Existence and Regularity for Boundary Cauchy Problems with Infinite Delay

where A is an unbounded operator on a Banach space (X, ‖⋅‖) of functions on [0,∞)with domainD(A),u(t) = u(t, ⋅) ∈ D(A) for each t ≥ 0, L : D(A) → ∂X is the operator defined by L(V(⋅)) = V(0) for V(⋅) ∈ D(A), and ∂X := {V(0); V(⋅) ∈ X} is a “boundary space.” For each t ≥ 0,Φ(t) is a bounded linear operator fromX to X. Equation (2) can be further transformed into a Cauchy problem. To do this, suppose that the domain D ≡ D(A) of A and ∂X are Banach spaces such that D is dense and continuously embedded in X. A ∈ B(D,X) and L ∈ B(D, ∂X). We make the following hypothesis.


Introduction
Consider the following problem: where ( , ) represents the density of the population of age at time , is the death rate, and ( ) is the number of newborns at time . Such models were introduced by Lotka in 1925 and have been studied by many authors. For a detailed discussion, we refer the reader to [1,2].
The problem (1) can be transformed into the following abstract boundary Cauchy problem: where A is an unbounded operator on a Banach space (X, ‖⋅‖) of functions on [0, ∞) with domain (A), ( ) = ( , ⋅) ∈ (A) for each ≥ 0, : (A) → X is the operator defined by (V(⋅)) = V(0) for V(⋅) ∈ (A), and X := {V(0); V(⋅) ∈ X} is a "boundary space. " For each ≥ 0, Φ( ) is a bounded linear operator from X to X. Equation (2) can be further transformed into a Cauchy problem. To do this, suppose that the domain ≡ (A) of A and X are Banach spaces such that is dense and continuously embedded in X. A ∈ ( , X) and ∈ ( , X). We make the following hypothesis.
(S2) is a surjection from to X. | ker( −A) has a continuous inverse for any ∈ ( ) (the resolvent set of ).

