SOLVABILITY FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

作者：Guo Yingxin^*

来源：Bulletin of the Australian Mathematical Society, 2009, 80(1): 125-138.

DOI：10.1017/S0004972709000124

• Caputo's fractional derivative
• fractional differential equation
• boundary-value problem
• nontrivial solution
• Leray-Schauder nonlinear alternative

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP)
D(alpha)u(t) + f(t, u(t)) + q(t) = 0, 0 < t < 1,
u(0) = 0, u(1) = beta u(eta)
where 1 < alpha <= 2, eta is an element of (0, 1), beta is an element of R = (-infinity, +infinity), beta eta(alpha-1) not equal 1, D(alpha) is the Riemann-Liouville differential operator of order alpha, and f : [0, 1] x R -> R is continuous, q(t) : [0, 1] -> [0, +infinity) is Lebesgue integrable. We give some sufficient conditions for the existence of nontrivial solutions to the above boundary-value problems. Our approach is based on the Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity on f which was essential for the technique used in almost all existed literature.

• 出版日期2009-8
• 单位曲阜师范大学