- Research Article
- Open Access

# Existence of Pseudo-Almost Automorphic Mild Solutions to Some Nonautonomous Partial Evolution Equations

- Toka Diagana
^{1}Email author

**2011**:895079

https://doi.org/10.1155/2011/895079

© Toka Diagana. 2011

**Received:**15 September 2010**Accepted:**29 October 2010**Published:**31 October 2010

## Abstract

We use the Krasnoselskii fixed point principle to obtain the existence of pseudo almost automorphic mild solutions to some classes of nonautonomous partial evolutions equations in a Banach space.

## Keywords

- Banach Space
- Fixed Point Theorem
- Mild Solution
- Interpolation Space
- Evolution Family

## 1. Introduction

*almost automorphic*mild solutions to the nonautonomous abstract differential equations

where for is a family of closed linear operators with domains satisfying Acquistapace-Terreni conditions, and the function is almost automorphic in uniformly in the second variable, was studied. For that, the author made extensive use of techniques utilized in [2], exponential dichotomy tools, and the Schauder fixed point theorem.

where
for
is a family of linear operators satisfying Acquistpace-Terreni conditions and
are pseudo-almost automorphic functions. For that, we make use of exponential dichotomy tools as well as the well-known *Krasnoselskii* fixed point principle to obtain some reasonable sufficient conditions, which do guarantee the existence of pseudo-almost automorphic mild solutions to (1.2).

The concept of pseudo-almost automorphy is a powerful generalization of both the notion of almost automorphy due to Bochner [3] and that of pseudo-almost periodicity due to Zhang (see [4]), which has recently been introduced in the literature by Liang et al. [5–7]. Such a concept, since its introduction in the literature, has recently generated several developments; see, for example, [8–12]. The question which consists of the existence of pseudo-almost automorphic solutions to abstract partial evolution equations has been made; see for instance [10, 11, 13]. However, the use of Krasnoselskii fixed point principle to establish the existence of pseudo-almost automorphic solutions to nonautonomous partial evolution equations in the form (1.2) is an original untreated problem, which is the main motivation of the paper.

The paper is organized as follows: Section 2 is devoted to preliminaries facts related to the existence of an evolution family. Some preliminary results on intermediate spaces are also stated there. Moreover, basic definitions and results on the concept of pseudo-almost automorphy are also given. Section 3 is devoted to the proof of the main result of the paper.

## 2. Preliminaries

Let be a Banach space. If is a linear operator on the Banach space , then, , , , , and stand, respectively, for its domain, resolvent, spectrum, null-space or kernel, and range. If is a linear operator, one sets for all .

If are Banach spaces, then the space denotes the collection of all bounded linear operators from into equipped with its natural topology. This is simply denoted by when . If is a projection, we set .

### 2.1. Evolution Families

This section is devoted to the basic material on evolution equations as well the dichotomy tools. We follow the same setting as in the studies of Diagana [1].

Assumption (H.1) given below will be crucial throughout the paper.

for , .

on associated with such that for all with , and

- (a)
for such that ;

- (b)
for where is the identity operator of ;

- (c)
is continuous for ;

- (d)
- (e)
for and with .

It should also be mentioned that the above-mentioned proprieties were mainly established in [16, Theorem 2.3] and [17, Theorem 2.1]; see also [15, 18]. In that case we say that generates the evolution family . For some nice works on evolution equations, which make use of evolution families, we refer the reader to, for example, [19–29].

Definition 2.1.

One says that an evolution family
has an *exponential dichotomy* (or is *hyperbolic*) if there are projections
that are uniformly bounded and strongly continuous in
and constants
and
such that

- (f)
;

- (g)
the restriction of is invertible (we then set );

- (h)
and for and .

are called Green function corresponding to and .

This setting requires some estimates related to . For that, we introduce the interpolation spaces for . We refer the reader to the following excellent books [30–32] for proofs and further information on theses interpolation spaces.

for all , where the fractional powers are defined in the usual way.

for .

for and , with the corresponding norms.

We have the following fundamental estimates for the evolution family .

Proposition 2.2 (see [33]).

Suppose that the evolution family has exponential dichotomy. For , , and , the following hold.

In addition to above, we also assume that the next assumption holds.

(H.2)The domain is constant in . Moreover, the evolution family generated by has an exponential dichotomy with constants and dichotomy projections for .

### 2.2. Pseudo-Almost Automorphic Functions

Let denote the collection of all -valued bounded continuous functions. The space equipped with its natural norm, that is, the sup norm is a Banach space. Furthermore, denotes the class of continuous functions from into .

Definition 2.3.

for each .

If the convergence above is uniform in , then is almost periodic in the classical Bochner's sense. Denote by the collection of all almost automorphic functions . Note that equipped with the sup-norm turns out to be a Banach space.

Among other things, almost automorphic functions satisfy the following properties.

Theorem 2.4 (see [34]).

If , then

- (i)
,

- (ii)
for any scalar ,

- (iii)
, where is defined by ,

- (iv)
the range is relatively compact in , thus is bounded in norm,

- (v)
if uniformly on , where each , then too.

