Existence of the Mild Solutions for Delay Fractional Integrodifferential Equations with Almost Sectorial Operators

and Applied Analysis 3 Definition 2.2 see 11 . The Riemann-Liouville derivative of order q with the lower limit zero for a function f ∈ AC 0,∞ can be written as Df t 1 Γ ( 1 − q) d dt ∫ t 0 t − s −qf s ds, t > 0, 0 < q < 1. 2.3 Definition 2.3 see 11 . The Caputo derivative of order q for a function f ∈ AC 0,∞ can be written as Df t D q( f t − f 0 ), t > 0, 0 < q < 1. 2.4 Remark 2.4. 1 If f t ∈ C1 0,∞ , then Df t 1 Γ ( 1 − q) ∫ t 0 t − s −qf ′ s ds I1−qf ′ t , t > 0, 0 < q < 1; 2.5 2 the Caputo derivative of a constant is equal to zero. We will need the following facts from the theory of measures of noncompactness and condensing maps see, e.g., 22, 23 . Definition 2.5. Let E be a Banach space, 2 the family of all nonempty subsets of E, A,≥ a partially ordered set and ν : 2 → A. If for every Ω ∈ 2: ν co Ω ν Ω , 2.6 then we say that ν is a measure of noncompactness MNC in E. A MNC ν is called: i monotone if Ω0,Ω1 ∈ 2, Ω0 ⊂ Ω1 implies ν Ω0 ≤ ν Ω1 ; ii nonsingular if ν {a0} ∪Ω ν Ω for every a0 ∈ E, Ω ∈ 2; iii invariant with respect to union with compact sets if ν {D} ∪ Ω ν Ω for every relatively compact set D ⊂ E, Ω ∈ 2. IfA is a cone in a normed space, we say that the MNC ν is; iv algebraically semiadditive if ν Ω0 Ω1 ≤ ν Ω0 ν Ω1 for each Ω0,Ω1 ∈ 2; v regular if ν Ω 0 is equivalent to the relative compactness of Ω; vi real ifA is 0, ∞ with the natural order. As an example of the MNC possessing all these properties, we may consider the Hausdorff MNC: χ Ω inf{ε > 0 : Ω has a finite ε-net}. 2.7 Now, let G : 0, h → 2 be a multifunction. It is called: i integrable, if it admits a Bochner integrable selection g : 0, h → E, g t ∈ G t for a.e. t ∈ 0, h ; 4 Abstract and Applied Analysis ii integrably bounded, if there exists a function ∈ L1 0, h , E such that ‖G t ‖ : sup{‖g‖ : g ∈ G t } ≤ t a.e. t ∈ 0, h . 2.8 We present the following assertion about χ-estimates for a multivalued integral 23, Theorem 4.2.3 . Proposition 2.6. For an integrable, integrably bounded multifunction G : 0, h → 2 , where X is a separable Banach space, let χ G t ≤ q t , for a.e. t ∈ 0, h , 2.9 where q ∈ L1 0, h . Then, χ ∫ t 0G s ds ≤ ∫ t 0 q s ds for all t ∈ 0, h . Let E be a Banach space, ν a monotone nonsingular MNC in E. Definition 2.7. A continuous map F : Y ⊆ E → E is called condensing with respect to a MNC ν or ν-condensing if for every bounded set Ω ⊆ Y which is not relatively compact, we have ν F Ω /≥ ν Ω . 2.10 The following fixed point principle see, e.g., 22, 23 will be used later. Theorem 2.8. Let M be a bounded convex closed subset of E and F : M → M a ν-condensing map. Then, Fix F {x : x F x } is nonempty. Theorem 2.9. Let V ⊂ E be a bounded open neighborhood of zero and F : V → E a ν-condensing map satisfying the boundary condition: x / λF x , 2.11 for all x ∈ ∂V and 0 < λ ≤ 1. Then, FixF is a nonempty compact set. To prove the main result, we will need the following generalization of Gronwall’s lemma for singular kernels 24, Lemma 7.1.1 . Lemma 2.10. Let x, y : 0, T → 0, ∞ be continuous functions. If y · is nondecreasing and there are constants a > 0 and 0 < θ < 1 such that x t ≤ y t a ∫ t 0 t − s −θx s ds, 2.12 then there exists a constant κ κ θ such that x t ≤ y t κa ∫ t 0 t − s −θy s ds, for each t ∈ 0, T . 2.13 Abstract and Applied Analysis 5 Next, we recall the knowledge of almost sectorial operator, for more details, we refer to 25, 26 . Let −1 < γ < 0 and Sμ with 0 < μ < π be the open sector: { z ∈ C \ {0} : ∣∣arg z∣∣ < μ}, 2.14and Applied Analysis 5 Next, we recall the knowledge of almost sectorial operator, for more details, we refer to 25, 26 . Let −1 < γ < 0 and Sμ with 0 < μ < π be the open sector: { z ∈ C \ {0} : ∣∣arg z∣∣ < μ}, 2.14 and Sμ its closure, that is, Sμ { z ∈ C \ {0} : ∣∣arg z∣∣ ≤ μ} ∪ {0}. 