Oscillation is one of the important branches in applied mathematics and can be induced or destroyed by the introduction of nonlinearity, delay, or a stochastic term. The oscillation of differential and difference equations contributes to many realistic applications, such as torsional oscillations, the oscillation of heart beats, sinusoidal oscillation, voltage-controlled neuron models, and harmonic oscillation with damping.
The empty sums and products were taken to be zero and one, respectively. Denote by he set of all natural numbers, the set of all real numbers, and the set of all positive real numbers. Define by and for any a, such that .
Definition 1 ([19,20]). The Euler gamma function is defined by:
Using its reduction formula, the Euler gamma function can also be extended to the half-plane except for
Definition 2 ([21]). The generalized falling function is defined by:
for those values of t and r such that the right-hand side of this equation makes sense. If is a nonpositive integer and is not a nonpositive integer, then we use the convention that
U. The generalized rising functits reduction on is defined by:
formula, the Euler gamma function can alose values of t and r so that the right-hand side of this equation is sensible. If t is a nonpositive integer, but is no be extended to the half-planet a nonpositive integer, then we use the convention that
Definition 3 ([22]). Lexcept for and
Definition 2 ([21]). . The gfirst-ordeneralized falling function isr forward (delta) and backward (nabla) differences of u are defined by:
respectively. The -order delta and nabla differences of u are defined recursively by
and:
respectively.
Definition 4 ([21]). Let and . Then, the -order delta fractional sum of u based at a is defined by:
Definition 5 ([21]). Let and . Then, the
f-or thosde values of t and r such that the right-hand side of this equation makes sense. If r nabla fractional sum of u based at a is defined by:
Definition 6 ([21]). Let , is a nonpositivnd choose integer and is not a nonpositive integer, then we use the convention tuch that . The . Th-orde generalized rising functionr Riemann–Liouville delta fractional difference of u is defined by:
Definition 7 ([21]). Let , and choose such that . Then, the
f-order those values of t and r so that the right-hand side of this equation is sensiblRiemann–Liouville nabla fractional difference of u is defined by:
Definition 8 ([23]). Le. If t , and is. a nonpositive integer, but Then, the is-order nCaputot a nonpositive integer, then we use t delta fractional difference of u is defined by:
whe convrention that . If , then:
Definition 39 ([2224]). Let and . Then, the . The first-order forward (delta) and backward (nabla)Caputo nabla fractional differences of u areis defined by:
where .
Consider the following higher-order nonlinear delta fractional difference equations involving the Riemann–Liouville and the Caputo operators of arbitrary order:
respectively. The -order delta and nabla differences of u are defined recursively by
and:
respectively.
Definition 4 ([21]). Let and . Then, the -order delta fractional sum of u based at a is defined by:
and:
Definition 5 ([21]). Let and . Then, the -order nabla fractional sum of u based at a is defined by:
Here, and choose such that
Definition 6 ([21]). Let ; ,and and choose sare continuous. A solution uch that of (1) (or (2)) is said to be oscillatory if for every natural number M, there exists such that . The ; otherwise, it is called non-oscillatory. An equation is said to be oscillatory if all of its solutions are oscillatory.
Let -order Riemann–Liouville delta fractional difference of u is defined by:
be continuous and , be positive real numbers. We make the following assumptions:
(A1) The functions satisfy the sign condition , , , ;
(A2)
Definition 7 ([21]). Let , and choose such that;
(A3) . Then, the -order Riemann–Liouville nabla fractional difference of u is defined by:
.
In [25], Senem et al. established some oscillation theorems given in the sequel.
Theorem 1 ([25]). Let (A1)–(A2) be satisfied with . If:
and:
for every sufficiently large T, where:
then Equation (1) is oscillatory.