Non-autonomous Inhomogeneous Boundary Cauchy Problems and Retarded Equations

In this paper we prove the existence and the uniqueness of classical solution of non-autonomous inhomogeneous boundary Cauchy problems, this solution is given by a variation of constants formula. Then, we apply this result to show the existence of solution of a retarded equation.


Introduction
Consider the following Cauchy problem with boundary conditions (IBCP ) This type of problems presents an abstract formulation of several natural equations such as retarded differential equations, retarded (difference) equations, dynamical population equations and neutral differential equations.
In the autonomous case (A(t) = A, L(t) = L, φ(t) = φ) the Cauchy problem (IBCP ) was studied by Greiner [2,3].He used a perturbation of domain of generator of semigroups, and showed the existence of classical solutions of (IBCP ) via variation of constants formula.In the homogeneous case (f = 0), Kellermann [6] and Nguyen Lan [8] have showed the existence of an evolution family (U (t, s)) t≥s≥0 as the classical solution of the problem (IBCP ).
The aim of this paper is to show well-posedness in the general case (f = 0) and apply this result to get a solution of a retarded equation.In Section 2 we prove the existence and uniqueness of the classical solution of (IBCP ).For that purpose, we transform (IBCP ) into an ordinary Cauchy problem and prove the equivalence of the two problems.Moreover, the solution of (IBCP ) is explicitly given by a variation of constants formula similar to the one given in [3] in the autonomous case.We note that the operator matrices method was also used in [4,8,9] for the investigation of inhomogeneous Cauchy problems without boundary conditions.Section 3 is devoted to an application to the retarded equation We introduce now the following basic definitions which will be used in the sequel.A family of linear (unbounded) operators (A(t)) 0≤t≤T on a Banach space X is called a stable family if there are constants A family of bounded linear operators (U (t, s)) 0≤s≤t on X is said an evolution family if (1) U (t, t) = I d and U (t, r)U (r, s) = U (t, s) for all 0 ≤ s ≤ r ≤ t, (2) the mapping (t, s) ∈ R 2 + : t ≥ s (t, s) −→ U (t, s) is strongly continuous.For evolution families and their applications to non-autonomous Cauchy problems we refer to [1,5,10].

