Advances in Difference Equations

, 2010:310951

# Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-Differential Equation

• José Paulo C. dos Santos
• Sandro M. Guzzo
• Marcos N. Rabelo
Open Access
Research Article
Part of the following topical collections:
1. Abstract Differential and Difference Equations

## Abstract

The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay is studied.

## Keywords

Periodic Solution Heat Flux Dense Subset Mild Solution Resolvent Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 1. Introduction

In this paper, we study the existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations modelled in the form

where and , , are closed linear operators; is a Banach space; the history , , belongs to some abstract phase space defined axiomatically are appropriated functions.

The study of abstract neutral equations is motivated by different practical applications in different technical fields. The literature related to ordinary neutral functional differential equations is very extensive and we refer the reader to Chukwu [1], Hale and Lunel [2], Wu [3], and the references therein. As a practical application, we note that the equation

arises in the study of the dynamics of income, employment, value of capital stock, and cumulative balance of payment; see [1] for details. In the above system, is a real number, the state , , are continuous functions matrices, is a constant matrix, represents the government intervention, and the private initiative. We note that by assuming the solution is known on , we can transform this system into an abstract system with unbounded delay described as (1.1).

Abstract partial neutral differential equations also appear in the theory of heat conduction. In the classic theory of heat conduction, it is assumed that the internal energy and the heat flux depend linearly on the temperature and on its gradient . Under these conditions, the classic heat equation describes sufficiently well the evolution of the temperature in different types of materials. However, this description is not satisfactory in materials with fading memory. In the theory developed in [4, 5], the internal energy and the heat flux are described as functionals of and . The next system, see for instance [6, 7, 8, 9], has been frequently used to describe this phenomenon,

In this system, is open, bounded, and with smooth boundary; ; represents the temperature in at the time ; is a physical constant , are the internal energy and the heat flux relaxation, respectively. By assuming that the solution is known on and , we can transform this system into an abstract system with unbounded delay described in the form (1.1).

Recent contributions on the existence of solutions with some of the previously enumerated properties or another type of almost periodicity to neutral functional differential equations have been made in [10, 11], for the case of neutral ordinary differential equations, and in [12, 13, 14, 15] for partial functional differential systems.

The purpose of this work is to study the existence of asymptotically almost periodic mild solutions for the neutral system (1.1). To this end, we study the existence and qualitative properties of an exponentially stable resolvent operator for the integro-differential system

There exists an extensive literature related to the existence and qualitative properties of resolvent operator for integro-differential equations. We refer the reader to the book by Gripenberg et al. [16] which contains an overview of the theory for the case where the underlying space has finite dimension. For abstract integro-differential equations described on infinite dimensional spaces, we cite the Prüss book [17] and the papers [18, 19, 20], Da Prato et al. [21, 22], and Lunardi [9, 23]. To finish this short description of the related literature, we cite the papers [24, 25, 26] where some of the above topics for the case of abstract neutral integro-differential equations with unbounded delay are treated.

To the best of our knowledge, the study of the existence of asymptotically almost periodic solutions of neutral integro-differential equations with unbounded delay described in the abstract form (1.1) is an untreated topic in the literature and this is the main motivation of this article.

To finish this section, we emphasize some notations used in this paper. Let and be Banach spaces. We denote by the space of bounded linear operators from into endowed with norm of operators, and we write simply when . By , we denote the range of a map , and for a closed linear operator , the notation represents the domain of endowed with the graph norm, , . In the case , the notation stands for the resolvent set of and is the resolvent operator of . Furthermore, for appropriate functions and , the notation denotes the Laplace transform of and the convolution between and , which is defined by . The notation stands for the closed ball with center at and radius in . As usual, represents the subspace of formed by the functions which vanish at infinity.

## 2. Preliminaries

In this work, we will employ an axiomatic definition of the phase space similar to that in [27]. More precisely, will denote a vector space of functions defined from into endowed with a seminorm denoted by and such that the following axioms hold.
1. (A)
If with is continuous on and , then for each the following conditions hold:
1. (i)

2. (ii)

3. (iii)

where is a constant, and are functions such that and are respectively continuous and locally bounded, and are independent of .

(A1) If is a function as in , then is a -valued continuous function on .
1. (B)

The space is complete.

(C2) If is a sequence in formed by functions with compact support such that uniformly on compact, then and as

Remark 2.1.

In the remainder of this paper, is such that

for every continuous and bounded; see [27, Proposition ] for details.

Definition 2.2.

Let be the -semigroup defined by on and on . The phase space is called a fading memory if as for each with .

Remark 2.3.

In this work, we suppose that there exists a positive such that

for each . Observe that this condition is verified, for example, if is a fading memory, see [27, Proposition ].

Example 2.4 (The phase space ).

Let , and let be a nonnegative measurable function which satisfies the conditions (g-5) and (g-6) in the terminology of [27]. Briefly, this means that is locally integrable, and there exists a nonnegative, locally bounded function on such that for all and , where is a set with Lebesgue measure zero. The space consists of all classes of functions such that is continuous on , Lebesgue-measurable, and is Lebesgue integrable on . The seminorm in is defined by

The space satisfies axioms (A), (A-1), and (B).  Moreover, when and , we can take , , and , for ; see [27, Theorem ] for details.

Now, we need to introduce some concepts, definitions, and technicalities on almost periodical functions.

Definition 2.5.

A function is almost periodic (a.p.) if for every , there exists a relatively dense subset of , denoted by , such that

Definition 2.6.

A function is asymptotically almost periodic (a.a.p.) if there exists an almost periodic function and such that .

The next lemmas are useful characterizations of a.p and a.a.p functions.

Lemma 2.7 (see [28, Theorem ]).

A function is asymptotically almost periodic if and only if for every there exist and a relatively dense subset of, denoted by , such that

In this paper, and are the spaces

endowed with the norms and respectively. We know from the result in [28] that and are Banach spaces.

Next, and are abstract Banach spaces.

Definition 2.8.

Let be an open subset of
1. (a)

A continuous function (resp., ) is called pointwise almost periodic (p.a.p.), (resp., pointwise asymptotically almost periodic (p.a.a.p.) if (resp., ) for every .

2. (b)
A continuous function is called uniformly almost periodic (u.a.p.), if for every and every compact there exists a relatively dense subset of , denoted by , such that

1. (c)
A continuous function is called uniformly asymptotically almost periodic (u.a.a.p.), if for every and every compact there exists a relatively dense subset of , denoted by , and such that

The next lemma summarizes some properties which are fundamental to obtain our results.

Lemma 2.9 (see [29, Theorem ]).

Let be an open set. Then the following properties hold.
1. (a)

If is p.a.p. and satisfies a local Lipschitz condition at , uniformly at , then is u.a.p.

2. (b)

If is p.a.a.p. and satisfies a local Lipschitz condition at , uniformly at , then is u.a.a.p.

3. (c)

If then. Moreover, if is a fading memory space and is such that and , then.

4. (d)

If is u.a.p. and is such that , then.

5. (e)

If is u.a.a.p and is such that , then.

## 3. Resolvent Operators

In this section, we study the existence and qualitative properties of an exponentially resolvent operator for the integro-differential abstract Cauchy problem

The results obtained for the resolvent operator in this section are similar to those that can be found, for instance, in the papers [21, 23, 30]. In this paper, we prove the necessary estimates for the proof of an existence theorem of asymptotically almost periodic solutions for (1.1). For the better comprehension of the subject, we will introduce the following definitions, hypothesis, and results.

We introduce the following concept of resolvent operator for integro-differential problem (3.1).

Definition 3.1.

A one-parameter family of bounded linear operators on is called a resolvent operator of (3.1) if the following conditions are verified.
1. (a)

Function is strongly continuous and for all .

2. (b)

for every ,
1. (c)

There exist constants such that for every

Definition 3.2.

A resolvent operator of (3.1) is called exponentially stable if there exist positive constants such that

In this work, we always assume that the following conditions are verified.

(H1) The operator is the infinitesimal generator of an analytic semigroup on , and there are constants , and such that and for all .

(H2) For all is a closed linear operator, , and is strongly measurable on for each . There exists such that exists for and for all and . Moreover, the operator valued function has an analytical extension (still denoted by ) to such that for all , and as .

(H3) There exist a subspace dense in and positive constants , such that , , and for every and all .

In the sequel, for, , and , set

and for, , the paths

with are oriented counterclockwise. In addition, is the set

We next study some preliminary properties needed to establish the existence of a resolvent operator for the problem (3.1).

Lemma 3.3.

There exists such that and the function is analytic. Moreover,
and there exist constants for such that
(3.10)

for every .

Proof.

Since
(3.11)
fixed there exists a positive number such that for . Consequently, the operator has a continuous inverse with . Moreover, for , we have
(3.12)
and for ,
(3.13)
which shows (3.7) and that . Now, from (3.7) we obtain and
(3.14)
Consequently,
(3.15)
the functions are analytic, and estimates (3.8), and (3.10) are valid. In addition, for , we can write
(3.16)

for sufficiently large. This proves (3.9) and completes the proof.

Observation 1.

If is a resolvent operator for (3.1), it follows from (3.3) that for all . Applying Lemma 3.3 and the properties of the Laplace transform, we conclude that is the unique resolvent operator for (3.1).

In the remainder of this section, and are numbers such that and . Moreover, we denote by a generic constant that represents any of the constants involved in the statements of Lemma 3.3 as well as any other constant that arises in the estimate that follows. We now define the operator family by

(3.17)

We will next establish that is a resolvent operator for (3.1).

Lemma 3.4.

The function is exponentially bounded in .

Proof.

If , from (3.17) and estimate (3.8), we get
(3.18)
On the other hand, using that is analytic on , for , we obtain
(3.19)

This complete the proof.

Lemma 3.5.

The operator function is exponentially bounded in .

Proof.

It follows from (3.9) that the integral in
(3.20)
is absolutely convergent in and defines a linear operator . Using that is closed, we can affirm that . From Lemma 3.3, is analytic and . If and , we have
(3.21)
For and , we get
(3.22)

From before and Lemma 3.4, we infer that is exponentially bounded in . The proof is finished.

Lemma 3.6.

The function is strongly continuous.

Proof.

It is clear from (3.17) that is continuous at for every . We next establish the continuity at . Let and be sufficiently large, using that
(3.23)

where represent the curve for .

For and , we get

(3.24)
Furthermore, it follows from (3.8), and assumption (H2) that
(3.25)
where is integrable for . From the Lebesgue dominated convergence theorem, we infer that
(3.26)
Let now be the curve for . Turning to apply Cauchy's theorem combining with the estimate
(3.27)
we obtain
(3.28)

we can affirm that for all , which completes and the proof since is dense in and is bounded on .

Notice that the sectors from Lemma 3.3, is analytic. Consider the contours

(3.29)
and oriented counterclockwise. By Cauchy theorem for , we obtain
(3.30)
The following estimate:
(3.31)
is the one responsible for the fact that the integral tends to as tend to in a similar way the integral tend to as tend to so that
(3.32)
For , we obtain
(3.33)

and proceeding as before, we obtain for all which ends the proof.

The following result can be proved with an argument similar to that used in the proof of the preceding lemma with changing by

Lemma 3.7.

The function is strongly continuous.

We next set .

Lemma 3.8.

The function has an analytic extension to , and
(3.34)

Proof.

For and , we can write where , and If , from (3.8) and (3.17), we obtain
(3.35)
Using that is analytic on , for , , we get
(3.36)

This property allows us to define the extension by this integral.

Similarly, the integral on the right hand side of (3.34) is also absolutely convergent in and strong, continuous on for . For ,

(3.37)

where is integrable for . From the Lebesgue dominated convergence theorem, we obtain that verifies (3.34). The proof is ended.

Lemma 3.9.

Proof.

Using that is analytic on and that the integrals involved in the calculus are absolutely convergent, we have
(3.38)

Theorem 3.10.

The function is a resolvent operator for the system (3.1).

Proof.

Let . From Lemma 3.9, for ,
(3.39)
which implies
(3.40)
Applying [31, Proposition , Corollary ], we get
(3.41)
which in turn implies that
(3.42)

Arguing as above but using the equality we obtain that (3.2) holds.

