Existence Results for Abstract Partial Neutral Integro-differential Equation with Unbounded Delay

In this paper we study the existence and regularity of mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay.


Introduction
In this paper we study the existence and regularity of mild solutions for a class of abstract partial neutral integro-differential equations in the form d dt (x(t) + f (t, x t )) = Ax(t) + t 0 B(t − s)x(s)ds + g(t, x t ), t ∈ I = [0, a], (1.1) where A : D(A) ⊂ X → X and B(t) : D(B(t)) ⊂ X → X, t ≥ 0, are closed linear operators; (X, • ) is a Banach space; the history x t : (−∞, 0] → X, x t (θ) = x(t + θ), belongs to some abstract phase space B defined axiomatically and f, g : I × B → X are appropriated functions.Abstract partial neutral integro-differential equations with unbounded delay arises, for instance, in the theory development in Gurtin & Pipkin [13] and Nunziato [28] for the description of heat conduction in materials with fading memory.In the classic theory of heat conduction, it is assumed that the internal energy and the heat flux depends linearly on the temperature u(•) and on its gradient ∇u(•).Under these conditions, the classic heat equation describes sufficiently well the evolution of the temperature in different types of materials.However, this description is not satisfactory in materials with fading memory.In the theory developed in [13,28], the internal energy and the heat flux are described as functionals of u and u x .The next system, see for instance [2,4,5,24], has been frequently used to describe this phenomena, In this system, Ω ⊂ R n is open, bounded and with smooth boundary; (t, x) ∈ [0, ∞) × Ω; u(t, x) represents the temperature in x at the time t; c 1 , c 2 are physical constants and k i : R → R, i = 1, 2, are the internal energy and the heat flux relaxation respectively.By assuming that the solution u(•) is known on (−∞, 0], we can transform this system into an abstract partial neutral integro-differential system with unbounded delay.The literature relative to ordinary neutral differential equations is very extensive and we refer the reader to [14] related this matter.Concerning partial neutral functional differential equations (B ≡ 0), we cite the pioneer Hale paper [15] and Wu & Xia [29,31] for equations with finite delay and Hernández et al. [16,17,18,19] for systems with unbounded delay.To the best of our knowledge, the study of the existence and qualitative properties of solutions of neutral integro-differential equations with unbounded delay described in the abstract form (1.1) are untreated topics in the literature and this, is the main motivation of this article.

Preliminaries
Throughout this paper, (X, • ) is a Banach space and A : D which is dense in X.To obtain our results, we assume that the abstract Cauchy problem has an associated resolvent operator of bounded linear operators (R(t)) t≥0 on X.
Definition 2.1 A family of bounded linear operators (R(t)) t≥0 from X into X is a resolvent operator family for system (2.1) if the following conditions are verified.
(2.4) Motivated by Grimmer [11], we adopt the following concepts of mild, classical and strong solution for the non-homogeneous system (2.4).
For additional details on the existence and qualitative properties of resolvent of operators for integro-differential systems, we refer the reader to [6,7,11,25] and the references therein.
In this work, we employ an axiomatic definition of the phase space B which is similar to that introduced in [23].Specifically, B will be a linear space of functions mapping (−∞, 0] into X endowed with a seminorm • B and verifying the following axioms. for every t ∈ [σ, σ + a) the following conditions hold: where (A1) For the function x(•) in (A), the function t → x t is continuous from [σ, σ + a) into B.
(B) The space B is complete.
To study the regularity of solutions, we consider the next axiom introduced in [22].
Let (Z, • Z ) and (W, • W ) be Banach spaces.In this paper, the notation L(Z, W ) stands for the Banach space of bounded linear operators from Z into W endowed with the uniform operator topology and we abbreviate this notation to L(Z) when Z = W .The notation, B r (x, Z) stands for the closed ball with center at x and radius r > 0 in Z.Additionally, for a bounded function γ : [0, a] → Z and t ∈ [0, a] we use the notation and we simplify this notation to γ t when no confusion about the space Z arises.

