Periodicity of a time-delay system of fractional order joining n-deviating arguments

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.


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, 2016;. 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, 2007;Peng, Liu, Zhou, and Huang, 2006;Sirma, Tunç, and Özlem, 2010). 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, 2012;Ibrahim and Jahangiri, 2015;Ibrahim, Ahmad, and Mohammed, 2016;Lizama and Poblete, 2011). Investigations in physics, engineering, biological sciences and other fields have recently CONTACT Rabha W. Ibrahim rabhaibrahim@yahoo.com 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, 2016;Feng, 2013;Wang and Zhang, 2009). In particular, a specific example has been provided in on how T-periodic solutions can be acquired via these theorems with (t) = 0. 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 and D 2μ denotes the Riemann-Liouville differential operator. System (1) can be represented analogy voltage transmission. In a powered problem, ϕ characterizes a damping or friction time, ϑ ı signifies the returning force, is a superficially useful force and ε k 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 (1) under some certain conditions. These results are supported by an example.

Setting
In this section, we provide background definitions and existing results that are essential to our arguments to establish the periodic solutions of Equation (1). We let χ be a real Banach space and : χ → χ be a completely continuous operator. The following preliminaries can be found in Burton (1985).
where N is a neighbourhood of u 0 , which satisfies that u 0 is the unique fixed point in N of .

Definition 2.2:
If there exits u 0 ∈ χ with u 0 = φ such that u 0 = αu 0 , α ∈ R then α is called an eigenvalue of operator and u 0 is called the eigenfunction of operator corresponding to α.
Definition 2.4: The Riemann-Liouville fractional integral is defined as follows: where denotes the gamma function. The Riemann-Liouville fractional derivative is defined as follows: where denotes the Gamma function (see Jumarie, 2013;Kilbas, Srivastava, and Trujillo, 2006;Podlubny, 1999;Tarasov, 2010). 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 : χ → χ is a completely continuous operator. If
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 G μ : ϒ → χ by It is clear that G μ is continuous and bounded. By utilizing G μ , we define an extended operator := κG μ : ϒ → ϒ. Thus, is completely continuous. According to Lemma 2.6, the set is bounded in ϒ, and hence, has a fixed point in ϒ. Therefore, Equation (1) has a periodic solution u such that u(0) = u(T). To prove that the periodic solution is bounded, we aim to apply Lemmas 2.7 and 2.8. Consider u ∈ ϒ, 0 < α < 1 achieving u = α u. Consequently, u(τ ) is a solution for Assume thatτ and τ are the maximum point and minimum point of u(τ ) on [0, T]. Then, we have By (H 4 ), it is clear that |u( )| ≤ d. Therefore, we obtain (12) consequently, we attain where • L 2 is the norm of L 2 [0, T]. By using the fact (see Kilbas et al., 2006), Equation (9) can be approximated as follows: If 1 − α = (2μ + 1), then u (τ ) + u(τ ) = 0 has a periodic solution. Therefore, by multiplying Equation (14) with u (τ ), integrating from 0 to T and taking into account that u is a periodic solution of Equation (14), we obtain Using (H 1 ), we obtain that Holder's inequality implies Since Joining Equations (17) and (19) with (H 2 ) and (H 3 ), we impose In view of Lemma 2.7, we conclude Consequently, we receive By (H 5 ) and Lemma 2.8, we observe Thus, applying Equations (21)-(23) in Equation (20), we arrive at which is equivalent to for sufficient > ; thus, we get This showed that the periodic solution of Equation (1) is bounded. This completes the proof.

Theorem 3.2: Assume that
Proof: Let , χ and ϒ be given as in Theorem 3.1. We define a differential operator as follows: Our aim is to apply Lemma 2.5. Therefore, we must show that the operator is convex in ϒ. By utilizing the assumptions (H 1 )-(H 7 ) and n k=1 ϑ k (τ , 0) = (τ ), a computation implies that But κ is arbitrary, this yields Thus, we get ( u) = (u). Since ( u) ≥ ( 0), we have which implies that is a continuously differentiable convex function. Hence, has at least one periodic fixed point, which corresponds to the solution of Equation (1). In the same manner of Theorem 3.1, and which is equivalent to This leads that the periodic solution of Equation (1) is bounded. This completes the proof.

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 ϕ(u ), which has upper and lower bounds (see (H 1 ) and (H 7 )). The delay is computed for the periodic solution by the argument functions ϑ k . In the future works, we shall discuss the periodicity of solutions of fractional differential systems, including the generalized term D 3μ by utilizing the continuation theorem.