Existence results of solutions for some fractional neutral functional integro-differential equations with infinite delay

In this paper, by means of the Banach fixed point theorem and the Krasnoselskii’s fixed point theorem, we investigate the existence of solutions for some fractional neutral functional integro-differential equations involving infinite delay. This paper deals with the fractional equations in the sense of Caputo fractional derivative and in the Banach spaces. Our results generalize the previous works on this issue. Also, an analytical example is presented to illustrate our results


Introduction
Differential equations with fractional order appear frequently in applications as the mathematical modeling of natural phenomena in the fields of sciences and engineering including fluid flow, economics, electrical networks, and etc. (see [1][2][3][4][5]).In fact, most of these equations are more accurate for the description of the property of phenomena, as compared with the corresponding integer-order models.Therefore, the study of such equations has attracted a great deal of attention of researchers that we refer to the monographs such as Miller and Ross [6], Podlubny [7], Kilbas et al. [8], Diethelm [9], and some articles [10,11].
The concept of the existence theory of solutions for fractional functional differential equations with infinite delay is increasing as a necessary district of scholarships [12][13][14][15][16][17].There are some papers dealing with this issue by using some techniques such as, fixed point theorems, the Leray-Schauder theory, method of steps, lower and upper solutions method and etc. [12,[15][16][17][18][19][20][21].In 2008, Benchohra et al. [19], investigated the existence of solutions for the following Riemann-Liouville fractional order functional differential equations with infinite delay using the Leray-Schauder fixed point theorem. and Also, Agarwal et al., studied the initial value problem of fractional neutral Caputo fractional derivative and established the existence results of solution of this problem by using Krasnoselskii's fixed point theorem [12].Ren et al. [23], by utilizing the Banach fixed point theorem, the Leray-Schauder fixed point theorem and the Krasnoselskii fixed point theorem, discussed the existence and uniqueness of mild solutions in α-norm to the following semilinear integro-differential evolution equations: where A is the infinitesimal generator of a compact semigroup.
Recently, Xie [24] and Dabas and Chauhan [25], analyzed the existence and uniqueness results for impulsive fractional integro-differential evolution equations with infinite delay, by means of Monch fixed point theorem and Kuratowski measure of noncompactness, respectively.
To close the gap, motivated and inspired by the works above, in this paper we investigate the existence of solutions for the following fractional neutral functional integral-differential equation: which is equipped with the new suitable conditions on functions f , g.Where c D α denotes the Caputo fractional derivative, f : [0, T] × B × R → R and g : [0, T] × B → R, are continuous functions, also B is a phase space of mapping (−∞, 0] into R which will be explained in Section 2. For x : (−∞, T] → R, we define x t (θ) = x(t + θ) for t ∈ [0, T] and −∞ < θ ≤ 0, as well as for k : with k 0 = sup 0≤t≤T t 0 k(t, s)ds .The main tools used in this paper are Banach fixed point theorem and the Krasnoselskii's fixed point theorem.
This paper is organized as follows.In Section 2, we provide some required notation and basic concepts.In Section 3, the existence of solutions for problem (4) is analyzed under the Banach fixed point theorem and the Krasnoselskii's fixed point theorem.As a last point, an application is given in Section 4 to illustrate our theoretical results.

Preliminaries
In this section, we introduce some primary components, definitions and notations from the fractional calculus and the phase space which are used in the sequel [7].
We consider C(J, R) as a Banach space of all continuous functions from J into R with the norm where |.| denotes a suitable complete norm on R. Also, L 1 (J, R) denotes the Banach space of measurable functions from J into R, which are Lebesgue integrable with the norm The fractional integral of order α > 0 of a function x ∈ L 1 (J, R) is defined as where Γ(.) is the Gamma function.
Let n − 1 < α ≤ n, the α-th Caputo derivative of x ∈ C(J, R) is defined as In this paper, to describe fractional neutral functional integro-differential equations with infinite delay, we assume an evident definition of the phase space (B, .B ) such that is a seminormed linear space of functions mapping (−∞, 0] into R and satisfies in the following fundamental axioms [19,26]: (A): for every x : (−∞, T] → R with x 0 ∈ B and t ∈ [0, T], the following conditions hold: Furthermore, the following notations are used in this paper, and