2.15 Let us recall the following definition. Definition 2.11. Let −1 < γ < 0 and 0 < ω < π/2. By Θγω X , we denote the family of all linear closed operators A : D A ⊂ X → X which satisfy: 1 σ A ⊂ Sω {z ∈ C \ {0} : | arg z| ≤ ω} ∪ {0}; 2 for every ω < μ < π , there exists a constant Cμ such that ‖R z;A ‖L X ≤ Cμ|z| , for all z ∈ C \ Sμ. 2.16 A linear operator Awill be called an almost sectorial operator on X if A ∈ Θγω X . Remark 2.12. Let A ∈ Θγω X , then the definition implies that 0 ∈ ρ A . Remark 2.13 see 15 . From 26 , note in particular that if A ∈ Θγω X , then A generates a semigroup T t with a singular behavior at t 0 in a sense, called semigroup of growth 1 γ . Moreover, the semigroup T t is analytic in an open sector of the complex plane C, but the strong continuity fails at t 0 for data which are not sufficiently smooth. We denote the semigroup associated with A by T t . For t ∈ S0 π/2−ω, T t e−tz A 1 2πi ∫ Γθ e−tzR z;A dz 2.17 forms an analytic semigroup of growth order 1 γ , where ω < θ < μ < π/2 − | arg t|, the integral contour Γθ : {R eiθ} ∪ {R e−iθ} is oriented counter-clockwise 15, 26 . We have the following proposition on T t 26, Theorem 3.9 . Proposition 2.14. LetA ∈ Θγω X with −1 < γ < 0 and 0 < ω < π/2. Then the following properties remain true: i T t is analytic in Sπ/2−ω and d dtn T t −A T t , for all t ∈ Sπ/2−ω; 2.18 6 Abstract and Applied Analysis ii the functional equation T s t T s T t for all s, t ∈ Sπ/2−ω holds; iii there exists a constant C0 C0 γ > 0 such that ‖T t ‖L X ≤ C0t−γ−1, for all t > 0; 2.19 iv if β > 1 γ , then D A ⊂ ΣT {x ∈ X : limt→ 0; t>0T t x x}. Clearly, we note that the condition ii of the Proposition 2.14 does not satisfy for t 0 or s 0. The relation between the resolvent operators of A and the semigroup T t is characterized by. Proposition 2.15 see 26, Theorem 3.13 . Let A ∈ Θγω X with −1 < γ < 0 and 0 < ω < π/2. Then for every λ ∈ C with Reλ > 0, one has R λ;A ∫∞ 0 e−λtT t dt. 2.20 Based on the work in 15 , we define operator families {Sq t }|t∈S0 π/2−ω and {Pq t }|t∈S0 π/2−ω by Sq t x ∫∞ 0 Ψq σ T σt xdσ, t ∈ Sπ/2−ω, x ∈ X, Pq t x ∫∞ 0 qσΨq σ T σt xdσ, t ∈ Sπ/2−ω, x ∈ X, 2.21 where Ψq z with 0 < q < 1 is a function of Wright-type cf. e.g., 15 as follows: Ψq z : ∞ ∑ n 0 −z n n!Γ (−qn 1 − q) 1 π ∞ ∑ n 1 −z n n − 1 ! ( nq ) sin ( nπq ) , z ∈ C. 2.22 We collect some basic properties onΨq z . For more details, we refer to 7, 11, 15, 25 . Proposition 2.16. For −1 < r̃ <∞, λ > 0, the following results hold: 1 Ψq t ≥ 0, t > 0; 2 ∫∞ 0 q/t q Ψq 1/t e−λtdt e−λ q ; 3 ∫∞ 0 Ψq t t dt Γ 1 r̃ /Γ 1 qr̃ . Theorem 2.17 see 15, Theorem 3.1 . For each fixed t ∈ S0 π/2−ω, Sq t and Pq t are linear and bounded operators on X. Moreover, for all t > 0, −1 < γ < 0, 0 < q < 1, ∥∥Sq t x∥∥ ≤ C0Γ (−γ) Γ ( 1 − q(1 γ)) t −q 1 γ ‖x‖, x ∈ X, ∥∥Pq t x∥∥ ≤ qC0Γ ( 1 − γ) Γ ( 1 − qγ) t −q 1 γ ‖x‖, x ∈ X. 2.23 Abstract and Applied Analysis 7 Theorem 2.18 see 15, Theorem 3.2 . For t > 0, Sq t and Pq t are continuous in the uniform operator topology. Moreover, for every r̃ > 0, the continuity is uniform on r̃,∞ . Remark 2.19 see 15, Theorem 3.4 . Let β > 1 γ . Then for all x ∈ D A , lim t→ 0; t>0 Sq t x x. 2.24and Applied Analysis 7 Theorem 2.18 see 15, Theorem 3.2 . For t > 0, Sq t and Pq t are continuous in the uniform operator topology. Moreover, for every r̃ > 0, the continuity is uniform on r̃,∞ . Remark 2.19 see 15, Theorem 3.4 . Let β > 1 γ . Then for all x ∈ D A , lim t→ 0; t>0 Sq t x x. 2.24 Next, we will present the definition of mild solution of problem 1.1 . According to Definitions 2.1–2.3, we can rewrite problem 1.1 in the equivalent integral equation: u t φ 0 1 Γ ( q ) ∫ t 0 t − s q−1[Au s f s, us a u s ]ds, t ∈ 0, T , u t φ t , t ∈ −r, 0 2.25 provided that the integral in 2.25 exists, where a u t ∫ t 0 g t, s, us ds. 2.26 Set û λ ∫∞ 0 e−λtu t dt, f̂ λ ∫∞ 0 e−λtf t, ut dt, ĝ λ ∫∞ 0 e−λta u t dt, 2.27 formally applying the Laplace transform to 2.25 , we get û λ 1 λ φ 0 1 λq Aû λ 1 λq [ f̂ λ ĝ λ ] , 2.28 then λ −A û λ λq−1φ 0 [ f̂ λ ĝ λ ] , 2.29 thus û λ λq−1 λ −A −1φ 0 λ −A −1 [ f̂ λ ĝ λ ] λq−1 ∫∞ 0 e−λ qsT s φ 0 ds ∫∞ 0 e−λ qsT s [ f̂ λ ĝ λ ] ds 2.30 provided that the integral in 2.30 exists. Then, using Proposition 2.16 2 , we have λq−1 ∫∞ 0 e−λ qsT s φ 0 ds q ∫∞ 0 λt q−1e− λt q T t φ 0 dt − 1 λ ∫∞ 0 ( d dt e− λt q ) T t φ 0 dt 8 Abstract and Applied Analysis ∫∞ 0 ∫∞ 0 q σq Ψq ( 1 σq ) e−λtσT t φ 0 dσdt ∫∞ 0 e−λt [∫∞ 0 q σq 1 Ψq ( 1 σq ) T ( t σq ) φ 0 dσ ] dt ∫∞ 0 e−λt [∫∞ 0 Ψq τ T tτ φ 0 dτ ] dt, ∫∞ 0 e−λ qsT s f̂ λ ds ∫∞ 0 e− λτ q qτq−1T τ (∫∞ 0 e−λtf t, ut dt ) dτ ∫∞ 0 ∫∞ 0 e−λτσ q σq 1 Ψq ( 1 σq ) qτq−1T τ (∫∞ 0 e−λtf t, ut dt ) dσdτ q2 ∫∞ 0 ∫∞ 0 e−λθ θq−1 σ2q 1 Ψq ( 1 σq ) T ( θ σq )(∫∞ 0 e−λtf t, ut dt ) dθdσ


