Startseite Solvability of Anti-periodic BVPs for Impulsive Fractional Differential Systems Involving Caputo and Riemann–Liouville Fractional Derivatives
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Solvability of Anti-periodic BVPs for Impulsive Fractional Differential Systems Involving Caputo and Riemann–Liouville Fractional Derivatives

  • Yuji Liu EMAIL logo
Veröffentlicht/Copyright: 9. Januar 2018
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

Sufficient conditions are given for the existence of solutions of anti-periodic value problems for impulsive fractional differential systems involving both Caputo and Riemann–Liouville fractional derivatives. We allow the nonlinearities p(t)f(t,x,y,z,w) and q(t)g(t,x,y,z,w) in fractional differential equations to be singular at t=0 and t=1. Both f and g may be super-linear and sub-linear. The analysis relies on some well known fixed point theorems. The initial value problem discussed may be seen as a generalization of some ecological models. An example is given to illustrate the efficiency of the main theorems. Many unsuitable lemmas in recent published papers are pointed out in order not to mislead readers. A conclusion section is given at the end of the paper.

Funding statement: Supported by the Natural Science Foundation of Guangdong province (No:S2011010001900) and the Foundation for High-level talents in Guangdong Higher Education Project.

Acknowledgements:

The author would like to thank the referees and the editors for their careful reading and some useful comments on improving the presentation of this paper.

References

[1] Q. Dai, H. Li, To study blowing-up solutions of a nonlinear system of fractional differential equations (in Chinese). Sci. Sin. Math. 42 (12) (2012) 1205–1212.10.1360/012011-800Suche in Google Scholar

[2] N. Kosmatov, Initial value problems of fractional order with fractional impulsive conditions, Results Math. 63(3) (2013), 1289–1310.10.1007/s00025-012-0269-3Suche in Google Scholar

[3] M. Kirane, S. A. Malik, The profile of blowing-up solutions to a nonlinear system of fractional differential equations, Nonlinear Anal.: TMA 73(12) (2010), 3723–3736.10.1016/j.na.2010.06.088Suche in Google Scholar

[4] M. Kirane, M. Medved, N. Tata, On the nonexistence of blowing-up solutions to a fractional functional-differential equation, Georgian Math. J. 19(1) (2012), 127–144.10.1515/gmj-2012-0006Suche in Google Scholar

[5] T. A. M. Langlands, B. I. Henry, S. L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions, J. Math. Biol. 59(6) (2009), 761–808.10.1007/s00285-009-0251-1Suche in Google Scholar PubMed

[6] C. Li, G. Chen, Chaos and hyperchaos in the fractional-order Rossler equations, Physica A 341(2004), 55–61.10.1016/j.physa.2004.04.113Suche in Google Scholar

[7] Y. Liu, New existence results for positive solutions of boundary value problems for coupled systems of multi-term fractional differential equations, Hacettepe J. Math. Stat. 45(2) (2016), 391–416.10.15672/HJMS.20164512499Suche in Google Scholar

[8] Y. Liu, B. Ahmad, R. P Agarwal, Existence of solutions for a coupled system of nonlinear fractional differential equations with fractional boundary conditions on the half-line, Adv. Differ. Equ. 46 (2013), 4618pages.10.1186/1687-1847-2013-46Suche in Google Scholar

[9] Z. M. Odibat, S. Momani, Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order, Int. J. Nonlinear Sci. Numer. Simul. 7(1) (2006), 27–34.10.1515/IJNSNS.2006.7.1.27Suche in Google Scholar

[10] Z. Wei, Q. Li, J. Che, Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl. 367(1) (2010), 260–272.10.1016/j.jmaa.2010.01.023Suche in Google Scholar

[11] X. Yang, D. Baleanu, M. Lazarević, M. Cajić, Fractal boundary value problems for integral and differential equations with local fractional operators, Thermal Sci. 19(2015), 959–966.10.2298/TSCI130717103YSuche in Google Scholar

