- Research Article
- Open Access
© T. Diagana and R. P. Agarwal. 2009
- Received: 12 March 2009
- Accepted: 3 July 2009
- Published: 17 August 2009
- Fractional Power
- Bounded Subset
- Interpolation Space
- Analytic Semigroup
- Uniform Continuity
where is a sectorial linear operator on a Banach space whose corresponding analytic semigroup is hyperbolic; that is, the operator are arbitrary linear (possibly unbounded) operators on , and are -pseudo almost automorphic for and jointly continuous functions.
The concept of pseudo almost automorphy, which is the central tool here, was recently introduced in literature by Liang et al. [1, 2]. The pseudo almost automorphy is a generalization of both the classical almost automorphy due to Bochner  and that of pseudo almost periodicity due to Zhang [4–6]. It has recently generated several developments and extensions. For the most recent developments, we refer the reader to [1, 2, 7–9]. More recently, in Diagana , the concept of -pseudo almost automorphy (or Stepanov-like pseudo almost automorphy) was introduced. It should be mentioned that the -pseudo almost automorphy is a natural generalization of the notion of pseudo almost automorphy.
In this paper, we will make extensive use of the concept of -pseudo almost automorphy combined with the techniques of hyperbolic semigroups to study the existence of pseudo almost automorphic solutions to the class of partial hyperbolic differential equations appearing in (1.3) and then to the -dimensional heat equation (1.1).
In this paper, as in the recent papers [10–12], we consider a general intermediate space between and . In contrast with the fractional power spaces considered in some recent papers by Diagana , the interpolation and Hölder spaces, for instance, depend only on and and can be explicitly expressed in many concrete cases. Literature related to those intermediate spaces is very extensive; in particular, we refer the reader to the excellent book by Lunardi , which contains a comprehensive presentation on this topic and related issues.
Existence results related to pseudo almost periodic and almost automorphic solutions to the partial hyperbolic differential equations of the form (1.3) have recently been established in [12, 15–18], respectively. Though to the best of our knowledge, the existence of pseudo almost automorphic solutions to the heat equation (1.1) in the case when the coefficients are -pseudo almost automorphic is an untreated original problem and constitutes the main motivation of the present paper.
Let be two Banach spaces. Let (resp., ) denote the collection of all -valued bounded continuous functions (resp., the class of jointly bounded continuous functions ). The space equipped with the sup norm is a Banach space. Furthermore, (resp., ) denotes the class of continuous functions from into (resp., the class of jointly continuous functions ).
Definition 2.1 (see ).
Definition 2.7 (see ).
Definition 2.8 (see ).
A weaker version of Definition 2.8 is the following.
The notion of -almost automorphy is a new notion due to N'Guérékata and Pankov .
Definition 2.12 (Bochner).
The function in Definition 2.12 is measurable but not necessarily continuous. Moreover, if is continuous, then is uniformly continuous. If the convergence above is uniform in , then is almost periodic. Denote by the collection of all almost automorphic functions . Note that equipped with the sup norm, , turns out to be a Banach space.
Definition 2.14 (see ).
2.3. Pseudo Almost Automorphy
A substantial result is the next theorem, which is due to Liang et al. .
Theorem 2.19 (see ).
We also have the following composition result.
Theorem 2.20 (see ).
This section is devoted to the notion of -pseudo almost automorphy. Such a concept is completely new and is due to Diagana .
Definition 3.1 (see ).
Theorem 3.2 (see ).
Theorem 3.3 (see ).
We have the following composition theorems.
Using the theorem of composition of functions of (see ) it is easy to see that .
Using the theorem of composition [2, Theorem 2.4] for functions of it is easy to see that .
The class of sectorial operators is very rich and contains most of classical operators encountered in literature.
see details in [23, Proposition 1.15, page 305]
the abstract Hölder spaces as well as the complex interpolation spaces ; see Lunardi  for details.
This section is devoted to the search of an almost automorphic solution to the partial hyperbolic differential equation (1.3).
In order to show that and are well defined, we need the next lemma whose proof can be found in Diagana .
Lemma 5.2 (see ).
Consequently the uniform limit ; see [21, Lemma 2.5] . Therefore, is pseudo almost automorphic.
