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

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

Let be a Banach space. In the recent paper by Diagana [1], the existence of almost automorphic mild solutions to the nonautonomous abstract differential equations

(1.1)

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.

In this paper we study the existence of pseudo-almost automorphic mild solutions to the nonautonomous partial evolution equations

(1.2)

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.

(H.1)The family of closed linear operators for on with domain (possibly not densely defined) satisfy the so-called Acquistapace-Terreni conditions, that is, there exist constants , , , and with such that

(2.1)

for , .

It should mentioned that (H.1) was introduced in the literature by Acquistapace et al. in [14, 15] for . Among other things, it ensures that there exists a unique evolution family

(2.2)

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) , and

  

  (2.3)

  for , ;

• (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 .

Under Acquistpace-Terreni conditions, the family of operators defined by

(2.4)

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.

Let be a sectorial operator on (for that, in assumption (H.1), replace with ) and let . Define the real interpolation space

(2.5)

which, by the way, is a Banach space when endowed with the norm . For convenience we further write

(2.6)

Moreover, let of . In particular, we have the following continuous embedding:

(2.7)

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

In general, is not dense in the spaces and . However, we have the following continuous injection:

(2.8)

for .

Given the family of linear operators for , satisfying (H.1), we set

(2.9)

for and , with the corresponding norms.

Now the embedding in (2.7) holds with constants independent of . These interpolation spaces are of class [32, Definition 1.1.1], and hence there is a constant such that

(2.10)

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.

• (i) There is a constant , such that

  

  (2.11)

• (ii) There is a constant , such that

  

  (2.12)

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.

A function is said to be almost automorphic if, for every sequence of real numbers , there exists a subsequence such that

(2.13)

is well defined for each , and

(2.14)

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.

A jointly continuous function is said to be almost automorphic in if is almost automorphic for all ( being any bounded subset). Equivalently, for every sequence of real numbers , there exists a subsequence such that

(2.15)

is well defined in and for each , and

(2.16)

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].

Define

(2.17)

Similarly, will denote the collection of all bounded continuous functions such that

(2.18)

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.

Suppose that belongs to , with being uniformly continuous on any bounded subset of uniformly in . Furthermore, one supposes that there exists such that

(2.19)

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 .

To study the existence of pseudo-almost automorphic solutions to (1.2), in addition to the previous assumptions, we suppose that the injection

(3.1)

is compact, and that the following additional assumptions hold:

(H.3). Moreover, for any sequence of real numbers there exist a subsequence and a well-defined function such that for each , one can find such that

(3.2)

whenever for , and

(3.3)

whenever for all , where with .

(H.4)  

• (a) The function is pseudo-almost automorphic in the first variable uniformly in the second one. The function is uniformly continuous on any bounded subset of for each . Finally,

  

  (3.4)

where and is a continuous, monotone increasing function satisfying

(3.5)

• (b) The function is pseudo-almost automorphic in the first variable uniformly in the second one. Moreover, is globally Lipschitz in the following sense: there exists for which

  

  (3.6)

for all and .

(H.5)The operator is invertible for each , that is, for each . Moreover, there exists such that

(3.7)

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.

A continuous function is said to be a mild solution to (1.2) provided that the function is integrable on , the function is integrable on and

(3.8)

for and for all .

Under assumptions (H.1), (H.2), and (H.5), it can be readily shown that (1.2) has a mild solution given by

(3.9)

for each .

We denote by and the nonlinear integral operators defined by

(3.10)

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.

For each , suppose that assumptions (H.1), (H.2) hold, and let be real numbers such that with . Then there are two constants such that

(3.11)

(3.12)

Proof.

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

Letting and using (H.2) and the above-mentioned approximate, we obtain

(3.13)

Now since as , it follows that there exists such that

(3.14)

Now, let . Using (2.11) and the fact , we obtain

(3.15)

In summary, there exists such that

(3.16)

for all with .

Let . Since the restriction of to is a bounded linear operator it follows that

(3.17)

for by using (2.12).

A straightforward consequence of Lemma 3.3 is the following.

Corollary 3.4.

For each , suppose that assumptions (H.1), (H.2), and (H.5) hold, and let be real numbers such that with . Then there are two constants such that

(3.18)

(3.19)

Proof.

We make use of (H.5) and Lemma 3.3. Indeed, for each ,

(3.20)

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.

We first show that . For that, let and be the integral operators defined, respectively, by

(3.21)

Now, using (2.11) it follows that for all ,

(3.22)

and hence

(3.23)

where .

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

Now

(3.24)

Now, using the continuity of and the Lebesgue Dominated Convergence Theorem we conclude that

(3.25)

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.

Let . Setting and using Theorem 2.10 it follows that . Let , where and . Let us show that . Indeed, since , for every sequence of real numbers there exists a subsequence such that

(3.26)

is well defined for each and

(3.27)

for each .

Set and for all .

Now

(3.28)

Using (2.11) and the Lebesgue Dominated Convergence Theorem, one can easily see that

(3.29)

Similarly, using (H.3) and [40] it follows that

(3.30)

Therefore,

(3.31)

Using similar ideas as the previous ones, one can easily see that

(3.32)

Again using (2.11) it follows that

(3.33)

Set

(3.34)

Since is translation invariant it follows that belongs to for each , and hence

(3.35)

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).

Let , and let , where

(3.36)

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.

Let , and let for each . Then we have

(3.37)

and hence

(3.38)

Similarly,

(3.39)

and hence

(3.40)

Let . Clearly,

(3.41)

Clearly,

(3.42)

Similarly,

(3.43)

Now

(3.44)

where is a positive constant.

Consequently, letting it follows that

(3.45)

where is a positive constant.

Therefore, for each such that

(3.46)

for all , then belongs to with

(3.47)

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.

The set is compactly contained in , that is, the canonical injection is compact, which yields

(3.48)

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.

The proof follows along the same lines as that of [2, Proposition 3.4]. Recalling that in view of Lemma 3.7, we have

(3.49)

for all , with , where are positive constants. Consequently, and yield and

(3.50)

where .

Therefore, there exists such that for all , the following hold:

(3.51)

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

Define

(3.52)

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.

Let . Again, using the composition of pseudo-almost automorphic functions (Theorem 2.10) it follows that is in whenever . In particular,

(3.53)

Now write , where and , that is, where

(3.54)

Clearly, . Indeed, since , for every sequence of real numbers there exists a subsequence such that

(3.55)

is well defined for each and

(3.56)

for each .

Set and for all .

Now

(3.57)

Using (3.18) and the Lebesgue Dominated Convergence Theorem, one can easily see that

(3.58)

Similarly, using (H.3) it follows that

(3.59)

Therefore,

(3.60)

Using similar ideas as the previous ones, one can easily see that

(3.61)

Now, let . Again from (3.18), we have

(3.62)

Now

(3.63)

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 .

Now, if we set it follows that

(3.64)

for all .

Choose such that

(3.65)

and let be the closed ball in of radius . It is then clear that

(3.66)

for all and hence .

To complete the proof we have to show that is a strict contraction. Indeed, for all

(3.67)

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).