On the other hand, by Lemma 3.8 we infer that . Next, we analyze the differentiability on . Let and for all we can choose such that

(3.43)
For and , there exists such that
(3.44)
Consequently, for we have that
(3.45)

which proves the existence of the right derivative of at zero and that This proves that resolvent equation (3.3) is valid for every and for every . This completes the proof.

Corollary 3.11.

If then the function is an exponentially stable resolvent operator for the system (3.1).

In the next result, we denote by the fractional power of the operator (see [32] for details).

Theorem 3.12.

Suppose that the conditions are satisfied. Then there exists a positive number such that
(3.46)

for all

Proof.

Let From [32, Theorem ], there exists such that
(3.47)
Since is a valued function, for all
(3.48)
where is independent of . From (3.48), we get for
(3.49)
On the other hand, using that is analytic on , for , we get
(3.50)
From the previous facts, we conclude that
(3.51)

which ends the proof.

Corollary 3.13.

If and , then there exists such that
(3.52)

In the remainder of this section, we discuss the existence and regularity of solutions of

(3.53)
(3.54)

where . In the sequel, is the operator function defined by (3.17). We begin by introducing the following concept of classical solution.

Definition 3.14.

A function , , is called a classical solution of (3.53)-(3.54) on if , the condition (3.54) holds and (3.53) is verified on .

The next result has been established in [30].

Theorem 3.15 ([30, Theorem ]).

Let . Assume that and is a classical solution of (3.53)-(3.54) on . Then
(3.55)

An immediate consequence of the above theorem is the uniqueness of classical solutions.

Corollary 3.16.

If are classical solutions of (3.53)-(3.54) on , then on .

Motivated by (3.55), we introduce the following concept.

Definition 3.17.

A function is called a mild solution of (3.53)-(3.54) if
(3.56)

## 4. Existence Result of Asymptotically Almost Periodic Solutions

In this section, we study the existence of asymptotically almost periodic mild solutions for the abstract integro-differential system (1.1). To establish our existence result, motivated by the previous section we introduce the following assumptions.

(P1) There exists a Banach space continuously included in such that the following conditions are verified.

1. (a)

For every , and . In addition, for every .

2. (b)
There are positive constants such that

1. (c)

There exists such that

(P2) The continuous function is p.a.a.p, and there exists a continuous function such that

(P3) The continuous function is p.a.a.p, and there exists a continuous function such that

Motivated by the theory of resolvent operator, we introduce the following concept of mild solution for (1.1).

Definition 4.1.

A function is called a mild solution of (1.1) on , if the functions andare integrable on for every and

Lemma 4.2.

Let condition hold and let be a function in . If is the function defined by then .

Proof.

Let . Let , be as in Lemma 2.7 and such that . For and , we get
which implies that

Now, from inequality (4.6) and Lemma 2.7, we conclude that is a.a.p. The proof is complete.

Lemma 4.3.

Assume that the condition is fulfilled. Let and let be the function defined by

Proof.

Let , be as in Lemma 2.7 and such that
for where . For and , we get
We obtain
(4.10)
which implies that
(4.11)

From inequality (4.11) and Lemma 2.7, we conclude that is a.a.p., which ends the proof.

Now, we can establish our existence result.

Theorem 4.4.

Assume that is a fading memory space and , and are held. If and for every , then there exists such that for each , there exists a mild solution, , of (1.1) on such that and .

Proof.

Let and be such that
(4.12)
where is the constant introduced in Remark 2.3. We affirm that the assertion holds for Let On the space
(4.13)
endowed with the metric , we define the operator by
(4.14)

where is the function defined by the relation and on . From the hypothesis , and , we obtain that is well defined and that Moreover, from Lemmas 4.2 and 4.3 it follows that .

Next, we prove that is a contraction from into . If and , we get

(4.15)
where the inequality has been used and represent the continuous inclusion of on . Thus, . On the other hand, for we see that
(4.16)

which shows that is a contraction from into The assertion is now a consequence of the contraction mapping principle. The proof is complete.

## 5. Applications

In this section, we study the existence of asymptotically almost periodic solutions of the partial neutral integro-differential system

for , , and Moreover, we have identified .

