- Research Article
- Open Access
Almost Automorphic and Pseudo-Almost Automorphic Solutions to Semilinear Evolution Equations with Nondense Domain
© B. de Andrade and C. Cuevas. 2009
- Received: 27 March 2009
- Accepted: 27 May 2009
- Published: 16 June 2009
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.
- Lipschitz Condition
- Mild Solution
- Bounded Continuous Function
- Automorphic Function
- Unbounded Linear Operator
In recent years, the theory of almost automorphic functions has been developed extensively (see, e.g., Bugajewski and N'guérékata , Cuevas and Lizama , and N'guérékata  and the references therein). However, literature concerning pseudo-almost automorphic functions is very new (cf. ). 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.  concerning the composition of pseudo-almost automorphic functions. The same authors in  have applied the results to obtain pseudo-almost automorphic solutions to semilinear differentail equations (see also ). On the other hand, in article by Blot et al. , the authors have obtained existence and uniqueness of pseudo-almost automorphic solutions to some classes of partial evolutions equations.
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  (see [10–12]). We note that pseudo-almost automorphic functions are more general and complicated than pseudo-almost periodic functions (cf. ).
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.
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. ).
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.
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 .
Lemma 2.3 (see ).
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.
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 ).
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 ).
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 ). 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 ).
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 ).
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
3.1. Almost Automorphic Solutions
The following property of convolution is needed to establish our result.
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.
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.
Let , and let be the function defined in Lemma 3.1. Then .
The proof is now completed.
Now, we are ready to state and prove the following result.
Then (1.1) has a unique pseudo-almost automorphic (mild) solution.
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.
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.
Therefore is a contraction.
The following consequence is now immediate.
If where is the constant in Lemma 2.8, then (1.1) has a unique pseudo-almost automorphic (mild) solution.
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.
Claudio Cuevas is partially supported by CNPQ/Brazil under Grant 300365/2008-0.
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