The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay

and Applied Analysis 3 [0, b]. If there exist positive constants a and α ∈ (0, 1) such that V(t) ≤ w(t) + a ∫ t 0 (t − s) V(s)ds, then there exists a constant K = K(α) such that V(t) ≤ w(t)+Ka∫t 0 w(s)(t− s) −α ds, for all t ∈ [0, b]. In this paper we use the alternative Leray-Schauder’s theorem and Banach’s contraction principle for getting the main results. These theorems can be found in [27, 28]. 3. Existence and Uniqueness In this section, we prove the existence results for (1) and (3) by using the alternative of Leray-Schauder’s theorem. Further, our results for the unique solution is based on the Banach contraction principle. Let us start by defining what we mean by a solution of (1). Let the space Ω = {y : (−∞, b] 󳨀→ R : y|(−∞,0] ∈B and y|[0,b] is continuous} . (11) A function y ∈ Ω is said to be a solution of (1) if y satisfies (1). For the existence results on (1), we need the following lemma. Lemma 4. Equation (1) is equivalent to the Volterra integral equation y (t) = n ∑ k=0 ( −α k ) n!t n−k (n − k)! I α−β+k y (t) + I α f (t, yt) , t ∈ J. (12) Proof. The proof is an immediate consequence of Proposition 2. To study the existence and uniqueness of solutions for (1), we transform (1) into a fixed-point problem. Consider the operator P : Ω → Ω defined by Py (t) = { L (I) y (t) + If (t, yt) , t ∈ [0, b] , φ (t) , t ∈ (−∞, 0] , (13) where, L (I) = n ∑ k=0 ( −α k ) n!t n−k (n − k)! I α−β+k . (14) Let x(⋅) : (−∞, b] → R be the function defined as x (t) = { 0, if t ∈ [0, b] , φ (t) , if t ∈ (−∞, 0] . (15) Then, we get x0 = φ. For each z ∈ C([0, b],R) with z(0) = 0, we denote by z the function defined as follows: z (t) = { z (t) , if t ∈ [0, b] , 0, if t ∈ (−∞, 0] . (16) If y(⋅) satisfies the integral equation y(t) = L(I)y(t) + I α f(t, yt), then we can decompose y(⋅) as y(t) = z(t) + x(t), −∞ < t ≤ b, which implies yt = zt + xt for every 0 ≤ t ≤ b, and the function z(⋅) satisfies z (t) =L (I) z (t) + I α f (t, zt + xt) , (17) set C0 = {z ∈ C([0, b],R) : z(0) = 0}, and let ‖ ⋅ ‖b be the seminorm in C0 defined by ‖z‖b = ‖z0‖B + sup{|z(t)| : 0 ≤ t ≤ b} = sup{|z(t)| : 0 ≤ t ≤ b}, z ∈ C0. C0 is a Banach space with norm ‖ ⋅ ‖b. Let the operator F : C0 → C0 be defined by Fz (t) =L (I) z (t) + I α f (t, zt + xt) , (18) where t ∈ [0, b]. The operator P has a fixed point equivalent to F that has a fixed point too. Theorem5. Assume thatf is a continuous function, and there exist p, q ∈ C(J,R) such that |f(t, u)| ≤ p(t)+q(t)‖u‖B, t ∈ J, u ∈B. Then, (1) has at least one solution on (−∞, b]. Proof. It is enough to show that the operator F : C0 → C0 defined as (18) satisfies the following: (i) F is continuous, (ii) F maps bounded sets into bounded sets in C0, (iii) F maps bounded sets into equicontinuous sets of C0, and (iv) F is completely continuous. (i) Let {zn} converges to z in C0, then 󵄩󵄩󵄩󵄩Fzn (t) − Fz (t) 󵄩󵄩󵄩󵄩 ≤ n ∑ k=0 󵄨󵄨󵄨󵄨( −α k ) 󵄨󵄨󵄨󵄨 n!t n−k (n − k)! I α−β+k 󵄨󵄨󵄨󵄨zn (t) − z (t) 󵄨󵄨󵄨󵄨 + I α 󵄨󵄨󵄨󵄨f (t, nt + xt) − f (t, zt + xt) 󵄨󵄨󵄨󵄨 ≤ n ∑ k=0 󵄨󵄨󵄨󵄨( −α k ) 󵄨󵄨󵄨󵄨 n! b n−k 󵄩󵄩󵄩󵄩zn − z 󵄩󵄩󵄩󵄩 (n − k)!Γ (α − β + k + 1) + b α 󵄩󵄩󵄩󵄩f (t, nt + xt) − f (t, zt + xt) 󵄩󵄩󵄩󵄩 Γ (α + 1) . (19) Hence, ‖Fzn(t) − Fz(t)‖ → 0 as zn → z, and thus f is continuous. (ii) For any λ > 0, let Bλ = {z ∈ C0 : ‖z‖b ≤ λ} be a bounded set. We show that there exists a positive 4 Abstract and Applied Analysis constant μ such that ‖Fz‖∞ ≤ μ. Let z ∈ Bλ, since f is a continuous function, we have for each t ∈ [0, b], |Fz (t)| ≤ n ∑ k=0 󵄨󵄨󵄨󵄨( −α k ) 󵄨󵄨󵄨󵄨 n!t n−k (n − k)!Γ (α − β + k)