Existence and regularity of mild solutions
In this section we study the existence and regularity of mild solutions of the abstract integro-differential system (1.1)-(1.2).Motivated by (2.5), we consider the following concept of mild solution.
To establish our results we introduce the following assumption.In the sequel, [D(A)] is the domain of A endowed with the graph norm.
(H 2 ) The function g : I × B → X verifies the following conditions.
(i) The function g(t, •) : B → X is continuous for every t ∈ I.
(ii) For every x ∈ X, the function g(•, x) : I → X is strongly measurable.
(iii) There are a function m g ∈ C(I; [0, ∞)) and a continuous non-decreasing function (H 5 ) The function g : R × B → X is continuous and there exists a continuous function .
for every t ∈ I and all ψ j ∈ B r (ϕ, B).
EJQTDE, 2009 No. 29, p. 6 Remark 3.1 We note that the condition (H 1 ) is linked to the integrability of the functions s → AR(t−s), τ → τ 0 B(τ −ξ)R(t−τ )dξ, since except trivial cases, these functions are not integrable.Consider, for instance, the case in which A is the infinitesimal generator of a C 0 -semigroup (T (t)) t≥0 on X and B(t) = 0 for every t ≥ 0. In this case R(t) = T (t) for t ∈ I, and by assuming that AR(•) ∈ L 1 (0, t; L(X)), we have that T (t)x − x = A t 0 T (s)ds = t 0 AT (s)ds, which implies that the semigroup is continuous at zero in the uniform operator topology and, as consequence, that A is a bounded linear operator on X.
On the other hand, if condition H 1 is verified and x ∈ C(I, Y ), then from the Bochner criterion for integrable functions and the estimates we infer that the functions s → AR(t − s)x(s) and s For similar remarks concerning condition (H 1 ), see [19,20] Remark 3.2 In the rest of this paper, y : (−∞, a] → X is the function defined by Now, we can establish our first existence result. ) and that the following condition is verified.
(a) For all t ∈ (0, a] and r > 0, there exit a compact set Then there exists a mild solution of (1.1) We note that from Remark 3.1 and condition H 3 it follows that ΓS(b) ⊂ S(b).
Next, we prove that Γ verifies the conditions of the Leray Schauder Alternative Theorem (see [9,Theorem 6.5.4]).Let (x n ) n∈N be a sequence in S(b) and x ∈ S(b) such that x n → x in S(b).From the phase space axioms, we have that the set as n → ∞.Now, a standard application of the Lebesgue dominated convergence Theorem permits to conclude that Γ is continuous.Now, we establish a priori estimates for the solutions of the integral equation z = λΓz, λ ∈ (0, 1).Let x λ be a solution of z = λΓz, λ ∈ (0, 1), and α λ (•) be the function defined by x λ (θ) .By using that x λ s B + y s B ≤ α λ (s), we find that Consequently, Denoting by β λ (t) the right hand side of (3.2), we obtain and ds.
To prove that Γ is completely continuous, we introduce the decomposition Γ = and (Γ i x) 0 = 0 for i = 1, 2, 3. From [21, Lemma 3.1] we infer that Γ 1 and Γ 3 are completely continuous.Next, we prove that Γ 2 is also completely continuous.In the sequel, for q > 0 we use the notations B q = B q (0, S(b)) and q * = (M a + K a MH) ϕ B +K a q.
The case t = 0 is trivial.Let 0 < ε < t < b.From the assumptions, we can fix numbers 0 = t which shows that Γ 2 B q is right equicontinuity at t ∈ (0, a).A similar procedure allows us to prove the right equicontinuity at zero and the left equicontinuity at t ∈ (0, b].Thus, Γ 2 B q is equicontinuous.This completes the proof that Γ 2 is completely continuous.In the next result, we employ the notations introduced in Remark 3.2.Theorem 3.2 Let conditions (H 1 ), (H 4 ) and (H 5 ) be hold.If L f (0)K(0) i c L(Y,X) < 1, then there exists a unique mild solution of (1.1)-(1.2) on [0, b] for some 0 < b ≤ a.
where the notation introduced in (2.6) have been used.Let Γ be the map introduced in the proof of Theorem 3.1.Next, we prove that Γ is a contraction from B ρ (0, S(b)) into B ρ (0, S(b)).For x ∈ S(b), we have that and hence, which shows that Γ is a contraction on B ρ (0, S(b)).This complete the proof.By using [26,Theorem 4.3.2],we also obtain the existence of mild solution.
