Fractional du Bois-Reymond lemma of order α ∈ (1 over 2; 1)

The main result of the paper is a fractional du Bois-Reymond lemma for functions of one variable with Riemann-Liouville derivatives of order α ∈ (1/2, 1). Proof of this lemma is based on a theorem on the integral representation of a function possessing the fractional derivative of order α ∈ (1/2, 1) and on a fractional variant of the theorem on the integration by parts. These theorems are derived in the paper, too. We apply the lemma to some fractional variational problems and obtain the appropriate Euler-Lagrange equations.