×

A survey and new investigation on \((n,n-k)\)-type boundary value problems for higher order impulsive fractional differential equations. (English) Zbl 1403.34002

In this paper, the author firstly takes a survey for some existing results on boundary value problems of higher order ordinary differential equations. Then, the author makes a review for some literature on the solvability of boundary value problems for impulsive fractional differential equations. Thirdly, the author considers existence results of solutions for three classes of \((n,n-p)\)-type boundary value problems of impulsive higher order fractional differential equations by using some standard fixed point theorems in Banach space. Some examples are presented to illustrate the efficiency of the obtained results. Finally, the author gives some future research topics on impulsive fractional differential equations.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A08 Fractional ordinary differential equations
34A37 Ordinary differential equations with impulses
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] B. Ahmad, A. Alsaedi, A. Assolami, Relationship between lower and higher order anti-periodic boundary value problems and existence results, J. Comput. Anal. Appl. 16(2)(2014), 210-219. · Zbl 1291.34040
[2] R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109(2010), 973-1033. · Zbl 1198.26004
[3] R. P. Agarwal, M. Benchohra, B. A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differ. Equ. Math. Phys. 44(2008), 1-21. · Zbl 1178.26006
[4] R.P. Agarwal, M. Bohner, P.J. Y.Wong, Positive solutions and eigenvalues of conjugate bound- ary value problems, Proc. Edinburgh Math. Soc. (Series 2), 42(2)(1999), 349-374. · Zbl 0934.34008
[5] R. Agarwal, S. Hristova, D. O’Regan, Stability of solutions to impulsive Liouville-Caputo fractional differential equations, Electron. J. Diff. Equ. 58 (2016), 1-22. · Zbl 1335.34009
[6] R.P. Agarwal, D. O’Regan, Positive solutions for (p; n-p) conjugate boundary value problems, J. Differ. Equ. 150(2)(1998), 462-473. · Zbl 0920.34027
[7] B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems involving fractinal differential equations, Nonlinear Anal. Hybrid Syst. 3(2009), 251-258. · Zbl 1193.34056
[8] B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal. Hybrid Syst. 4(2010), 134-141. · Zbl 1187.34038
[9] B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid Syst. 3 (2009), 251-258. · Zbl 1193.34056
[10] B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order. Nonl. Anal. Hybrid Syst. 4(2010), 134-141. · Zbl 1187.34038
[11] C.J. Chyan, J. Henderson, Positive solutions of 2m th-order boundary value problems, Appl. Math. Letters, 15(6)(2002), 767-774. · Zbl 1019.34019
[12] J. M. Davis, P. W. Eloe, J. Henderson, Triple positive solutions and dependence on higher order derivatives, J. Math. Anal. Appl. 237(2)(1999), 710-720. · Zbl 0935.34020
[13] J.M. Davis, J. Henderson, P.J.Y.Wong, General Lidstone problems: multiplicity and symmetry of solutions, J. Math. Anal. Appl. 251(2)(2000), 527-548. · Zbl 0966.34023
[14] J.M. Davis, J. Henderson, Triple positive solutions for (k; nk) conjugate boundary value problems, Math. Slovaca, 51(3)(2001), 313-320. · Zbl 0996.34017
[15] J.M. Davis, J. Henderson, P.K. Rajendra, Eigenvalue intervals for nonlinear right focal problems, Appl. Anal. 74(1-2)(2000), 215-231. [16] P. W. Eloe, J. Henderson, Singular nonlinear (k; n - k) conjugate boundary value problems, J. Differ. Equ. 133(1) (1997), 136-151. · Zbl 1031.34083
[16] P.W. Eloe, J. Henderson, Singular nonlinear (nC1; 1) conjugate boundary value problems, Georgian Math. J. 4(5)(1997), 401-412. · Zbl 0882.34029
[17] P.W. Eloe, J. Henderson, Positive solutions for (n - 1; 1) conjugate boundary value problems, Nonlinear Anal. TMA, 28(10)(1997), 1669-1680. · Zbl 0871.34015
