Almost Automorphic and Pseudo-Almost Automorphic Solutions to Semilinear Evolution Equations with Nondense Domain

Abstract

We study the existence and uniqueness of almost automorphic (resp., pseudo-almost automorphic) solutions to a first-order differential equation with linear part dominated by a Hille-Yosida type operator with nondense domain.

1. Introduction

In recent years, the theory of almost automorphic functions has been developed extensively (see, e.g., Bugajewski and N'guérékata [1], Cuevas and Lizama [2], and N'guérékata [3] and the references therein). However, literature concerning pseudo-almost automorphic functions is very new (cf. [4]). It is well known that the study of composition of two functions with special properties is important and basic for deep investigations. Recently an interesting article has appeared by Liang et al. [5] concerning the composition of pseudo-almost automorphic functions. The same authors in [6] have applied the results to obtain pseudo-almost automorphic solutions to semilinear differentail equations (see also [7]). On the other hand, in article by Blot et al. [8], the authors have obtained existence and uniqueness of pseudo-almost automorphic solutions to some classes of partial evolutions equations.

In this work, we study the existence and uniqueness of almost automorphic and pseudo-almost automorphic solutions for a class of abstract differential equations described in the form

(1.1)

where is an unbounded linear operator, assumed to be Hille-Yosida (see Definition 2.5) of negative type, having the domain , not necessarily dense, on some Banach space is a continuous function, where . The regularity of solutions for (1.1) in the space of pseudo-almost periodic solutions was considered in Cuevas and Pinto [9] (see [10–12]). We note that pseudo-almost automorphic functions are more general and complicated than pseudo-almost periodic functions (cf. [5]).

The existence of almost automorphic and pseudo-almost automorphic solutions for evolution equations with linear part dominated by a Hille-Yosida type operator constitutes an untreated topic and this fact is the main motivation of this paper.

2. Preliminaries

Let be Banach spaces. The notations and stand for the collection of all continuous functions from into and the Banach space of all bounded continuous functions from into endowed with the uniform convergence topology. Similar definitions as above apply for both and We recall the following definition (cf. [7]).

Definition 2.1.

A continuous function is called almost automorphic if for every sequence of real numbers there exists a subsequence such that is well defined for each and , for each Since the range of an almost automorphic function is relatively compact, then it is bounded. Almost automorphic functions constitute a Banach space, when it is endowed with the supremum norm.

A continuous function is called almost automorphic if is almost automorphic in uniformly for all in any bounded subset of . is the collection of those functions.

A continuous function (resp., ) is called pseudo-almost automorphic if it can be decomposed as where (resp., ) and is a bounded continuous function with vanishing mean value, that is,

(2.1)

(resp., is a bounded continuous function with

(2.2)

uniformly for in any bounded subset of ). Denote by (resp., ) the set of all such functions. In both cases above, and are called, respectively, the principal and the ergodic terms of .

We define

(2.3)

Remark 2.2.

is a Banach space, where is the supremum norm (see [6]).

Lemma 2.3 (see [13]).

Let be an almost automorphic function in for each and assume that satisfies a Lipschitz condition in uniformly in . Let be an almost automorphic function. Then the function defined by is almost automorphic.

Lemma 2.4 (see [5, 7]).

Let and assume that is uniformly continuous in any bounded subset uniformly in . If , then the function belongs to .

We recall some basic properties of extrapolation spaces for Hille-Yosida operators which are a natural tool in our setting. The abstract extrapolation spaces have been used from various purposes, for example, to study Volterra integro differential equations and retarded differential equations (see [14]).

Definition 2.5.

Let be a Banach space, and let be a linear operator with domain . One says that is a Hille-Yosida operator on if there exist and a positive constant such that and The infinimum of such is called the type of . If the constant can be chosen smaller than zero, is called of negative type.

Let be a Hille-Yosida operator on , and let; , and let be the operator defined by . The following result is well known.

Lemma 2.6 (see [12]).

The operator is the infinitesimal generator of a -semigroup on with for . Moreover, and , for .

For the rest of paper we assume that is a Hille-Yosida operator of negative type on . This implies that , that is, . We note that the expression defines a norm on . The completion of , denoted by , is called the extrapolation space of associated with . We note that is an intermediary space between and and that (see [12]). Since , we have that which implies that has a unique bounded linear extension to . The operator family is a -semigroup on , called the extrapolated semigroup of . In the sequel, is the generator of .