International Journal of Differential Equations
It is easy to see that (4) is a form of the following abstract Cauchy problem: ( ) = ( ( ) + 1 ( , ( ))) + 2 ( , ( )) , ≥ 0, where is the infinitesimal generator of a 0 -semigroup on a general Banach space , is a bounded linear operator satisfying certain conditions, and 1 , 2 : R + × → . In this way, the problem of solving (1) or (2) is transformed to that of solving (5). Equations of the form like (5) were considered in [3][4][5]. An important tool used is the multiplicative perturbation which was first studied by Desch and Schappacher [3] in 1989 for 0 -semigroup. In recent years, this type of perturbations has been further developed and applied by many authors (cf., e.g., Engel and Nagel [6], Piskarëv and Shaw [7]). In this paper, our proof will also be based on an application of the multiplicative perturbation theorem.
Equation (2) has been considered in [8,9] for the cases ( ) = ∫ , respectively. Suppose that B is a linear space of functions from (−∞, 0] to X. Then these two cases can be viewed as a function from B to X. That says that ( ) depends on the "history" of . Thus, for such functions , (2) becomes a retarded Cauchy problem.
The following abstract retarded Cauchy problem has been considered by many authors (see [10][11][12][13] and the references therein): ( ) = ( ) + ( , ) , ≥ 0, where generates a 0 -semigroup (⋅) on , P is a linear space of functions from (−∞, 0] to satisfying some axiom which will be described later, is a function from [0, ∞) × P to , and, for a solution function : R → and for every ≥ 0, the function : (−∞, 0] → , defined by is required to belong to P. The theory of partial differential equations with infinite delay has attracted widespread attention. In [14][15][16], the variation-of-constant formula is used to study existence of solutions, regularity, existence of periodic solution, and stability for (6) when the delay is finite. In [10], a similar argument is used to solve (6) when an operator (not necessarily densely defined) satisfies the Hille-Yosida condition (maybe nondensely defined) and the delay is infinite. For a detailed discussion about infinite delay equations, we refer the reader to [13].
The main purpose of this paper is to consider the following more general boundary Cauchy problem with infinite delay: where 1 is a function from R + × P to X and 2 is a function from R + × P to X. 1 and 2 may be nonlinear. This abstract boundary delay problem has been studied by Piazzera [8] in some special cases. The case without delay also has been studied in [9]. Similar to the way that (2) is transformed into the form of (5), we can transform (9) into the following generalized retarded abstract Cauchy problem with delay: where 1 , 2 are functions from R + × P to . It is a generalization of (5) (and hence of (2)) as well as of (6).
In Section 2, we show the uniqueness and existence of solution of (10). It will be solved by using a variationof-constant formula similar to (8). The obtained result (Theorem 7) can be viewed as a partial generalization of [8,9].
Then we apply Theorem 7 in Section 3 to investigate an age dependent population equation for the situation that the birth process depends on the past of the population, as the following system describes: This equation contains as particular cases those equations that are considered in the recent papers [8,9]. Finally, we study in Section 4 regularity of mild solutions of (10). The property about equilibrium will be studied. The precise definition of equilibrium will be specified later. In [10], it is shown that the equilibrium of the solution semigroup associated with (6) is locally exponentially stable when its linearized solution around this equilibrium is exponentially bounded. We extend this result to a special case of (10).
In the rest of this paper, we suppose that satisfies condition ( ) with respect to (⋅) and the function satisfies the corresponding properties. Next, we make the hypotheses about for = 1, 2.
is called a mild solution of (12) on [0, ] if satisfies the following conditions: First, we show the uniqueness and existence of mild solutions to (12). Proof. By assumptions on 1 and 2 , there is a constant > 0 independent of ∈ [0, ] such that for ∈ [0, ] and 1 , 2 ∈ P. Moreover, we define the following real number: be a Banach space equipped with the norm Let Then ( ) is a closed subset of ( ). Note that the closed set ( ) and the operator are dependent on and . From the definition of , one can see that the fixed point of is a mild solution of (12) on [0, ]. Furthermore, if = and has a unique fixed point, then the fixed point is the unique solution to (12) from the definition of and the proof is completed. So, it is sufficient to show that has a unique fixed point in ( ). The unique fixed point will be found step by step. First, we show that there is an ∈ (0, ] such that has a unique fixed point. This fact will be shown by finding an ∈ (0, ] such that is a contraction. Suppose that V 1 , V 2 ∈ ( ). For ∈ [0, ], by the definition of , assumption of and hypotheses (A1) and (A2), it follows that So, by the assumption on , there exist ∈ N and ∈ (0, ] such that = and ( ( ) + ) < 1 for each ∈ [0, ]. On the other hand, ( V 1 − V 2 )( ) = ( ) − ( ) = 0 for all ≤ 0. It follows that is a contraction on ( ). Hence has a unique fixed point 1 ∈ ( ) by the contraction mapping principle. If = , then the proof is completed. Next, if 2 ≤ , then the previous argument will be repeated. Let us define the function : (−∞, 0] by Now, we can define the closed set ( ) of ( ) and define the operator from ( ) to ( ) by for each V ∈ ( ) and ∈ [0, ]. Repeating the previous argument, has a unique fixed point 2 in ( ). Define : Then, we show that is a fixed point of 2 on (2 ). If 0 ≤ ≤ , then ( ) = 1 ( ) = ( 1 )( ), so that it follows from (20) that In particular, for = , it becomes If ≤ ≤ 2 , let = + ℎ; then by (26) one has Hence is a fixed point of 2 in (2 ). Since 1 and 2 are the unique points in ( ) and ( ), respectively, it follows that is the unique fixed point in (2 ). This argument can be repeated until = . At the end, we can find the unique fixed point of on ( ).
International Journal of Differential Equations 5 Next, we want to give a sufficient condition for the existence of classical solution to (12). To do this, we need the differentiability of mild solutions. We give the following more restrictive conditions.
: R×P → is continuously differentiable and the derivatives 1 , 2 satisfy the following Lipschitz conditions: there is a constant > 0 such that for ∈ [0, ∞) and 1 , 2 ∈ P, where denotes the derivative with respect to the th variable.
The following lemmas are needed.

