Existence results on nonautonomous partial functional differential equations with state-dependent infinite delay

. The aim of this work is to establish the existence of mild solutions for some nondensely nonautonomous partial functional differential equations with state-dependent infinite delay in Banach space. We assume that, the linear part is not necessarily densely defined and generates an evolution family under the hyperbolique conditions. We use the classic Shauder Fixed Point Theorem, the Nonlinear Alternative Leray-Schauder Fixed Point Theorem and the theory of evolution family, we show the existence of mild solutions. Secondly, we obtain the existence of mild solution in a maximal interval using Banach’s Fixed Point Theorem which may blow up at the finite time, we show that this solution depends continuously on the initial data under the global Lipschitz condition on the second argument of F and we get the existence of global mild solution. We propose some model arising in dynamic population for the application of our results.


Introduction
Partial differential equations play a crucial role in providing mathematical answers to natural phenomena and they continue to be an indispensable tool in scientific investigations of real-world problems. The future behaviors of many phenomenas are therefore supposed to be described by the solutions of an ordinary or partial differential equations. These have long played important roles in the history of mathematical modeling and will undoubtedly continue to serve as indispensable tools in future investigations. They are encountered in a variety of problems in physics, chemistry, biology, medicine, economics, engineering, climate and disease modeling and many others.
In this work, we study the existence of mild solutions for the following partial functional differential equation with state-dependent infinite delay x ρ(t,xt) ); t ∈ J := [0, b], in a Banach space (X, ∥·∥). Here (A(t)) t≥0 is a given family of closed linear operators in X with non necessarily dense domain and satisfying the hyperbolic conditions (A 1 ) through (A 3 ) introduce by Tanaka in [45,46] which will be specified later. The phase space B is a linear space of functions mapping (−∞, 0] into X satisfying some Axioms which will be described in the sequel. F : J × B is continuous and ρ : J × B → (−∞, b] are appropriate functions. The history x t (t ≥ 0), represents the mapping defined from (−∞, 0] into X by For the nonautonomous dynamical systems, the basic law of evolution is not static in the sense that the environment change with time. Parameters in real-world situations and particularly in the life sciences are rarely constant over time. The theory of nonautonomous dynamical systems is a well-developed and successful mathematical framework to describe time-varying phenomena. Its applications in the life sciences range from simple predator-prey models to complicated signal traduction pathways in biological cells, in physics from the motion of a pendulum to complex climate models, and beyond that to further fields as diverse as chemistry (reaction kinetics), economics, engineering, sociology, demography, and biosciences. Nonautonomous differential equations has received the great attention see for instance the works [22,26,28,40,42,47,51] and some recent works [9,[37][38][39]. For some applications, we refer the reader to the handbook by Peter E. Kloeden and Christian Pötzsche [44]. Note that when A(t) := A is independent of t, the theory of partial functional differential equations was studied by several authors. Hernández et al. [34] studied the existence of mild solutions of Equation (1.1) by using the classical C 0 -semigroup theory. Later on, Belmekki et al. [12] obtained the existence results of the following partial functional differential equations with state-dependent delay: where the operator A satisfies the usual Hille-Yosida condition except the density of D(A) in X. They obtained their results by using the variation of constants formula which is given in terms of integrated semigroups. In the autonomous case where ρ(t, x t ) = t, we refer the reader to Adimy et al [2], K. Ezzinbi et al [23,24], Hale and Lunel [30], G. F. Webb. [48,49], Wu [50], and the papers [2, 3, 13, 14, 16-18, 18, 36].
The literature related to partial nonautonomous functional differential equations with delay for which ρ(t, ψ) = t is very extensive and we refer the reader to the papers in [9,13,25,37,38,40,47] concerning this case. Recently Kpoumié et al in [9], investigate several results on the existence of solutions of the following nonautonomous equation : Nondensely nonautonomous partial functional differential equations with state-dependent infinite delay where (A(t)) t≥0 is a given family of closed linear operators on a Banach space (X, ∥ · ∥) not necessarily densely defined satisfying the hyperbolic conditions, B is a linear space of functions mapping (−∞, 0] to X satisfying some Axioms and F a continuous function defined on [0, +∞) × B with values in X. In this context, they have studied the local existence of the mild solutions which may blow up at the finite time, the global existence of mild solutions are given and under sufficient conditions, the existence of the strict solutions have been obtained.
Functional differential equations with state-dependent delay appear frequently in applications as models of equations and for this reason the study of this type of equation has attracted attention in recent years and more than ten years ago we refer the reader to the handbook by Cañada et al. [5], the book [19], the papers [6,8,11,12,20,26,27,31,32] and the references therein. In [39], we investigated the existence of mild solutions of the following nonautonomous equation: where (A(t)) t≥0 is a given family of closed linear operators on a Banach space (X, ∥ · ∥) not necessarily densely defined and satisfying the hyperbolic conditions (A 1 ) through (A 3 ) introduced by Tanaka in [46] which will be specified in Section 2. F is a given function defined on [0, +∞) × X with values in X, the initial data ρ : [−r; 0] → X is a continuous function, ρ is a positive bounded continuous function on X and r is the maximal delay defined by r = sup x∈X ρ(x) .
In this paper, we study the existence of at least one mild solutions where the family of closed linear operators on a Banach space is not necessarily densely defined. Note that there are many examples where evolution equations are not densely defined. One can refer to [1,4,21] for references and discussion on this subject. Our work is motivated by [9,34]. The results obtained is a continuation of work done by Hernãndez et al in [34], Belmekki et al. [12] and Kpoumié et al in [39].
In the whole of this work we employ an axiomatic definition for the phase space B due to Hale and Kato [29]. We assume that B is a normed linear space of functions mapping (−∞, 0] to X endowed with a normed | · | B and satisfying the following Axioms: There exist a positive constant H and functions K(·); M (·) : [0, +∞) → [0; +∞), with K continuous and M locally bounded, and the are independent of x, such that for any σ ∈ R and a > 0, if x is a function mapping (−∞, σ + a[ into X, a > 0, such that x σ ∈ B, and x(·) is continuous on [σ, σ + a[, then for every t in [σ, σ + a[ the following conditions hold : For examples and more details on phase space, see the book by Y. Hino, S. Murakami and T. Naito [35]. The organization of this work is as follows: in Section 2, we recall some results on nonautonomous evolution family with nondensely domain theory that will be used to develop our main results. In Section 3, we use the variant of Shauder's Fixed Point Theorem and the nonlinear alternative of Leray-Schauder's to prove the existence of at least one mild solution. In Section 4, we propose an application to some models with state dependent delay.

