Existence of Solutions for Fractional Evolution Equations with Infinite Delay and Almost Sectorial Operator

This paper discusses a class of semilinear fractional evolution equations with infinite delay and almost sectorial operator on infinite interval in Banach space. By using the properties of analytic semigroups and Schauder’s fixed-point theorem, this paper obtains the existence of mild solutions of the fractional evolution equation. Moreover, this paper also discusses the existence of mild solution when the analytic semigroup lacks compactness by Kuratowski measures of noncompactness and Darbo–Sadovskii fixed-point theorem.


Introduction
Fractional differential models play a very important role in describing many complex phenomena such as chaotic system [1], fluid flow [2,3], anomalous diffusion [4][5][6][7], and so on. Compared with the classical partial differential models such as [8][9][10][11][12][13][14][15][16][17][18][19], the biggest advantage of models with fractional derivatives is their global property and history memory. Delay is short for time delay, which exists widely in the objective world. In the differential equation model with delay, the function depends not only on the current state but also on the past time state, so it is more suitable to describe the process with time memory. is property of delay is very similar to that of fractional derivatives. So many researchers introduced fractional derivatives into differential equations with delay [20][21][22][23][24]. Evolution equation, which is a general appellation for some partial differential equations with time variable, is mainly used to describe the time-dependent state and process. Common evolution equations include the wave equation, the heat equation, Schrodinger equation, KdV equation, Navier-Stokes equation, and so on. By using the operator semigroup theory, some partial differential evolution equations can be represented to some abstract ordinary differential equations (ODEs) in some special functional spaces. At present, the research on integer-order evolution equations has been relatively perfect [25,26], but the research on fractional-order evolution equations is still in the preliminary stage. e existence of solutions for fractional evolution equations is also the basis of the following study. e mild solution of integer-order evolution equations is defined by the constant variation method, which cannot be directly extended to fractional-order evolution equations.
Li [20] studied the following fractional evolution equations with almost sectorial operator on finite interval: where c D q t is the Caputo fractional derivative operator, the evolution operator A is an almost sectorial operator, and B is a phase space. x t is the element of B defined by x t (θ) � x(t + θ), θ ∈ (− ∞, 0]. Here, x t (·) represents the history of state up to the present time.
Baliki et al. [22] discussed a second-order evolution equation with infinite delay and obtained the existence and attractivity of mild solutions by Schauder's fixed point as follows: { } 0≤t<∞ is a family of linear closed operators, x t (θ) � x(t + θ), θ ∈ (− ∞, 0], and B is a phase space. e existence of mild solutions for fractional evolution equations and evolution equations with infinite delay has been discussed in several papers (see [20,21]). However, we find that most of the previous papers discuss the fractional evolution equations in the conventional spaces of continuous function on finite or infinite interval and in Banach space on finite interval. To our knowledge, no paper is devoted to the existence of mild solutions with infinite delay and almost sectorial operator on infinite interval on Banach space.
In this paper, we consider the following fractional evolution problem with infinite time delay: where c D q 0+ is the Caputo fractional derivative operator, the evolution operator A is an almost sectorial operator, f is a given function which will be introduced later, and B is a phase space. For any continuous function x and any t ≥ 0, x t is the same as in equation (1) which represents the history of state up to the present time. e rest of this paper is organized as follows. In Section 2, we recall some definitions, propositions, notations, and lemmas. In Section 3, the main results of this paper are obtained. We consider two cases: the semigroup Q(t) generated by operator A with compactness and without compactness. For the case that Q(t) is compact, we construct a special Banach space B ′ and obtain the existence of global mild solution by using Schauder's fixed-point theorem. For the case that Q(t) is not compact, we expand the result of eorem 1.2.4 in Guo et al. [27] from any compact interval to infinite interval (see Lemma 10) and obtain the existence of global mild solution by applying Kuratowski measures of noncompactness theory and Darbo-Sadovskii fixed-point theorem.

Preliminaries
In this section, we introduce some notations, definitions, lemmas, and preliminary facts that will be used in the rest of this paper. Let (E, | · |) be a Banach space. Denote B(E) as the space of all bounded linear operators from E to itself with norm ‖·‖ B(E) .

