ABSTRACT
In this study, we generalize a time-delay system joining n-deviating arguments by utilizing the concept of the Riemann–Liouville fractional calculus. The class of equations is taken in view of the Rayleigh-type equation. Our tool is based on the fixed-point theorems. The existence of the outcomes is delivered under some certain conditions. The application is illustrated in the sequel.
1. Introduction
Time-delay schemes or dead-time schemes, transmissible schemes are categories of differential equations (ordinary, partial and fractional) in which the derivative of the unidentified function at a positive time is given in the relations of the values of the function at former times. Usually such equations are imposed with deviating arguments. The applications of this problem are increasing and appearing in many fields such as dynamic systems and its relatives (Luo, Wang, Liang, Wei, and Alsaadi, Citation2016; Luo, Wang, Wei, and Alsaadi, Citation2016).
The most important study of time-delay systems is the periodicity of the solution. The periodic solutions of Rayleigh equations with two deviating arguments have been studied by many authors (Huang, He, Huang, and Tan, Citation2007; Peng, Liu, Zhou, and Huang, Citation2006; Sirma, Tunç, and Özlem, Citation2010). These studies mentioned that periodic solutions have desired properties in differential equations, and are among the most vital research directions in the theory of differential equations. Such solutions are also vital in dynamical systems, with applications ranging from celestial mechanics to biology and finance. The class of fractional differential equations (FDEs) is the most important modifications of the field of ODEs to an arbitrary (non-integer) order to model complex phenomena (Ibrahim, Citation2012; Ibrahim and Jahangiri, Citation2015; Ibrahim, Ahmad, and Mohammed, Citation2016; Lizama and Poblete, Citation2011). Investigations in physics, engineering, biological sciences and other fields have recently demonstrated the application of FDEs in the dynamics of many systems. FDEs with delay can also describe natural phenomena more accurately than those without delay.
The periodic solutions have also been studied for the different generalizations of the Rayleigh equation. Some studies are introduced to discuss the existence of solutions of certain ordinary differential equations with deviating arguments (see Chaddha and Pandey, Citation2016; Feng, Citation2013; Wang and Zhang, Citation2009). In particular, a specific example has been provided in on how T-periodic solutions can be acquired via these theorems with
. The present study follows this direction. The active performance of the Rayleigh equation has been generally examined because of its request in many practical fields, such as mechanics, physics and the engineering method grounds. A similar method is observed in the case of an improved additional voltage. The Rayleigh system with n-deviating arguments is in the form of
(1) where
and
denotes the Riemann–Liouville differential operator. System (Equation1
(1) ) can be represented analogy voltage transmission. In a powered problem, ϕ characterizes a damping or friction time,
signifies the returning force, ϱ is a superficially useful force and
is the time pause of the returning force. The periodicity of solutions trends in vibration and noise engineering. Therefore, we aim to establish the existence of the periodic solutions of Equation (Equation1
(1) ) under some certain conditions. These results are supported by an example.
2. Setting
In this section, we provide background definitions and existing results that are essential to our arguments to establish the periodic solutions of Equation (Equation1(1) ). We let χ be a real Banach space and
be a completely continuous operator. The following preliminaries can be found in Burton (Citation1985).
Definition 2.1.
If is an isolated fixed point of Λ, then the fixed-point index at
of Λ is defined by
(2) where
is a neighbourhood of
which satisfies that
is the unique fixed point in
of Λ.
Definition 2.2.
If there exits with
such that
then α is called an eigenvalue of operator Λ and
is called the eigenfunction of operator Λ corresponding to α.
Definition 2.3.
Let be continuous. The function
is periodic on
if
(3)
Definition 2.4.
The Riemann–Liouville fractional integral is defined as follows:
where Γ denotes the gamma function. The Riemann–Liouville fractional derivative is defined as follows:
where Γ denotes the Gamma function
see Jumarie, Citation2013; Kilbas, Srivastava, and Trujillo, Citation2006; Podlubny, Citation1999; Tarasov, Citation2010).
Lemma 2.5
Schauder fixed-point theorem
Every continuous function from a convex compact subset of a Banach space to itself has at least one fixed point.
