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 [[bar.d].sup.(0)].

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, Eqs.

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