Every phenomenon in the world in one way or other is nonlinear in nature. Thus, the better understanding of these phenomena can be obtained from models constructed via nonlinear equations. The analytical solution of nonlinear equations is not always possible to obtain as in the case of linear equations. However, approximate solutions can be obtained for the nonlinear equations, which provide a better understanding of the behavior of the equations. In the case of nonlinear equations, without actually solving the equations, one can very well answer questions such as the existence of solutions, whether the system is stable, whether it can be controlled, whether the system is chaotic, or whether it exhibits periodicity. Thus, this direct method of analyzing the system behavior can be useful and help engineers in their research. Scientists and researchers are very much interested in the qualitative properties such as the oscillation, stability, controllability, bifurcation, chaos, and so on.
2. Preliminaries
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 . The generalized rising function is defined by:
for those values of t and r so that the right-hand side of this equation is sensible. If t is a nonpositive integer, but is not a nonpositive integer, then we use the convention that
Definition 3 ([22]). Let and . The first-order 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 -order nabla fractional sum of u based at a is defined by:
Definition 6 ([21]). Let , and choose such that . The -order Riemann–Liouville delta fractional difference of u is defined by:
Definition 7 ([21]). Let , and choose such that . Then, the -order Riemann–Liouville nabla fractional difference of u is defined by:
Definition 8 ([23]). Let , and . Then, the -order Caputo delta fractional difference of u is defined by:
where . If , then:
Definition 9 ([24]). Let and . Then, the -order Caputo nabla fractional difference of u is defined by:
where .
3. Oscillation
3.1. Oscillatory Behavior of Delta Fractional Difference Equations
Consider the following higher-order nonlinear delta fractional difference equations involving the Riemann–Liouville and the Caputo operators of arbitrary order:
and:
Here, and choose such that ; and are continuous. A solution u of (1) (or (2)) is said to be oscillatory if for every natural number M, there exists such that ; otherwise, it is called non-oscillatory. An equation is said to be oscillatory if all of its solutions are oscillatory.
Let be continuous and , be positive real numbers. We make the following assumptions:
(A1) The functions satisfy the sign condition , , , ;
(A2) ;
(A3) .
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.
Theorem 2 ([25]). Let and (A1)–(A3) be satisfied with . If
and:
for every sufficiently large T, where G is defined as in Theorem 1, then every bounded solution of Equation (1) is oscillatory.
Theorem 3 ([25]). Let (A1) and (A2) be satisfied with . If:
and:
for every sufficiently large T, where G is defined as in Theorem 1, then Equation (2) is oscillatory.
Theorem 4 ([
25])
. Let and (A1)–(A3) be satisfied with . If:
and:
for every sufficiently large T, where G is defined as in Theorem 1, then every bounded solution of Equation (2) is oscillatory.
Following the work in [
25], Li et al. [
26] investigated the oscillation of forced delta fractional difference equations with the damping term of the form:
where ; , and such that:
and for .
Theorem 5 ([
26])
. For , suppose that:
and:
where M is a constant and:
Then, Equation (3) is oscillatory.
Theorem 6 ([
26])
. For , suppose that:
and:
where M is a constant and V is defined as in Theorem 5. Then, Equation (3) is oscillatory.
In this line, Seçer et al. [
27] investigated the oscillation of the following nonlinear delta fractional difference equations:
for , Here, , and are the quotients of two odd positive numbers such that , , and are positive sequences,
is continuous, and:
Theorem 7 ([
27])
. If there exists a positive sequence ϕ such that:
then Equation (4) is oscillatory. Here:
and,
Theorem 8 ([
27])
. Let ϕ be a positive sequence. Furthermore, we assume that there exists a double sequence such that:
If:
Then, Equation (4) is oscillatory.
If we choose the double sequence:
we have the following corollary.
Corollary 1 ([
27])
. Under the conditions of Theorem 8 and: