Mild solutions for some nonautonomous evolution equations with state-dependent delay governed by equicontinuous evolution families

. In this work, we study the existence solutions and the dependence continuous with the initial data for some nondensely nonautonomous partial functional differential equations with state-dependent delay in Banach spaces. We assume that the linear part is not necessarily densely defined, satisfies the well-known hyperbolic conditions and generate a noncompact evolution family. Our existence results are based on Sadovskii fixed point Theorem. An application is provided to a reaction-diffusion equation with state-dependent delay.


Introduction
Partial differential equations with delay are important for investigating some problems raised from natural phenomena.They have been successfully used to study a number of areas of biological, physical, engineering applications, and such equations have received much attention in recent years.It is generally known that taking into account the past states of the model, in addition to the present one, makes the model more realistic.This leads to the so called functional differential equations.In recent years, nonlinear evolution equations with Djeunankam et al.
In 1970, Kato in [12] initiated a study of the evolution family solution of hyperbolic linear evolution equations of the form x ′ (t) = A(t)x(t), t ≥ s, x(s) = x s ∈ X.
(1.1) in a Banach space X.Some fundamental and basic results about the well posedness and dynamical behavior of equation (1.1) were established under the so called stability condition, ((B 2 ) in Section 2).The autors focus on the nonautonomous linear case.
In 2011, Belmekki et al investigated in [5] several results on the existence of solutions of the initial value problem for a new class of abstract evolution equations with state-dependent delay in Banach space X, where f : [0; +∞) × X −→ X is a suitable nonlinear function, the initial data φ : [−r; 0] −→ X is a continuous function, ρ is a positive bounded continuous function defined on X and r is the maximal delay given by r = sup x∈X ρ(x).The autors focus on the case where the differential operator in the main part is nondensely define and independent of time t in [0, a].Here the equation is autonomous partial functional differential equations with state-dependent delay.Their approach is based on a nonlinear alternative of Leray-Schauder and integrated semigroup (S(t)) t≥ which is considered to be compact for t > 0.
In 2019, Kpoumie et al. investigated in [15] several results on the existence of solutions of the following nonautomous equations: in a Banach space (X, ∥.∥), where the family of closed linear operator (A(t)) t≥0 on X is not necessarily densely, satisfying the hyperbolic conditions (B 1 ) through (B 3 ) and φ : [−r; 0] −→ X the continuous function.
Their approach is based on a nonlinear alternative of Leray-Schauder under the assumption of the compactness of evolution family generated by (A(t)) t≥0 .They get the existence of mild solution under the Carathéodory condition on f.
In 2019, Chen and al. investigated in [7] several results on the existence of solutions of the nonautonomous parabolic evolution equations with non-instantaneous impulses in Banach space E: (1.4) by introducing the concepts of mild and classical solutions, where A : Their results are based on Sadovskii fixed point Nonautonomous evolution equations with state-dependent delay governed by equicontinuous evolution families Theorem and they consider that evolution family is noncompact.Therefore, it is for great significance and interesting to study the nonautonomous evolution equation where the family of closed linear operator (A(t)) t≥0 on X is not necessarily densely define and generates the noncompact evolution famillies.Driven by the above aspects, we will investigate the existence of mild solutions and the dependent continuous on the initial data of the following nonautonomous partial functional differential equations with state-dependent delay governed by noncompact evolution families of the form in a Banach space (X, ∥.∥), where the family of closed linear operator (A(t)) t≥0 on X is not necessarily densely and satisfying the hyperbolic conditions (B 1 ) through (B 3 ) introduced by Kato in [12] that will be specified later.f : [0; +∞) × X −→ X is a suitable nonlinear function satisfying some conditions which will be specified later.The initial data φ : [−r; 0] −→ X is a continuous function and ρ is a positive bounded continuous function on X.The constant r is the maximal delay defined by r = sup x∈X ρ(x).
We point out that the work of this paper is the following of [5,7,12,15].But under appropriate circonstances, evolutionary families are not compact.Our work is organized as follows: First, we recall some preliminary results about the evolution family generated by (A(t)) t≥0 and recall also some preliminary results concern Kuratowski measure.Second, we use the alternative of Sadovskii fixed point Theorem to prove the existence of at least one mild solution and the dependent continuous on initial data.Third, we propose an application to illustrate the main result.

