SOLVABILITY FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP)where 1<α≤2,η∈(0,1),β∈ℝ=(−∞,+∞),βηα−1≠1,Dαis the Riemann–Liouville differential operator of orderα, andf:[0,1]×ℝ→ℝ is continuous,q(t):[0,1]→[0,+∞) 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 onfwhich was essential for the technique used in almost all existed literature.