A FRACTIONAL FUNDAMENTAL LEMMA AND A FRACTIONAL INTEGRATION BY PARTS FORMULA - APPLICATIONS TO CRITICAL POINTS OF BOLZA FUNCTIONALS AND TO LINEAR BOUNDARY VALUE PROBLEMS

作者：Bourdin Loic^*; Idczak Dariusz

来源：Advances in Differential Equations, 2015, 20(3-4): 213-232.

DOI：10.57262/ade/1423055200

In this paper, we first identify some integrability and regularity issues that frequently occur in fractional calculus of variations. In particular, it is well-known that Riemann-Liouville derivatives make boundary singularities emerge. The major aim of this paper is to provide a framework ensuring the validity of the fractional Euler-Lagrange equation in the case of a Riemann-Liouville derivative of order alpha is an element of (0,1). For this purpose, we consider the set of functions possessing p-integrable Riemann-Liouville derivatives and we introduce a class of quasi-polynomially controlled growth Lagrangian.
In the first part of the paper, we prove a new fractional fundamental (du Bois-Reymond) lemma and a new fractional integration by parts formula involving boundary terms. The proof of the second result is based on an integral representation of functions possessing Riemann-Liouville derivatives. In the second part of the paper, we give not only a necessary optimality condition of Euler-Lagrange type for fractional Bolza functionals, but also necessary optimality boundary conditions. Finally, we give an additional application of our results: we prove an existence result for solutions of linear fractional boundary value problems. This last result is based on a Hilbert structure and the classical Stampacchia theorem.

• 出版日期2015-4