# 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.

## 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.

on associated with such that for all with , and

- (a)
- (b)
- (c)
- (d)
- (e)

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

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 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.

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]).

- (i)
- (ii)
- (iii)
- (iv)
- (v)

Definition 2.5.

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.

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)

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.

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)
- (2)
- (3)

We need the following new technical lemma.

Lemma 3.3.

Proof.

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

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.

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

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.

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.

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.

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

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.

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 .

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

## References

- Diagana T: Almost automorphic mild solutions to some classes of nonautonomous higher-order differential equations. Semigroup Forum. In pressGoogle Scholar
- Goldstein JA, N'Guérékata GM:
**Almost automorphic solutions of semilinear evolution equations.***Proceedings of the American Mathematical Society*2005,**133**(8):2401-2408. 10.1090/S0002-9939-05-07790-7MathSciNetView ArticleMATHGoogle Scholar - Bochner S:
**Continuous mappings of almost automorphic and almost periodic functions.***Proceedings of the National Academy of Sciences of the United States of America*1964,**52:**907-910. 10.1073/pnas.52.4.907MathSciNetView ArticleMATHGoogle Scholar - Diagana T:
*Pseudo Almost Periodic Functions in Banach Spaces*. Nova Science, New York, NY, USA; 2007:xiv+132.MATHGoogle Scholar - Liang J, Zhang J, Xiao T-J:
**Composition of pseudo almost automorphic and asymptotically almost automorphic functions.***Journal of Mathematical Analysis and Applications*2008,**340**(2):1493-1499. 10.1016/j.jmaa.2007.09.065MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J, Zhang J:
**Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces.***Semigroup Forum*2008,**76**(3):518-524. 10.1007/s00233-007-9011-yMathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Zhu X-X, Liang J:
**Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(11):4079-4085. 10.1016/j.na.2008.08.018MathSciNetView ArticleMATHGoogle Scholar - Cieutat P, Ezzinbi K:
**Existence, uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces.***Journal of Mathematical Analysis and Applications*2009,**354**(2):494-506. 10.1016/j.jmaa.2009.01.016MathSciNetView ArticleMATHGoogle Scholar - Diagana T:
**Existence of pseudo-almost automorphic solutions to some abstract differential equations with***-pseudo-almost automorphic coefficients.**Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(11):3781-3790. 10.1016/j.na.2008.07.034MathSciNetView ArticleMATHGoogle Scholar - Ezzinbi K, Fatajou S, N'guérékata GM:
**Pseudo-almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(4):1641-1647. 10.1016/j.na.2008.02.039MathSciNetView ArticleMATHGoogle Scholar - Ezzinbi K, Fatajou S, N'Guérékata GM:
**Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces.***Journal of Mathematical Analysis and Applications*2009,**351**(2):765-772. 10.1016/j.jmaa.2008.11.017MathSciNetView ArticleMATHGoogle Scholar - Liang J, N'Guérékata GM, Xiao T-J, Zhang J:
**Some properties of pseudo-almost automorphic functions and applications to abstract differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(7):2731-2735. 10.1016/j.na.2008.03.061MathSciNetView ArticleMATHGoogle Scholar - Diagana T:
**Pseudo-almost automorphic solutions to some classes of nonautonomous partial evolution equations.***Differential Equations & Applications*2009,**1**(4):561-582.MathSciNetView ArticleMATHGoogle Scholar - Acquistapace P, Flandoli F, Terreni B:
**Initial-boundary value problems and optimal control for nonautonomous parabolic systems.***SIAM Journal on Control and Optimization*1991,**29**(1):89-118. 10.1137/0329005MathSciNetView ArticleMATHGoogle Scholar - Acquistapace P, Terreni B:
**A unified approach to abstract linear nonautonomous parabolic equations.***Rendiconti del Seminario Matematico della Università di Padova*1987,**78:**47-107.MathSciNetMATHGoogle Scholar - Acquistapace P:
**Evolution operators and strong solutions of abstract linear parabolic equations.***Differential and Integral Equations*1988,**1**(4):433-457.MathSciNetMATHGoogle Scholar - Yagi A:
**Abstract quasilinear evolution equations of parabolic type in Banach spaces.