  We study dynamic minimization problems of the calculus of variations with
generalized Lagrangian functionals that depend on a general linear operator $K$
and 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. Main results are illustrated with special cases, when $K$
is a general kernel operator and, in particular, with $K$ 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].