Preliminary results
Our notations in this section are the usual in the theory of evolution equations.In particular, we denote by C(E, F ) the space of continuous functions from E into F and C 2 (E, F ) denotes the space of twice continuously differentiable functions from E into F .We mention here some results on nonautonomous differential equations with nondense domaine.We cite [12,13,16,18,19].We recall some properties and Theorems.
In the whole of this work, we assume the following hyperbolic assumptions: (B 1 ) D(A(t)) := D independent of t and not necessarily densely defined (D ⊊ X) .(B 2 ) The family (A(t) t≥0 is stable in the sense that there are constants M ≥ 1 and We follow by recall the classical result which gives us the existence and explicit formula of the evolution family generated by (A(t)) t≥0 due to Kato [12].Let λ > 0, 0 ≤ s ≤ t and x ∈ D, Theorem 2.1.[1,12] Assuming the three conditions (B 1 ) − (B 3 ).Then the limit exists for x ∈ D and 0 ≤ s ≤ t, where the convergence is uniform on Γ := {(t, s) : 0 ≤ s ≤ t}.Moreover, the family {U (t, s) : (t, s) ∈ Γ} satisfies the following properties: for all 0 ≤ s ≤ t, where D(t) is defined by Then there exists M ≥ 1 and ω ∈ R such that Remark 2.3.Since (B 2 ), λ > ω and hence for (2.1), we get that ω is non positive.And by using Corollary 2.2, we have ∥U (t, s)x∥ ≤ M ∥x∥ for each x ∈ D and 0 ≤ s ≤ t.
In the following, we give some results on the existence of solutions for the following nondensely nonautonomous partial functional differential equation where f : [0, a] −→ X is a function.The following Theorem gives us the generalized variation of constants formula of equation (2.2).
Nonautonomous evolution equations with state-dependent delay governed by equicontinuous evolution families Definition 2.6.[12] For x 0 ∈ D, a continuous function x : [0, a] −→ X is called a mild solution of equation (2.2) if it satisfies the equation (2.3).We introduce some basic definitions and properties of the Kuratowski noncompactness measure, this will be used to demonstrate our main result.Definition 2.7.[4,7] The Kuratowski measure of noncoampactness µ(.) defined on bounded set V of Banach space E is Definition 2.8.[4,7] Consider a Banach space X, and a nonempty subset Theorem 2.9.[4,7] Let E be a Banach space and U, V ⊆ E be bounded.The following properties are satisfied: , where X is another Banach space.
For more details about properties of the Kuratowski measure of noncompactness, we refer to the monographs of Bana's and Goebel [4] and Deimling [7].

Existence of mild solution
In this section, we try our self to prove the existence of global mild solutions for equation (1.5) using the equicontinuity of {U (t, s) : 0 ≤ s ≤ t < +∞}.We begin by define the mild solution that correspond to the definition in (1.5) and denote C r := C[−r, 0] with r > 0.
Definition 3.1.Let φ ∈ C r such that φ(0) ∈ D. We say that a continuous function x : (−r; +∞) −→ X is a mild solution of the equation (1.5), if it satisfies the following equation Firstly we study the local mild solution of equation (1.5).To obtain our result, we consider the following assumptions : The nonlinear function f : [0; ∞) × X −→ X is continuous; and for some r > 0 there exist a constant δ 1 > 0 and ϕ r ∈ L 1 ([0, a], R + ) such that for all t ∈ [0, a] and u ∈ C([−r, a], X) satisfying ∥u∥ ≤ r, ∥f (t, u)∥ ≤ ϕ r (t) and lim sup (H 2 ) There exists positive constant L 1 such that for any countable set D ⊂ X, Proof.Our proof is based on Sadoskii's fixed Point Theorem.
. By the strongly continuity of the evolution family {U (t, s) : 0 ≤ s ≤ t ≤ a}, we get for 0 ≤ s ≤ t ≤ a that:
Case 1: Assume that We claim that there exists a constant R > 0 such that G(B R ) ⊂ B R where From (H 1 ), we consider the definition of the operator G, the continuity of ρ and (H 2 ) hypothesis.We get for any t ∈ [0; a] that: And by using (H 1 ) we have: By using the Lebesgue dominated convergence theorem, we get that: Consequently, Gu n → Gu as n → +∞.So the operator G is continuous in B R .We claim that the operator G : B R −→ B R is equicontinuous.For all u ∈ B R , 0 < t 1 < t 2 ≤ a and ε > 0 small enough, by using (H 3 ) , we get that: → 0 as t 2 → t 1 and ε → 0.
We claim that the operator For any B ⊂ B R , B is bounded.By using Theorem 2.10, there exists a countable set And since the operator G : B R −→ B R is equicontinuous, by the Theorem 2.12 we get that (3.4) By using the definition of the operator G 1 , we get that (G 1 u)(t) = U (t, 0)φ(0) for all u ∈ B and 0 ≤ t ≤ a. Therefore G 1 (B)(t) = {U (t, 0)φ(0)} for t ∈ [0; a].From the definition of µ, we have Nonautonomous evolution equations with state-dependent delay governed by equicontinuous evolution families µ(G 1 (B)(t)) = 0 for all t ∈ [0; a] and according to the Theorem 2.12, we get µ(G 1 (B)) = 0.By using Theorem 2.9, Theorem 2.11, the assumptions (H 1 ) and the definition of G 2 , we have We know that A ⊂ B, and using Theorem 2.9, The inequality (3.9) proves that the operator G : B R −→ B R is condensing.From the Theorem 2.13, the problem (1.5) has at least one local mild solution defined on [−r, a].
We know that 4M L1a Therefore, the problem (1.5) has at least one local mild solution on [−r, a].

Application
In this section, we apply our results to the following non-autonomous partial differential equation of evolution.Nonautonomous evolution equations with state-dependent delay governed by equicontinuous evolution families We get that M = 1.

(H 3 )Theorem 3 . 2 .
We assume that the evolution family U (t, s) t≥s≥0 is equicontinuous i.e for any s ≥ 0, the function t −→ U (t, s) is continuous by operator norm for t ∈ (s; +∞).Let a > 0 and assume that the family of linear operators A(t) t≥0 satisfies the hyperbolic conditions (B 1 )-(B 3 ), the assumptions (H 1 ) -(H 3 ) and φ(0) ∈ D. Then the problem (1.5) has at least one local mild solution defined on [−r, a].Moreover, the mild solution depends continuously on the initial data.