Periodicity and positivity of a class of fractional differential equations

Fractional differential equations have been discussed in this study. We utilize the Riemann–Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is described in a fractional periodic Sobolev space. Positivity of outcomes is considered under certain requirements. We develop and extend some recent works. An example is constructed.

of differential equations of arbitrary order. The British mathematical physicist Lord Rayleigh introduced an equation of the form (Strutt 1877) to model a clarinet reed oscillation; which is showed by Wang and Zhang (2009). This equation was named after Lord Rayleigh, who studied equations of this type in relation to problems in acoustics. In the years of 1977 and 1985 respectively, Gaines and Mawhin (1977) have been imposed some continuation theorems and employed them to demonstrate the existence of periodic solutions to ordinary differential equations (ODEs). A specific example was given in Yang (2015, p. 99) to introduce, how T-periodic solutions can be obtained by using the established theorems for the differential equation of the form Newly, investigators discussed the existence of periodic solutions to Rayleigh equations and extended Rayleigh equations by considering or ignoring the concept of delay. Various new results concerning the existence of periodic solutions to the mentioned equations have been presented. Wang and Yan (2000) established the existence of periodic solutions of the non-autonomous Rayleigh equation of the type: Zhou and Tang (2007) have studied the existence of periodic solutions for a kind of nonautonomous Rayleigh equations of retarded type: Wang and Zhang (2009) investigated the following Rayleigh type equation: In this study, we consider a Rayleigh-type equation with state-dependent delay of the form and its conducive formal where ϕ and ϑ are 2π-periodic in t, ϕ(t, 0) = ϑ(t, 0) = 0 for t ∈ R, �, p ∈ C(R, R), ε, p are 2π -periodic in t, such that p has the property: and D µ is the Riemann-Liouville fractional differential operator.
x ′′ (t) + f (x ′ (t)) + g(t, x(t)) = e(t). (1) We have imposed two contributed theorems on the existence of periodic solutions of Eq. (1). Our main aim is to generalize, modify and extend the outcomes of the works given in Wang and Yan (2000), Zhou andTang (2007), Tunç (2014). In addition, this effort is a contribution to the subject in the literature and it may be useful for researchers who work on the qualitative behaviors of solutions. Positivity of solutions is investigated under some requests. Our method is based on the idea of the continuation partition theorem of degree theory. Applications are illustrated in the sequel.

Setting
In this paper, we need the following setting. For the sake of convenience, let that is, the Sobolev space W k,p (J ) is defined as with the order of the Sobolev space (W k,p (J ))k ∈ N.
In the sequel, we assume that k = 1. Hence, we deal with the fractional periodic Sobolev space of a continuous integrable function u(t), t ∈ [0, 2π ] Note that the above space is formulated as a Banach space.
Definition 1 Let X , . be a Banach space. Then φ : R → X is called periodic if φ is continuous, and for each ε>0 such that for a number t with the property that for each t ∈ R.
Fractional order integral and differentiation were obtained by Leibniz. To analyze phenomena having singularities of type t µ , the concept of fractional calculus is utilized. The fractional order operator is a nonlocal operator. Due to this property, fractional calculus is employed to study memories of Brownian motion, which is thought to be beneficial in mathematical sciences.

Definition 3
The Riemann-Liouville fractional derivative defined as follows: The periodicity of the class of fractional differential equations is studied in various spaces, Agarwal et al. studied the periodicity of various classes of fractional differential equations by assuming the mild solution (Agarwal et al. 2010), Ibrahim and Jahangiri (2015) imposed a periodicity method by applying some special transforms for fractional differential equations, and recently, Rakkiyappan et al. (2016) introduced the periodicity by utilizing fractional neural network model. Extra studies in fractional calculus can be located in Khorshidi et al. (2015).

Results
The subsequent Lemma plays a key function for showing the periodicity of Eq. (1). In the sequel, we assume that u ∈ W := W 1,p ([0, 2π ]).

Lemma 1 Suppose that u(t) is a continuous and differentiable T-periodic function with
Then together with the estimate From the above two inequalities and the Definition 3., we obtain that Hence, we complete the proof.
Lemma 1 shows the boundedness of the fractional differential operator by the norm of fractional space. This result allows us to investigate the periodicity of the solutions. If the differential equation satisfies the initial condition u(t • ) = 0, then, we can attain We have the following main results: Theorem 1 Suppose that there exist constants, with the validity such that the following conditions hold: • (H3) uϑ(t, u(t − ε(t, u)))>0; • |ϑ(t, u(t − ε(t, u)))|>η 1 |u| + κ for all t ∈ J , |u|>d; (H4) ϑ(t, u(t − ε(t, u)))>η 2 u − m for all t ∈ J , u ≤ −d.

If then Eq.
(1) has at least one 2π-periodic solution.
where By the inequality one can calculate that This completes the proof of Theorem 1. Theorem 1 shows that the solution is bounded by its fractional derivative in a fractional space. Therefore, Lemma 1 and Theorem 1 imply the periodicity of the solution in a bounded domain. Our next result illustrates different types of assumptions to get the periodicity and positivity of Eq. (2) and hence Eq. (1).
If then Eq.
Proof We again consider the auxiliary equation, Eq.

one can have
This completes the proof of Theorem 2. It is clear that the solution in Theorem 2 is positive as well as periodic.