Existence Results for a Class of Fractional Differential Equations with Periodic Boundary Value Conditions and with Delay

and Applied Analysis 3 where G 1,α (t, s) = {{{{{{{{{ {{{{{{{{{ { Γ (α) E α,α (t α ) E α,α ((1 − s) α ) t α−1 (1 − s) α−1 1 − Γ (α) E α,α (1) +(t − s) α−1 E α,α ((t − s) α ) , 0 ≤ s ≤ t ≤ 1, Γ (α) E α,α (t α ) E α,α ((1 − s) α ) t α−1 (1 − s) α−1 1 − Γ (α) E α,α (1) , 0 ≤ t ≤ s ≤ 1, (14) is given in Section 4. Let x(⋅) : [−τ, 1] → R be the function defined by x (t) = { 0, t ∈ (0, 1) , φ (t) , t ∈ [−τ, 0] . (15) For each z ∈ C 1−α ([0, 1],R) with lim t→0 +t 1−α z(t) = c we denote z the function defined by z (t) = { z (t) , t ∈ (0, 1) , 0, t ∈ [−τ, 0] . (16) If u(⋅) satisfies the integral equation,


Introduction
In mathematics delay differential equations are a type of differential equation in which the derivative of unknown function at a certain time is given in terms of the values of the function at previous times.
While physical events such as acceleration and deceleration take little time compared to the times needed to travel most distances, times involved in biological processes such as gestation and maturation can be substantial when compared to the data-collection times in most population studies.Therefore, it is often imperative to explicitly incorporate these process times into mathematical models of population dynamics.These process times are often called delay times, and the models that incorporate such delay times are referred as delay differential equation models [1,2].
Recently Benchoohra et al. [28] studied existence of solutions for a class of fractional differential equations with infinite delay; namely, where   is the standard Riemann-Liouville fractional derivative and  satisfies some assumptions.First, in this paper we consider nonlinear delayed fractional differential equations: associated with boundary conditions where   is the standard Riemann-Liouville fractional derivative and  is a continuous function.Here   (⋅) represents the properitoneal state from time − up to time  which is defined by   () = ( + ), − ≤  ≤ 0. We proved the uniqueness of existence solutions for (2) with periodic boundary condition (3) under some further conditions.For investigating to establish an existence theorem, we also consider a class of nonlinear delayed fractional differential equations of the form with periodic boundary condition The paper has been organized as follows.In Section 2 we give basic definitions and preliminary.Unique solution of ( 2)-( 3) under some conditions is proved in Section 3. The existence solution of ( 4)-( 5) under some assumptions has been presented in Section 4.

Preliminaries
For the convenience of the readers, we firstly present the necessary definitions from the fractional calculus theory and functional analysis.These definitions and results can be found in the literature [3,7,38].
Let [0, 1] be the Banach space of all continuous real functions defined on [0, 1] with the norm Let   [0, 1],  ≥ 0, be the space of all functions  such that   () ∈ [0, 1] which is a Banach space when endowed with the norm Definition 1.For a function  defined on an interval [, ], the Riemann-Liouville fractional integral of  of order  > 0 is defined by and Riemann-Liouville fractional derivative of () of order  > 0 defined by provided that the right-hand side of the pervious equation is pointwise defined on (, +∞). We Definition 3. The beta function is usually defined by and we have also the following expression for the beta function: Theorem 4 (Arzela-Ascoli's theorem).A subset of [, ] is compact if and only if it is closed, bounded, and equicontinuous.
Theorem 5 (Banach's fixed point theorem).Consider a metric space  = (, ), where  ̸ = 0. Suppose that  is complete and  :  →  is a contraction on .Then  has precisely one fixed point.

Uniqueness of Solution
In this section we prove (2) with boundary condition (3) and another condition on  has a unique solution.Before proving, we need to introduce some notations that will be provided in the following.

Existence of Solution
In this section, by using Krasnoselskii's theorem, we discuss the existence solution of (4) under some assumptions on  and further conditions.Before proving this theorem, we prove the following lemma which will be used in the next theorem.

Lemma 8. Consider the following nonlinear fractional differential equation of the form
with periodic boundary conditions where ℎ is a continuous function.Then the periodic boundary value problem (28)-( 29) is equivalent to an integral equation given by () = ∫ where Proof.We consider the following fractional differential equation: with where lim  → 0 +  1− () = .Laplace transform of (31) yields from which and the inverse Laplace transform gives the solution Hence we have Therefore, which leads to since Γ() , (1) ̸ = 1 we have Then the solution of the problem ( 28)-( 29) is given by This completes the proof.Now we prove our main result using Lemma 8 and two more assumptions which follow next.
Proof.(i) Note that by Lemma 8, (4)-( 5) is equivalent to integral equation (17).Define  1 ,  2 : Ω → Ω by For  ∈  we have, (ii) We will prove that  1 is a contraction: Then we have (50) Abstract and Applied Analysis 7 Then by Arzela-Ascoli's theorem we conclude that  2 is compact.By using Krasnoselskii's theorem there exists  ∈  such that  is a fixed point of .This completes the proof.

Conclusions
We considered two types of nonlinear delay fractional differential equations (FDE) with periodic boundary conditions involving Remann-Liouville fractional derivative possessing with a lower terminal at 0. In order to obtain the results in this paper, we have shown the existence and the uniqueness of solution for a class of nonlinear delayed FDE by Banach contraction principle.Then using Krasnoselskii's fixed point theorem we established an existence theorem for a different type of the equation that we have proven its uniqueness theorem.