Let be another Banach space.

Definition 2.5.

for all and .

The collection of such functions will be denoted by .

For more on almost automorphic functions and related issues, we refer the reader to, for example, [1, 4, 9, 13, 34–39].

uniformly in , where is any bounded subset.

Definition 2.6 (see Liang et al. [5, 6]).

A function
is called pseudo-almost automorphic if it can be expressed as
, where
and
. The collection of such functions will be denoted by
*.*

The functions
and
appearing in Definition 2.6 are, respectively, called the *almost automorphic* and the *ergodic perturbation* components of
.

Definition 2.7.

A bounded continuous function
belongs to
whenever it can be expressed as
, where
and
. The collection of such functions will be denoted by
*.*

An important result is the next theorem, which is due to Xiao et al. [6].

Theorem 2.8 (see [6]).

The space equipped with the sup norm is a Banach space.

The next composition result, that is Theorem 2.9, is a consequence of [12, Theorem 2.4].

Theorem 2.9.

for all and .

Then the function defined by belongs to provided .

We also have the following.

Theorem 2.10 (see [6]).

If belongs to and if is uniformly continuous on any bounded subset of for each , then the function defined by belongs to provided that .

## 3. Main Results

Throughout the rest of the paper we fix , real numbers, satisfying with .

is compact, and that the following additional assumptions hold:

whenever for all , where with .

(H.4)

- (a)

- (b)

for all and .

To study the existence and uniqueness of pseudo-almost automorphic solutions to (1.2) we first introduce the notion of a mild solution, which has been adapted to the one given in the studies of Diagana et al. [35, Definition 3.1].

Definition 3.1.

for and for all .

for each .

The main result of the present paper will be based upon the use of the well-known fixed point theorem of Krasnoselskii given as follows.

Theorem 3.2.

Let be a closed bounded convex subset of a Banach space . Suppose the (possibly nonlinear) operators and map into satisfying

- (1)
for all , then ;

- (2)
the operator is a contraction;

- (3)
the operator is continuous and is contained in a compact set.

Then there exists such that .

We need the following new technical lemma.

Lemma 3.3.

Proof.

Let . First of all, note that for all , such that and .

for all with .

for by using (2.12).

A straightforward consequence of Lemma 3.3 is the following.

Corollary 3.4.

Proof.

Equation (3.19) has already been proved (see the proof of (3.12)).

Lemma 3.5.

Under assumptions (H.1), (H.2), (H.3), and (H.4), the mapping is well defined and continuous.

Proof.

where .

It remains to prove that is continuous. For that consider an arbitrary sequence of functions which converges uniformly to some , that is, .

and hence as .

The proof for is similar to that of and hence omitted. For , one makes use of (2.12) rather than (2.11).

Lemma 3.6.

Under assumptions (H.1), (H.2), (H.3), and (H.4), the integral operator defined above maps into itself.

Proof.

for each .

Set and for all .

for each .

One completes the proof by using the well-known Lebesgue dominated convergence theorem and the fact as for each .

The proof for is similar to that of and hence omitted. For , one makes use of (2.12) rather than (2.11).

Clearly, the space equipped with the norm is a Banach space, which is the Banach space of all bounded continuous Hölder functions from to whose Hölder exponent is .

Lemma 3.7.

Under assumptions (H.1), (H.2), (H.3), (H.4), and (H.5), maps bounded sets of into bounded sets of for some , where , are the integral operators introduced previously.

Proof.

where is a positive constant.

where is a positive constant.

for all , where depends on .

The proof of the next lemma follows along the same lines as that of Lemma 3.6 and hence omitted.

Lemma 3.8.

The integral operator maps bounded sets of into bounded sets of .

Similarly, the next lemma is a consequence of [2, Proposition 3.3].

Lemma 3.9.

is compact, too.

Theorem 3.10.

Suppose that assumptions (H.1), (H.2), (H.3), (H.4), and (H.5) hold, then the operator defined by is compact.

Proof.

where .

In view of the above, it follows that is continuous and compact, where is the ball in of radius with .

for all .

Lemma 3.11.

Under assumptions (H.1), (H.2), (H.3), (H.4), and (H.5), the integral operators and defined above map into itself.

Proof.

for each .

Set and for all .

as for every . One completes the proof by using the Lebesgue's dominated convergence theorem.

The proof for is similar to that of except that one makes use of (3.19) instead of (3.18).

Theorem 3.12.

Under assumptions (H.1), (H.2), (H.3), (H.4), and (H.5) and if is small enough, then (1.2) has at least one pseudo-almost automorphic solution.

Proof.

We have seen in the proof of Theorem 3.10 that is continuous and compact, where is the ball in of radius with .

for all .

for all and hence .

and hence is a strict contraction whenever is small enough.

Using the Krasnoselskii fixed point theorem (Theorem 3.2) it follows that there exists at least one pseudo-almost automorphic mild solution to (1.2).

## Declarations

### Acknowledgment

The author would like to express his thanks to the referees for careful reading of the paper and insightful comments.

## Authors’ Affiliations

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