A sum operator method for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems

作者：Zhai Chengbo^*; Yan Weiping; Yang Chen

来源：Communications in Nonlinear Science & Numerical Simulation, 2013, 18(4): 858-866.

DOI：10.1016/j.cnsns.2012.08.037

• Riemann-Liouville fractional derivative
• Fractional differential equation
• Positive solution
• Existence and uniqueness
• Fixed point theorem of a sum operator

In this paper, we are concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problems given by
-D(0+)(alpha)u(t) = f(t, u(t)) + g(t, u(t)), 0 < t < 1, 3 < alpha <= 4,
where D-0+(alpha) is the standard Riemann-Liouville fractional derivative, subject either to the boundary conditions u(0) = u'(0) = u ''(0) = u ''(0) or u(0) = u'(0) = u ''(0) = 0, u ''(1) - beta u ''(eta) for eta, beta eta(alpha-3) epsilon (0, 1). Our analysis relies on a fixed point theorem of a sum operator. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. Two examples are given to illustrate the main results.

• 出版日期2013-4
• 单位山西大学