Open Access

Weighted -Asymptotically -Periodic Solutions of a Class of Fractional Differential Equations

Advances in Difference Equations20102011:584874

Received: 23 September 2010

Accepted: 8 December 2010

Published: 14 December 2010


We study the existence of weighted -asymptotically -periodic mild solutions for a class of abstract fractional differential equations of the form , where is a linear sectorial operator of negative type.

1. Introduction

-asymptotically -periodic functions have applications to several problems, for example in the theory of functional differential equations, fractional differential equations, integral equations and partial differential equations. The concept of -asymptotic -periodicity was introduced in the literature by Henríquez et al. [1, 2]. Since then, it attracted the attention of many researchers (see [110]). In Pierri [10] a new -asymptotically -periodic space was introduced. It is called the space of weighted -asymptotically -periodic (or -asymptotically -periodic) functions. In particular, the author has established conditions under which a -asymptotically -periodic function is asymptotically -periodic and also discusses the existence of -asymptotically -periodic solutions for an integral abstract Cauchy problem. The author has applied the results to partial integrodifferential equations.

We study in this paper sufficient conditions for the existence and uniqueness of a weighted -asymptotically -periodic (mild) solution to the following semi-linear integrodifferential equation of fractional order

where , is a linear densely defined operator of sectorial type on a complex Banach space and is an appropriate function. Note that the convolution integral in (1.1) is known as the Riemann-Liouville fractional integral [11]. We remark that there is much interest in developing theoretical analysis and numerical methods for fractional integrodifferential equations because they have recently proved to be valuable in various fields of sciences and engineering. For details, including some applications and recent results, see the monographs of Ahn and MacVinish [12], Gorenflo and Mainardi [13] and Trujillo et al. [1416] and the papers of Agarwal et al. [1723], Cuesta [11, 24], Cuevas et al. [5, 6], dos Santos and Cuevas [25], Eidelman and Kochubei [26], Lakshmikantham et al. [2730], Mophou and N'Guérékata [31], Ahmed and Nieto [32], and N'Guérékata [33]. In particular equations of type (1.1) are attracting increasing interest (cf. [5, 11, 24, 34]).

The existence of weighted -asymptotically -periodic (mild) solutions for integrodifferential equation of fractional order of type (1.1) remains an untreated topic in the literature. Anticipating a wide interest in the subject, this paper contributes in filling this important gap. In particular, to illustrate our main results, we examine sufficient conditions for the existence and uniqueness of a weighted -asymptotically -periodic mild solution to a fractional oscillation equation.

2. Preliminaries and Basic Results

In this section, we introduce notations, definitions and preliminary facts which are used throughout this paper. Let and be Banach spaces. The notation stands for the space of bounded linear operators from into endowed with the uniform operator topology denoted , and we abbreviate to and whenever . In this paper denotes the Banach space consisting of all continuous and bounded functions from into with the norm of the uniform convergence. For a closed linear operator we denote by the resolvent set and by the spectrum of (that is, the complement of in the complex plane). Set the resolvent of for .

2.1. Sectorial Linear Operators and the Solution Operator for Fractional Equations

A closed and linear operator is said sectorial of type μ if there are and such that the spectrum of is contained in the sector and , for all .

In order to give an operator theoretical approach for the study of the abstract system we recall the following definition.

Definition 2.1 (see [17]).

Let be a closed linear operator with domain in a Banach space . One calls the generator of a solution operator for (1.1)-(1.2) if there are and a strongly continuous function such that and , for all . In this case, is called the solution operator generated by . By [35, Proposition 2.6], . We observe that the power function is uniquely defined as , with .

We note that if is a sectorial of type μ with , then is the generator of a solution operator given by , , where is a suitable path lying outside the sector (cf. [11]). Recently, Cuesta [11, Theorem 1] proved that if is a sectorial operator of type for some and , then there exists such that

Remark 2.2.

In the remainder of this paper, we always assume that is a a sectorial of type and , are the constants introduced above.

2.2. Weighted -Asymptotically -Periodic Functions

We recall the following definitions.

Definition 2.3 (see [1]).

A function is called -asymptotically -periodic if there exists such that . In this case, we say that is an asymptotic period of .

Throughout this paper, represents the space formed for all the -valued -asymptotically -periodic functions endowed with the uniform convergence norm denoted . It is clear that is a Banach space (see [1, Proposition 3.5]).

Definition 2.4 (see [10]).

Let . A function is called weighted -asymptotically -periodic (or -asymptotically -periodic) if .

In this paper, represents the space formed by all the -asymptotically -periodic functions endowed with the norm

Proposition 2.5.

The space is a Banach space.


Let be a Cauchy sequence in . From the definition of , there exists such that in . Next, we prove that in .

By noting that is a Cauchy sequence, for given there exists such that , for all , which implies
Under the above conditions, for and we see that

which implies that for and as .

To conclude the proof we need to show that . Let as above. Since , there exits such that for all . Now, by using that , for we get

which implies that . This completes the proof.

Definition 2.6.

A function is called uniformly -asymptotically -periodic on bounded sets if for every bounded subset , the set is bounded and , uniformly for . If we say that is uniformly -asymptotically -periodic on bounded sets (see [1]).

To prove some of our results, we need the following lemma.

Lemma 2.7.

Let . Assume is uniformly -asymptotically -periodic on bounded sets and there is such that

If , then the function belongs to .


Using the fact that is bounded, it follows that . For be given, we select such that
for all and . Then, for we see that

which proves the assertion.

Lemma 2.8.

Let . Let and be the function defined by

If as and , then .


From the estimate , it follows that . For be given we select such that
for all . Under these conditions, for we have that

which completes the proof.

3. Existence of Weighted -Asymptotically -Periodic Solutions

In this section we discuss the existence of weighted -asymptotically -periodic solutions for the abstract system (1.1)-(1.2). To begin, we recall the definition of mild solution for (1.1)-(1.2).

Definition 3.1 (see [5]).

A function is called a mild solution of the abstract Cauchy problem (1.1)-(1.2) if

Now, we can establish our first existence result.

Theorem 3.2.

Assume is a uniformly -asymptotically -periodic on bounded sets function and there is a mesurable bounded function such that

If , then there exits a unique -asymptotically -periodic mild solution of (1.1)-(1.2). Suppose, there is a function such that and , for every and all . If is such that as , then is weighted -asymptotically -periodic.


Let be the operator defined by
We show initially that is -valued. Since , as , it is sufficient to show that the function is -valued. Let . Using the fact that is a bounded function, it follows that . For be given, we select a constant such that
Then, for we see that

which implies that as , and hence . Moreover, from the above estimate it is easy to infer that , for all , is a contraction and there exists a unique -asymptotically -periodic mild solution of (1.1)-(1.2).

Next, we prove that last assertion. Let be the function defined by . For , we get
Concerning the quantities and , we note that
Using the estimates (3.7) in (3.6), we see that
where is a positive constant independent of . Finally, by using the Gronwall-Bellman inequality we infer that

which shows that . This completes the proof.

Example 3.3.

We set , with . Let be a function such that , for all and let be defined by , . We observe that

whence is -asymptotically -periodic on bounded sets. By Theorem 3.2 we conclude that if , then there is a unique -asymptotically -periodic mild solution of (1.1)-(1.2). Moreover .

Theorem 3.4.

Let . Assume , as and
where is the constant introduced in Lemma 2.8.Then there is a unique weighted -asymptotically -periodic mild solution of


The proof is based in Lemmas 2.7 and 2.8. Let be the map defined by
We show initially that is -valued. From the estimate

we have that .

Let . From Lemma 2.7, we have that is a weighted -asymptotically -periodic function and by Lemma 2.8 we obtain that . Thus, the map is -valued. In order to prove that is a contraction, we note that for and ,
so that,
On the another hand, for we see that
from which we obtain that
By noting that is a linear operator for all and combining (3.16) and (3.18) we obtain that

for all , which shows that is a contraction on and hence there is a unique -asymptotically -periodic mild solution. The proof is complete.

