Method of upper and lower solutions for fractional differential equations

In this paper, we show the existence and uniqueness of solutions for the boundary-value problems of fractional differential equations, using the upper and lower solutions method and monotone iterative algorithm. An example is also included to illustrate our results.

[1]  Xiping Liu,et al.  Multiple solutions for fractional differential equations with nonlinear boundary conditions , 2010, Comput. Math. Appl..

[2]  Xiping Liu,et al.  Three nonnegative solutions for fractional differential equations with integral boundary conditions , 2011, Comput. Math. Appl..

[3]  Yang Liu,et al.  A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann-Liouville derivative , 2012, Appl. Math. Lett..

[4]  Mei Jia,et al.  Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order , 2008 .

[5]  A. Kilbas,et al.  Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions , 2005 .

[6]  Samir Ladaci,et al.  Fractional order adaptive high-gain controllers for a class of linear systems , 2008 .

[7]  Najla Yazidi,et al.  Monotone method for singular Neumann problem , 2002 .

[8]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[9]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[10]  Maslina Darus,et al.  Subordination and superordination for univalent solutions for fractional differential equations , 2008 .

[11]  Jianhua Shen,et al.  Boundary value problems involving upper and lower solutions in reverse order , 2009 .

[12]  Alan D. Freed,et al.  On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity , 1999 .

[13]  Tadeusz Jankowski,et al.  Solvability of three point boundary value problems for second order differential equations with deviating arguments , 2005 .

[14]  Sihua Liang,et al.  Positive solutions for boundary value problems of nonlinear fractional differential equation , 2009 .

[15]  Patrick Habets,et al.  Two-Point Boundary Value Problems: Lower and Upper Solutions , 2012 .

[16]  Wenjie Zuo,et al.  On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations , 2004 .

[17]  Shuqin Zhang,et al.  The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order , 2011, Comput. Math. Appl..

[18]  Saad Zagloul Rida,et al.  On the solution of the fractional nonlinear Schrödinger equation , 2008 .

[19]  J. H. Barrett,et al.  Differential Equations of Non-Integer Order , 1954, Canadian Journal of Mathematics.

[20]  Baofeng Wu,et al.  Existence and uniqueness of solution for fractional differential equations with integral boundary conditions. , 2009 .

[21]  Fangfei Li,et al.  Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order , 2011, Appl. Math. Comput..