Abstract
In this paper by using the minimal principle and Morse theory, we prove the existence of solutions to the following Kirchhoff type fractional differential equation:
{M(integral(R)(vertical bar(-infinity)D(t)(alpha)u(t)vertical bar(2) + b(t)vertical bar u(t)vertical bar(2))dt)
center dot(D-t(infinity)alpha((-infinity)D(t)(alpha)u(t) + b(t)u(t)) = f(t,u(t)), t is an element of R,
u is an element of H-alpha(R),
where alpha is an element of (1/2,1), D-t(infinity)alpha and D--infinity(t)alpha are the right and left inverse operators of the corresponding Liouville-Weyl fractional integrals of order a respectively, Ha is the classical fractional Sobolev Space, u is an element of R, b: R -> R, inf (t is an element of R)b(t) > 0, f:R x R -> R Caratheodory function and M: R+ -> R+ is a function that satisfy some suitable conditions.