Lemma 2.6
Leray–Schauder fixed-point theorem
Suppose that χ is a real Banach space and is a completely continuous operator. If
(4) is bounded, then Λ has a fixed point
where
(5)
Lemma 2.7.
Suppose and
. Then
Lemma 2.8.
Assume that is a constant,
is periodic with period T, and
. Then, for any
that is periodic with period T, then
3. Findings
The following assumptions are used in this study:
(
)
(
(
(
(
(
(
Theorem 3.1.
Let –
hold. If
and
then (Equation1
(1) ) has at least one periodic solution.
Proof.
Our aim is to apply Lemma 2.6. Let
Then χ and ϒ are real Banach spaces with the norm
We define an operator
by
(8) It is clear that
is continuous and bounded. By utilizing
we define an extended operator
Thus, Λ is completely continuous. According to Lemma 2.6, the set
is bounded in ϒ, and hence, Λ has a fixed point in ϒ. Therefore, Equation (Equation1
(1) ) has a periodic solution u such that
. To prove that the periodic solution is bounded, we aim to apply Lemmas 2.7 and 2.8. Consider
achieving
. Consequently,
is a solution for
(9) Assume that
and
are the maximum point and minimum point of
. Then, we have
and thus, there exists
with
(10) which yields
(11) By
, it is clear that
. Therefore, we obtain
(12) consequently, we attain
(13) where
is the norm of
. By using the fact (see Kilbas et al., Citation2006),
Equation (Equation9
(9) ) can be approximated as follows:
(14) If
, then
has a periodic solution. Therefore, by multiplying Equation (Equation14
(14) ) with
integrating from 0 to T and taking into account that u is a periodic solution of Equation (Equation14
(14) ), we obtain
(15) Using
, we obtain that
(16) Holder's inequality implies
(17) Since
are T-periodic, differentiable, and
(18) then, we obtain
(19) Joining Equations (Equation17
(17) ) and (Equation19
(19) ) with
and
, we impose
(20) In view of Lemma 2.7, we conclude
(21) Consequently, we receive
(22) By
and Lemma 2.8, we observe
(23) Thus, applying Equations (Equation21
(21) )–(Equation23
(23) ) in Equation (Equation20
(20) ), we arrive at
(24) which is equivalent to
for sufficient
thus, we get
(25) This showed that the periodic solution of Equation (Equation1
(1) ) is bounded. This completes the proof.
Theorem 3.2.
Assume that –
hold. If
and
Proof.
Let and ϒ be given as in Theorem 3.1. We define a differential operator Θ as follows:
Our aim is to apply Lemma 2.5. Therefore, we must show that the operator Λ is convex in ϒ. By utilizing the assumptions
–
and
, a computation implies that
But κ is arbitrary, this yields
(26) Thus, we get
. Since
, we have
which implies that Λ is a continuously differentiable convex function. Hence, Λ has at least one periodic fixed point, which corresponds to the solution of Equation (Equation1
(1) ). In the same manner of Theorem 3.1, and
we have
(27) which is equivalent to
for sufficient thus, we get
(28) This leads that the periodic solution of Equation (Equation1
(1) ) is bounded. This completes the proof.
4. An example
Consider and
; in the following fractional system:
(29) such that
and
. This implies
and if
we get
then in view of Theorem 3.1, the system (Equation29
(29) ) has at least one periodic solution in
. Note that if
, Theorem 3.1 is failing because
.
Now, we let . It is clear that
. Hence, under the above data, we conclude that
which yields, by Theorem 3.2, the system (Equation29
(29) ) has at least one periodic solution.
5. Conclusions
By utilizing various types of fixed-point theorems, we showed that a system of FDEs-type Rayleigh equation of n-arguments has at least one periodic solution. We imposed two cases of this system, depending on the n-arguments (homogenous and non-homogenous). The conditions are little different in both cases. The main function of this system is , which has upper and lower bounds (see
and
). The delay is computed for the periodic solution by the argument functions
. In the future works, we shall discuss the periodicity of solutions of fractional differential systems, including the generalized term
by utilizing the continuation theorem.
Authors' contributions
The authors jointly worked on deriving the results and approved the final manuscript.
Acknowledgments
The authors thank the referees for giving useful suggestions for improving the work.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCiD
R.W. Ibrahim http://orcid.org/0000-0001-9341-025X
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