Solutions for Impulsive FractionalOrder Differential Equations Based on Critical Point Theorem

doi:10.3969/j.cnki.jdxb.2016.03.004

Abstract

Abstract:

The Critical-Point Theory and other mathematical analysis techniques for the boundary value problems are applied for the discussion of the existence and multiplicity of solutions for the Dirichlet boundary value problems for a class of impulsive fractional order differential equations.The sufficient conditions for the existence and multiplicity are obtained.

Key words: impulsive, fractional order；differential equations, variational methods, multiple solutions

YANG Fei，CHEN Guoping. Solutions for Impulsive FractionalOrder Differential Equations Based on Critical Point Theorem[J]. Journal of Jishou University（Natural Sciences Edition), DOI: 10.3969/j.cnki.jdxb.2016.03.004.

References

[1] AGARWAL RAVI P，AHMAD BASHIR.Existence Theory for Anti-Periodic Boundary Value Problems of Fractional Differential Equations and Inclusions[J].Computers and Mathematics with Applications，2011，62(3):1 200-1 214.
[2] TARASOV VASILY E.Fractional Dynamics:Applications of Fractional Calculus to Dynamics of Particles，Fields and Media[M].北京：高等教育出版社，2011:89-112
[3] WANG Jinrong，ZHOU Yong，FECKAN MICHAL.Nonlinear Impulsive Problems for Fractional Differential Equations and ULam Stability[J].Computers and Mathematics with Applications，2012，64(10):3 389-3 405.
[4] GUO Tianliang，JIANG Wei.Impulsive Problems for Fractional Differential Equations with Boundary Value Conditions[J].Computers and Mathematics with Applications，2012，64:3 281-3 291.
[5] CHEN Yi，TANG Xianhua.Solvability of Sequential Fractional Order Multi-Point Boundary Value Problems at Resonance[J].Applied Mathematics and Computation，2012，218:7 638-7 648.
[6] KILBAS ANATOLY A，SRIVASTAVA HARI M，TRUJILLO JUAN J.Theory and Applications of Fractional Differential Equations[M]∥Noorth-Holland Mathematics Studies.Amsterdam:Elsevier Science Ltd.，2006：105-146.
[7] ZHANG Ziheng，YUAN Rong.An Application of Variational Methods to Dirichlet Boundary Value Problem with Impulses[J].Nonlinear Analysis:Real World Applications，2010，11:155-162.
[8] JIAO Feng，ZHOU Yong.Existence of Solutions for a Class of Fractional Boundary Value Problems via Critical Point Theory[J].Computers and Mathematics with Applications，2011，62.
[9] JIA Mei，LIU Xiping.Multiplicity of Solutions for Integral Boundary Value Problems of Fractional Differential Equations with Upper and Lower Solutions[J].Applied Mathematics and Computation，204，232(1):313-323.
[10] JIAO Feng，ZHOU Yong.Existence Results For Fractional Boundary Value Problem var Critical Point Theory[J].International Journal of Bifurcation and Chaos，2012，22(4).
[11] ZHANG Quanguo，SUN Hongrui，LI Yaning.Existence of Solution for a Fractional Advection Dispersion Equation in 〖WTHZ〗R〖WTBZ〗n[J].Applied Mathematical Modelling，2014，38:4 062-4 075.
[12] SUN Hongrui，ZHANG Quanguo.Existence of Solutions for a Fractional Boundary Value Problem via the Mountain Pass Method and an Iterative Technique[J].Computers and Mathematics with Applications，2012，64:3 436-3 443.
[13] XIE Wenzhe，XIAO Jing，LUO Zhiguo.Existence of Solutions for Fractional Boundary Value Problem with Nonlinear Derivative Dependence[J].Abstract and Applied Analysis，2014，Article ID 812910，8 Pages：1-8.
[14] CHEN Jing，TANG X H.Existence and Multiplicity of Solutions for Some Fractional Boundary Value Problem via Critical Point Theory[J].Abstract and Applied Analysis，2012，Article ID 648635，21 Pages：1-21.
[15] BIN Ge.Multiple Solutions for a Class of Fractional Boundary Value Problems[J].Abstract and Applied Analysis，2012，Article ID 468980，16 Pages：1-16.
[16] NEMAT NYAMORADI.Infinitely Many Solutions for a Class of Fractional Boundary Value Problems with Dirichlet Boundary Conditions[J].Mediterranean Journal Mathematics，2014，11(1):75-87.
[17] GABRIELE BONANNO，ROSANA RODRGUEZ-LPEZ，STEPAN TERSIAN.Multiple Solutions to Boundary Value Problem for Impulsive Fractional Differential Equations[J].Fractional Calculus and Applied Analysis，2014，17(3):1 016-1 038.
[18] NEMAT NYAMORADI，ROSANA RODRGUEZ-LPEZ.On Boundary Value Problems for Impulsive Fractional Differential Equations[J].Applied Mathmatics and Computation，2015，271:874-892.
[19] RABINOWITZ PAUL H.Minimax Methods in Critical Point Theory with Applications to Differential Equations[M].American:American Mathmatical Society，1986：5-6.
[20] NIETO JUAN J，O’REGAN DONAL.Variational Approach to Impulsive Differential Equations[J].Nonlinear Analysis:Real World Applications，2009，10(2):680-690.