In  Almeida studied certain problems of calculus of variations
that are dependent upon a Lagrange function on a Caputo-type fractional derivative; sufficient and necessary conditions of the first- and second-order are presented.
Mathematically, formulated in the view of the equations of (1)-(2), this particularly given problem setting is stated as the simplest problem of the calculus of variations
for the objective functionals likewise :
In the discrete time setting (see e.g., -,  and ) and in the time scale setting (see e.g., , , ), the question of characterizing the nonnegativity and positivity of F"([bar.x], [bar.u]; [eta], v) was intensively studied when F itself is symplectic (i.e., of the form (D) below), or when we are in the calculus of variations
Saunders, "Thirty years of the inverse problem in the calculus of variations
," Reports on Mathematical Physics, vol.
In order to solve this problem, we will resort to the calculus of variations
Li-Jost, Calculus of Variations
, Cambridge University Press, Cambridge, UK, 2008.
Bohner, "Calculus of variations
on time scales," Dynamic Systems and Applications, vol.
of Tokyo) introduces researchers and graduate students to the stochastic calculus of variation
, also called Malliavin calculus, for processes with jumps--that is both pure jump processes and jump-diffusions.
Introduction to the Calculus of Variations
and Control with Modern Applications provides the fundamental background required to develop rigorous necessary conditions that are the starting points for theoretical and numerical approaches to modern variational calculus and control problems.
Keywords: Calculus of variations
, Euler-Lagrange equation, boundedness of the solution.
Results on differential equations and the calculus of variations
with fractional operators of variable order can be found in [15, 16] and references therein.
Lions, The concentration-compactness principle in the calculus of variations
. The locally compact case.