Nonautonomous evolution family with nondense domain
In this section, we recall some notations, definitions and preliminary facts concerning our work. Throughout this paper we used the results which are detailed in [43,45,46]. We assume that B(X) is the Banach space of all bounded linear operators from X to itself. In this work, we assume the following hyperbolic assumptions: (A 1 ) D(A(t)) := D independent of t and not necessarily densely defined.
(A 2 ) The family (A(t)) t≥0 is stable that means there are constants M ≥ 1 and w ∈ R such that: We recall here the classical result which gives us the existence and explicit formula of the evolution family generated by (A(t)) t≥0 due to Oka and Tanaka [43] and Tanaka [46]. Theorem 2.1. (Oka and Tanaka [43]; Tanaka [46]) Assume that (A(t)) t≥0 satisfies conditions (A 1 ) -(A 3 ). Then the limit exists for x ∈ D and t ≥ s ≥ 0, where the convergence is uniform on Γ := {(t, s) : t ≥ s ≥ 0}. Moreover, the family {U (t, s) : (t, s) ∈ Γ} satisfies the following properties: vi) for all x ∈ D(s) and t ≥ s ≥ 0, the function t → U (t, s)x is continuously differentiable with: Let λ > 0, t ≥ s ≥ 0 and x ∈ X. We define U λ (t, s) by: Nondensely nonautonomous partial functional differential equations with state-dependent infinite delay Remark 2.1. For x ∈ X, λ > 0 and t ≥ r ≥ s ≥ 0 one can see that We consider the following nonautonomous linear evolution equation: Theorem 2.2. (Tanaka [46]) Assume that (A 1 )-(A 3 ) hold. Let x 0 ∈ D and f ∈ L 1 ([0, a], X). Then the limit exists uniformly for t ∈ [0, a] and x is a continuous function on [0, a].
Definition 2.1. (Tanaka [46]) For x 0 ∈ D, a continuous function x : [0, a] → X is called a mild solution of the initial value of Equation (2.1) if x satisfies the following equation: In the whole of this work, we assume that (A 1 ) -(A 3 ) are true and w > 0.

