SOLVABILITY FOR A COUPLED SYSTEM OF PERTURBED IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH PERIODIC AND ANTI-PERIODIC BOUNDARY CONDITIONS

Wei Zhang; Jinbo Ni

doi:10.11948/20210052

2021 Volume 11 Issue 6

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Wei Zhang, Jinbo Ni. SOLVABILITY FOR A COUPLED SYSTEM OF PERTURBED IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH PERIODIC AND ANTI-PERIODIC BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2876-2894. doi: 10.11948/20210052

Citation:

Wei Zhang, Jinbo Ni. SOLVABILITY FOR A COUPLED SYSTEM OF PERTURBED IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH PERIODIC AND ANTI-PERIODIC BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2876-2894. doi: 10.11948/20210052

SOLVABILITY FOR A COUPLED SYSTEM OF PERTURBED IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH PERIODIC AND ANTI-PERIODIC BOUNDARY CONDITIONS

Wei Zhang^1,2, ,,
Jinbo Ni¹

1.
School of mathematics and big data, Anhui University of Science and Technology, Huainan, Anhui, 232001, China
2.
School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China

Corresponding author: Email: zhangwei_azyw@163.com (W. Zhang)
Fund Project: The authors were supported by the Key Program of University Natural Science Research Fund of Anhui Province (KJ2020A0291)

Abstract

Abstract

In this paper, we first provide a refinement result for the abstract continuation theorem for k-set contractions. The new version of the theorem is equivalent to the usual one and it better adapts to study the existence of solutions for nonlinear differential equations. Then we discuss a new class of coupled system of implicit fractional boundary value problem. The nonlinear terms of equations involving perturbations, and the boundary conditions are constituted by periodic and anti-periodic boundary conditions. Based on the abstract continuation theorem for k-set contractions, an interesting existence result is obtained. Finally, an example is constructed for illustrating the application of our main results.
- Implicit fractional differential equation /
- periodic and anti-periodic boundary conditions /
- abstract continuation theorem /
- k-set contraction /
- coupled system
MSC: 34A08, 34B15

