In this work we prove the existence of solutions for the fractional differential equation tDα∞(-∞Dαtu(t))+L(t)u(t)=∇W(t,u(t)), u ∈α(ℝ,ℝN). Hwhere α ∈ (1/2, 1). Assuming L is coercive at infinity we show that this equation has at least one nontrivial solution. © 2013 Texas State University - San Marcos.