Existence of mild solutions
In this section, we use some Fixed Point Theorems and the Kuratowski's measure of noncompactness to establish the existence of mild solutions of Equation (1.1). In this work, we always assume that ρ : We say that a continuous function x : (−∞, b] → X is a mild solution of Equation (1.1) if x satisfies the following equation We introduce the Kuratowski's measure of noncompactness α(·) of bounded sets K on a Banach space Y which is defined by: Some basic properties of α(·) are given in the following Lemma.
The terminology and notations employed in this work coincide with those generally used in functional analysis. In particular, for Banach spaces (X, ∥ · ∥), (Y, ∥ · ∥), the notation L(X, Y ) stands for the Banach space of bounded linear operators from X into Y, and we abbreviate this notation to L(X) when X = Y . Moreover B r (z, X) denotes the open ball with center at z and radius r > 0 in X and for a bounded function x : J → X and 0 ≤ t ≤ b we employ the notation ∥x∥ X,t for ∥x∥ X,t := sup ∥x(θ)∥. We will simply write ∥x∥ t when no confusion arises.
To prove our main result we will use the following variant of Schauder's Theorem see Radu Precup [41] and the Nonlinear Alternative of Leray-Schauder see A. Granas [27] or W. Arendt [10].   Let (E, d) be a non empty complete metric space and a mapping T : E → E such that T p is a strict contraction (p ∈ N ⋆ ). Then T admits a unique fixed pointx in E (i.e. T (x) =x) and the sequence (x n ) n define by x n = T (x n−1 ) with x 0 ∈ E, converges tox.
If Ψ is constant, then from it follows that Let us consider the following assumptions: Nondensely nonautonomous partial functional differential equations with state-dependent infinite delay (C 2 ). The function F : J × B → X satisfies the following properties.
In the sequel, we prove the existence of mild solution of equation (1.1).

Theorem 3.4.
Let Ω be a nonempty open subset of B and the function F : Proof. We use the classic Schauder's Fixed Point Theorem.
Step 1. Let φ ∈ Ω be such that φ(0) ∈ D. Then, there exists a constants r > 0, By virtue of Axioms (B 1 ) − (i) and (B 2 ), y t ∈ B and t → y t is a continuous function. Then for γ ∈ (0, r) there Let a be a constant such that: Let us introduce the following space: The restriction of y to (∞, a] is an element of F a . In fact ∥y t − φ∥ B ≤ γ for all t ∈ [0, b 1 ] whereas γ < r then ∥ỹ t − φ∥ B ≤ r for all t ∈ [0, a] thus y ∈ F a . Therefore F a is nonempty.
For all u ∈ F a , we have Then F a is bounded. By using the triangular inequality in B it is clear that λp + (1 − λ)q ∈ F a for any p, q ∈ F a , with λ ∈ [0, 1].
Then F a is convex. Now we prove that F a is closed. To prove that, consider a convergent sequence (ũ n t ) n∈N of F a which converges toũ t . We want to show thatũ t ∈ F a .
Nondensely nonautonomous partial functional differential equations with state-dependent infinite delay To continue our proof, we need the following Lemma.
Lemma 3.4. Let φ ∈ B such that φ t ∈ B for every t ∈ ρ − . Assume that there exists a locally bounded function Proof.
Consider the mapping K defined on F a by: From definition (3.1), theorem (2.1) and the assumptions on φ, we infer that (Kx)(·) is well defined. We claim that K(F a ) ⊂ F a . In fact, Axiom (C 2 ) implies that for every x ∈ F a , the mapping s → F (s,x ρ(s,x) ) is continuous on [0, a]. Hence this mapping v := Kx is continuous on [0, a]. In the other hand, One has On one hand, by Axiom (B 1 ) − (iii), we have for any t ∈ [0, a], Therefore v ∈ F a . We have proved that F a is a nonempty, bounded, convex and closed subset of F a : Now we want to prove that K is a completely continuous operator.
Next, we will show now that the range of K ; Range(K) := {Ku, u ∈ F a }, is relatively compact in F a . By the Arzela-Ascoli theorem, it suffices to prove that Range(K)(t) is relatively compact in X for each t ∈ [0, a] , and Range(K) is equicontinuous on [0, a].
Step 3. The set of fonctions Range(K)(t) of is relatively compact on F a . To prove this assertion, it is sufficient to show that the set (Ku)(t) − U (t, 0)φ(0) : u ∈ F a is relatively compact.
Since Range(K)(t 0 ) is relatively compact and . There exists a compact set G such that: Thus, we get lim Using similar argument for 0 ≤ t ≤ t 0 ≤ b, we conclude that Range(K) is equicontinuous. Then by Arzelá-Ascoli's Theorem, Range(K) is retlatively compact. Since K is continuous by Step 2, we can conclude that K is a completely continuous operator. The existence of at least one a mild solution for Equation ( Proof. Let E = C(J, X) and K : E → E be the operator defined by (3.3). In order to use Leray Schauder Alternative Theorem. We claim that the set we obtain that and using the nondecreasing character of V , we have : Hence Using relation (3.5), we get This implies that, the set of functions {ν(·) : 0 < µ < 1} is bounded in C(J : X). Thus the set {x(·) : 0 < µ < 1} is also bounded in C(J : X) since We obtain the completely continuous property of K by proceeding as in the proof of Theorem 3.4. Since E is convex and 0 ∈ E, then the Nonlinear Alternative Leray-Schauder's Fixed Point Theorem guaranties the existence of at least one mild solution for Equation (1.1). Arguing as in the proof of Theorem 3.2 we can prove that K is completely continuous. Then by the Nonlinear Alternative Leray-Schauder's Fixed Point Theorem the exists at least one mild solution for Equation (1.1). □

