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This paper is concerned with the existence and uniqueness of solutions for a coupled system of fractional differential equations with nonlocal and integral boundary conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. The results are explained with the aid of examples. The case of nonlocal strip conditions is also discussed.
Wydawca
Czasopismo
Rocznik
Tom
Strony
91--108
Opis fizyczny
Bibliogr. 36 poz.
Twórcy
autor
- 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
autor
- 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
autor
- 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
autor
- 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
Bibliografia
- [1] Podlubny I. Fractional Differential Equations, Academic Press, San Diego, 1999. ISBN: 0125588402.
- [2] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. ISBN: 0444518320.
- [3] Sabatier J, Agrawal OP, Machado JAT. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007. ISBN: 1402060416, 9781402060410.
- [4] Zhao X, Yang HT, He YQ. Identification of constitutive parameters for fractional viscoelasticity, Commun. Nonlinear Sci. Numer. Simul. 2014; 19 (1): 311-322. doi: 10.1016/j.cnsns.2013.05.019.
- [5] Yang XJ, Srivastava HM. An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives, Commun. Nonlinear Sci. Numer. Simulat. 2015; 29: 499-504. doi: 10.1016/j.cnsns.2015.06.006.
- [6] Yang XJ, Baleanu D, Srivastava HM. Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 2015; 47: 54-60. URL http://dx.doi.org/10.1016/j.aml.2015.02.024.
- [7] Zhang YD, Chen S, Wang SH, Yang JF, Phillips P. Magnetic resonance brain image classification based on weighted-type fractional Fourier transform and nonparallel support vector machine, Int. J. Imag. Syst. Tech. 2015; 25: 317-327. doi: 10.1002/ima.22144.
- [8] Zhang Y, Yang X, Cattani C, Rao RV, Wang S, Phillips P. Tea category identification using a novel fractional Fourier entropy and Jaya algorithm, Entropy. 2016; 18: 77. doi: 10.3390/e18030077.
- [9] Bitsadze A, Samarskii A. On some simple generalizations of linear elliptic boundary problems, MR0247271 Russian Acad. Sci. Dokl. Math. 1969; 10: 398-400.
- [10] Byszewski L. Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 1991; 162 (2): 494-505. doi: 10.1016/0022-247X(91)90164-U.
- [11] Ahmad B, Alsaedi A, Alghamdi BS. Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions, Nonlinear Anal. Real World. Appl. 2008; 9: 1727-1740. doi: 10.1016/j.nonrwa.2007.05.005.
- [12] Čiegis R, Bugajev A. Numerical approximation of one model of the bacterial self-organization, Nonlinear Anal. Model. Control 2012; 17 (3): 253-270.
- [13] Ahmad B, Nieto JJ. Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. 2011; pp. 36. doi: 10.1186/1687-2770-2011-36.
- [14] Alsaedi A, Ntouyas SK, Ahmad B. Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions, Abstr. Appl. Anal. 2013; Art. ID 869837, 17 pages. URL http://dx.doi.org/10.1155/2013/869837.
- [15] Graef JR, Kong L, Kong Q. Application of the mixed monotone operator method to fractional boundary value problems, Fract. Differ. Calc. 2011; 2: 554-567. doi: 10.7153/fdc-02-06.
- [16] Akyildiz FT, Bellout H, Vajravelu K, Van Gorder RS. Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces, Nonlinear Anal. Real World Appl. 2011; 12: 2919-2930. URL http://dx.doi.org/10.1016/j.nonrwa.2011.02.017.
- [17] Bai ZB, Sun W. Existence and multiplicity of positive solutions for singular fractional boundary value problems, Comput. Math. Appl. 2012; 63 (9): 1369-1381. URL http://dx.doi.org/10.1016/j.camwa.2011.12.078.
- [18] Cabada A, Wang G. Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 2012; 389 (1): 403-411. doi: 10.1016/j.jmaa.2011.11.065.
