Existence and Regularity for Boundary Cauchy Problems with Infinite Delay

Abstract

The aim of this work is to investigate a class of boundary Cauchy problems with infinite delay. We give some sufficient conditions ensuring the uniqueness, existence, and regularity of solutions. For illustration, we apply the result to an age dependent population equation, which covers some special cases considered in some recent papers.

1. 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 is an unbounded operator on a Banach space of functions on with domain , for each , is the operator defined by for , and is a “boundary space.” For each , is a bounded linear operator from to  .

Equation (2) can be further transformed into a Cauchy problem. To do this, suppose that the domain of and are Banach spaces such that is dense and continuously embedded in . and . We make the following hypothesis.(S1) generates a -semigroup on where denotes the kernel of .(S2) is a surjection from to  . has a continuous inverse for any (the resolvent set of ).If assumptions (S1) and (S2) hold, then the operator is continuous from to  , and for all the operator satisfies At least formally, we can rewrite (2) as

It is easy to see that (4) is a form of the following abstract Cauchy problem: where is the infinitesimal generator of a -semigroup on a general Banach space , is a bounded linear operator satisfying certain conditions, and .

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 [3–5]. An important tool used is the multiplicative perturbation which was first studied by Desch and Schappacher [3] in 1989 for -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 and , respectively. Suppose that is a linear space of functions from to  . Then these two cases can be viewed as a function from to  . 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 [10–13] and the references therein): where generates a -semigroup on , is a linear space of functions from to   satisfying some axiom which will be described later, is a function from to  , and, for a solution function and for every , the function , defined by is required to belong to .

The theory of partial differential equations with infinite delay has attracted widespread attention. In [14–16], 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 is a function from to   and is a function from to  . and 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 , are functions from 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 variation-of-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).

2. Solutions to (10)

Let be a Banach space. Throughout this paper, is the infinitesimal generator of -semigroup on with domain and for some , , and . In this paper, we assume that the phase space is a Banach space consisting of some functions from to   and satisfies the following axioms, which were introduced first by Hale and Kato in [17].(A1)There exist a positive constant and functions , , with continuous and locally bounded, such that, for any and , if , and is continuous on , then for every the following conditions hold:(i),(ii),(iii).(A2)For each function in (A1), is a -valued continuous function on .

The objective of this section is devoted to investigate well-posedness results for the Cauchy problem:

Definition 1. Let generate a -semigroup on . One says that satisfies condition () with respect to if there is a continuous, nondecreasing function with such that and for each and .

The following are important examples of operators that satisfy the condition in Definition 1:(a) is a bounded linear operator from to  , the Banach space endowed with the graph norm .(b) is a bounded linear operator from to  , where denotes the Favard space of given by and endowed with the norm .These conclusions can be found in [3, 4].

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 .(H1) is continuous and satisfies a Lipschitz condition; that is, there is a constant such that  for and .

Definition 2. Let . A continuous function is called a mild solution of (12) on if satisfies the following conditions: (i) for ,(ii)  for  .

Definition 3. Let . A continuous function is called a classical solution of (12) on if satisfies the following conditions: (i) for ,(ii) and satisfies (12),(iii).First, we show the uniqueness and existence of mild solutions to (12).

Theorem 4. Suppose that and satisfy hypothesis (H1). Then (12) has a unique mild solution on for each .

Proof. By assumptions on and , there is a constant independent of such that for and . Moreover, we define the following real number: where , , , and and are the functions defined in hypothesis (A1). Note that is independent of . Let and let be a Banach space equipped with the norm Let Then is a closed subset of .
Note that it follows from (A2) and (H1) that , , are continuous in on . Then, since , satisfies condition (), we have (see the proof of Theorem 2.2 in [7]). Thus we can define by 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 . 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 such that has a unique fixed point. This fact will be shown by finding an such that is a contraction. Suppose that . For , by the definition of , assumption of and hypotheses (A1) and (A2), it follows that So, by the assumption on , there exist and such that and for each . On the other hand, for all . It follows that is a contraction on . Hence has a unique fixed point by the contraction mapping principle.
If , then the proof is completed. Next, if , then the previous argument will be repeated. Let us define the function by Since implies with and , it follows from the hypotheses (A1)(i) that .
Now, we can define the closed set of and define the operator from to   by for each and . Repeating the previous argument, has a unique fixed point in . Define by Then, we show that is a fixed point of on . If , then , so that it follows from (20) that In particular, for , it becomes
If , let ; then by (26) one has Hence is a fixed point of in . Since and are the unique points in and , respectively, it follows that is the unique fixed point in . This argument can be repeated until . At the end, we can find the unique fixed point of on .

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.(B)If is a Cauchy sequence in and if converges compactly to on (i.e., for each compact subset of , is convergent uniformly to ), then and , as .(C)For a sequence in , if as , then , as , for each .(H2) is continuously differentiable and the derivatives , satisfy the following Lipschitz conditions: there is a constant such that  for and , where denotes the derivative with respect to the th variable.