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References

[1]	B. Ahmad, N. Alghamdi, A. Alsaedi and S. K. Ntouyas, A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions, Fract. Calc. Appl. Anal., 2019, 22(3), 601-618. doi: 10.1515/fca-2019-0034 CrossRef Google Scholar
[2]	B. Ahmad, A. Alsaedi, S. K. Ntouyas and J. Tariboon, Hadamard-type fractional differential equations, inclusions and inequalities, Springer, Cham, 2017. Google Scholar
[3]	M. Ahmad, J. Jiang, A. Zada, S. O. Shah and J. Xu, Analysis of coupled system of implicit fractional differential equations involving Katugampola-Caputo fractional derivative, Complexity, 2020, 2020, Art. ID 9285686. Google Scholar
[4]	N. Ahmad, Z. Ali, K. Shah, A. Zada and G. ur Rahman, Analysis of implicit type nonlinear dynamical problem of impulsive fractional differential equations, Complexity, 2018, 2018, Art. ID 6423974. Google Scholar
[5]	Z. Ali, A. Zada and K. Shah, Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem, Bound. Value Probl., 2018. DOI: 10.1186/s13661-018-1096-6. CrossRef Google Scholar
[6]	Z. Ali, A. Zada and K. Shah, On Ulam's stability for a coupled systems of nonlinear implicit fractional differential equations, Bull. Malays. Math. Sci. Soc., 2019, 42(5), 2681-2699. doi: 10.1007/s40840-018-0625-x CrossRef Google Scholar
[7]	Asma, G. U. Rahman and K. Shah, Mathematical analysis of implicit impulsive switched coupled evolution equations, Results Math., 2019, 74(4), Art. No. 142. doi: 10.1007/s00025-019-1066-z CrossRef Google Scholar
[8]	D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional calculus. Models and numerical methods, World Scientific Publishing, Singapore, 2017. Google Scholar
[9]	J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel Dekker, New York, 1980. Google Scholar
[10]	M. Benchohra, S. Bouriah and M. A. Darwish, Nonlinear boundary value problem for implicit differential equations of fractional order in banach spaces, Fixed Point Theory, 2017, 18(2), 457-470. doi: 10.24193/fpt-ro.2017.2.36 CrossRef Google Scholar
[11]	M. Benchohra, S. Bouriah and J. R. Graef, Nonlinear implicit differential equations of fractional order at resonance, Electron. J. Differential Equations., 2016, 2016(324), 1-10. Google Scholar
[12]	M. Benchohra, S. Bouriah and J. R. Graef, Boundary value problems for nonlinear implicit Caputo-Hadamard-type fractional differential equations with impulses, Mediterr. J. Math., 2017, 14(5), Art. No. 206. doi: 10.1007/s00009-017-1012-9 CrossRef Google Scholar
[13]	M. Benchohra, S. Bouriah and J. J. Nieto, Existence of periodic solutions for nonlinear implicit Hadamard's fractional differential equations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM., 2018, 112(1), 25-35. doi: 10.1007/s13398-016-0359-2 CrossRef Google Scholar
[14]	M. Benchohra and M. S. Souid, L¹-solutions for implicit fractional order differential equations with nonlocal conditions, Filomat, 2016, 30(6), 1485-1492. doi: 10.2298/FIL1606485B CrossRef Google Scholar
[15]	T. Chen, W. Liu and Z. Hu, A boundary value problem for fractional differential equation with p-Laplacian operator at resonance, Nonlinear Anal., 2012, 75(6), 3210-3217. doi: 10.1016/j.na.2011.12.020 CrossRef Google Scholar
[16]	T. Chen, W. Liu and J. Liu, Existence of solutions for some boundary value problems of fractional p-Laplacian equation at resonance, Bull. Belg. Math. Soc., Simon Stevin, 2013, 20(3), 503-517. Google Scholar
[17]	H. A. Fallahgoul, S. M. Focardi and F. J. Fabozzi, Fractional calculus and fractional processes with applications to financial economics. Theory and application, Elsevier/Academic Press, London, 2017. Google Scholar
[18]	H. Fang and J. Li, On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl., 2001, 259(1), 8-17. doi: 10.1006/jmaa.2000.7340 CrossRef Google Scholar
[19]	P. M. Fitzpatrick and W. V. Petryshyn, Galerkin methods in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations, Trans. Amer. Math. Soc., 1978, 238, 321-340. doi: 10.1090/S0002-9947-1978-0513094-2 CrossRef Google Scholar
[20]	M. Frigon and T. Kaczynski, Boundary value problems for systems of implicit differential equations, J. Math. Anal. Appl., 1993, 179(2), 317-326. doi: 10.1006/jmaa.1993.1353 CrossRef Google Scholar
[21]	A. E. Garcia and J. T. Neugebauer, Solutions of boundary value problems at resonance with periodic and antiperiodic boundary conditions, Involve, 2019, 12(1), 171-180. doi: 10.2140/involve.2019.12.171 CrossRef Google Scholar
[22]	D. Guo, Y. J. Cho and J. Zhu, Partial ordering methods in nonlinear problems, Nova Science Publishers, New York, 2004. Google Scholar
[23]	S. Heikkilä, First order discontinuous implicit differential equations with discontinuous boundary conditions, Nonlinear Anal., 1997, 30(3), 1753-1761. doi: 10.1016/S0362-546X(97)00250-2 CrossRef Google Scholar