To represent this system in the abstract form (1.1), we choose the spaces and ; see Example 2.4 for details. We also consider the operators , , given by , for Moreover, has discrete spectrum, the eigenvalues are , with corresponding eigenvectors , and the set of functions is an orthonormal basis of and for . For from [32] we can define the fractional power of is given by where In the next theorem, we consider . We observe that and for from [33, Proposition ], we obtain that is a sectorial operator satisfying Moreover, it is easy to see that conditions (H2)-(H3) in Section 3 are satisfied with , and is the space of infinitely differentiable functions that vanish at and . Under the above conditions, we can represent the system

in the abstract form

We define the functions by

where
1. (i)

the functions are continuous and ;

2. (ii)
the functions , are measurable, for all and

Moreover, are bounded linear operators, , , and a straightforward estimation using (ii) shows that and

for all . This allows us to rewrite the system (5.1) in the abstract form (1.1) with

Theorem 5.1.

Assume that the previous conditions are verified. Let and such that then there exists a mild solution of (5.1) with .

Proof.

For from , we obtain

since . By using a similar procedure as in the proofs of Lemma 3.3 and Theorem 3.10, we obtain the existence of resolvent operator for (5.2). From the hypothesis, we obtain by the Lemma 3.3, Corollaries 3.11 and 3.13, the assumption is satisfied. From Theorem 4.4, the proof is complete.

## Notes

### Acknowledgment

José Paulo C. dos Santos is partially supported by FAPEMIG/Brazil under Grant CEX-APQ-00476-09.