Introduction
Fractional differential and integrodifferential equations have received increasing attention during recent years and have been studied extensively see, e.g., 1-15 and references therein since they are playing an increasingly important role in engineering, physics, electrolysis chemical, fractional biological neurons, condensate physics, statistical mechanics, and so on.
We mention that much of the previous research on the fractional equations was done provided that the operator in the linear part is the infinitesimal generator of a strongly continuous operator semigroup, a compact semigroup, or an analytic semigroup, or is a Hille-Yosida operator see, e.g., 1-4, 8-10, 12-14 . However, as presented in 15, Examples 1.1 and 1.2 , for which the resolvent operators do not satisfy the required estimate to be a sectorial operator. Von Wahl in 21 first introduced examples of almost sectorial operators which are not sectorial. Recently, the study of evolution equations involving almost sectorial operators has been investigated extensively. However, much less is known about the fractional evolution equations with almost sectorial operators see 15 and the references therein .
In this paper, we are concerned with the following fractional integrodifferential equations: Au t f t, u t t 0 g t, s, u s ds, t ∈ 0, T , u t φ t , t ∈ −r, 0 , where T > 0, 0 < q < 1, and 0 < r < ∞. The fractional derivative is understood here in the Caputo sense. X is a separable Banach space. A is an almost sectorial operator to be introduced later.
For any continuous function v defined on −r, T and any t ∈ 0, T , we denote by u t the element of C −r, 0 , X defined by u t θ u t θ , θ ∈ −r, 0 . Our paper is organized as follows. In Section 2, we give out some preliminaries about fractional order operator, measure of noncompactness, and almost sectorial operators. The existence result will be established in Section 3. In Section 4, an example is given to show the application of the abstract result.