- [19] Wang G, Ahmad B, Zhang L, Agarwal RP. Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. App. Math. 2013; 249: 51-56. doi: 10.1016/j.cam.2013.02.010.
- [20] Ahmad B, Ntouyas SK, Alsaedi A. A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng. 2013; Art. ID 320415, pp. 9.
- [21] O'Regan D, Stanek S. Fractional boundary value problems with singularities in space variables, Nonlinear Dynam. 2013; 71 (4): 641-652. doi: 10.1007/s11071-012-0443-x.
- [22] Graef JR, Kong L, Wang M. Existence and uniqueness of solutions for a fractional boundary value problem on a graph, Fract. Calc. Appl. Anal. 2014; 17: 499-510. doi: 10.2478/s13540-014-0182-4.
- [23] Wang G, Liu S, Zhang L. Eigenvalue problem for nonlinear fractional differential equations with integral boundary conditions, Abstr. Appl. Anal. 2014; Art. ID 916260, 6 pp. URL http://dx.doi.org/10.1155/2014/916260.
- [24] Punzo F, Terrone G. On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Anal. 2014; 98: 27-47. doi: 10.1016/j.na.2013.12.007.
- [25] Zhai C, Xu L. Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter, Commun. Nonlinear Sci. Numer. Simul. 2014; 19: 2820-2827. URL http://dx.doi.org/10.1155/2007/10368.
- [26] Zhang L, Ahmad B, Wang G. Successive iterations for positive extremal solutions of nonlinear fractional differential equations on a half-line, Bull. Aust. Math. Soc. 2015; 91 (1): 116-128 doi: 10.1017/S0004972714000550.
- [27] Ahmad B, Ntouyas SK. Nonlocal fractional boundary value problems with slit-strips boundary conditions, Fract. Calc. Appl. Anal. 2015; 18 (1): 261-280. URL https ://doi.org/10.1515/fca-2015-0017.
- [28] Ahmad B, Nieto JJ. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 2009; 58 (9): 1838-1843. URL http://dx.doi.org/10.1016/j.camwa.2009.07.091.
- [29] Su X. Boundary value problem for a coupled system of nonlinear fractional differential equations, Apr. Math. Lett. 2009; 22 (l): 64-69. URL http://dx.doi.org/10.1016/j.aml.2008.03.001.
- [30] Sun J, Liu Y, Liu G. Existence of solutions for fractional differential systems with antiperiodic boundary conditions, Comput. Math. Appl. 2012; 64 (6): 1557-1566. URL http://dx.doi.org/10.1016/j.camwa.2011.12.083.
- [31] Faieghi M, Kuntanapreeda S, Delavari H, Baleanu D. LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear Dynam. 2013; 72 (1): 301-309. doi: 10.1007/s11071-012-0714-6.
- [32] Senol B, Yeroglu C. Frequency boundary of fractional order systems with nonlinear uncertainties. Franklin Inst. 2013; 350 (7): 1908-1925. doi: 10.1016/j.jfranklin.2013.05.010.
- [33] Henderson J, R. Luca R. Existence and multiplicity of positive solutions for a system of fractional boundary value problems, Bound. Value Probl. 2014: 60, pp. 17. doi: 10.1186/1687-2770-2014-60.
- [34] Kirane M, Ahmad B, Alsaedi A, Al-Yami M. Non-existence of global solutions to a system of fractional diffusion equations, Acta Appl. Math. 2014; 133 (1): 235-248. doi: 10.1007/s10440-014-9865-4.
- [35] Ahmad B, Ntouyas SK. Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput. 2015; 266 (C): 615-622. doi: 10.1016/j.amc.2015.05.116.
- [36] Granas A, Dugundji J. Fixed Point Theory, Springer-Verlag, New York, 2003. ISBN: 978-1-4419-1805-5, 978-0-387-21593-8.
Uwagi
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-03fdba19-a76c-4309-ba33-7215f269ab00