[12] A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results and problems-I, Appl. Anal. 78 (2001), 153 –192.10.1080/0003681021000022032Suche in Google Scholar

[13] A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results and problems-II, Appl. Anal. 81(2) (2002), 435–493.10.1080/0003681021000022032Suche in Google Scholar

[14] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006.Suche in Google Scholar

[15] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, USA, 1999.Suche in Google Scholar

[16] F. Gao, X. Yang, Fractional Maxwell fluid with fractional derivative without singular kernel, Therm. Sci. 20(S3) (2016), 873–879.10.2298/TSCI16S3871GSuche in Google Scholar

[17] X. Yang, Some new applications for heat and fluid flows via fractional derivatives without singular kernel, Thermal Sci. 20(S3) (2016), 833–839.10.2298/TSCI16S3833YSuche in Google Scholar

[18] A. Yang, Y. Han, J. Li, W. Liu, On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel, Thermal Sci. 20(S3) (2016), 717–721.10.2298/TSCI16S3717YSuche in Google Scholar

[19] X. Yang, H. Srivastava, J. Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Thermal Sci. 20(2016), 753–756.10.2298/TSCI151224222YSuche in Google Scholar

[20] H. Ergoren, A. Kilicman, Some existence results for impulsive nonlinear fractional differential equations with closed boundary conditions, Abstract and Applied Analysis, Volume 2012, Article ID 387629, 15pages.10.1155/2012/387629Suche in Google Scholar

[21] X. Yang, J. Machado, D. Baleanu, C. Cattani, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos: Interdisciplin. J. Nonlinear Sci. 26(8) (2016), 110–118.10.1063/1.4960543Suche in Google Scholar PubMed

[22] X. Yang, J. Machado, J. Hristov, Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow, Nonlinear Dyn. 84(2016), 3–7.10.1007/s11071-015-2085-2Suche in Google Scholar

[23] J. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod Phys. B 20(10) (2006), 1141–99.10.1142/S0217979206033796Suche in Google Scholar

[24] J.H. He, Variational iteration method: a kind of non-linear analytical technique, some examples, Int. J. Nonlinear Mech. 34(4) (1999), 699–708.10.1016/S0020-7462(98)00048-1Suche in Google Scholar

[25] J.H. He, X.H. Wu, Construction of solitary solution and compaction-like solution by variational iteration method, Chaos Soliton Fract. 29(2006), 108–13.10.1016/j.chaos.2005.10.100Suche in Google Scholar

[26] M. Belmekki, Juan J. Nieto, Rosana Rodriguez-Lopez, Existence of periodic solution for a nonlinear fractional differential equation, Bound. Value Prob. 2009 (2009), Article ID 324561, 18pages.10.1155/2009/324561Suche in Google Scholar

[27] M. Benchohra, J. Graef, S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal., 87(2008), 851–863.10.1080/00036810802307579Suche in Google Scholar

[28] G. L. Karakostas, Positive solutions for the Ф-Laplacian when Ф is a sup-multiplicative-like function, Electron. J. Differ. Equ. 68(2004), 1–12.Suche in Google Scholar

[29] X. Li, X. Liu, M. Jia, Y. Li, S. Zhang, Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval, Math. Meth. Appl. Sci. 40(6) (2017), 1892–1904.10.1002/mma.4106Suche in Google Scholar

[30] K. S. Miller, S. G. Samko, Completely monotonic functions, Integr. Transf. Spec. Funct. 12(2001), 389–402.10.1080/10652460108819360Suche in Google Scholar

[31] J. J. Nieto, Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Appl. Math. Lett. 23(2010), 1248–1251.10.1016/j.aml.2010.06.007Suche in Google Scholar

[32] J. J. Nieto, Comparison results for periodic boundary value problems of fractional differential equations, Fractional Differ. Equ. 1(2011), 99–104.10.7153/fdc-01-05Suche in Google Scholar

