Differential and Integral Equations

Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition – Application to fractional variational problems

Loïc Bourdin, Tatiana Odzijewicz, and Delfim F.M. Torres

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Abstract

We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and are defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler-Lagrange type. The main results are illustrated with special cases, when $K$ is a general kernel operator and, in particular, with $K$ being the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in [Abstr. Appl. Anal. 2012, Art. ID 871912; doi:10.1155/2012/871912].

Article information

Source
Differential Integral Equations, Volume 27, Number 7/8 (2014), 743-766.

Dates
First available in Project Euclid: 6 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.die/1399395751

Mathematical Reviews number (MathSciNet)
MR3200762

Zentralblatt MATH identifier
1340.26012

Subjects
Primary: 26A33: Fractional derivatives and integrals 49J05: Free problems in one independent variable

Citation

Bourdin, Loïc; Odzijewicz, Tatiana; Torres, Delfim F.M. Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition – Application to fractional variational problems. Differential Integral Equations 27 (2014), no. 7/8, 743--766. https://projecteuclid.org/euclid.die/1399395751


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