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Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition - application to fractional variational problems. (English) Zbl 1340.26012
This paper gives an answer to the question, posed in [T. Odzijewicz et al., Abstr. Appl. Anal. 2012, Article ID 871912, 24 p. (2012; Zbl 1242.49019)], on the existence of solutions to fractional variational problems. Fractional Euler-Lagrange equations are necessary optimality conditions for optimizers. However, in the absence of existence, the necessary conditions for extremality are vacuous. Therefore, this paper is essential for the development of a theory of the fractional calculus of variations. The authors provide sufficient conditions ensuring the existence of a minimizer for generalized Lagrangian functionals in the case of bounded-time intervals. They also prove a necessary optimality condition of Euler-Lagrange type. Finally, the main results are illustrated through examples with special cases of general kernel operators and, in particular, of fractional integrals (Riemann-Liouville and Hadamard).

26A33 Fractional derivatives and integrals
49J05 Existence theories for free problems in one independent variable
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