which stems from the fact that exp(xt) is an eigenvector of the derivative operator
[D.sub.x] to eigenvalue t.
Then the modified Caputo's derivative operator
of f is defined by
where [lambda] > 0 is a parameter, 1 < [alpha] [less than or equal to] 2, n - 1 < [beta] [less than or equal to] n, n [greater than or equal to] 2, [mathematical expression not reproducible] is the Riemann-Liouville derivative operator
, [mathematical expression not reproducible] is the Caputo fractional derivative operator
, [[phi].sub.p] is the p-Laplacian operator defined by [[phi].sub.p](s) = [[absolute value of s].sup.p-2]s, [([[phi].sub.p]).sup.-1] = [[phi].sub.q], 1/p + 1/q = 1, p > 1, [eta] [member of] (0, 1), and a > 0 and satisfies 1 - [a.sup.p-1] [[eta].sup.[alpha]-1] > 0, and f: [0,1] x [0, +[infinity]) [right arrow] [0, +[infinity]) is a continuous function.
277]) A covariant derivative operator
on [pi] is an R-bilinear map [nabla]: X(N) x r([pi]) [right arrow] [GAMMA]([pi]), (X, s) [right arrow] [[nabla].sub.X]s, such that:
where [d.sup.(0).sub.i,j] is the (i,j) element of discrete exterior derivative operator
The new momentum operator representation in (6), along with the correspondence principle [p.sub.[mu]] [??] i[[partial derivative].sub.[mu]], leads to a deformation of the derivative operator
in configuration space
Then, from Theorem 3 and the closability of the Malliavin derivative operator
, for some constant [epsilon] < 1,
where [mathematical expression not reproducible] denotes the higher-order terms with the Landau notation o(*) associated with the asymptotic behaviour of the function E (considering the variable quantities (U, [lambda])) and [phrase omitted][mathematical expression not reproducible] is the first-order partial derivative operator
associated with the function E with respect to [lambda] (resp., U) at point [[lambda].sup.k.sub.n+1] (resp., [U.sup.k.sub.n+1]) for fixed and constant displacement U (resp., mechanical load A) at point [U.sup.k.sub.n+1] (resp., [[lambda].sup.k.sub.n+1]) which is kth iterative variable of incremental time [t.sub.n+1].
[mathematical expression not reproducible] denotes Caputo's fractional derivative operator
[11-13], [mathematical expression not reproducible], since Caputo's fractional derivative allows us to couple the fractional differential equations with initial conditions in the traditional form [mathematical expression not reproducible].
Caputo or Riemann-Liouville fraction derivative operator
Baleanu and Mustafa present their own findings and some by others during the past few years into fractional calculus, which is used to study the fractional order integral and derivative operator
over real and complex domains.
The delta derivative operator
[[DELTA].sub.[phi]]: [K.sup.*] [right arrow] [K.sup.*] is defined by