Existence of solutions for a three-point Hadamard fractional resonant boundary value problem

• Autorzy: Gholami Yousef

Identyfikatory

DOI

10.1515/jaa-2021-2084

• Języki publikacji: EN

Abstrakty

EN

This article focuses on the creation of an existence theorem for a fully nonlinear Hadamard fractional boundary value problem subject to special three-point boundary conditions. By making use of the coincidence degree theory, it is proved that our governing problem makes resonance, that is, the linear part of the differential operator is non-invertible (equally, the corresponding linear problem has at least one nontrivial solution). Constructing some hypotheses on the linear part of the differential operator, nonlinearities and boundary conditions, we give an existence criterion for at least one solution of the fractional-order resonant boundary value problem under study. At the end, a numerical example is presented to illustrate the obtained theoretical results.

Słowa kluczowe

EN

fractional differential equations; Hadamard fractional derivative; nonlinear boundary value problems; multi-point boundary conditions; resonance; existence; coincidence degree theory

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2023
• Tom: Vol. 29, nr 1
• Strony: 31--47
• Opis fizyczny: Bibliogr. 54 poz., wykr.

Twórcy

autor

Gholami Yousef

• Department of Applied Mathematics, Sahand University of Technology, P.O. Box: 51335-1996, Tabriz, Iran