[33] K. Shah and R. Ali Khan, Iterative scheme for a coupled system of fractional-order differential equations with three-point boundary conditions Mathematical Methods in the Applied Sciences, to appear, 4 AUG 2016, DOI: 10.1002/mma.4122.Suche in Google Scholar

[34] X. Wang, C. Bai, Periodic boundary value problems for nonlinear impulsive fractional differential equations, Electr. J. Qualitative Theory Differ. Equ. 3(2011), 1–15.10.14232/ejqtde.2011.1.3Suche in Google Scholar

[35] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equation, Electron. J. Diff. Eqns. 36(2006), 1–12.10.1016/j.camwa.2009.06.034Suche in Google Scholar

[36] A. Alsaedi, S. Aljoudi, B. Ahmad, Existence of solutions for Riemann-Liouvillle type coupled systems of fractional integro-differential equations and boundary conditions, Electron. J. Diff. Equ., 2016(211) (2016), 1–14.Suche in Google Scholar

[37] M. Chaieb, A. Dhifli, M. Zribi Positive solutions for systems of competitive fractional differential equations, Electron. J. Diff. Equ. Vol. 2016(133) (2016), 1–13.10.1186/s13662-016-0813-7Suche in Google Scholar

[38] R. Caponetto, G. Dongola, L. Fortuna, Frational order systems Modeling and control applications, World Scientific Series on nonlinear science, Ser. A, Vol. 72, World Scientific, Publishing Co. Pvt. Ltd. Singapore, 2010.10.1142/7709Suche in Google Scholar

[39] K. Diethelm, Multi-term fractional differential equations, multi-order fractional differential systems and their numerical solution. J. Eur. Syst. Autom. 42(2008), 665–676.10.3166/jesa.42.665-676Suche in Google Scholar

[40] M. S. Tavazoei, M. Haeri, Chaotic attractors in incommensurate fractional order systems, Physica D, 327(2008), 2628–2637.10.1016/j.physd.2008.03.037Suche in Google Scholar

[41] M. S. Tavazoei, M. Haeri, Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Anal. 69(2008), 1299–1320.10.1016/j.na.2007.06.030Suche in Google Scholar

[42] L. J. Guo, Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems, Chin. Phys. 14(2005), 1517–1521.10.1088/1009-1963/14/8/007Suche in Google Scholar

[43] W. H. Deng, C. P. Li, Chaos synchronization of the fractional Lu system, Physica A 353(2005), 61–72.10.1016/j.physa.2005.01.021Suche in Google Scholar

[44] I. Petras, Chaos in the fractional-order Volta’s system: modeling and simulation. Nonlinear Dyn. 57(2009), 157–170.10.1007/s11071-008-9429-0Suche in Google Scholar

[45] I. Petras, Fractional-Order Feedback Control of a DC Motor, J. Electr. Eng. 60(2009), 117–128.Suche in Google Scholar

[46] S. Das, P.K.Gupta, A mathematical model on fractional Lotka-Volterra equations, J. Theor. Biol. 277(2011), 1–6.10.1016/j.jtbi.2011.01.034Suche in Google Scholar PubMed

[47] N Ozalp, I Koca, A fractional order nonlinear dynamical model of interpersonal relationships, Adv. Differ. Equ. 189 (2012), 18pages.10.1186/1687-1847-2012-189Suche in Google Scholar

[48] C. Sulem, P. Sulem, The Nonlinear Schrödinger Equation: self Focusing and Wave Collapse Springer, Berlin, 2000.Suche in Google Scholar

[49] C. C. Tisdell, Basic existence and a priori bound results for solutions to systems of boundary value problems for fractional differential equations, Electron. J. Diff. Equ. 2016(84) (2016), 1–9.10.14232/ejqtde.2016.1.92Suche in Google Scholar

[50] K. M. Furati, M. Kirane, Necessary conditions for the existence of global solutions to systems of fractional differential equations, Fractional Calculus Appl. Anal. 11(2008), 281–298.Suche in Google Scholar

