Existence of solution a fractional differential equation

In this paper, we study the existence of nontrivial solution for the fractional differential equation of order α with three point boundary conditions having the following form D α u ( t ) = f ( t , v ( t ) , D ν v ( t ) ) , t ∈ ( 0 , T ) u ( 0 ) = 0 , u ( T ) = a u ( ξ ) , where 1 < α < 2 , ν , a > 0 , ξ ∈ ( 0 , T ) , T α − 1 + a ξ α − 1 ≠ 0 . D is the standard Riemann-Liouville fractional derivative operator and f ∈ C ( [ 0 , 1 ] × R 2 , R ) . Applying the Leray-Schauder nonlinear alternative we prove the existence of at least one solution. As an application, we also given some examples to illustrate the results obtained.