In the rest of this section, we discuss the regularity of the mild solutions of (1.1)-(1.2).Motivated by the definitions 2.3 and 2.4, we introduce the following concepts of solution.We need to introduce some additional notations.Let (Z, • Z ) and (W, • W ) be Banach spaces.For a function γ : I → Z and h ∈ R, we represent by ∂ h γ the function defined by where R(ζ(t, z), s, w) Z → 0 as (s, z) = |s| + w W → 0. For x ∈ X, we will use the notation χ x for the function χ x : (−∞, 0] → X given for χ x (θ) = 0 for θ < 0 and χ x (0) = x.We also define the operator functions S(t) Remark 3.3 In the sequel, we always assume that the operators A, R(t), B(s) commutes.
The proof of the next Lemma is a tedious routine based in the use of the phase space axioms.We choose to omit it.Lemma 3.1 Let conditions (H 1 ) and (H 6 ) be hold.Assume that the functions R( Proof: From Lemma 3.1, condition (H 6 ) and the fact that the functions B(θ), θ ∈ I, are uniformly bounded in L([D(A)], X)), we infer that the functions v 1 and d dt f (t, u t ) ∈ L 1 (0, b; X).Under these conditions, we know from Lemma 2.1 that the mild solution w( x(0) = ϕ(0) + f (0, ϕ), Proof: Using that Df (0, ϕ) ≡ 0, we can fix positive constants 0 < b 1 ≤ b, L and r such that u s B ≤ r for all s ∈ [0, b 1 ], ] : X) be the solution of the integral equation where The existence and uniqueness of z(•) is a consequence of the contraction mapping principle, the condition µ < 1 and the fact that d + dt y t | t=0 (0) = Aϕ(0).We omit details. .By using the notations introduced in (3.4) and remarking that h∂ h u ξ = u ξ+h − u ξ , we find that where H 1 (t, h) → 0 uniformly on [0, b 1 ] as h → 0. Since f, g are functions of class C 1 , it follows that (h,h∂ h us) h R(f, s, u s , h, h∂ h u s ) Y → 0 and (h,h∂ h us) h R(g, s, u s , h, h∂ h u s ) → 0 uniformly on [0, b 1 ] as h → 0, which allows re-write the last inequality in the form where H 2 (t, h) → 0 uniformly on [0, b 1 ] as h → 0. Now, from axiom (A)(iii) and the fact that µ < 1 we obtain Thus, to prove that u (•) = z(•) it is sufficient to show that u h −ϕ h − z 0 B → 0 as h → 0. Consider the decomposition u = y + z 1 + z 2 , where (z i ) 0 = 0, i = 1, 2, and From condition (c), we obtain that the Lagrange mean value Theorem for differentiable functions, for 0 < θ < h < b 1 we find that where [(0, ϕ), (θ, u θ )] represent the segment of line between (0, ϕ) and (θ, u θ ).This allows to conclude that From the above remarks, the functions S(t) = −Af (t, u t ) − t 0 B(t − s)f (s, u s )ds + g(t, u t ) and t → t 0 R(t − θ)S(θ)ds belong to C 1 ([0, b 1 ] : X), which implies from [11,Corollary 2] that v(•) is a classical solution of (3.5).Finally, the uniqueness of mild solution of (3.5) implies that v(t) = u(t) + f (t, u t ), which allows to conclude that u(•) is a classical solution of (1.1)-(1.2).

Applications
In this section we consider some applications of our abstract results.We begin considering the particular case which X is finite dimensional.

Ordinary neutral integro-differential equations
The literature for neutral systems for which x(t) ∈ R n is very extensive and for this type of systems our results are easily applicable.In fact, in this case, the functions A, B(s) are bounded linear operators, R(•) ∈ C([0, a], L(X)), R(t) is compact for every t ≥ 0 and the condition H 1 is verified with Y = X = R n .The following results are reformulations of Theorems 3.1 and 3.2 respectively.which arises in the study of the dynamics of income, employment, value of capital stock, and cumulative balance of payment, see [3] for details.In this system, λ is a real number, the state u(t) ∈ R n , C(•), B(•) are n × n matrix continuous functions, A is a constant n × n matrix, p(•) represents the government intervention and q(•) the private initiative.
are closed linear operators defined on a common domain D(A) EJQTDE, 2009 No. 29, p. 2

(H 1 )
There exists a Banach space (Y, • Y ) continuously included in X such that the following conditions are verified.EJQTDE, 2009 No. 29, p. 5 (a) For every t