Define the power of A as
where Γ θ � R + e iθ ∪ R + e − iθ is an appropriate path oriented counterclockwise and ω < θ < μ. en, the linear power space X β ≔ D(A β ) can be defined and X β is a Banach space with the graph norm ‖x‖ β � |A β x|, x ∈ D(A β ).
Next, let us introduce the semigroup associated with A. If A is an almost sectorial operator, then A generates an analytic semigroup Q(t) of growth order 1 + c as follows: where Furthermore, Q(t) satisfies the following properties.
and there exists a constant By Theorem 3.13 in Periago [28], if A is an almost sectorial operator, then for every λ ∈ C with Re λ > 0, Let X be the following set: where x [0,+∞) is the restriction of x on [0, +∞) and k is a constant.
In this paper, we use an axiomatic definition of the phase space B. (B, ‖·‖ B ) is a seminormed linear space of functions mapping (− ∞, 0] into E and satisfies the following axioms which are introduced by Hale and Kato in [30].
Definition 3 (see [31,32]). e Caputo fractional derivative of order q > 0 of the function f: (0, +∞) ⟶ E is given by where n is the smallest integer greater than or equal to q, provided that the right side is well defined on (0, +∞).
Lemma 1 (see [31,32]). For all f, g ∈ L q ((0, Next, we will introduce the mild solution of equation (3). Shu et al. [33] define the mild solution of equation (3) as where S q (t) and P q (t) have the following expressions and Γ is an appropriate path in ρ(− A).
Using the properties of the Mittag-Leffler function (for more details, we refer the readers to [32]), where Γ is the same path as in (4) (see [32]), the above operators S q (t) and P q (t) can be represented as the generalized Mittag-Leffler-type functions: Moreover, Wang et al. [29] and Zhou et al. [34][35][36] introduced the function of Wright-type M q (z): Mathematical Problems in Engineering 3 (19) and obtained another expression of S q (t), P q (t): In fact, these three expressions ( (16)- (20)) are equivalent in the case that t > 0 and A ∈ Θ c ω (E). erefore, in this paper, we use the same expression of S q (t), P q (t) as Wang et al. in [29] and Zhou et al. in [34][35][36]. en, the global mild solution of problem (3) is given in the following definition.
Lemma 2 (see [29]). For any fixed t > 0, S q (t) and P q (t) are linear and bounded operators and there exist constants C s and C p such that for all x ∈ E, Lemma 3 (see [29]). For t > 0, operators S q (t) and P q (t) are continuous in the uniform operator topology. Moreover, for every r > 0, the continuity is uniform on [r, +∞).

Main Results
In this section, our main purpose is to establish sufficient conditions for the existence of global mild solutions to problem (3) in X. Assume that: where p(t) is a nonnegative and continuous function on [0, +∞) and here exists a big enough k > 0 such that (i) For any t ≥ 0, In order to obtain the existence of global mild solution of problem (3), we transform it into a fixed-point problem. For any ϕ(0) ∈ X β , define the operator T: X ⟶ X as Let z(t): R ⟶ X be the function and x(t) � y(t) + z(t), t ∈ R. It is easy to know that x(t) satisfies (21) if and only if Define the set B ′ ≔ y ∈ X: y 0 � 0 ∈ B endowed with seminorm ‖·‖ b : Consequently, the operator T: X ⟶ X having a fixed point in X is equivalent to the operator T: B ′ ⟶ B ′ having a fixed point in B ′ . Lemma 6. Assume that condition (H) is valid; then, there exists a constant r > 0 such that where M satisfies sup t>0 ‖S q (t)ϕ(0)‖ β ≤ M. Consider B r ≔ y ∈ B ′ , ‖y‖ b ≤ r ; then, for any ϕ(0) ∈ X β , the operator T: B r ⟶ B r is continuous.
Next, we will prove the continuity of T. Let y n (t) ∞ n�1 ∈ B r and ‖y n − y‖ b ⟶ 0 as n ⟶ ∞ for any t ≥ 0. en, for any t > 0, by the continuity of f, which implies that ‖Ty n (t) − Ty(t)‖ b ⟶ 0 as n ⟶ ∞. erefore, the continuity of T is proved.

Mathematical Problems in Engineering
Proof.
Proof. We aim to prove this theorem by using Schauder's fixed-point theorem. In view of Lemma 6, T: B r ⟶ B r and T is continuous, so we just need to prove that for any bounded subset V ⊂ B r , TV is relatively compact in X. en, it is easy to prove that TV satisfies all conditions in Lemma 8.
Consider Lemma 6; we have proved that ‖Ty‖ b � sup t≥0 e − kt ‖Ty(t)‖ β ≤ r for any y ∈ B r which implies Ty, y ∈ B r is uniformly bounded. By Lemma 7, Ty, y ∈ B ′ is equicontinuous on any compact interval [0, T] of [0, +∞) and e − kt ‖Ty(t)‖ β < ε for any t ≥ T and y ∈ B ′ . en, it remains to show that It is easy to know that V(0) � 0 { } is compact in X. Let t ∈ [0, T) be fixed and for any ε ∈ (0, t), δ > 0, we define an operator T δ ε on V by the formula where y ∈ V. Under the compactness of Q(ε q δ)(ε q δ > 0) and the boundedness of we obtain that the set V δ ε (t) � (T δ ε y)(t), y ∈ V is relatively compact in X for any ε ∈ (0, t) and δ > 0. Moreover, for any y ∈ V, t > 0, we have (47)
Hence, T: B r ⟶ B r is a completely continuous operator. So, by Schauder's fixed-point theorem, T has at least one fixed point in B r which implies that problem (3) has at least one global mild solution in B r .

e Case at Q(t) Is Not Compact.
In this section, we assume that Q(t) is not compact. In the following, α and α B′ denote the Kuratowski measures of noncompactness of bounded sets in X β and in B ′ . For more details about Kuratowski measures of noncompactness, we refer the readers to [27]. Assume that: (H * ) ere exists m(t) ∈ L([0, +∞), [0, +∞)) such that I q 0+ m exists and for any bounded and for any t ≥ 0, Lemma 9 (see [27]). If V ⊂ C(J, E) is bounded and equicontinuous, then α(V(t)) is continuous and where J is any compact interval of [0, +∞).

Remark 2. A bounded and continuous operator
(63) Consider Lemma 9 and condition (H * ); let any t ≥ 0 be fixed, and for the above ε > 0, we have

Conclusions
In this paper, we investigated a class of fractional evolution equations with infinite delay and almost sectorial operator on unbounded domains in Banach space. We considered the case of compact semigroups and noncompact semigroups and obtained sufficient conditions of the existence of global mild solutions.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.