The proof is similar to that of Lemma 5.3 and hence omitted, though here we make use of the approximation (5.4) rather than (5.5).
We require the following assumption.
We have the following.
Under the previous assumptions including (H.3), then the -dimensional heat equation (1.1) has a unique pseudo almost automorphic solution whenever is small enough.
- 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: 1493–1499. 10.1016/j.jmaa.2007.09.065MATHMathSciNetView ArticleGoogle 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-yMATHMathSciNetView ArticleGoogle 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.52.4.907MATHMathSciNetView ArticleGoogle Scholar
- Zhang CY: Pseudo-almost-periodic solutions of some differential equations. Journal of Mathematical Analysis and Applications 1994, 151: 62–76.View ArticleGoogle Scholar
- Zhang CY: Pseudo almost periodic solutions of some differential equations. II. Journal of Mathematical Analysis and Applications 1995, 192: 543–561. 10.1006/jmaa.1995.1189MATHMathSciNetView ArticleGoogle Scholar
- Zhang CY: Integration of vector-valued pseudo-almost periodic functions. Proceedings of the American Mathematical Society 1994, 121: 167–174. 10.1090/S0002-9939-1994-1186140-8MATHMathSciNetView ArticleGoogle 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.2008.07.034MATHMathSciNetView ArticleGoogle Scholar
- Ezzinbi K, Fatajou S, N'Guérékata GM: Pseudo almost automorphic solutions to some neutral partial functional differential equations in banach space. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(1–2):674–684. 10.1016/j.na.2008.10.100View ArticleGoogle 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.2008.03.061MATHMathSciNetView ArticleGoogle Scholar
- Boulite S, Maniar L, N'Guérékata GM: Almost automorphic solutions for hyperbolic semilinear evolution equations. Semigroup Forum 2005, 71: 231–240. 10.1007/s00233-005-0524-yMATHMathSciNetView ArticleGoogle Scholar
- Diagana T, N'Guérékata GM: Pseudo almost periodic mild solutions to hyperbolic evolution equations in intermediate banach spaces. Applicable Analysis 2006, 85(6–7):769–780. 10.1080/00036810600708499MATHMathSciNetView ArticleGoogle Scholar
- Diagana T: Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Electronic Journal of Qualitative Theory of Differential Equations 2007, 2007(3):1–12.View ArticleGoogle Scholar
- Diagana T: Pseudo Almost Periodic Functions in Banach Spaces. Nova Science, New York, NY, USA; 2007.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.Google Scholar
- Diagana T, Henríquez N, Hernàndez E: Almost automorphic mild solutions to some partial neutral functional-differential equations and applications. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(5):1485–1493. 10.1016/j.na.2007.06.048MATHMathSciNetView ArticleGoogle Scholar
- Diagana T, N'Guérékata GM: Almost automorphic solutions to some classes of partial evolution equations. Applied Mathematics Letters 2007, 20(4):462–466. 10.1016/j.aml.2006.05.015MATHMathSciNetView ArticleGoogle Scholar
- Diagana T: Existence and uniqueness of pseudo-almost periodic solutions to some classes of partial evolution equations. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(2):384–395. 10.1016/j.na.2005.11.031MATHMathSciNetView ArticleGoogle Scholar
- Diagana T: Existence of almost automorphic solutions to some neutral functional differential equations with infinite delay. Electronic Journal of Differential Equations 2008, 2008(129):1–14.MathSciNetGoogle Scholar
- Pankov A: Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, Mathematics and Its Applications (Soviet Series). Volume 55. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1990.Google Scholar
- Diagana T: Stepanov-like pseudo almost periodic functions and their applications to differential equations. Communications in Mathematical Analysis 2007, 3(1):9–18.MATHMathSciNetGoogle Scholar
- Diagana T: Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 69(12):4277–4285.MathSciNetView ArticleGoogle Scholar
- N'Guérékata GM, Pankov A: Stepanov-like almost automorphic functions and monotone evolution equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(9):2658–2667. 10.1016/j.na.2007.02.012MATHMathSciNetView ArticleGoogle Scholar
- Engel KJ, Nagel R: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics. Volume 194. Springer, New York, NY, USA; 2000.Google Scholar
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.