Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Inﬁnite Delay

: The existence of solutions for a class of nonlinear neutral Hadamard-type fractional integrodifferential equations with inﬁnite delay is researched in this paper. By constructing an appropriate normed space and utilizing the Banach contraction principle, Krasnoselskii’s ﬁxed point theorem, we obtain some sufﬁcient conditions for the existence of solutions. Finally, we provide an example to illustrate the validity of our main results. delay; ﬁxed point theory


Introduction
The main purpose of this paper is to study the existence of solutions for the following Hadamard-type neutral fractional integro-differential equation with infinite delay written by 1 + e i (t, u t )] = a(t) f (t, u t , t 1 g(t, s, u s )ds), t ∈ J = [1, T], u(t) = φ(t), t ∈ (−∞, 1], (1) where T > 1, 0 < α ≤ 1 and β i > 0(i = 1, 2, . . . , m) are some given real constants. H D α 1 + denotes the Hadamard-type fractional derivative of order α, H J β i 1 + stands the Hadamard fractional integral of order β i . e i : J × B h → R, f : J × B h × R → R and g : J × J × B h → R are given continuous functions satisfying some assumptions that will be introduced later. The function a(t) ∈ C[J, R] and it is not identically zero on any subinterval of [1, T]. We also assume that u t : (−∞, 0] → B h with u t (θ) = u(t + θ) (θ ≤ 0) belongs to an abstract phase space B h which is defined in Preliminaries. φ(t) : (−∞, 1] → R is a given continuous initial function with φ(1) = 0.
Fractional calculus, especially fractional differential equation is an advantageous mathematical model used to describe properties of various processes and applications in lots of fields. Therefore, the fractional differential equations are widely concerned and studied by many people. There have been many papers dealing with fractional differential equations with different conditions such as impulses [1][2][3][4][5][6], time delays [1,3,4,7,8], implicit conditions [9], algebraic conditions [10], stochastic conditions [4], neutral conditions [1,2,6,10,11], boundary value conditions [5,[7][8][9]12], and so on. In 1892, Hadamard [13] proposed a new kind of fractional calculus. The definition of this fractional calculus contains the logarithmic function of any index in the integral kernel. This kind of fractional calculus is later called Hadamard fractional calculus. Hadamard fractional calculus has been widely concerned and studied by many scholars since it was put forward. In the monographs [14,15], the authors introduced the basic knowledge and results of Hadamard's fractional calculus and differential equation in detail. Some researchers have studied and explored the dynamic properties of Hadamard fractional calculus and differential equations. For example, the Hadamard fractional integration operators and the semigroup property were discussed in [16]. The Mellin's transformation of Hadamard fractional calculus was investigated in [17]. Some papers on Hadamard fractional differential boundary value problems [5,[18][19][20][21][22][23], integrodifferential equations [5,18,24], neutral equations [24] and coupled systems [5,25] have been published. Compared with the research results of Riemann-Liouville and Caputo fractional differential equations, the study on Hadamard fractional differential equation are relatively rare. Therefore, it is valuable and challenging to study the dynamic properties of Hadamard fractional differential equations.
In addition, the research on system (1) is also inspired by the literature [24]. The author studied the existence and uniqueness results of solutions for the following system where T > 1, 0 < α ≤ 1 and β i > 0(i = 1, 2, . . . , m) are some given real constants. H D α denotes the left-side Hadamard fractional derivative of order α, I β i stands the Riemann-Liouville fractional integral of order The author established the uniqueness of solutions by the Banach contraction principle and derived the existence of solutions by the Leray-Schauder alternative.
The rest of the paper is organized as follows. In Section 2, we recall some useful definitions, notations and lemmas. The main results are proved in Section 3. In Section 4, an example is given to demonstrate the application of our main results. Finally, we conclude the importance of studying problem (1) and our obtained results in Section 5.

Preliminaries
In this section, we introduce some necessary definitions, notations, lemmas and preliminary facts, which are required in the sequel.
First of all, we present the abstract phase B h similar to [26]. Here we assume that h : , it is easy to verify that X is a Banach space.

Definition 1 ([14]). The left-sided Hadamard fractional integral of order
provided the integral exists.