Theorem 7.
Let P satisfy axiom (B) or (C). Assume that 1 and 2 satisfy assumptions (H1) and (H2). In addition, assume that ∈ P is continuously differentiable with ∈ P, (0) + Proof. Consider the following equation: A similar argument as in the proof of Theorem 4 shows that there is a unique solution to (31) on [0, ]. Define the function by We first show that if there is an > 0 such that = on [0, ], then is a classical solution of (12) on [0, ]. In fact, in this case, is a differentiable mild solution. Denote ( ) = ( , ) for = 1, 2. It is easy to see that is continuously differentiable. Using integration by parts, we can write So, from the definition of mild solution, it follows that ( ) + Furthermore, by the assumption, (0) + 1 (0) ∈ ( ), and Definition 1, we see that ( ) + 1 ( ) ∈ ( ) for each ≥ 0. Hence, On the other hand, we see that International Journal of Differential Equations So, Differentiating both sides, we obtain Finally, comparing ( * ) with ( * * ), we see that is a classical solution on [0, ].
(44) For = 1, by (13), we have and similarly where By the assumption on , we can choose ∈ N and > 0 such that = and < 1.
for 0 ≤ ≤ . By a standard argument and using Gronwall's inequality, we get = on [0, ]. So, we have derived that is continuously differentiable on [0, ], and hence is a classical solution of (12) on [0, ]. If = , then the proof is completed. If ≤ = +ℎ ≤ 2 , from the definition (Definition 2(i)) of we see that

Application to Age Dependent Population Equations
In this section, the results in the previous section will be applied to age dependent population equations.

Moreover, is a bounded linear operator from R to Fav( ).
According to Theorem 12, we know that A, , and satisfy assumptions (S1) and (S2). Now, we suppose that > 0 in the rest of this section.
In the rest of this section, we suppose that the following conditions on the functions 1 , 2 , , and hold.
for ∈ R + . Now, we are going to verify that all assumptions of Theorem 7 are satisfied.

Lemma 15. 2 satisfies hypotheses (H1) and (H2).
Proof. In view of Lemma 14, it suffices to show that the operator 2 satisfies hypotheses (H1) and (H2). Suppose that ≥ 0 and 1 , 2 ∈ P. Using the estimate in the proof of Lemma 13, we see that Next, from the definition of 2 , it is easy to see that Since is a linear transformation, it follows that ‖ 2 ( 2 ( , 1 )) − 2 ( 2 ( , 2 ))‖ = 0. So, 2 satisfies hypotheses (H1) and (H2). The proof is completed. Proof. Let ≥ 0. By the assumption of (III)(a) and the Mean Value Theorem, we know that there is an between and +ℎ such that The continuity of → ( / ) ( , ⋅) in implies that the last term goes to 0 as ℎ → 0. So one can see that is continuously differentiable in with ( )(⋅) = ( / ) ( , ⋅). Moreover, ∈ P by assumption (III)(a) and the definition of P. Next, from assumption (III)(b), one can derive that Hence, by (III)(a), this implies that (0) − 1 (0, ) ∈ ( ). Finally, by using assumption (III)(b) one can derive that The proof is completed.
Consequently, in view of Lemmas 13-16, we can apply Theorem 7 to obtain the following theorem.

Solution Semigroups and Regularity
In this section, the regularity of the mild solution for ( ) = ( ( ) + 1 ( )) + 2 ( ) , ≥ 0, will be found. Throughout this section, we suppose that 1 and 2 satisfy the following condition: (H3) : P → satisfies a Lipschitz condition; that is, there is a constant > 0 such that for 1 , 2 ∈ P.
By Theorem 4, we know that (76) has a unique mild solution (⋅, ) on [0, ∞) for each ∈ P. Hence, we can define the nonlinear operator ( ) on P by for each ∈ P and ≥ 0.
Now, we will focus on the stability near an equilibrium of the nonlinear semigroup (⋅) on P. The following assumption is needed.
When (⋅) is a linear semigroup, this definition reduces to the usual definition of exponential stability of 0 -semigroups: Theorem 21 (see [19]). Let (⋅) be a nonlinear strongly continuous semigroup in a Banach space . Assume that ∈ is an equilibrium of (⋅) such that ( ) is Fréchet differentiable at for each ≥ 0, with ( ) the Fréchet derivative at of U(t). Then, (⋅) is a strongly continuous semigroup of bounded linear operators on . Moreover, if (⋅) is exponentially stable, then is an exponentially stable equilibrium of (⋅).
Since (H4) implies that 0 is an equilibrium of the semigroup (⋅) in Theorem 19, by Theorem 21, we have the following consequence.