Main result
In this section, we study the existence of solutions Eq. ( 4), to demonstrate our results.we list the following assumptions: (H 1 ): Let f : J × B × R → R be a continuous function and for each (t, x t , Kx(t)), (t, y t , (H 2 ): Let g : J × B → R be a continuous function and for each (t, x t ), (t, y t ) ∈ J × B, there exist (H 3 ): The constants I α L > 0 and λ 1 < 1 are determined by Proof.Assume that x is a solution of (4), therefore, for each t ∈ J, we have Applying the Riemann-Liouville fractional integral operator on both sides, we obtain Using the initial condition, we get and x is a solution of the integral equation (6).
Assume that the hypotheses (H 1 ) − (H 3 ) are satisfied.Therefore, the problem (4) has a unique solution.
It is evident that x satisfies in Eq. ( 6) if and only if y 0 = 0 and also, the function y(.) satisfies in the following equation, and let .B be the seminorm in B defined by ) is a Banach space.We define the operator Λ : B → B as follows From the assumption (A − ii), we get the following estimate, On the other hand, since the functions f , g are continuous and ϕ t B ≤ η, therefore, there exist the constants γ 2 , γ 3 , such that and considering the set For every y ∈ D R 1 , by means of (H 1 ), (H 2 ), (A − ii) and triangle inequality, we get Now, we show that Recalling (H 3 ), ( 9) and ( 10), we get Next, we shall show that Λ is a contraction mspping.For u(t), v(t) ∈ D R Therefore, where λ 1 is given in (H 3 ).Finally, we deduce Λ has a unique fixed point by means of the contraction mapping principle.
Utilize of fixed point theorems is a suitable tool for proving the existence and uniqueness of different equations (for instance see [15,16,27,28] and the references therein).For this purpose, in the following we will use of the Krasnoselskii fixed point theorem.
(Krasnoselskii's Fixed Point Theorem [12,29]).Let X be a Banach space, E be a bounded closed convex subset of X, and let S, U be maps of E into X such that Sx + Uy ∈ E for every pair x, y ∈ E. If S is a contraction and U is completely continuous, then the equation has at least one solution on E. Now, we present the subsequent assumptions: (H 5 ): The constant I α P > 0 is determined by Assume that the hypotheses (H 2 ) and (H 4 ) − (H 5 ) are satisfied.Then, the problem (4) has at least one solution.
We define the set D R 2 = y ∈ B : y B ≤ R 2 , so D R 2 is a closed, bounded and convex set of Banach space B. Also, we consider operators Λ 1 and Λ 2 on D R 2 as Next, we are going to show that Λ 1 + Λ 2 has a fixed point in D R 2 .Since the proof of the theorem is long, we split it into several steps.
For every u(t), v(t) ∈ D R 2 and t ∈ [0, T], by means of (H 2 ), we get Step 3. Λ 2 is a completely continuous operator.The continuity of f implies that the operator Λ 2 is continuous, we show that Λ 2 is uniformly bounded on D R Finally, we prove that Λ 2 v(t) : v(t) ∈ D R 2 is equicontinuous.For every 0 ≤ t 1 ≤ t 2 ≤ T, we obtain As t 1 → t 2 the right-hand side of the above inequality tends to zero.It means that Λ 2 v(t) : v(t) ∈ D R 2 is equicontinuous.Also, the results of steps 1-3, together with the Arzela-Ascoli theorem imply that Λ 2 is a completely continuous operator.The conclusion of the theorem holds by using the Krasnoselskii's fixed point theorem.

Application
To illustrate the application of the obtained results, we consider the following example, For phase space, we choose h(s) = e 2s , s < 0, then l = 1 2 .Also, we give Hence, the equation ( 11) can be written in the abstract form of the equation ( 4).Now, for t ∈ [0, 1], φ 1 , φ 2 ∈ B, and  Thus the conditions (H 1 ) − (H 5 ) are fulfilled.We realize that the equation ( 11) has a unique solution on [0, 1].

Conclusion
In this paper, we discussed the existence results for a class of fractional neutral functional integro-differential equations with time-dependent delay.Using the Banach fixed point theorem and the Krasnoselskii's fixed point theorem some results are presented.The new conditions are assumed in our works which we can generalize for another problems in the future.To validate the obtained theoretical results, we analyzed one example.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 June 2017 doi:10.20944/preprints201706.0090.v1
It is clear that the operator Λ has a fixed point if and only if Λ has a fixed point.So, our aim is to show that the operator Λ has a fixed point.