The following lemmas are needed.

Lemma 5 (see [13]). Let satisfy axiom (B) and let , be a continuous function such that is continuous for . Then for .

Lemma 6 (see [18]). Let satisfy axiom (C) and let , , be a continuous function. Then for all , the function is continuous and for .

Theorem 7. Let satisfy axiom (B) or (C). Assume that and satisfy assumptions (H1) and (H2). In addition, assume that is continuously differentiable with , , and . If is the unique mild solution of (12) on , then is continuously differentiable on . Furthermore, is a classical solution of (12) on .

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 . Define the function by
We first show that if there is an such that on , then is a classical solution of (12) on . In fact, in this case, is a differentiable mild solution. Denote for . 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, , and Definition 1, we see that for each . Hence,
On the other hand, we see that So, Differentiating both sides, we obtain Finally, comparing with , we see that is a classical solution on .
Next, we show that there does exist an such that on . Recall the integrated semigroup generated by ; that is, for each . One can obtain that for . Here denotes the identity map. Therefore, by the closedness of and the assumption on , becomes for , where denotes the identity map. By Lemma 5 or Lemma 6, we obtain By the elementary properties of , for . Moreover, using integration by parts and simple computation, one can derive that for and . Consequently, by (36)–(40), satisfies for . On the other hand, by hypotheses (H1) and (H2), there exists such that for . Moreover, we define the following real number: where and is the function defined in hypothesis (A1). Therefore, by (41) and definition of , one can obtain that
For , by (13), we have and similarly
Therefore where By the assumption on , we can choose and such that and .
For , Consequently, from (44), (47), and (49), one can derive that for . By a standard argument and using Gronwall's inequality, we get on . So, we have derived that is continuously differentiable on , and hence is a classical solution of (12) on .
If , then the proof is completed. If , from the definition (Definition 2(i)) of we see that It follows that is a mild solution of Repeating the previous argument, we can show that is differentiable on . Hence, is a classical solution of (12) on . This argument can be repeated until . At the end, one can show that is a classical solution of (12) on .

3. Application to Age Dependent Population Equations

In this section, the results in the previous section will be applied to age dependent population equations.

Theorem 8 (see [13]). Let be a Banach space and let be a fixed number. Suppose that denotes the space endowed with the norm ; then satisfies assumptions (A1), (A2), (B), and (C).

Let with norm , and let be the phase space endowed with the norm for a fixed . Let us consider the following system: where , , , and is a fixed real number.

Remark 9. The linear cases for and have been considered by many authors. In [8], Piazzera considers the case that is defined by , , and , where and . In [9], the authors consider the case that and , where , , and .

In the first step, we rewrite (53) in operator theoretic form on the Banach space . We define the operator with domain . The boundary operator is defined by . Then with domain generates a -semigroup on . Before doing the next step, we give the following definition and theorem which characterize the condition () in some special cases.

Definition 10. Let generate a -semigroup on a Banach space (, ). The Favard class is the space .

It is known that becomes a Banach space if we define for . The following theorem, which gives an important example of operators satisfying condition (), can be found in [3, 7].

Theorem 11. Let generate a -semigroup on a Banach space . If , then satisfies condition () with respect to .

Theorem 12 (see [3]). Let . Then and is given by for and . Moreover, is a bounded linear operator from to  .

According to Theorem 12, we know that , , and satisfy assumptions (S1) and (S2). Now, we suppose that in the rest of this section.

For rewriting (53), we define the function where denotes the characterization function and , and define the linear functional: It is easy to see that and for . Furthermore, by Theorem 12, it follows that . So, satisfies condition () with respect to by Theorem 11. We introduce the following notations: (i), , , and ;(ii) is defined by ; here we assume that for each and ;(iii) is defined by  where and are the functions in (53);(iv) and .Using these notations, we can rewrite (53) as Indeed, condition , means , , . To see that the differential equation in (58) is equivalent to the first two lines of (53), we first suppose that is a solution of (58). By the definition of and the definition of in Theorem 12, we see that which is the equation in the first line of (53). Moreover, condition means that and so , that is, the second line of (53). Hence is a solution of (53).

Conversely, it is easy to see that is a solution of (58) wherever is a solution of (53).

In the rest of this section, we suppose that the following conditions on the functions , , , and hold.(I)Suppose that a function satisfies the following conditions:(a)For each , the function . is continuously differentiable with respect to the first and second variables.(b) There are and such that for , , and .(II) is differentiable. There is a function such that for all and , and is integrable on .(III)(a), , is continuous in and .(b) satisfies for .

Now, we are going to verify that all assumptions of Theorem 7 are satisfied.

Lemma 13. satisfies hypotheses (H1) and (H2).

Proof. From the definition of and assumption (II), it follows that for . Obviously, . Since is a linear transformation, it follows that Consequently, satisfies hypotheses (H1) and (H2).

Lemma 14. The function satisfies hypotheses (H1) and (H2).