To complete this paper, we examine the existence and uniqueness of weighted -asymptotically -periodic mild solutions for the following fractional differential equation
with boundary conditions

where and . In what follows we consider the space and let be the operator given by with domain , . It is well known that is sectorial of type negative.

Proposition 3.5.

Let satisfying conditions of Lemma 2.8 and let . If is small enough, then the problems (3.20)–(3.22) has a unique -asymptotically -periodic mild solution.


Problem (3.20)–(3.22) can be expressed as an abstract fractional differential equation of the form (3.12), where , for . We define
We have the following estimates:
estimate (3.25), we get
Since we obtain that . Moreover, we have the inequality

If we choose small enough, we have that condition (3.11) is fulfilled. By Theorem 3.4, the problems (3.20)–(3.22) has a unique -asymptotically -periodic (mild) solution. This finishes the proof.



C. Cuevas thanks the Department of Mathematics of Universidad de La Frontera, where this project was started. The authors are grateful to the referees for their valuable comments and suggestions. C. Cuevas is partially supported by CNPQ/Brazil under Grant 300365/2008-0.

Authors’ Affiliations

Departamento de Matemática, Universidade Federal de Pernambuco
Departamento de Física e Matemática da Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo
Departamento de Matemática y Estadística, Universidad de La Frontera