[51] M. Rehman, R. Khan, A note on boundary value problems for a coupled system of fractional differential equations, Comput. Math. Appl. 61(2011), 2630–2637.10.1016/j.camwa.2011.03.009Suche in Google Scholar

[52] J. Wang, H. Xiang, Z. Liu, Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations, Int. J. Differ. Equ. 2010 (2010), Article ID 186928, 12pages.10.1155/2010/186928Suche in Google Scholar

[53] A. Yang, W. Ge, Positive solutions for boundary value problems of N-dimension nonlinear fractional differential systems, Bound. Value Prob. 2008, article ID 437453, 20 pages.10.1155/2008/437453Suche in Google Scholar

[54] S. Deng, B. Guo, Generalized homoclinic solutions of a coupled Schrödinger system under a small purterbation, J. Dyn. Diff. Equat. 24(2012), 761–776.10.1007/s10884-012-9274-1Suche in Google Scholar

[55] C. Sulem, P. Sulem, The Nonhnear Schrödinger Equation: self Focusing and Wave Collapse Springer, Berlin, 2000.Suche in Google Scholar

[56] L. Vaquez, L. Streite, Nonlinear Klein-Gordon and Schrödinger systems: Tbeory and Applictions, World Scientific, Singapore, 1997.10.1142/9789814531429Suche in Google Scholar

[57] R. P. Agarwal, M. Benchohra, B. A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differential Equations Math. Phys. 44(2008), 1–21.10.1134/S0012266108010011Suche in Google Scholar

[58] R. Dehghant and K. Ghanbari, Triple positive solutions for boundary value problem of a nonlinear fractional differential equation, Bull. Iran. Math. Soc. 33(2007), 1–14.Suche in Google Scholar

[59] E. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electr. J. Qualitative Theory Differ. Equ. 3(2008), 1–11.10.14232/ejqtde.2008.1.3Suche in Google Scholar

[60] Y. Liu, Existence of Solutions of a New Class of Impulsive Initial Value Problems of Singular Nonlinear Fractional Differential Systems, Int. J. Nonlinear Sci. Numer. Simul. 17(7–8) (2016), 343–353.10.1515/ijnsns-2013-0044Suche in Google Scholar

[61] Y. Liu, Solvability of multi-point boundary value problems for multiple term Riemann-Liouville fractional differential equations, Comput. Math. Appl. 64(4) (2012), 413–431.10.1016/j.camwa.2011.12.004Suche in Google Scholar

[62] Y. Liu, Studies on BVPs for IFDEs involved with the Riemann-Liouville type fractional derivatives, Nonautonomous Dyn. Syst. 3(1) (2016), 42–84.10.1515/msds-2016-0004Suche in Google Scholar

[63] Y. Liu, Studies on impulsive differential models with multi-term Riemann-CLiouville fractional derivatives, J. Appl. Math. Comput. 52(1) (2016), 529–565.10.1007/s12190-015-0953-zSuche in Google Scholar

[64] Y. Liu, Existence of global solutions of impulsive IVPs of singular fractional differential systems on half line, Fractional Differ. Calculus 6(1) (2016), 35–56.10.7153/fdc-06-03Suche in Google Scholar

[65] Z. Liu, X. Li, Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 18(6) (2013), 1362–1373.10.1016/j.cnsns.2012.10.010Suche in Google Scholar

[66] Z. Liu, L. Lu, I. Szanto, Existence of solutions for fractional impulsive differential equations with p-Laplacian operator, Acta Mathematica Hungarica 141(3) (2013), 203–219.10.1007/s10474-013-0305-0Suche in Google Scholar

[67] A. M. Nakhushev, The Sturm-Liouville Problem for a Second Order Ordinary Differential equations with fractional derivatives in the lower terms, Dokl. Akad. Nauk SSSR 234(1977), 308–311.Suche in Google Scholar

[68] S. Z. Rida, H.M. El-Sherbiny and A. Arafa, On the solution of the fractional nonlinear Schr\"(o)dinger equation, Phys. Lett. A 372(2008), 553–558.10.1016/j.physleta.2007.06.071Suche in Google Scholar