• https://orcid.org/0000-0002-9460-9997

Bibliografia

• [1] B. Ahmad, S. K. Ntouyas and A. Alsaedi, New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions, Bound. Value Probl. 2013 (2013), Paper No. 275.
• [2] A. Ahmadova, I. T. Huseynov, A. Fernandez and N. I. Mahmudov, Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 97 (2021), Article ID 105735.
• [3] A. Ahmadova, I. T. Huseynov and N. I. Mahmudov, Controllability of fractional stochastic delay dynamical systems, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 46 (2020), no. 2, 294-320.
• [4] A. Ahmadova and N. I. Mahmudov, Langevin differential equations with general fractional orders and their applications to electric circuit theory, J. Comput. Appl. Math. 388 (2021), Paper No. 113299.
• [5] A. Amara, S. Etemad and S. Rezapour, Topological degree theory and Caput-Hadamard fractional boundary value problems, Adv. Difference Equ. 369 (2020), Paper No. 369.
• [6] T. M. Atanackovic, S. Pilipovic, B. Stankovic and D. Zorica, Fractional Calculus with Applications in Mechanics, Wiley, New York, 2014.
• [7] A. Babakhani and V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003), no. 2, 434-442.
• [8] Z. Bai, Solvability for a class of fractional m-point boundary value problem at resonance, Comput. Math. Appl. 62 (2011), no. 3, 1292-1302.
• [9] Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equations, J. Math. Anal. Appl. 311 (2005), 495-505.
• [10] Z. Bai and Y. Zhang, The existence of solutions for a fractional multi-point boundary value problem, Comput. Math. Appl. 60 (2010), no. 8, 2364-2372.
• [11] D. Baleanu, Z. B. Guvenc and J. A. T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Dordrecht, 2010.
• [12] D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer, New York, 2012.
• [13] A. Benham and N. Kosmatov, Multiple positive solutions of a fourth-order boundary value problem, Mediterr. J. Math. 14 (2017), no. 2, Paper No. 78.
• [14] K. Buvaneswari, P. Karthikeyan and D. Baleanu, On a system of fractional coupled hybrid Hadamard differential equations with terminal conditions, Adv. Difference. Equ. 2020 (2020), Paper No. 419.
• [15] L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), no. 3, 743-748.
• [16] S. Etemad, S. Rezapour and F. M. Saker, On a fractional Caputo-Hadamard problem with boundary value conditions via different orders of the Hadamard fractional operators, Adv. Difference. Equ. 2020 (2020), Paper No. 272.
• [17] R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977.
• [18] S. Gao, Q. Wang and B. Wu, Existence and global exponential stability of periodic solutions for coupled control systems on networks with feedback and time delays, Commun. Nonlinear Sci. Numer. Simul. 63 (2018), 72-87.
• [19] L.-J. Guo, J.-P. Sun and Y.-H. Zhao, Existence of positive solutions for nonlinear third-order three-point boundary value problems, Nonlinear Anal. 68 (2008), no. 10, 3151-3158.
• [20] X. Hao, M. Zuo and L. Liu, Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities, Appl. Math. Lett. 82 (2018), 24-31.
• [21] J. Henderson and R. Luca, Existence of positive solutions for a system of semipositone fractional boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Paper No. 22.
• [22] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [23] I. T. Huseynov, A. Ahmadova, A. Fernandez and N. I. Mahmudov, Explicit analytical solutions of incommensurate fractional differential equation systems, Appl. Math. Comput. 390 (2021), Paper No. 125590.
• [24] C. Kiataramkul, S. K. Ntouyas, J. Tariboon and A. Kijjathanakorn, Generalized Sturm-Liouville and Langevin equations via Hadamard fractional derivatives with anti-periodic boundary conditions, Bound. Value Probl. 2016 (2016), Paper No. 217.
• [25] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Nath. Stud. 204, Elsevier, Amsterdam, 2006.
• [26] V. Kiryakova, Generalized Fractional Calculus and Applications, John Wiley & Sons, New York, 1994.
• [27] N. Kosmatov, A symmetric solution of a multipoint boundary value problem at resonance, Abstr. Appl. Anal. 2006 (2006), Article ID 54121.
• [28] N. Kosmatov, Multi-point boundary value problems on time scales at resonance, J. Math. Anal. Appl. 323 (2006), no. 1, 253-266.
• [29] N. Kosmatov, A boundary value problem of fractional order at resonance, Electron. J. Differential Equations 2010 (2010), Paper No. 135.
• [30] N. Kosmatov, A singular non-local problem at resonance, J. Math. Anal. Appl. 394 (2012), no. 1, 425-431.
• [31] N. Kosmatov and W. Jiang, Resonant functional problems of fractional order, Chaos Solitons Fractals 91 (2016), 573-579.
• [32] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. Lond. Math. Soc. (2) 63 (2001), no. 3, 690-704.
• [33] R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), no. 4, 673-688.
• [34] Y. Li, J. Qin and B. Li, Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays, Neurocomputing 292 (2018), no. 31, 91-103.
• [35] S. Liang and J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Anal. 71 (2009), no. 11, 5545-5550.
• [36] S. Liang and J. Zhang, Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval, Math. Comput. Model. 54 (2011), no. 5-6, 1334-1346.
• [37] W. Liu, L. Liu and Y. Wu, Existence of solutions for integral boundary value problems of singular Hadamard-type fractional differential equations on infinite interval, Adv. Difference. Equ. 2020 (2020), Paper No. 274.
• [38] Y. Liu and X. Liu, Existence of solutions of resonant boundary value problems for fractional differential equations, Diff. Equ. Cont. Proc. (2012), no. 3, 70-90.
• [39] N. I. Mahmudov, I. T. Huseynov, N. A. Aliev and F. A. Aliev, Analytical approach to a class of Bagley-Torvik equations, TWMS J. Pure Appl. Math. 11 (2020), no. 2, 238-258.
• [40] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, World Scientific, Singapore, 2010.
• [41] K. S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractioal Differential Equation, John Wiley, New York, 1993.
• [42] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
• [43] I. Petras, Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation, Springer, Berlin, 2011.
• [44] I. Podlubny, Fractional Differential Equations, Math. Sci. Appl. 19, Academic Press, New York, 1999.
• [45] S. N. Rao, M. Singh and M. Z. Meetei, Multiplicity of positive solutions for Hadamard fractional differential equations with p-Laplacian operator, Bound. Value Probl. 2020 (2020), Paper No. 43.
• [46] S. S. Ray, Fractional Calculus with Applications for Nuclear Reactor Dynamics, CRC Press, Boca Raton, 2016.
• [47] C.-S. Sin and L. Zheng, Existence and uniqueness of global solutions of Caputo-type fractional differential equations, Fract. Calc. Appl. Anal. 19 (2016), no. 3, 765-774.
• [48] J. Wang, The existence of positive solutions for the one-dimensional p-Laplacian, Proc. Amer. Math. Soc. 125 (1997), no. 8, 2275-2283.
• [49] W. Yukunthorn, S. Suantai, S. K. Ntouyas and J. Tariboon, Boundary value problems for impulsive multi-order Hadamard fractional differential equations, Bound. Value Probl. 2015 (2015), Paper No. 48.
• [50] R. Zafar, M. ur Rehman and M. Shams, On Caputo modification of Hadamard-type fractional derivative and fractional Taylor series, Adv. Difference Equ. 2000 (2020), Paper No. 219.
• [51] Y. Zhang and Z. Bai, Existence of solutions for nonlinear fractional three-point boundary value problems at resonance, J. Appl. Math. Comput. 36 (2011), no. 1-2, 417-440.
• [52] W. Zhang and W. Liu, Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval, Bound. Value Probl. 2018 (2018), Paper No. 134.
• [53] K. Zhao, Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays, Dyn. Syst. 30 (2015), no. 2, 208-223.
• [54] X. Zhao and W. Ge, Unbounded solutions for a fractional boundary value problems on the infinite interval, Acta Appl. Math. 109 (2010), no. 2, 495-505.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).