  1. Henríquez HR, Pierri M, Táboas P:On -asymptotically -periodic functions on Banach spaces and applications. Journal of Mathematical Analysis and Applications 2008,343(2):1119-1130. 10.1016/j.jmaa.2008.02.023MathSciNetView ArticleMATHGoogle Scholar
  2. Henríquez HR, Pierri M, Táboas P:Existence of -asymptotically -periodic solutions for abstract neutral equations. Bulletin of the Australian Mathematical Society 2008,78(3):365-382. 10.1017/S0004972708000713MathSciNetView ArticleMATHGoogle Scholar
  3. Agarwal RP, de Andrade B, Cuevas C: On type of periodicity and ergodicity to a class of integral equations with infinite delay. Journal of Nonlinear and Convex Analysis 2010,11(2):309-333.MathSciNetMATHGoogle Scholar
  4. Caicedo A, Cuevas C: -asymptotically -periodic solutions of abstract partial neutral integro-differentail equations. Functional Differential Equations 2010,17(1-2):387-405.MathSciNetMATHGoogle Scholar
  5. Cuevas C, de Souza JC: -asymptotically -periodic solutions of semilinear fractional integro-differential equations. Applied Mathematics Letters 2009,22(6):865-870. 10.1016/j.aml.2008.07.013MathSciNetView ArticleMATHGoogle Scholar
  6. Cuevas C, César de Souza J:Existence of -asymptotically -periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Analysis: Theory, Methods & Applications 2010,72(3-4):1683-1689. 10.1016/ ArticleMathSciNetMATHGoogle Scholar
  7. Cuevas C, Lizama C: -asymptotically -periodic solutions for semilinear Volterra equations. Mathematical Methods in the Applied Sciences 2010,33(13):1628-1636.MathSciNetMATHGoogle Scholar
  8. de Andrade B, Cuevas C: -asymptotically -periodic and asymptotically -periodic solutions to semi-linear Cauchy problems with non-dense domain. Nonlinear Analysis: Theory, Methods & Applications 2010,72(6):3190-3208. 10.1016/ ArticleMATHGoogle Scholar
  9. Nicola SHJ, Pierri M:A note on -asymptotically periodic functions. Nonlinear Analysis: Real World Applications 2009,10(5):2937-2938. 10.1016/j.nonrwa.2008.09.011MathSciNetView ArticleMATHGoogle Scholar
  10. Pierri M:On -asymptotically -periodic functions and applications. submittedGoogle Scholar
  11. Cuesta E: Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Discrete and Continuous Dynamical Systems. Series A 2007, 2007: 277-285.MathSciNetMATHGoogle Scholar
  12. Anh VV, Mcvinish R: Fractional differential equations driven by Lévy noise. Journal of Applied Mathematics and Stochastic Analysis 2003,16(2):97-119. 10.1155/S1048953303000078MathSciNetView ArticleMATHGoogle Scholar
  13. Gorenflo R, Mainardi F: Fractional calculus: integral and differential equations of fractional order. In Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), CISM Courses and Lectures. Volume 378. Edited by: Carpinteri A, Mainardi F. Springer, Vienna, Austria; 1997:223-276.View ArticleGoogle Scholar
  14. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
  15. Nigmatullin RR, Trujillo JJ: Mesoscopic fractional kinetic equations versus a Riemann-Liouville integral type. In Advances in Fractional Calculus. Springer, Dordrecht, The Netherlands; 2007:155-167.View ArticleGoogle Scholar
  16. Rivero M, Trujillo JJ, Velasco MP: On deterministic fractional models. In New Trends in Nanotechnology and Fractional Calculus Applications. Springer, New York, NY, USA; 2010:123-150.View ArticleGoogle Scholar
  17. Agarwal RP, de Andrade B, Cuevas C: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Analysis: Real World Applications 2010, 11: 3532-3554. 10.1016/j.nonrwa.2010.01.002MathSciNetView ArticleMATHGoogle Scholar
  18. Agarwal RP, de Andrade B, Cuevas C: On type of periodicity and ergodicity to a class of fractional order differential equations. Advances in Difference Equations 2010, 2010:-25.Google Scholar
  19. Agarwal RP, Belmekki M, Benchohra M: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Advances in Difference Equations 2009, 2009:-47.Google Scholar
  20. Agarwal RP, Benchohra M, Hamani S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Applicandae Mathematicae 2010,109(3):973-1033. 10.1007/s10440-008-9356-6MathSciNetView ArticleMATHGoogle Scholar
  21. Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Analysis: Theory, Methods & Applications 2010,72(6):2859-2862. 10.1016/ ArticleMATHGoogle Scholar
  22. Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Computers & Mathematics with Applications 2010,59(3):1095-1100.MathSciNetView ArticleMATHGoogle Scholar
  23. Agarwal RP, Cuevas C, Soto H, El-Gebeily M: Asymptotic periodicity for some evolution equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications. In pressGoogle Scholar
  24. Cuesta E, Lubich C, Palencia C: Convolution quadrature time discretization of fractional diffusion-wave equations. Mathematics of Computation 2006,75(254):673-696. 10.1090/S0025-5718-06-01788-1MathSciNetView ArticleMATHGoogle Scholar
  25. dos Santos JPC, Cuevas C: Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations. Applied Mathematics Letters 2010,23(9):960-965. 10.1016/j.aml.2010.04.016MathSciNetView ArticleMATHGoogle Scholar
  26. Eidelman SD, Kochubei AN: Cauchy problem for fractional diffusion equations. Journal of Differential Equations 2004,199(2):211-255. 10.1016/j.jde.2003.12.002MathSciNetView ArticleMATHGoogle Scholar
  27. Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):3337-3343. 10.1016/ ArticleMATHGoogle Scholar
  28. Lakshmikantham V, Devi JV: Theory of fractional differential equations in a Banach space. European Journal of Pure and Applied Mathematics 2008,1(1):38-45.MathSciNetMATHGoogle Scholar
  29. Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2677-2682. 10.1016/ ArticleMATHGoogle Scholar
  30. Lakshmikantham V, Vatsala AS: Theory of fractional differential inequalities and applications. Communications in Applied Analysis 2007,11(3-4):395-402.MathSciNetMATHGoogle Scholar
  31. Mophou GM, N'Guérékata GM: Existence of the mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 2009,79(2):315-322. 10.1007/s00233-008-9117-xMathSciNetView ArticleMATHGoogle Scholar
  32. Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
  33. N'Guérékata GM: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1873-1876. 10.1016/ ArticleMATHGoogle Scholar
  34. Hilfe R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.Google Scholar
  35. Keyantuo V, Lizama C: On a connection between powers of operators and fractional Cauchy problems. submittedGoogle Scholar


© Claudio Cuevas et al. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.