[18] G. Feltrin, F. Zanolin, Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems, arXiv preprint arXiv:1503.04954, 2015. · Zbl 1345.34031
[19] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, 2000. · Zbl 0998.26002
[20] J. Henderson and A. Ouahab, Impulsive differential inclusions with fractional order, Comput. Math. Appl. 59(2010), 1191-1226. · Zbl 1200.34006
[21] J. Henderson, A. Ouahab, Impusive differential inclusions with fractioanl order, Comput. Math. Appl. 59(2010), 1191-1226. · Zbl 1200.34006
[22] J. Henderson, W. Yin, Singular (k; n - k) boundary value problems between conjugate and right focal, J. Comput. Appl. Math. 88(1)(1998), 57-69. · Zbl 0901.34026
[23] V.A. Il’in, E.I. Moiseev, An a priori bound for a solution of the problem conjugate to a nonlocal boundary-value problem of the first kind, Differ. Equ. 24(5)(1988), 795-804. · Zbl 0641.34016
[24] D. Jiang, H. Liu, Existence of positive solutions to (k; n - k) conjugate boundary value prob- lems, Kyushu J. Math. 53(1)(1999), 115-125. · Zbl 0928.34020
[25] N. Kosmatov, On a singular conjugate boundary value problem with infinitely many solutions, Math. Sci. Res. Hot-Line, 4(2000), 9-17. · Zbl 1010.34014
[26] L. Kong, T. Lu, Positive solutions of singular (n; n-k) conjugate boundary value problem, J. Appl. Math. Bioinformatics, 5(1)(2015), 13-24. · Zbl 1354.34047
[27] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
[28] Y. Liu, Global existence of solutions for a system of singular fractional differential equations with impulse effects, J. Appl. Math. and Informatics, 33(3-4)(2015), 327-342.
[29] Y. Liu, On piecewise continuous solutions of higher order impulsive fractional differential equations and applications, Appl. Math. Comput. 287(2016), 3849.
[30] Y. Liu, Bifurcation techniques for a class of boundary value problems of fractional impulsive differential equations, J. Nonlinear Sci. Appl. 8 (2015), 340-353. · Zbl 1319.34013
[31] V. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [33] X. Li, F. Chen, and X. Li, Generalized anti-periodic boundary value problems of impulsive fractional differential equations, Commun. Nonl. Sci. Numer. Simul. 18(1)(2013), 28-41.
[32] Y. Liu, W. Ge, Periodic boundary value problems for n-th order ordinary differential equations with a p-Laplacian, J. Anal. Math. 16(2005), 1-22. · Zbl 1088.34013
[33] X. Lin, D. Jiang, X. Li, Existence and uniqueness of solutions for singular (k; n-k) conjugate boundary value problems, Comput. Math. Appl. 52(3)(2006), 375-382. · Zbl 1143.34306
[34] Z. Liu, X. Li, Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations, Commun. Nonl. Sci. Numer. Simul. 18 (6)(2013), 1362-1373. · Zbl 1283.34005
[35] P. Li, H. Shang, Impulsive problems for fractional differential equations with nonlocal boundary value conditions, Abst. Appl. Anal. 2014 (2014), Article ID 510808, 13 pages.
[36] X. Liu, Y. Zhang, H. Shi, Existence and nonexistence results for a fourth-order discrete neu- mann boundary value problem, Stud. Sci. Math. Hungarica, 51(2)(2014), 186-200.
[37] J. Mawhin, Topological degree methods in nonlinear boundary value problems, in: CBMS Regional Conference Series in Mathematics 40, American Math. Soc., Providence, R.I., 1979. · Zbl 0414.34025
[38] R. Ma, Positive solutions for semipositone (k; n - k) conjugate boundary value problems, J. Math. Anal. Appl. 252(1)(2000), 220-229. · Zbl 0979.34012
[39] F. Mainardi, Fractional Calculus: Some basic problems in continuum and statistical mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics, 291-348, CISM Courses and Lectures 378, Springer, Vienna, 1997. · Zbl 0917.73004
[40] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New, York, 1993. · Zbl 0789.26002
[41] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5(2002), 367-386. · Zbl 1042.26003
[42] I. Podlubny, Fractional Differential Equations. Mathmatics in Science and Engineering, Vol. 198, Academic Press, San Diego, California, USA, 1999.
