
[14]	M. Al-Qurashi, S. Rashid, Y. Karaca, Z. Hammouch, D. Baleanu, Y. M. Chu, Achieving more precise bounds based on double and triple integral as proposed by generalized proportional fractional operators in the Hilfer sense, Fractals, 29 (2021), 2140027. doi: 10.1142/S0218348X21400272. doi: 10.1142/S0218348X21400272
[15]	P. O. Mohammed, T. Abdeljawad, F. Jarad, Y. M. Chu, Existence and uniqueness of uncertain fractional backward difference equations of Riemann-Liouville type, Math. Probl. Eng., 2020 (2020), 6598682. doi: 10.1155/2020/6598682. doi: 10.1155/2020/6598682
[16]	J. M. Shen, S. Rashid, M. A. Noor, R. Ashraf, Y. M. Chu, Certain novel estimates within fractional calculus theory on time scales, AIMS Math., 5 (2020), 6073–6086. doi: 10.3934/math.2020390. doi: 10.3934/math.2020390
[17]	N. Benkhettou, A. Hammoudi, D. F. M. Torres, Existence and uniqueness of solution for a fractional Riemann-Liouville initial value problem on time scales, J. King Saud Univ. Sci., 28 (2016), 87–92. doi: 10.1016/j.jksus.2015.08.001. doi: 10.1016/j.jksus.2015.08.001
[18]	M. Malik, V. Kumar, Existence, stability and controllability results of coupled fractional dynamical system on time scales, Bull. Malays. Math. Sci. Soc., 43 (2020), 3369–3394. doi: 10.1007/s40840-019-00871-0. doi: 10.1007/s40840-019-00871-0
[19]	I. Area, A. Canada, J. $\acute{A}$. Cid, D. Franco, E. Liz, R. L. Pouso, et al., Nonlinear analysis and boundary value problems, Santiago de Compostela: Springer, 2018.
[20]	K. Mekhalfi, D. F. M. Torres, Generalized fractional operators on time scales with application to dynamic equations, Eur. Phys. J-Spec. Top., 226 (2017), 3489–3499. doi: 10.1140/epjst/e2018-00036-0. doi: 10.1140/epjst/e2018-00036-0
[21]	M. R. Sidi Ammi, D. F. M. Torres, Existence and uniqueness results for a fractional Riemann$-$Liouville nonlocal thermistor problem on arbitrary time scales, J. King Saud Univ. Sci., 30 (2018), 381–385. doi: 10.1016/j.jksus.2017.03.004. doi: 10.1016/j.jksus.2017.03.004
[22]	A. Souahi, A. Guezane-Lakoud, R. Khaldi, On some existence and uniqueness results for a class of equations of order $0 < \alpha\leq1$ on arbitrary time scales, Int. J. Differ. Equ., 2016 (2016), 7327319. doi: 10.1155/2016/7327319. doi: 10.1155/2016/7327319
[23]	D. Vivek, K. Kanagarajan, S. Sivasundaram, On the behavior of solutions of fractional differential equations on time scale via Hilfer fractional derivatives, Fract. Calc. Appl. Anal., 21 (2018), 1120–1138. doi: 10.1515/fca-2018-0060. doi: 10.1515/fca-2018-0060
[24]	M. Yeung, D. Walton, Data point reduction for NC tool path generation on over-determined data set, Control. Eng. Pract., 1 (1993), 117–122. doi: 10.1016/0967-0661(93)92181-3. doi: 10.1016/0967-0661(93)92181-3
[25]	P. Cordaro, J. Hounie, Local solvability in $C_c^\infty$ of over-determined systems of vector fields, J. Funct. Anal., 87 (1989), 231–249. doi: 10.1016/0022-1236(89)90009-8. doi: 10.1016/0022-1236(89)90009-8
[26]	F. An, Y. Cao, B. Liu, Optimized decentralized filtered-x least mean square algorithm for over-determined systems with periodic disturbances, J. Sound Vib., 491 (2021), 115763. doi: 10.1016/j.jsv.2020.115763. doi: 10.1016/j.jsv.2020.115763
[27]	E. Y. Belay, W. Godah, M. Szelachowska, R. Tenzer, ETH$-$GM21: A new gravimetric geoid model of Ethiopia developed using the least-squares collocation method, J. Afr. Earth Sci., 183 (2021), 104313. doi: 10.1016/j.jafrearsci.2021.104313. doi: 10.1016/j.jafrearsci.2021.104313