Preliminaries
Throughout this paper, we denote by X a separable Banach space with norm · . For a linear operator A, we denote by D A the domain of A, by ρ A the resolvent set of A, and by R z; A zI − A −1 , z ∈ ρ A the resolvent of A. Moreover, we denote by L X the Banach space of all linear and bounded operators on X and by C a, b , X the space of all X-valued continuous functions on a, b with the supremum norm as follows: x C a,b ,X sup{ x t : t ∈ a, b }, for any x ∈ C a, b , X .

2.1
Moreover, we abbreviate u L p 0,T ,R with u L p for any u ∈ L p 0, T , R . Let us recall the following known definitions. For more details, see 7, 11 .
Definition 2.1 see 11 . The fractional integral of order q with the lower limit zero for a function f ∈ AC 0, ∞ is defined as Definition 2.3 see 11 . The Caputo derivative of order q for a function f ∈ AC 0, ∞ can be written as 2 the Caputo derivative of a constant is equal to zero.
We will need the following facts from the theory of measures of noncompactness and condensing maps see, e.g., 22, 23 .
iii invariant with respect to union with compact sets if ν {D} ∪ Ω ν Ω for every relatively compact set D ⊂ E, Ω ∈ 2 E . If A is a cone in a normed space, we say that the MNC ν is; vi real if A is 0, ∞ with the natural order.
As an example of the MNC possessing all these properties, we may consider the Hausdorff MNC: χ Ω inf{ε > 0 : Ω has a finite ε-net}.

2.7
Now, let G : 0, h → 2 E be a multifunction. It is called: Abstract and Applied Analysis ii integrably bounded, if there exists a function ∈ L 1 0, h , E such that We present the following assertion about χ-estimates for a multivalued integral 23, Theorem 4.2.3 .

Proposition 2.6. For an integrable, integrably bounded multifunction
Let E be a Banach space, ν a monotone nonsingular MNC in E.
The following fixed point principle see, e.g., 22, 23 will be used later. x / λF x , 2.11 for all x ∈ ∂V and 0 < λ ≤ 1. Then, Fix F is a nonempty compact set.
To prove the main result, we will need the following generalization of Gronwall's lemma for singular kernels 24, Lemma 7.1.1 . Lemma 2.10. Let x, y : 0, T → 0, ∞ be continuous functions. If y · is nondecreasing and there are constants a > 0 and 0 < ϑ < 1 such that then there exists a constant κ κ ϑ such that x t ≤ y t κa t 0 t − s −ϑ y s ds, for each t ∈ 0, T .