Lemma 2 ([14]
). If α > 0, β > 0, then the following properties hold: Lemma 3 (Banach contraction principle [27]). If E is a real Banach space and F : E → E is a contraction mapping, then F has a unique fixed point in E.
Lemma 4 (Krasnoselskii's fixed point theorem [28]). Let X 1 be a nonempty closed convex subset of a Banach space (X, · ). Let P, Q be two operators such that (i) Px + Qy ∈ X 1 for any x, y ∈ X 1 ; (ii) P is a contraction mapping; (iii) Q is continuous and compact.
Then there exists a z ∈ X 1 such that z = Pz + Qz.
= 0 be some given functions, and 0 < α ≤ 1, β > 0 and T > 1 be some constants. Then a function u(t) ∈ X is a solution of the following linear system if and only if u(t) ∈ X is a solution of the integral equation as follows: Proof. Assume that u(t) ∈ X is a solution of (2). When t ∈ J = [1, T], according to the first equation of (2) and Lemma 1, we have In the light of the existence of u(1) = 0, we have c 1 = 0. When t ∈ (−∞, 1], then u(t) = φ(t). Thus, we derive which indicates that u(t) ∈ X is also a solution of (3). Conversely, if u(t) ∈ X is a solution of (3), noting that the above process is completely reversible, then we know that u(t) ∈ X is also a solution of (2). The proof is completed.

For any
it follows from Lemma 5 that an operator F : X → X defined by Then solving the problem (1) is equivalent to finding the fixed point of the operator F defined by (4). Next we shall present and prove our main results. To this end, the following assumptions are needed later.
(H 1 ) The functions e i : J × B h → R(i = 1, 2, . . . , m) are continuous, and there exist some constants k i > 0 such that (H 2 ) The function a(t) ∈ C(J, R) is not identically zero on any subinterval of [1, T], and there exists a constant M > 0 such that for any t ∈ J, |a(t)| ≤ M. (H 3 ) The function f : J × B h × R → R is continuous, and there exist some constants n 1 , n 2 > 0 such that (H 4 ) The function g : J × J × B h → R is continuous, and there exists a constant q > 0 such that Theorem 1. Assume that (H 1 )-(H 5 ) hold. Then the problem (1) has a unique solution u * ∈ X.
Proof. Now we are applying Lemma 3 to prove that F : X → X defined by (4) has a unique fixed point. Actually, for all t ∈ (−∞, T], u, u ∈ X , when t ∈ (−∞, 1], we have which implies that When t ∈ [1, T], it follows from (4) and In view of (H 5 ), (6) and (7), we conclude that F defined as (4) is a contraction mapping. Thus, F has a unique fixed point u * (t) ∈ X. Consequently, the problem (1) has a unique solution u * (t) ∈ X. The proof of Theorem 1 is completed.
(H 8 ) The function g : J 2 × B h → R is a continuous function, and there exists a positive constant Q such that Proof. Given a constant r > 0, define clear that Ω is a nonempty closed convex subset of the Banach space X. We also define the operator F : X → X as (4). Here we split F = F 1 + F 2 such that, for any t ∈ (−∞, T], u ∈ X, When t ∈ (−∞, 1], for all u ∈ Ω, we have When t ∈ [1, T], for all u ∈ Ω, it follows from (4), (H 2 ) and (H 6 )-(H 9 ) that We derive that Fu = F 1 u + F 2 u ∈ Ω from (10) and (11), that is, the condition (i) of Lemma 4 holds. Similarly, we can also prove that F 1 , F 2 : Ω → Ω ⊂ X, and F 1 is uniformly bounded. Next we only need to prove that F 1 is equicontinuous, and F 2 is a contraction mapping. In fact, for all u, u ∈ Ω, when t ∈ (−∞, 1], which implies that When t ∈ [1, T], it follows from (9), (H 1 ) and (H 5 ) that (13) and (14) means that F 2 is a contraction mapping. Therefore, the condition (ii) of Lemma 4 holds. Finally, we show that F 1 is equicontinuous. For all u ∈ Ω, t 1 , t 2 ∈ (−∞, T] with t 1 < t 2 , when −∞ < t 1 < t 2 ≤ 1, from (8),

Conclusions
As a useful mathematical model, fractional-order differential calculus is used to describe properties of various processes and applications in lots of fields such as blood flow problems, chemical engineering, porous medium, aerodynamics, polymer rheology, population dynamics, and so on. Compared with Caputo or Riemann-Liouville type fractional differential equations, the study on Hadamard-type fractional differential equations is more difficult and complicated. The study on Hadamard-type fractional differential equations involving the time delays will be more challenging. Therefore, we mainly study the existence of solutions of the problem (1) in this paper. By applying the Banach contraction principle and Krasnoselskii's fixed point theorem, some new existence criteria of solutions are obtained.