[43] I. Rachunkova, S. Stanek, A singular boundary value problem for odd-order differential equa- tions, J. Math. Anal. Appl. 291(2)(2004), 741-756. · Zbl 1056.34032
[44] P. Shi, L. Dong, Studies on anti-periodic boundary value problems for two classes of special second order impulsive differential equations, Math. Meth. Appl. Sci. 37(1)(2014), 123-135. · Zbl 1288.34029
[45] Y. Tian and Z. Bai, Existence results for three-point impulsive integral boundary value problems involving fractinal differential equations, Comput. Math. Appl. 59(2010), 2601-2609. · Zbl 1193.34007
[46] S. Tian, W. Gao, Positive solutions of singular (k; n - k) conjugate eigenvalue problem, J. Appl. Math. Bioinformatics, 5(2)(2015), 85-97. · Zbl 1354.34048
[47] M. ur-Rehman, P.W. Eloe, Existence and uniqueness of solutions for impulsive fractional differential equations, Appl. Math. Comput. 224 (2013), 422-431. · Zbl 1334.34019
[48] P. J. Y. Wong, Triple positive solutions of conjugate boundary value problems. Comput. Math. Appl. 36(9)(1998), 19-35. · Zbl 0936.34018
[49] X. Wang, Existence of solutions for nonlinear impulsive higher order fractional differential equations, Electron. J. Qual. Theo. Differ. Equ. 80(2011), 1-12.
[50] P.J.Y. Wong, R.P. Agarwal, Singular differential equations with (n, p) boundary conditions, Math. Comput. Modelling, 28(1)(1998), 37-44. · Zbl 1076.34507
[51] G. Wang, B. Ahmad, L. Zhang, J.J. Nieto, Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Comput. Math. Appl. 62 (2011), 1389-1397. · Zbl 1228.34021
[52] G.Wang, B. Ahmad, L. Zhang, On impulsive boundary value problems of fractional differential equations with irregular boundary conditions, Abst. Appl. Anal. Volume 2012, Article ID 356132, 18 pages. · Zbl 1253.34018
[53] J. R. Wang, Y. Yang, W. Wei, Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces, Opuscula Math. 30(2010), 361-381. · Zbl 1232.34013
[54] C. Yuan, Multiple positive solutions for (n-1; 1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Electron. J. Qual. Theo. Differ. Equ. 2010, 36: 1-12. · Zbl 1210.34008
[55] X. Yang, Existence of solutions for 2n-order boundary value problem, Appl. Math. Comput., 237(1)(2003), 77-87. · Zbl 1033.34027
[56] A. Yang, J. Henderson, Jr. C. Nelms, Extremal points for a higher-order fractional boundary- value problem, Electron. J. Differ. Equ. 161(2015), 1-12. · Zbl 1334.34023
[57] Y. Yang, J. Zhang, Nontrivial solutions on a kind of fourth-order Neumann boundary value problems, Appl. Math. Comput. 218(13) (2012), 7100-7108. · Zbl 1259.65123
[58] K. Zhao, P. Gong, Positive solutions for impulsive fractional differential equations with gener- alized periodic boundary value conditions, Adv. Differ. Equ. 225(2014), 1-15.
[59] W. Zhou, X. Liu, Existence of solution to a class of boundary value problem for impulsive fractional differential equations. Adv. Differ. Equ. 12(2014), 1-12. · Zbl 1351.34013
[60] W. Zhou, X. Liu, and J. Zhang, Some new existence and uniqueness results of solutions to semilinear impulsive fractional integro-differential equations, Adv. Differ. Equ. 38(2015), 1-16. · Zbl 1398.34116
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.