[69] Z. Wei, W. Dong, J. Che, Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative, Nonlinear Anal.: Theor. Meth. Appl. 73(2010), 3232–3238.10.1016/j.na.2010.07.003Suche in Google Scholar

[70] S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252(2000), 804–812.10.1006/jmaa.2000.7123Suche in Google Scholar

[71] V. Gupta, J. Dabas, Functional impulsive differential equation of order α ∈ (1,2) with nonlocal initial and integral boundary conditions, Math. Meth. Appl. Sci. 40(7) (2017), 2409–2420.10.1002/mma.4147Suche in Google Scholar

[72] Y. Liu, Existence of global solutions of impulsive IVPs of singular fractional differential systems on half line, Fractional Differ. Calculus 6(1) (2016), 35–56.10.7153/fdc-06-03Suche in Google Scholar

[73] Y. Liu, Global Existence of Solutions of a Class of Singular Fractional Differential Equations with Impulse Effects, Matematika 32(1) (2016), 13–29.10.11113/matematika.v32.n1.758Suche in Google Scholar

[74] Y. Liu, Existence of solutions of BVPs for a class of IFDEs on half line involving Hardamard fractional derivatives, J. Nonlinear Funct. Anal. 2016 (2016), Article ID 26 pages.Suche in Google Scholar

[75] Y. Liu, P.J.Y. Wong, Global existence of solutions for a system of singular fractional differential equations with impulse effects, J. Appl. Math. Inform. 33(3–4) (2015), 327–342.10.14317/jami.2015.327Suche in Google Scholar

[76] X. Yang, Y. Liu, Existence of solutions of IVPs of singular multi-term fractional differential equations with impulse effects, Differ. Equ. Control Processes 2(2016), 72–120.Suche in Google Scholar

[77] Y. Liu, Solvability of impulsive (n, n-p) boundary value problems for higher order fractional differential equations, Math. Sci. 10(3) (2016), 71–81.10.1007/s40096-016-0180-2Suche in Google Scholar

[78] Y. Liu, On piecewise continuous solutions of higher order impulsive fractional differential equations and applications, Appl. Math. Comput. 287(2016), 38–49.10.1016/j.amc.2016.03.041Suche in Google Scholar

[79] Y. Liu, Solvability of impulsive periodic boundary value problems for higher order fractional differential equations, Arab. J. Math. 5(2016), 195–214.10.1007/s40065-016-0153-1Suche in Google Scholar

[80] Y. Liu, Piecewise continuous solutions of initial value problems of singular fractional differential equations with impulse effects, Acta Mathematica Scientia 36(5) (2016), 1492–1508.10.1016/S0252-9602(16)30085-6Suche in Google Scholar

[81] Y. Liu, New boundary value problems for higher order impulsive fractional differential equations and their solvability, Fractional Differ. Calculus 7(1) (2017), 1–121.10.7153/fdc-07-01Suche in Google Scholar

[82] Y. Liu, S. Li, Periodic boundary value problems of singular fractional differential equations with impulse effects, Malaya J. Math. 3(4) (2015), 423–490.10.26637/mjm304/006Suche in Google Scholar

[83] R. Agarwal, S. Hristova, D. O’Regan, Stability of solutions to impulsive Caputo fractional differential equations, Electron. J. Diff. Equ. 58(2016), 1–22.Suche in Google Scholar

[84] R. Agarwal, M. Benchohra, S. Hamani, et al., Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half line, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18(2) (2011), 235–244.Suche in Google Scholar

[85] A. Arara, M. Benchohra, N. Hamidi and J. J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Anal., 72(2) (2010), 580–586.10.1016/j.na.2009.06.106Suche in Google Scholar

[86] F. Chen and Y. Zhou, Attractivity of fractional functional differential equations, Comput. Math. Appl. 62(3)(2011), 1359–1369.10.1016/j.camwa.2011.03.062Suche in Google Scholar