Well-posedness of Cauchy problem with boundary coditions
Let D, X and Y be Banach spaces, D densely and continuously embedded in X, consider families of operators A(t) ∈ L(D, X), L(t) ∈ L(D, Y ) and φ(t) ∈ L(X, Y ), 0 ≤ t ≤ T .In this section we will use the operator matrices method in order to prove the existence of classical solution for the non-autonomous Cauchy problem with inhomogeneous boundary conditions it means that we will transform this Cauchy problem into an ordinary homogeneous one.
In all this section we consider the following hypotheses : (H 1 ) t −→ A(t)x is continuously differentiable for all x ∈ D; (H 2 ) the family (A 0 (t)) 0≤t≤T , A 0 (t) := A(t)| kerL(t) , is stable, with (M 0 , ω 0 ) constants of stability; (H 3 ) the operator L(t) is surjective for every t ∈ [0, T ] and t −→ L(t)x is continuously differentiable for all x ∈ D; (H 4 ) t −→ φ(t)x is continuously differentiable for all x ∈ X; (H 5 ) there exist constants γ > 0 and ω ∈ R such that and u satisfies (IBCP ).If (IBCP ) has a classical solution, we say that it is well-posed.
We recall the following results which will be used in the sequel.
As consequences of this lemma we have are the projections in D onto ker(λ − A(t)) and D(A 0 (t)) respectively.In order to get the homogenization of (IBCP ), we introduce the Banach space Let A φ (t) be a matrix operator defined on E by To the family A φ (•) we associate the homogeneous Cauchy problem In the following proposition we give an equivalence between solutions of (IBCP ) and those of (N CP ).
(ii) Let u be a classical solution of (IBCP ) with initial value u 0 .Then, the function . (2.1) hence the equation (2.1) yields to The initial value condition is obvious.The assertion (ii) is obvious.
Now we return to the study of the Cauchy problem (N CP ).For that aim, we recall the following result.
Theorem 2.1.( [11], Theorem 1.3) Let (A(t)) 0≤t≤T be a stable family of linear operators on a Banach space X such that i) the domain D := (D(A(t)), • D ) is a Banach space independent of t, ii) the mapping t −→ A(t)x is continuously differentiable in X for every x ∈ D.
Then there is an evolution family (U (t, s)) 0≤s≤t≤T on D. Moreover U (t, s) has the following properties : (1) U (t, s)D(s) ⊂ D(t) for all 0 ≤ s ≤ t ≤ T , where D(r) is defined by the mapping t −→ U (t, s)x is continuously differentiable in X on [s, T ] and In order to apply Theorem 2.1, we need the following lemma.
Proof..For t ∈ [0, T ], we write A φ (t) as Since A φ (t) is a perturbation of A(t) by a linear bounded operator on E, hence, in view of a perturbation result ([10], Thm.5.2.3) it is sufficient to show the stability of (A(t)) 0≤t≤T .
Let λ > ω 0 and x f y , we have On the other hand, for As a consequence, we get ¿From (2.2) and (2.3) , we obtain that the resolvent of A(t) is given by Hence, by a direct computation one can obtain, for a finite sequence 0 ≤ t ¿From the hypothesis (H 5 ), we conclude that ||L λ,t || ≤ γ(λ − ω) −1 for all t ∈ [0, T ] and λ > ω.Define ω 1 = max(ω 0 , ω).Therefore, by using (H 2 ), we obtain for , where M := max(M, M γ).Thus the lemma is proved.Now we are ready to state the main result.
Theorem 2.2.Let f be a continuously differentiable function on [0, T ] onto Y .
Then, for all initial value u 0 ∈ D, such that L(s)u 0 = φ(s)u 0 + f (s), the Cauchy problem (IBCP ) has a unique classical solution u.Moreover, u is given by the variation of constants formula where (U (t, s)) t≥s≥0 is the evolution family generated by A 0 (t) and f (t, u(t)) := φ(t)u(t) + f (t).
Proof.First, for the existence of U (t, s) we refer to [7].Since (A φ (t)) 0≤t≤T is stable and from assumptions (H 1 ), (H 3 ) and (H 4 ), (A φ (t)) 0≤t≤T satisfies all conditions of Theorem 2.1, then there exists an evolution family U φ (t, s) such that, for all initial value u0 f 0 ∈ D(A φ (s)), the function is a classical solution of (N CP ).Therefore, from (i) of Proposition 2.1, u 1 is a classical solution of (IBCP ).The uniqueness of u 1 comes from the uniqueness of the solution of (NCP) and Proposition 2.1.
Definition 3.1.A function v : [s − r, T ] −→ E is said a solution of (RE), if it is continuously differentiable, K(t)v t is well defined, ∀t ∈ [0, T ] and v satisfies (RE).
In this section we are interested in the study of the retarded equation (RE), we will apply the abstract result obtained in the previous section in order to get a solution of (RE).As a first step, we show that this problem can be written as a boundary Cauchy one.More precisely, we show in the following theorem that solutions of (RE) are equivalent to those of the boundary Cauchy problem Where A(t) is defined by Proof.i) Let u be a classical solution of (IBCP ) , then from Definition 2.1, v is continuously differentiable.On the other hand, (3.1) and (3.3) implies that u verifies the translation property Hence v satisfies (RE).
This theorem allows us to concentrate our self on the problem (IBCP ) .So, it remains to show that the hypotheses (H 1 ) − (H 5 ) are satisfied.
The hypotheses (H 1 ), (H 3 ) are obvious and (H 4 ) can be obtained from the assumptions on the operator K(t).