[24]	R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000. Google Scholar
[25]	W. Jiang, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 2011, 74(5), 1987-1994. doi: 10.1016/j.na.2010.11.005 CrossRef Google Scholar
[26]	W. Jiang, Solvability of fractional differential equations with p-Laplacian at resonance, Appl. Math. Comput., 2015, 260, 48-56. Google Scholar
[27]	W. Jiang, J. Qiu and C. Yang, The existence of solutions for fractional differential equations with p-Laplacian at resonance, Chaos, 2017, 27(3), 032102. doi: 10.1063/1.4979367 CrossRef Google Scholar
[28]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. Google Scholar
[29]	L. Kong and M. Wang, Existence of solutions for a second order discrete boundary value problem with mixed periodic boundary conditions, Appl. Math. Lett., 2020, 102, 106138. doi: 10.1016/j.aml.2019.106138 CrossRef Google Scholar
[30]	L. Kong and M. Wang, Multiple and particular solutions of a second order discrete boundary value problem with mixed periodic boundary conditions. Electron. J. Qual. Theory Differ. Equ., 2020. DOI: 10.14232/ejqtde.2020.1.47. CrossRef Google Scholar
[31]	Y. Li, Positive periodic solutions for a periodic neutral differential equation with feedback control, Nonlinear Anal. Real World Appl., 2005, 6(1), 145-154. doi: 10.1016/j.nonrwa.2004.08.002 CrossRef Google Scholar
[32]	H. V. Long and N. Dong, An extension of Krasnoselskii's fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty, J. Fixed Point Theory Appl., 2018, 20(1), Art. No. 37. doi: 10.1007/s11784-018-0507-8 CrossRef Google Scholar
[33]	S. Lu, On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments, J. Math. Anal. Appl., 2003, 280(2), 321-333. doi: 10.1016/S0022-247X(03)00049-0 CrossRef Google Scholar
[34]	R. L. Magin, Fractional calculus in bioengineering, Begell House Publishers, Inc., Connecticut, 2006. Google Scholar
[35]	A. D. Mali and K. D. Kucche, Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations, Math. Methods Appl. Sci., 2020, 43(15), 8608-8631. doi: 10.1002/mma.6521 CrossRef Google Scholar
[36]	J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, Providence, R. I., 1979. Google Scholar
[37]	H. Ngo Van and V. Ho, A survey on the initial value problems of fuzzy implicit fractional differential equations, Fuzzy Sets and Systems, 2020, 400, 90-133. doi: 10.1016/j.fss.2019.10.012 CrossRef Google Scholar
[38]	W. V. Petryshyn, Solvability of various boundary value problems for the equation x" = f(t, x, x', x") - y, Pacific J. Math., 1986, 122(1), 169-195. doi: 10.2140/pjm.1986.122.169 CrossRef f(t, x, x', x") - y" target="_blank">Google Scholar
[39]	W. V. Petryshyn and Z. Yu, Existence theorems for higher order nonlinear periodic boundary value problems, Nonlinear Anal., 1982, 6(9), 943-969. doi: 10.1016/0362-546X(82)90013-X CrossRef Google Scholar
[40]	Samina, K. Shah, R. A. Khan and D. Baleanu, Study of implicit type coupled system of non-integer order differential equations with antiperiodic boundary conditions, Math. Methods Appl. Sci., 2019, 42(6), 2033-2042. doi: 10.1002/mma.5496 CrossRef Google Scholar
[41]	K. Shah, A. Ali and S. Bushnaq, Hyers-Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions, Math. Methods Appl. Sci., 2018, 41(17), 8329-8343. doi: 10.1002/mma.5292 CrossRef Google Scholar
[42]	H. Sheng, Y. Chen and T. Qiu, Fractional processes and fractional-order signal processing. Techniques and applications, Springer, London, 2012. Google Scholar
[43]	J. V. D. Sousa and E. C. de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the Ψ-Hilfer operator, J. Fixed Point Theory Appl., 2018, 20(3), Art. No. 96. doi: 10.1007/s11784-018-0587-5 CrossRef Google Scholar
[44]	D. Vivek, K. Kanagarajan and E. M. Elsayed, Nonlocal initial value problems for implicit differential equations with Hilfer-Hadamard fractional derivative, Nonlinear Anal. Model. Control., 2018, 23(3), 341-360. doi: 10.15388/NA.2018.3.4 CrossRef Google Scholar
[45]	D. Vivek, K. Kanagarajan and E. M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 2018, 15(1), Art. No. 15. doi: 10.1007/s00009-017-1061-0 CrossRef Google Scholar
[46]	J. Wang, A. Zada and H. Waheed, Stability analysis of a coupled system of nonlinear implicit fractional anti-periodic boundary value problem, Math. Methods Appl. Sci., 2019, 42(18), 6706-6732. doi: 10.1002/mma.5773 CrossRef Google Scholar
[47]	D. Xue, Fractional-order control systems: Fundamentals and numerical implementations, De Gruyter, Berlin, 2017. Google Scholar
[48]	W. Zhang, W. Liu and T. Xue, Existence and uniqueness results for the coupled systems of implicit fractional differential equations with periodic boundary conditions, Adv. Difference Equ., 2018. DOI: 10.1186/s13662-018-1867-5. CrossRef Google Scholar