[87] C. Kou, H. Zhou, Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis, Nonlinear Anal.: Theory, Meth. Appl. 74(17) (2011), 5975–5986.10.1016/j.na.2011.05.074Suche in Google Scholar

[88] S. Liang, J. Zhang, Existence of three positive solutions of m-point boundary value problems for some nonlinear fractional differential equations on an infinite interval, Comput. Math. Appl. 61(11) (2011), 3343–3354.10.1016/j.camwa.2011.04.018Suche in Google Scholar

[89] S. Liang and J. Zhang, Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval, Math. Comput. Modell. 54(5–6) (2011), 1334–1346.10.1016/j.mcm.2011.04.004Suche in Google Scholar

[90] X. Su, Solutions to boundary value problem of fractional order on unbounded domains in a Banach space, Nonlinear Anal. 74(8) (2011), 2844–2852.10.1016/j.na.2011.01.006Suche in Google Scholar

[91] X. Su and S. Zhang, Unbounded solutions to a boundary value problem of fractional order on the half-line, Comput. Math. Appl. 61(4) (2011), 1079–1087.10.1016/j.camwa.2010.12.058Suche in Google Scholar

[92] X. Zhao, W. Ge, Some results for fractional impulsive boundary value problems on infinite intervals, Appl. Math. 56(4) (2011), 371–387.10.1007/s10492-011-0021-4Suche in Google Scholar

[93] Y. Liu, Existence of solutions of a class of impulsive initial value problems of singular nonlinear fractional differential systems, Int. J. Nonlinear Sci. Numer. Simul. 17(7–8) (2016), 343–353.10.1515/ijnsns-2013-0044Suche in Google Scholar

[94] Y. Liu, Bifurcation techniques for a class of boundary value problems of fractional impulsive differential equations, J. Nonlinear Sci. Appl. 8(4) (2015), 340–353.10.22436/jnsa.008.04.07Suche in Google Scholar

[95] W. Zou, X. Liu, Existence of solution to a class of boundary value problem for impulsive fractional differential equations, Adv. Differ. Equ. 2014(12)(2014), 12pages.10.1186/1687-1847-2014-12Suche in Google Scholar

[96] X. Zhang, C. Zhu, Z. Wu, Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Prob. 2013(80) (2013), 23pages.10.1186/1687-2770-2013-80Suche in Google Scholar

[97] K. Zhao, Impulsive integral boundary value problems of the higher-order fractional differential equation with eigenvalue arguments, Adv. Differ. Equ. 2015(382) (2015), 16pages.10.1186/s13662-015-0725-ySuche in Google Scholar

[98] M. Belmekki, J. J. Nieto and R. R. Lopez, Existence of periodic solution for a nonlinear fractional differential equation, Bound. value prob. (1)(2009), 18pages.10.1155/2009/324561Suche in Google Scholar

[99] M. A. E. Herzallah, Mild and strong solutions to new types of fractional order nonlinear equations with periodic boundary conditions, Indian J. Pure Appl. Math. 43(6)(2012), 619–635.10.1007/s13226-012-0037-9Suche in Google Scholar

[100] M. A. E. Herzallah and D. Baleanu, Existence of periodic mild solution for a nonlinear fractional differential equation, Comput. Math. Appl. 64(10) (2012), 3059–3064.10.1016/j.camwa.2011.12.060Suche in Google Scholar

[101] J. Wang, Z. Lin, On the impulsive fractional anti-periodic BVP modelling with constant coefficients, J. Appl. Math. Comput. 46(2014), 107–121.10.1007/s12190-013-0740-7Suche in Google Scholar

[102] Y. Zhang and J. Wang, Nonlocal cauchy problems for a class of implicit impulsive fractional relaxation differential systems, J. Appl. Math. Comput. 52(2016), 323–343.10.1007/s12190-015-0943-1Suche in Google Scholar