2.13
Abstract and Applied Analysis 5 Next, we recall the knowledge of almost sectorial operator, for more details, we refer to 25, 26 . Let −1 < γ < 0 and S 0 μ with 0 < μ < π be the open sector: z ∈ C \ {0} : arg z < μ , 2.14 and S μ its closure, that is, Let us recall the following definition.
Definition 2.11. Let −1 < γ < 0 and 0 < ω < π/2. By Θ γ ω X , we denote the family of all linear closed operators A : D A ⊂ X → X which satisfy: γ ω X , then A generates a semigroup T t with a singular behavior at t 0 in a sense, called semigroup of growth 1 γ. Moreover, the semigroup T t is analytic in an open sector of the complex plane C, but the strong continuity fails at t 0 for data which are not sufficiently smooth.
We have the following proposition on T t 26, Theorem 3.9 .
Proposition 2.14. Let A ∈ Θ γ ω X with −1 < γ < 0 and 0 < ω < π/2. Then the following properties remain true: i T t is analytic in S 0 π/2−ω and d n dt n T t −A n T t , for all t ∈ S 0 π/2−ω ; 2.18 ii the functional equation T s t T s T t for all s, t ∈ S 0 π/2−ω holds; iii there exists a constant C 0 C 0 γ > 0 such that Clearly, we note that the condition ii of the Proposition 2.14 does not satisfy for t 0 or s 0.
The relation between the resolvent operators of A and the semigroup T t is characterized by. Proposition 2.15 see 26, Theorem 3.13 . Let A ∈ Θ γ ω X with −1 < γ < 0 and 0 < ω < π/2. Then for every λ ∈ C with Re λ > 0, one has 2.20 Based on the work in 15 , we define operator families {S q t }| t∈S 0 where Ψ q z with 0 < q < 1 is a function of Wright-type cf. e.g., 15 as follows:

2.22
We collect some basic properties on Ψ q z . For more details, we refer to 7, 11, 15, 25 .

2.23
Abstract and Applied Analysis

2.31
Similarly, we have

2.32
Thus, from 2.30 -2.32 , we obtain t − s q−1 σΨ q σ T t − s q σ a u s dσ ds dt.

2.33
Abstract and Applied Analysis 9 We invert the last Laplace transform to obtain

2.34
Then from the above induction, when φ 0 ∈ D A β with β > 1 γ, we can give the following definition of the mild solution of 1.1 .
Definition 2.20. A continuous function u : −r, T → X satisfying the equation: is called a mild solution of 1.1 .
Remark 2.21 see 15 , Remark 4.1 . 1 In general, since the operator S q t is singular at t 0, solutions to problem 1.1 are assumed to have the same kind of singularity at t 0 as the operator S q t . 2 When φ 0 ∈ D A β with β > 1 γ, it follows from Remark 2.19 that the mild solution is continuous at t 0.

Main Result
Throughout this section, let A be an operator in the class Θ γ ω X and −1 < γ < 0, 0 < ω < π/2. Moreover, we require the following assumptions: Hf 1 f : 0, T × C −r, 0 , X → X satisfies f ·, y : 0, T → X is measurable for all y ∈ C −r, 0 , X and f t, · : C −r, 0 , X → X is continuous for a.e. t ∈ 0, T , and there exists a positive function μ · ∈ L p 0, T , R p > −1/ qγ > 1/q > 1 such that for almost all t ∈ 0, T ; 10 Abstract and Applied Analysis 2 there exists a nondecreasing function η ∈ L p 0, T , R such that for any bounded set D ⊂ C −r, 0 , X : Hg 1 g : Λ × C −r, 0 , X → X and g t, s, · : C −r, 0 , X → X is continuous for a.e. t, s ∈ Λ, and for each y ∈ C −r, 0 , X , the function g ·, ·, y : Proof. We define the operator F : C −r, T , X → C −r, T , X in the following way:

3.6
It is clear that the operator F is well defined. We define φ t φ t , t∈ −r, 0 , S q t φ 0 , t ∈ 0, T . and u t φ t , t ∈ −r, 0 . Let F : C −r, T , X → C −r, T , X be an operator defined by Fv t 0 for t ∈ −r, 0 and

3.10
Clearly the operator F has a fixed point is equivalent to F having one. We define C : {v ∈ C −r, T , X : v 0 0} ⊂ C −r, T , X . Next we will prove that F has a fixed point on C.
Let {v n } n∈N be a sequence such that v n → v in C as n → ∞. Since f satisfies Hf 1 and g satisfies Hg 1 , for almost every t ∈ 0, T and t, s ∈ Λ, we get f t, v n t φ t −→ f t, v t φ t , as n −→ ∞, g t, s, v n s φ s −→ g t, s, v s φ s , as n −→ ∞.

3.12
Therefore, a v φ t ≤ Since v n → v in C, it follows that there exists ε > 0 such that v n − v 0,T ≤ ε for n sufficiently large. Moreover, noting that v n t − v t −r,0 ≤ v n − v 0,T , we have

3.16
It follows from the Lebesgue's Dominated Convergence Theorem that so, we can take the appropriate L to satisfy 3.22 and 3.23 . Next, we show that the operator F is ν-condensing on every bounded subset of C.