[28]	M. Eguchi, K. Wada, Over-determined estimation of pulse transfer function model by least-squares method, IFAC Proc. Vol., 24 (1991), 823–828. doi: 10.1016/S1474-6670(17)52451-9. doi: 10.1016/S1474-6670(17)52451-9
[29]	S. Shang, Y. Tian, Z. Bai, Y. Yue, Infinitely many solutions for second-order impulsive differential inclusions with relativistic operator, Qual. Theor. Dyn. Syst., 20 (2020), 47. doi: 10.1007/s12346-021-00481-x. doi: 10.1007/s12346-021-00481-x
[30]	Z. Bai, S. Sun, Z. Du, Y. Q. Chen, The green function for a class of Caputo fractional differential equations with a convection term, Fract. Calc. Appl. Anal., 23 (2020), 787–798. doi: 10.1515/fca-2020-0039. doi: 10.1515/fca-2020-0039
[31]	W. Lian, Z. Bai, Eigenvalue criteria for existence of positive solutions to fractional boundary value problem, J. Funct. Space., 2020 (2020), 8196918. doi: 10.1155/2020/8196918. doi: 10.1155/2020/8196918
[32]	H. Wang, Boundary value problems for a class of first-order fuzzy delay differential equations, Mathematics-Basel, 8 (2020), 683. doi: 10.3390/math8050683. doi: 10.3390/math8050683
[33]	H. Wang, Monotone iterative method for boundary value problems of fuzzy differential equations, J. Intell. Fuzzy Syst., 30 (2016), 831–843. doi: 10.3233/IFS-151806. doi: 10.3233/IFS-151806
[34]	H. Wang, Two-point boundary value problems for first$-$order nonlinear fuzzy differential equation, J. Intell. Fuzzy Syst., 30 (2016), 3335–3347. doi: 10.3233/IFS-152081. doi: 10.3233/IFS-152081
[35]	X. Tian, Y. Zhang, Fractional time-scales Noether theorem with Caputo $\Delta$ derivatives for Hamiltonian systems, Appl. Math. Comput., 393 (2021), 125753. doi: 10.1016/j.amc.2020.125753. doi: 10.1016/j.amc.2020.125753
[36]	C. J. Song, Y. Zhang, Noether theorem for Birkhoffian systems on time scales, J. Math. Phys., 56 (2015), 102701. doi: 10.1063/1.4932607. doi: 10.1063/1.4932607
[37]	J. Zhou, Y. Li, Variational approach to a class of second order Hamiltonian systems on time scales, Acta Appl. Math., 117 (2012), 47–69. doi: 10.1007/s10440-011-9649-z. doi: 10.1007/s10440-011-9649-z
[38]	J. Zhou, Y. Li, Variational approach to a class of $p-$Laplacian systems on time scales, Adv. Differ. Equ., 2013 (2013), 297. doi: 10.1186/1687-1847-2013-297. doi: 10.1186/1687-1847-2013-297
[39]	X. Tian, Y. Zhang, Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems, Roy. Soc. Open Sci., 6 (2019), 191248. doi: 10.1098/rsos.191248. doi: 10.1098/rsos.191248
[40]	E. Shivanian, To study existence of at least three weak solutions to a system of over-determined Fredholm fractional integro-differential equations, Commun. Nonlinear Sci., 101 (2021), 105892. doi: 10.1016/j.cnsns.2021.105892. doi: 10.1016/j.cnsns.2021.105892
[41]	D. F. M. Torres, Cauchy's formula on nonempty closed sets and a new notion of Riemann-Liouville fractional integral on time scales, Appl. Math. Lett., 121 (2021), 107407. doi: 10.1016/j.aml.2021.107407. doi: 10.1016/j.aml.2021.107407
[42]	R. P. Agarwal, V. Otero-Espinar, K. Perera, D. R. Vivero, Basic properties of Sobolev's spaces on time scales, Adv. Differ. Equ., 2006 (2006), 38121. doi: 10.1155/ADE/2006/38121. doi: 10.1155/ADE/2006/38121
[43]	X. Hu, Y. Li, Fractional Sobolev space on time scales and its application to a fractional boundary value problem on time scales, J. Funct. Space., 2021 (2021), In Press.
[44]	G. Bonanno, S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1–10. doi: 10.1080/00036810903397438. doi: 10.1080/00036810903397438

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)