Fractional fundamental lemma and fractional integration by parts formula -- Applications to critical points of Bolza functionals and to linear boundary value problems

TL;DR

This paper develops fundamental lemmas and integration by parts formulas for fractional calculus, applying them to derive optimality conditions for fractional variational problems and to establish existence results for linear fractional boundary value problems.

Contribution

It introduces a fractional fundamental lemma and a fractional integration by parts formula, enabling new analysis tools for fractional variational calculus and boundary value problems.

Findings

Derived a fractional fundamental lemma.

Established a fractional integration by parts formula.

Proved existence of solutions for linear fractional boundary value problems.

Abstract

In the first part of the paper, we prove a fractional fundamental (du Bois-Reymond) lemma and a fractional variant of the integration by parts formula. The proof of the second result is based on an integral representation of functions possessing Riemann-Liouville fractional derivatives, derived in this paper too. In the second part of the paper, we use the previous results to give necessary optimality conditions of Euler-Lagrange type (with boundary conditions) for fractional Bolza functionals and to prove an existence result for solutions of linear fractional boundary value problems. In the last case we use a Hilbert structure and the Stampacchia theorem.