## References

1. 1.
Chukwu EN: Differential Models and Neutral Systems for Controlling the Wealth of Nations, Advances in Mathematics for Applied Sciences. Volume 54. World Scientific Publishing, River Edge, NJ, USA; 2001:xvi+513.Google Scholar
2. 2.
Hale J, Lunel SM: Introduction to Functional-Differential Equations. Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993.
3. 3.
Wu J: Theory and Applications of Partial functional-Differential Equations, Applied Mathematical Sciences. Volume 119. Springer, New York, NY, USA; 1996:x+429.
4. 4.
Gurtin ME, Pipkin AC: A general theory of heat conduction with finite wave speeds. Archive for Rational Mechanics and Analysis 1968,31(2):113-126. 10.1007/BF00281373
5. 5.
Nunziato J: On heat conduction in materials with memory. Quarterly of Applied Mathematics 1971, 29: 187-204.
6. 6.
Cannarsa P, Sforza D: Global solutions of abstract semilinear parabolic equations with memory terms. Nonlinear Differential Equations and Applications 2003,10(4):399-430. 10.1007/s00030-003-1004-2
7. 7.
Clément Ph, Nohel J: Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels. SIAM Journal on Mathematical Analysis 1981,12(4):514-535. 10.1137/0512045
8. 8.
Clément P, Philippe J: Global existence for a semilinear parabolic Volterra equation. Mathematische Zeitschrift 1992,209(1):17-26. 10.1007/BF02570816
9. 9.
Lunardi A: On the linear heat equation with fading memory. SIAM Journal on Mathematical Analysis 1990,21(5):1213-1224. 10.1137/0521066
10. 10.
Man NM, Minh NV: On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis 2004,3(2):291-300. 10.3934/cpaa.2004.3.291
11. 11.
Yuan R: Existence of almost periodic solutions of neutral functional-differential equations via Liapunov-Razumikhin function. Zeitschrift für Angewandte Mathematik und Physik 1998,49(1):113-136. 10.1007/s000330050084
12. 12.
Diagana T, Henríquez HR, Hernández EM: Almost automorphic mild solutions to some partial neutral functional-differential equations and applications. Nonlinear Analysis: Theory, Methods & Applications 2008,69(5-6):1485-1493. 10.1016/j.na.2007.06.048
13. 13.
Hernández E, Henríquez H: Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay. Journal of Mathematical Analysis and Applications 1998,221(2):499-522. 10.1006/jmaa.1997.5899
14. 14.
Hernández E, Pelicer M: Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations. Applied Mathematics Letters 2005,18(11):1265-1272. 10.1016/j.aml.2005.02.015
15. 15.
Hernández E, Diagana T: Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional-differential equations and applications. Journal of Mathematical Analysis and Applications 2007,327(2):776-791. 10.1016/j.jmaa.2006.04.043
16. 16.
Gripenberg G, Londen S-O, Staffans O: Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications. Volume 34. Cambridge University Press, Cambridge, UK; 1990:xxii+701.
17. 17.
Prüss J: Evolutionary Integral Equations and Applications, Monographs in Mathematics. Volume 87. Birkhäuser, Basel, Switzerland; 1993:xxvi+366.
18. 18.
Grimmer R: Resolvent operators for integral equations in a Banach space. Transactions of the American Mathematical Society 1982,273(1):333-349. 10.1090/S0002-9947-1982-0664046-4
19. 19.
Grimmer RC, Kappel F: Series expansions for resolvents of Volterra integro-differential equations in Banach space. SIAM Journal on Mathematical Analysis 1984,15(3):595-604. 10.1137/0515045
20. 20.
Grimmer RC, Pritchard AJ: Analytic resolvent operators for integral equations in Banach space. Journal of Differential Equations 1983,50(2):234-259. 10.1016/0022-0396(83)90076-1
21. 21.
Da Prato G, Iannelli M: Existence and regularity for a class of integro-differential equations of parabolic type. Journal of Mathematical Analysis and Applications 1985,112(1):36-55. 10.1016/0022-247X(85)90275-6
22. 22.
Da Prato G, Lunardi A: Solvability on the real line of a class of linear Volterra integro-differential equations of parabolic type. Annali di Matematica Pura ed Applicata 1988,150(1):67-117. 10.1007/BF01761464
23. 23.
Lunardi A: Laplace transform methods in integro-differential equations. Journal of Integral Equations 1985,10(1–3):185-211.
24. 24.
Henriquez H, Hernández E, dos Santos JPC: Asymptotically almost periodic and almost periodic solutions for partial neutral integrodifferential equations. Zeitschrift für Analysis und ihre Anwendungen 2007,26(3):363-375. 10.4171/ZAA/1329
25. 25.
Hernández E, dos Santos JPC: Existence results for partial neutral integro-differential equation with unbounded delay. Applicable Analysis 2007,86(2):223-237. 10.1080/00036810601148216
26. 26.
Henriquez HR, Hernández E, dos Santos JPC: Existence results for abstract partial neutral integro-differential equation with unbounded delay. Electronic Journal of Qualitative Theory of Differential Equations 2009, 29: 1-23.Google Scholar
27. 27.
Hino Y, Murakami S, Naito T: Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics. Volume 1473. Springer, Berlin, Germany; 1991:x+317.Google Scholar
28. 28.
Zaidman S: Almost-Periodic Functions in Abstract Spaces, Research Notes in Mathematics. Volume 126. Pitman (Advanced Publishing Program), Boston, Mass, USA; 1985:iii+133.Google Scholar
29. 29.
Yoshizawa T: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Volume 14. Springer, New York, NY, USA; 1975:vii+233.
30. 30.
Grimmer R, Prüss J: On linear Volterra equations in Banach spaces. Computers & Mathematics with Applications 1985,11(1–3):189-205. 10.1016/0898-1221(85)90146-4
31. 31.
Arendt W, Batty C, Hieber M, Neubrander F: Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics. Volume 96. Birkhäuser, Basel, Switzerland; 2001:xii+523.
32. 32.
Pazy A: Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.
33. 33.
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.Google Scholar

© José Paulo C. dos Santos et al. 2010

## Authors and Affiliations

• José Paulo C. dos Santos
• 1
Email author
• Sandro M. Guzzo
• 2
• Marcos N. Rabelo
• 3
1. 1.Departamento de Ciências Exatas, Universidade Federal de AlfenasRua Gabriel Monteiro da SilvaAlfenas-MGBrazil
2. 2.Colegiado do curso de Matemática-UNIOESTERua UniversitáriaCascavel-PRBrazil
3. 3.Departamento de MatemáticaUniversidade Federal de PernambucoRecife-PEBrazil