Lemma 2.7 (see [12]).

Under the previous conditions, the following properties are verified.

(i) and for every .

(ii)The operator is the unique continuous extension of , and is an isometry from into .

(iii)If , then ( exists and . In particular, and .

(iv)The space is dense in . Thus, the extrapolation space is also the completion of and . Moreover, is an extension of to . In particular, if , then and .

Lemma 2.8 (see [12]).

Let . Then the following properties are valid.

(i), for every .

(ii) where is independent of and .

(iii)The linear operator defined by is continuous.

(iv), for every .

(v) is the unique bounded mild solution in of

3. Existence Results

3.1. Almost Automorphic Solutions

The following property of convolution is needed to establish our result.

Lemma 3.1.

If is an almost automorphic function and is given by

(3.1)

then .

Proof.

Let be a sequence of real numbers. There exist a subsequence and a continuous functions such that converges to and converges to for each Since

(3.2)

Using the Lebesgue dominated convergence theorem, it follows that converges to for each . Proceeding as previously, one can prove that converges to for each . This completes the proof.

Theorem 3.2.

Assume that is an almost automorphic function in for each and assume that satisfies a Lipschitz condition in uniformly in . If where is the constant in Lemma 2.8, then (1.1) has a unique almost automorphic mild solution which is given by

(3.3)

Proof.

Let be a function in , from Lemma 2.3 the function is in From Lemma 2.8 and taking into account Lemma 3.1, the equation

(3.4)

has a unique solution in , which is given by

(3.5)

It suffices now to show that the operator has a unique fixed point in For this, let and be in and we can infer that

(3.6)

This proves that is a contraction, so by the Banach fixed point theorem there exists a unique such that This completes the proof of the theorem.

3.2. Pseudo-Almost Automorphic Solutions

To prove our next result, we need the following result.

Lemma 3.3.

Let , and letbe the function defined in Lemma 3.1. Then .

Proof.

It is clear that . If , where and . From Lemma 3.1 To complete the proof, we show that . For we see that

(3.7)

The preceding estimates imply that

(3.8)

The proof is now completed.

Now, we are ready to state and prove the following result.

Theorem 3.4.

Assume that is a pseudo-almost automorphic function and that there exists a bounded integrable function satisfying

(3.9)

Then (1.1) has a unique pseudo-almost automorphic (mild) solution.

Proof.

Let be a function in , from Lemma 2.4 the function belongs to . From Lemmas 2.8 and 3.3, (3.4) has a unique solution in which is given by (3.5). Let and be in , then we have

(3.10)

hence,

(3.11)

In general, we get

(3.12)

Hence, since for sufficiently large, by the contraction principle has a unique fixed point This completes the proof.

A different Lipschitz condition is considered in the following result.

Theorem 3.5.

Let be a pseudo-almost automorphic function. Assume that verifies the Lipschitz condition (3.9) with a bounded continuous function. Let If there is a constant such that for all where is the constant in Lemma 2.8, then (1.1) has a unique pseudo-almost automorphic (mild) solution.

Proof.

We define the map on by (3.5). By Lemmas 2.4 and 3.3, is well defined. On the other hand, we can estimate

(3.13)

Therefore is a contraction.

The following consequence is now immediate.

Corollary 3.6.

Let be a pseudo-almost automorphic function. Assume that verifies the uniform Lipschitz condition:

(3.14)

If where is the constant in Lemma 2.8, then (1.1) has a unique pseudo-almost automorphic (mild) solution.

3.3. Application

In this section, we consider a simple application of our abstract results. We study the existence and uniqueness of pseudo-almost automorphic solutions for the following partial differential equation:

(3.15)

with boundary initial conditions

(3.16)

Let , and let the operator be defined on by with domain

(3.17)

It is well known that is a Hille-Yosida operator of type-1 with domain nondense (cf. [15]). Equation (3.15) can be rewritten as an abstract system of the form (1.1), where ,

(3.18)

for all and . By [5, Example  2.5], is a pseudo-almost automorphic function. If we assume that then, by Corollary 3.6, (3.15) has a unique pseudo-almost automorphic mild solution.