Bifurcation techniques for a class of boundary value problems of fractional impulsive differential equations
-
2371
Downloads
-
3892
Views
Authors
Yansheng Liu
- Department of Mathematics, Shandong Normal University, Jinan, 250014, P. R. China.
Abstract
This paper investigates the existence of positive solutions for a class of boundary value problems (BVP)
of fractional impulsive differential equations and presents a number of new results. First, by constructing
a novel transformation, the considered impulsive system is convert into a continuous system. Second,
using a specially constructed cone, the Krein-Rutman theorem, topological degree theory, and bifurcation
techniques, some sufficient conditions are obtained for the existence of positive solutions to the considered
BVP. Finally, an example is worked out to demonstrate the main result.
Share and Cite
ISRP Style
Yansheng Liu, Bifurcation techniques for a class of boundary value problems of fractional impulsive differential equations, Journal of Nonlinear Sciences and Applications, 8 (2015), no. 4, 340--353
AMA Style
Liu Yansheng, Bifurcation techniques for a class of boundary value problems of fractional impulsive differential equations. J. Nonlinear Sci. Appl. (2015); 8(4):340--353
Chicago/Turabian Style
Liu, Yansheng. "Bifurcation techniques for a class of boundary value problems of fractional impulsive differential equations." Journal of Nonlinear Sciences and Applications, 8, no. 4 (2015): 340--353
Keywords
- Positive solutions
- bifurcation techniques
- fractional differential equations with impulse
- boundary value problems.
MSC
References
-
[1]
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.
-
[2]
B. Ahmada, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Analysis: Hybrid Systems, 4 (2010), 134-141.
-
[3]
Z. Bai, H. Lv, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505.
-
[4]
M. Choisy, J. F. Guegan, P. Rohani, Dynamics of infectious diseases and pulse vaccination: Teasing apart the embedded resonance effects, Physica D., 22 (2006), 26-35.
-
[5]
K. Diethelm , The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type, LNM 2004, Springer (2010)
-
[6]
A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett., 18 (2005), 729-32.
-
[7]
M. Fečkan, 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.
-
[8]
S. Gao, L. Chen, J. J. Nieto, A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 6037-6045.
-
[9]
D. Guo, Nonlinear Functional Analysis, Jinan: Shandong Science and Technology Press, in Chinese (2001)
-
[10]
J. Jiang, L. Liu, Y. Wu, Positive solutions to singular fractional differential system with coupled boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3061-3074.
-
[11]
D. Jiang, C. Yuan, The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application, Nonlinear Anal.-TMA, 72 (2010), 710-719.
-
[12]
A. A. Kilbas, J. J. Trujillo, Differential equations of fractional order: methods, results and problems I, Appl. Anal., 78 (2001), 153-192.
-
[13]
A. A. Kilbas, J. J. Trujillo, Differential equations of fractional order: methods, results and problems II, Appl. Anal., 81 (2002), 435-493.
-
[14]
V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)
-
[15]
Y. Liu, D. O'Regan, Bifurcation techniques for Lidstone boundary value problems, Nonlinear Anal.-TMA, 68 (2008), 2801-2812.
-
[16]
Y. Liu, D. O'Regan, Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1769-1775.
-
[17]
R. Ma, J. Xu, Bifurcation from interval and positive solutions of a nonlinear fourthorder boundary value problem, Nonlinear Anal.-TMA, 72 (2010), 113-122.
-
[18]
R. Ma, B. Yang, Z. Wang , Positive periodic solutions of first-order delay differential equations with impulses, Appl. Math. Comput., 219 (2013), 6074-6083.
-
[19]
I. Podlubny , Fractional Differential Equations , Mathematics in Science and Engineering, Academic Press, New York (1999)
-
[20]
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integral And Derivatives (Theory and Applications), Gordon and Breach, Switzerland (1993)
-
[21]
K. Schmitt , Positive solutions of semilinear elliptic boundary value problem, Kluwer, Dordrecht, (1995), 447-500.
-
[22]
K. Schmitt, R. C. Thompson, Nonlinear Analysis and Differential Equations: An Introduction, University of Utah Lecture Note, Salt Lake City (2004)
-
[23]
C, Tian, Y, Liu, Multiple positive solutions for a class of fractional singular boundary value problems, Mem. Differential Equations Math. Phys., 56 (2012), 115-131.
-
[24]
H. Wang, Existence results for fractional functional differential equations with impulses, J. Appl. Math. Comput., 38 (2012), 85-101.
-
[25]
X. Wang , Impulsive boundary value problem for nonlinear differential equations of fractional order, Computers & Math. Appl., 62 (2011), 2383-2391.
-
[26]
J. Xu, R. Ma, Bifurcation from interval and positive solutions for second order periodic boundary value problems, Appl. Math. Comput., 216 (2010), 2463-2471.
-
[27]
S. T. Zavalishchin, A. N. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers Group, Dordrecht (1997)