Proof. Suppose that and . From assumption (I)(a), it follows that is differentiable with respect to the first variable. By assumption (I)(b) and the definition of norm , it follows that So, hypotheses (H1) and (H2) hold.
Let be a fixed element of . Define by for . Obviously, is well defined and linear. We show that . Let with . Because of assumption (I)((a) and (b)) and the Mean Value Theorem, for each pair , there exists between and such that Hence as . So, we obtain that . Moreover, from assumption (I)(b) and the definition of , it is easy to see that Consequently, satisfies hypothesis (H2).

Lemma 15. satisfies hypotheses (H1) and (H2).

Proof. In view of Lemma 14, it suffices to show that the operator satisfies hypotheses (H1) and (H2). Suppose that and . Using the estimate in the proof of Lemma 13, we see that Next, from the definition of , it is easy to see that Since is a linear transformation, it follows that . So, satisfies hypotheses (H1) and (H2). The proof is completed.

Lemma 16. is continuously differentiable with , , and .

Proof. Let . By the assumption of (III)(a) and the Mean Value Theorem, we know that there is an between and such that The continuity of in implies that the last term goes to 0 as . So one can see that is continuously differentiable in with . Moreover, by assumption (III)(a) and the definition of . Next, from assumption (III)(b), one can derive that Hence, by (III)(a), this implies that . 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.

Theorem 17. Under assumptions (I)–(III), (58) admits a unique classical solution.

4. Solution Semigroups and Regularity

In this section, the regularity of the mild solution for will be found. Throughout this section, we suppose that and satisfy the following condition: (H3) satisfies a Lipschitz condition; that is, there is a constant such that  for .By Theorem 4, we know that (76) has a unique mild solution on for each . Hence, we can define the nonlinear operator on by for each and .

Theorem 18. Under hypotheses () and (),    is a nonlinear strongly continuous semigroup on ; that is,(i), where denotes the identity map,(ii) for each ,(iii) is a continuous function. Furthermore,(iv)for each , is a continuous function,(v)for each and , the following translation property holds: (vi)there exist constants and such that  for and .

Proof. (i), (ii), and (v) are easy to see from the definition of . (iii) is obtained from hypothesis (A2) and the definition of . (iv) follows from (vi). Hence it remains to show (vi). By assumption (H3) on and , there is a constant such that for . Moreover, we define the following real number: where and and are the functions defined in (A1). Let . Use and to denote and , respectively. By assumption (13) on and hypotheses (A1) and (H3), it follows that and so for . We can choose so small that and it follows that for . By Gronwall’s inequality, it follows that where Since, by (ii) and (86), for , where , then   and are desired constants. The proof is completed.

Now, we will focus on the stability near an equilibrium of the nonlinear semigroup on . The following assumption is needed.(H4) is continuously Fréchet differentiable with respect to and .

Suppose that and satisfy hypothesis (H4) with and being linear operators on . Then, by Theorem 4, the equation has a unique mild solution. Let denote the solution semigroup on associated with (89). Then is a -semigroup.

Theorem 19. Suppose that and satisfy hypotheses (H3) and (H4) with and . Then the Fréchet derivative at of the nonlinear semigroup , associated with (76), is equal to the semigroup associated with (89).

Proof. First, we show that, for any , is differentiable with respect to for each . First, since is a -semigroup, there exists a constant such that for . Fix a and let be arbitrary. Since the uniqueness of solution and assumption imply , it is sufficient to find a such that for each . By assumptions of and , there is a constant such that for . Moreover, we define the following real number: where and is the function defined in (A1). Let and denote the mild solutions of (76) and (89), respectively. Let . For all , we have We can choose so small that . Hence, the last inequality implies that for .
Since is differentiable at 0 and with Fréchet derivative , from the definition of Fréchet derivative and linearity of , for each , there exists a such that for and , and so for . Let . Then for given there exists a such that for and . Similarly, for and . Consequently, for and ; that is, is Fréchet differentiable for .
If , then, by (ii) and (iv) of Theorem 18, it follows that for with . Thus, for given , letting , we have shown that for . So, the mapping is Fréchet differentiable for and the Fréchet derivative equals the map . Repeating this argument, we can get the conclusion.

Definition 20 (see [19]). Let be a strongly continuous semigroup on a Banach space . A point is called an equilibrium of if for all . An equilibrium is said to be exponentially stable if there exist , , and such that for and .

When is a linear semigroup, this definition reduces to the usual definition of exponential stability of -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 , 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 is an equilibrium of the semigroup in Theorem 19, by Theorem 21, we have the following consequence.

Theorem 22. Suppose that and satisfy hypotheses (H3) and (H4). If is exponentially stable on , that is, there exist constants and such that for and , then zero is an exponentially stable equilibrium of on ; that is, there exist , , and such that for and .

Remark 23. Theorems 19 and 22 for the special case that and satisfies the Hille-Yosida condition can be found in [10].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.