Global existence of mild solutions and Blowing up phenomena
Let us give the following local Lipschitz condition on the nonlinear part F of Equation (1.1): (C 4 ) For each α > 0 there exists a positive constant r 0 (α) such that for φ, ψ ∈ B with |φ| B , |ψ| B ≤ α, we have: Contrarily to the previous results, if we replace conditions ((C 2 ) by condition (C 4 ), the following local existence results hold.
We prove now that K is strict contraction in F a (φ) and for this end, we consider x, z ∈ F a (φ). For t ∈ [0, a], we have Following the same reasoning, we can see that We can repeat the previous argument, and we obtain Since (K b M e wb r0(α)) n a n n! → 0 as n → +∞ then ∃n ∈ N such that (K b M e wb r0(α)) n a n n! < 1. It follows that K n is strict contraction and by the Banach fixe point theorem, we deduce there ∃!x ∈ F a such that K n x = x. Thus K n x = x implies that K n+1 x = Kx on the other hand K n (Kx) = K(x) it follows that K(x) is a fixed point of K n and since fixed point is unique then we get K(x) = x. Equation (1.1) has a unique mild solution x(., φ) which is defined on the interval (−∞, a]. This is true for all a > 0, then x(., φ) is a global solution of Equation (1.1) on R.
Next, we prove that the solution depends continuously on initial data. Let φ ∈ B and t ∈ [0, a[ be fixed. Show that x(·, φ) is continuous in the sense of φ. ∀ε > 0, look for k(a) > 0 such that for ψ ∈ B and |φ − ψ| B ≤ ε implies that Nondensely nonautonomous partial functional differential equations with state-dependent infinite delay We have by Lemme 3.3 using the Bellman-Gronwall Lemma it follows that Hence we can write It is clear that (M b + ζ + HK b M e wa ) e K b M e wa r0(α)θ > 0 hence, we deduce the continuous dependence on the initial data. □ Corollary 4.1. Assume that (C 4 ) holds. Let q 1 and q 2 be continuous fonctions from R + to R + such that Then, for ϕ ∈ B such that ϕ(0) ∈ D, Equation (1.1) has a unique mild solution which is defined on R.  We assume that a max < +∞ and lim sup

Application
For illustration of our previous result, we propose to study the following model.
Hence, the family of linear operators (A(t)) t≥0 on X satisfies the assumptions (A 1 ) -(A 3 ).
2) v 0 is uniformly continuous and bounded with respect to θ ∈ R − , uniformly with respect to x[0, π].
Nondensely nonautonomous partial functional differential equations with state-dependent infinite delay Then, the existence of mild solutions can be deduced from a direct application of Theorem 3.5 and we have the following result.