The notion of the phase space B plays an important role in the study of both qualitative and quantitative theories for functional differential equations.A common choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato [26].
Our approach is based on the Banach fixed-point theorem and on the nonlinear alternative of Leray-Schauder type [27,28].The organization of the paper is as follows.
In Section 2, we present some basic mathematical tools used in the paper.The main results are presented in Section 3. Section 4 is dedicated to our conclusions.

Preliminaries
In this section, we present some basic notations and properties which are used throughout this paper.First of all, we will explain the phase space B introduced by Hale and Kato [26] We notice that in this paper, we select  = 0 and  = R; thus (iii) can be converted to See [28] for examples of the phase space B satisfying all axioms (B 1 )-(B 4 ).
Let R + = (0,+∞) and  0 (R + ) be the space of all continuous real function on R + .Consider also the space  0 (R) ≥0 of all continuous real functions on R ≥0 which later identifies with the class of all  ∈  0 (R + ) such that lim  → 0 + () = (0 + ) ∈ R. By (, R), we denote the Banach space of all continuous functions from  into R with the norm ‖‖ ∞ := sup{|()| :  ∈ }, where | ⋅ | is a suitable complete norm on R.
The most common notation for th order derivative of a real-valued function (), which is defined in an interval denoted by (, ), is    ().Here, the negative value of  corresponds to the fractional integral.Definition 1.For a function  defined on an interval [, ], the Riemann-Liouville fractional integral of  of order  > 0 is defined by [1,6] and the Riemann-Liouville fractional derivative of () of order  > 0 reads as If  is continuous on [, ], then    () is integrable,  1− ()| = = 0, and Proposition 2. Let  be continuous on [0, ] and  a nonnegative integer, then where Proof.(i) can be found in [6, page 53], and (ii) is an immediate consequence of (7), and (i).
Lemma 3 (see [29]).Let V : [0, ] → [0, ∞) be a real function and (⋅) a nonnegative, locally integrable function on In this paper we use the alternative Leray-Schauder's theorem and Banach's contraction principle for getting the main results.These theorems can be found in [27,28].

Existence and Uniqueness
In this section, we prove the existence results for ( 1) and ( 3) by using the alternative of Leray-Schauder's theorem.Further, our results for the unique solution is based on the Banach contraction principle.Let us start by defining what we mean by a solution of (1).Let the space A function  ∈ Ω is said to be a solution of (1) if  satisfies (1).
For the existence results on (1), we need the following lemma.

Lemma 4. Equation (1) is equivalent to the Volterra integral equation
Proof.The proof is an immediate consequence of Proposition 2.
Proof.The solution of ( 1) is equivalent to the solution of the integral equation (17).Hence, it is enough to show that the operator  :  0 →  0 , satisfies the Banach fixed-point theorem.Consider , V ∈  0 and for each  ∈ [0, ], we have Hence, ‖() − (V)‖  ≤  ‖ −  * ‖  , and then  is a contraction.Therefore,  has a unique fixed point by Banach's contraction principle. where In analog to Theorem 5, we consider the operator  * :  0 →  0 defined by By using (H2) and Theorem 5, the operator  * is continuous and completely continuous.Now, it is sufficient to show that there exists an open set  * ⊆  0 with  ̸ =  * () for  ∈ (0, 1) and  ∈  * .
The unique solution of (3), under some conditions, is studied in the following theorem which is the result of the Banach contraction mapping.
such that  is defined in Theorem 6.
Proof.The proof is a similar process Theorem 6.

Conclusions
In this paper, the existence and the uniqueness of solutions for the nonlinear fractional differential equations with infinite delay comprising standard Riemann-Liouville derivatives have been discussed in the phase space.Leray-Schauder's alternative theorem and the Banach contraction principle were used to prove the obtained results.Further generalizations can be developed to some other class of fractional differential equations such as L()() = (,   ), where L() =    − ∑ −1 =1   ()  − , 0 <  1 < ⋅ ⋅ ⋅ <   < 1,   () = ∑   =0     , and   is nonnegative integer.