[103] Y. Liu, Survey and new results on boundary value problems of singular fractional differential equations with impulse effects, Electron. J. Diff. Equ. 296(2016), 1–177.Suche in Google Scholar

[104] X. Zhang X, T. Shu, Z. Liu, W. Ding, H. Peng and J. He, On the concept of general solution for impulsive differential equations of fractional-order q∈ (2,3), Open Math. 14(1) (2016), 452–473.10.1515/math-2016-0042Suche in Google Scholar

[105] M. Fekan, Y. Zhou, J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17(2012), 3050–3060.10.1016/j.cnsns.2011.11.017Suche in Google Scholar

[106] G. Wang, B. Ahmad, L. Zhang, A coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain, Abstr. Appl. Anal. 2012, Article ID 248709, 11pages.10.1155/2012/248709Suche in Google Scholar

[107] H. Akca, R. Alassar, Y. M. Shebadeh, Neural Networks: Modelling with Impulsive Differential Equations, 5–10 July 2004, Antalya, Turkey-Dynamical Systems and Applications, Proceedings, pp. 32–47.Suche in Google Scholar

[108] J. Lou, L. Chen, T. Ruggeri, An impulsive differential model on post exposure prophylaxis to HIV-1 exposed individual, J. Biol. Syst. 17(4) (2009), 659–683.10.1142/S0218339009002934Suche in Google Scholar

[109] J. Lou, Y. Lou, J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, J. Math. Biol. 65(4) (2012), 623–652.10.1007/s00285-011-0474-9Suche in Google Scholar PubMed PubMed Central

[110] C. Bai, Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance, Electron. J. Qual. Theory Differ. Equ. 89(2011), 1–19.10.14232/ejqtde.2011.1.89Suche in Google Scholar

[111] C. Bai, Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance, J. Appl. Math. Comput. 39(1–2) (2012), 421–443.10.1007/s12190-012-0537-0Suche in Google Scholar

[112] X. Wang, Impulsive boundary value problem for nonlinear differential equations of fractional order, Comput. Math. Appl. 62(5) (2011), 2383–2391.10.1016/j.camwa.2011.07.026Suche in Google Scholar

[113] A. Granas, J. Dugundji, Fixed point theory, Springer–Verlag, New York, 2003.10.1007/978-0-387-21593-8Suche in Google Scholar

[114] J. Mawhin, Topological degree methods in nonlinear boundary value problems, in: NSFCBMS Regional Conference Series in Math., American Math. Soc. Providence, RI, 1979.10.1090/cbms/040Suche in Google Scholar

[115] R. Agarwal, S. Hristova, D. O’Regan, Stability of solutions to impulsive Caputo fractional differential equations, Electron. J. Diff. Equ. 58(2016), 1–22.Suche in Google Scholar

[116] M. A. M. Alwash, Composition vonditions for two–dimensional polynomial systems, Differential Differ. Equ. Appl, 5(1) (2013), 1–12.10.7153/dea-05-01Suche in Google Scholar

[117] P. J. Torres, Existence of closed solutions for a 6polynomial first order differential equation, J. Math. Anal. Appl. 328(2007), 1108–1116.10.1016/j.jmaa.2006.05.078Suche in Google Scholar

[118] Y. Xu, Z. He, The short memory principle for solving Abel differential equation of fractional order, Comput. Math. Appl. 62 (12) (2011), 4796–4805.10.1016/j.camwa.2011.10.071Suche in Google Scholar

[119] P. K. Singh and T. Som, Fractional Ecosystem Model and Its Solution by Homotopy Perturbation Method, Int. J. Ecosyst. 2(5) (2012), 140–149.10.5923/j.ije.20120205.06Suche in Google Scholar

[120] E. Zeidler, Nonlinear functional analysis and its applications, I: Fixed point theorems, Springer-Verlag New York Inc., 1986.10.1007/978-1-4612-4838-5Suche in Google Scholar

Published Online: 2018-1-9
Published in Print: 2018-4-25

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 7.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2017-0009/html?lang=de
Button zum nach oben scrollen