In this paper, we study the oscillatory behavior of solutions for a type of generalized proportional fractional differential equations with forcing and damping terms. Several oscillation criteria are established for the proposed equations in terms of Riemann-Liouville and Caputo settings. The results of this paper generalize some existing theorems in the literature. Indeed, it is shown that for particular choices of parameters, the obtained conditions in this paper reduce our theorems to some known results. Numerical examples are constructed to demonstrate the effectiveness of the our main theorems. Furthermore, we present and illustrate an example which does not satisfy the assumptions of our theorem and whose solution demonstrates nonoscillatory behavior.
Fractional calculus is a mathematical branch investigating the properties of derivatives and integrals of non-integer orders. The significance of this subject falls in the fact that the fractional derivative has the feature of nonlocal nature. This property makes these derivatives suitable to simulate more physical phenomena such as earthquake vibrations, polymers, and so forth; see, for example, References [1,2,3,4,5,6,7,8,9,10] and the references cited therein.
In recent years, there have appeared different types of fractional derivatives. However, it has been realized that most of these derivatives lose some of their basic properties that classical derivatives have such as the product rule and the chain rule. Fortunately, Khalil et al.  defined a new well-behaved fractional derivative, called the “conformable fractional derivative”, which depends entirely on the classical limit definition of the derivative. Thereafter, researchers developed the conformable derivative and obtained different results exposing its features [12,13,14]. Recently, Jarad et al.  introduced the generalized proportional fractional (GPF) derivative of Caputo and Riemann-Liouville type involving exponential functions in their kernels. The GPF derivative not only preserves classical properties but also verifies semi group property and of nonlocal behavior. For recent results involving GPF derivative, one can refer to References [16,17,18].
In 2012, Grace et al.  initiated the study of oscillation theory for fractional differential equations. Thereafter, many researchers have investigated the oscillatory properties of fractional differential equations; see for instance References [20,21,22,23,24,25]. In 2019, Aphithana et al.  studied forced oscillatory properties of solutions to the conformable initial value problem of the form
where , ,p, , , are continuous functions, is the left conformable derivative of order of x, in the Riemann-Liouville setting and is the left conformable integral operator of order , , .
They also studied the forced oscillation of conformable initial value problems in the Caputo setting of the form
where , , and is the left conformable derivative of order of x, in the Caputo setting.
In 2020, Sudsutad et al.  established some oscillation criteria for the following generalized proportional fractional differential equation
with , is the generalized proportional fractional derivative operator of order , , in the Riemann-Liouville setting and is the generalized proportional fractional integral operator.
In this paper, motivated by the above papers, we establish some sufficient conditions for forced oscillation criteria of all solutions of the generalized proportional fractional (GPF) initial value problem with damping term in the Riemann-Liouville type of the form:
where , , is the left GPF derivative of order of y, in the Riemann-Liouville setting and is the left GPF integral of order , , , and p, , , .
Moreover, we study the forced oscillation criteria of all solutions of the GPF initial value problem with damping term in the Caputo type of the form
where , , is the left GPF derivative of order of y, in the Caputo setting and , and is the proportional derivative defined in Reference .
We claim that the results of this paper improve and generalize previously existing oscillation results in Reference .
The solution y of problem (1) (respectively (2)) is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is called nonoscillatory. An equation is called oscillatory if all its solutions are oscillatory.
In this section, we provide some basic definitions and results which will be used throughout this paper. For the justifications and proofs, the reader can consult References [13,15].
For , let the functions be continuous such that for all we have
and , , , .
Then, Anderson et al.  defined the modified conformable differential operator of order by
provided that the right-hand side exists at and The derivative given in (4) is called a proportional derivative. For more details about the control theory of the proportional derivatives and its component functions and , we refer the reader to .
Of special interest, we shall restrict ourselves to the case when and .
Therefore, (4) becomes
Notice that and . It is clear that the derivative (5) is somehow more general than the conformable derivative which does not tend to the original function as tends to 0.
To find the associated integral to the proportional derivative in (5), we solve the following equation
The above equation is a first order linear differential equation and its solution is given by
Define the proportional integral associated to by
where we accept that
 Let f be defined on and differentiable on and . Then, we have
 For and , we define the left GPF integral of f by
where is the left Riemann-Liouville fractional integral of order α.
The right GPF integral ending at b, however, can be defined by
where is the right Riemann-Liouville fractional integral of order α.
 For and , we define the left GPF derivative of f by
The right GPF derivative ending at b is defined by
If we let in Definition 4, then one can obtain the left and right Riemann-Liouville fractional derivatives as in . Moreover, it is clear that
 (Partial Conformable Derivatives). Let , and let the functions be continuous and satisfy (3). Given a function such that exists for each fixed , define the partial differential operator via
 (Conformable Exponential Function). Let , the points with , and let the function be continuous. Let be continuous and satisfy (3), with and Riemann integrable on . Then the exponential function with respect to in (4) is defined to be
Using (4) and (14), we have the following basic results.
 (Basic Derivatives). Let the conformable differential operator be given as in (4), where . Let the function be continuous. Let be continuous and satisfy (3), with and Riemann integrable on . Assume the functions f and g are differentiable as needed. Then
for and fixed , the exponential function satisfies
Using Proposition 1 (ii) with , and , we get , it is easy to verify that is a nonoscillatory solution of (51). Figure 1 demonstrates the solution .
In this paper, the oscillatory behavior of solutions of generalized proportional fractional initial value problem is studied. Forced and damped oscillation results are obtained via GPF operators in the frame of Riemann-Liouville and Caputo settings. The main theorems of this paper improve and generalize some existing oscillation theorems reported in the literature. In particular, for the choice of , our contributions obtained using GPF operators cover the results discussed in Reference  which are obtained via conformable operators. At the end, we presented some numerical examples with particular values of parameters to illustrate the validity of the proposed results. Interestingly, we provided an example demonstrating that the failure of any condition forces the existence of a nonoscillatory solution. This justifies the advantage of our findings.
We believe that the results of this paper are of great importance for the audience of interested researchers. Several types of oscillation conditions could be generalized by considering respective equations within GPF derivatives.
All authors contributed equally and significantly to this paper. All authors have read and approved the final version of the manuscript.
This research received no external funding.
J. Alzabut would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17. The second author was supported by DST-INSPIRE Scheme (No.DST/INSPIRE Fellowship/2018/IF180260) New Delhi, India. The third author was supported by DST-FIST Scheme (Grant No. SR/FST/MSI-115/2016), New Delhi, India. The fourth author was partially supported by Navamindradhiraj University Research Fund (NURS), Navamindradhiraj University, Thailand.
Conflicts of Interest
The authors declare no conflict of interest.
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The nonoscillatory behavior of the solution .
The nonoscillatory behavior of the solution .