***Bollettino dell'Unione Matematica Italiana B*1991,**5**(2):341-368.MathSciNetMATHGoogle Scholar - Yagi A:
**Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups. II.***Funkcialaj Ekvacioj*1990,**33**(1):139-150.MathSciNetMATHGoogle Scholar - Arendt W, Chill R, Fornaro S, Poupaud C:
**-maximal regularity for non-autonomous evolution equations.***Journal of Differential Equations*2007,**237**(1):1-26. 10.1016/j.jde.2007.02.010MathSciNetView ArticleMATHGoogle Scholar - Arendt W, Batty CJK:
**Almost periodic solutions of first- and second-order Cauchy problems.***Journal of Differential Equations*1997,**137**(2):363-383. 10.1006/jdeq.1997.3266MathSciNetView ArticleMATHGoogle Scholar - Chicone C, Latushkin Y:
*Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs*.*Volume 70*. American Mathematical Society, Providence, RI, USA; 1999:x+361.View ArticleMATHGoogle Scholar - Ding H-S, Liang J, N'Guérékata GM, Xiao T-J:
**Pseudo-almost periodicity of some nonautonomous evolution equations with delay.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(5):1412-1418. 10.1016/j.na.2006.07.026MathSciNetView ArticleMATHGoogle Scholar - Liang J, Nagel R, Xiao T-J:
**Nonautonomous heat equations with generalized Wentzell boundary conditions.***Journal of Evolution Equations*2003,**3**(2):321-331.MathSciNetMATHGoogle Scholar - Liang J, Xiao T-J:
**Solutions to nonautonomous abstract functional equations with infinite delay.***Taiwanese Journal of Mathematics*2006,**10**(1):163-172.MathSciNetMATHGoogle Scholar - Maniar L, Schnaubelt R:
**Almost periodicity of inhomogeneous parabolic evolution equations.**In*Evolution Equations, Search ResultsLecture Notes in Pure and Applied Mathematics*.*Volume 234*. Dekker, New York, NY, USA; 2003:299-318.Google Scholar - Schnaubelt R:
**Asymptotically autonomous parabolic evolution equations.***Journal of Evolution Equations*2001,**1**(1):19-37. 10.1007/PL00001363MathSciNetView ArticleMATHGoogle Scholar - Schnaubelt R:
**Asymptotic behaviour of parabolic nonautonomous evolution equations.**In*Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics*.*Volume 1855*. Springer, Berlin, Germany; 2004:401-472.View ArticleGoogle Scholar - Xiao T-J, Liang J:
**Existence of classical solutions to nonautonomous nonlocal parabolic problems.***Nonlinear Analysis, Theory, Methods and Applications*2005,**63**(5–7):e225-e232.View ArticleMATHGoogle Scholar - Xiao T-J, Liang J, van Casteren J:
**Time dependent Desch-Schappacher type perturbations of Volterra integral equations.***Integral Equations and Operator Theory*2002,**44**(4):494-506. 10.1007/BF01193674MathSciNetView ArticleMATHGoogle Scholar - Amann H:
*Linear and Quasilinear Parabolic Problems, Monographs in Mathematics*.*Volume 89*. Birkhäuser, Boston, Mass, USA; 1995:xxxvi+335.View ArticleGoogle Scholar - Engel K-J, Nagel R:
*One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics*.*Volume 194*. Springer, New York, NY, USA; 2000:xxii+586.MATHGoogle Scholar - Lunardi A:
*Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications*.*Volume 16*. Birkhäuser, Basel, Switzerland; 1995:xviii+424.MATHGoogle Scholar - Baroun M, Boulite S, Diagana T, Maniar L:
**Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations.***Journal of Mathematical Analysis and Applications*2009,**349**(1):74-84. 10.1016/j.jmaa.2008.08.034MathSciNetView ArticleMATHGoogle Scholar - N'Guerekata GM:
*Almost Automorphic and Almost Periodic Functions in Abstract Spaces*. Kluwer Academic/Plenum Publishers, New York, NY, USA; 2001:x+138.View ArticleMATHGoogle Scholar - Diagana T, Hernández E, Rabello M:
**Pseudo almost periodic solutions to some non-autonomous neutral functional differential equations with unbounded delay.***Mathematical and Computer Modelling*2007,**45**(9-10):1241-1252. 10.1016/j.mcm.2006.10.006MathSciNetView ArticleMATHGoogle Scholar - Diagana T:
**Stepanov-like pseudo almost periodic functions and their applications to differential equations.***Communications in Mathematical Analysis*2007,**3**(1):9-18.MathSciNetMATHGoogle Scholar - Diagana T:
**Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(12):4277-4285. 10.1016/j.na.2007.10.051MathSciNetView ArticleMATHGoogle Scholar - Diagana T: Existence of almost automorphic solutions to some classes of nonautonomous higher-order differential equations. Electronic Journal of Qualitative Theory of Differential Equations 2010, (22):1-26.Google Scholar
- Diagana T:
**Existence of pseudo-almost automorphic solutions to some abstract differential equations with****-pseudo-almost automorphic coefficients.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(11):3781-3790. 10.1016/j.na.2008.07.034MathSciNetView ArticleMATHGoogle Scholar - Baroun M, Boulite S, N'Guérékata GM, Maniar L:
**Almost automorphy of semilinear parabolic evolution equations.***Electronic Journal of Differential Equations*2008,**2008**(60):1-9.MathSciNetMATHGoogle Scholar

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