Boundary Value Problems
VolumeΒ 2009Β (2009), Article IDΒ 182527, 19 pages
doi:10.1155/2009/182527
Research Article

Existence of Pseudo Almost Automorphic Solutions for the Heat Equation with 𝑆 𝑝 -Pseudo Almost Automorphic Coefficients

Toka Diagana1Β and Ravi P. Agarwal2

1Department of Mathematics, Howard University, 2441 6th Street NW, Washington, DC 20005, USA
2Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL-32901, USA

Received 12 March 2009; Accepted 3 July 2009

Academic Editor: VeliΒ Shakhmurov

Copyright Β© 2009 Toka Diagana and Ravi P. Agarwal. 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.

Abstract

We obtain the existence of pseudo almost automorphic solutions to the 𝑁 -dimensional heat equation with 𝑆 𝑝 -pseudo almost automorphic coefficients.

1. Introduction

Let Ξ© βŠ‚ 𝐑 𝑁 ( 𝑁 β‰₯ 1 ) be an open bounded subset with 𝐢 2 boundary πœ• Ξ© , and let 𝐗 = 𝐋 2 ( Ξ© ) be the space square integrable functions equipped with its natural β€– β‹… β€– 𝐋 2 ( Ξ© ) topology. Of concern is the study of pseudo almost automorphic solutions to the 𝑁 -dimensional heat equation with divergence terms πœ•  ξ‚€  ξ‚€   πœ• 𝑑 πœ‘ + 𝐹 𝑑 , 𝐡 πœ‘  ξ‚„ = 𝚫 πœ‘ + 𝐺 𝑑 , 𝐡 πœ‘ , 𝑑 ∈ 𝐑 , π‘₯ ∈ Ξ© πœ‘ ( 𝑑 , π‘₯ ) = 0 , 𝑑 ∈ 𝐑 , π‘₯ ∈ πœ• Ξ© , ( 1 . 1 ) where the symbols  𝐡 and Ξ” stand, respectively, for the first- and second-order differential operators defined by  𝐡 ∢ = 𝑁  𝑗 = 1 πœ• πœ• π‘₯ 𝑗 , 𝚫 = 𝑁  𝑗 = 1 πœ• 2 πœ• π‘₯ 2 𝑗 , ( 1 . 2 ) and the coefficients 𝐹 , 𝐺 ∢ 𝐑 Γ— 𝐇 1 0 ( Ξ© ) ↦ 𝐋 2 ( Ξ© ) are 𝑆 𝑝 -pseudo almost automorphic.

To analyze (1.1), our strategy will consist of studying the existence of pseudo almost automorphic solutions to the class of partial hyperbolic differential equations 𝑑 [ ] 𝑑 𝑑 𝑒 ( 𝑑 ) + 𝑓 ( 𝑑 , 𝐡 𝑒 ( 𝑑 ) ) = 𝐴 𝑒 ( 𝑑 ) + 𝑔 ( 𝑑 , 𝐢 𝑒 ( 𝑑 ) ) , 𝑑 ∈ 𝐑 , ( 1 . 3 ) where 𝐴 ∢ 𝐷 ( 𝐴 ) βŠ‚ 𝐗 ↦ 𝐗 is a sectorial linear operator on a Banach space 𝐗 whose corresponding analytic semigroup ( 𝑇 ( 𝑑 ) ) 𝑑 β‰₯ 0 is hyperbolic; that is, 𝜎 ( 𝐴 ) ∩ 𝑖 𝐑 = βˆ… , the operator 𝐡 , 𝐢 are arbitrary linear (possibly unbounded) operators on 𝐗 , and 𝑓 , 𝑔 are 𝑆 𝑝 -pseudo almost automorphic for 𝑝 > 1 and jointly continuous functions.

Indeed, letting 𝐴 πœ‘ = Ξ” πœ‘ for all πœ‘ ∈ 𝐷 ( 𝐴 ) = 𝐇 1 0 ( Ξ© ) ∩ 𝐇 2 ( Ξ© ) ,  𝐡 πœ‘ = 𝐡 πœ‘ = 𝐢 πœ‘ for all πœ‘ ∈ 𝐇 1 0 ( Ξ© ) , and 𝑓 = 𝐹 and 𝑔 = 𝐺 , one can readily see that (1.1) is a particular case of (1.3).

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 [3] 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 [7], the concept of 𝑆 𝑝 -pseudo almost automorphy (or Stepanov-like pseudo almost automorphy) was introduced. It should be mentioned that the 𝑆 𝑝 -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 𝑆 𝑝 -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 𝑁 -dimensional heat equation (1.1).

In this paper, as in the recent papers [10–12], we consider a general intermediate space 𝐗 𝛼 between 𝐷 ( 𝐴 ) and 𝐗 . In contrast with the fractional power spaces considered in some recent papers by Diagana [13], the interpolation and Hölder spaces, for instance, depend only on 𝐷 ( 𝐴 ) and 𝐗 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 [14], 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 𝑓 , 𝑔 are 𝑆 𝑝 -pseudo almost automorphic is an untreated original problem and constitutes the main motivation of the present paper.

2. Preliminaries

Let ( 𝐗 , β€– β‹… β€– ) , ( 𝐘 , β€– β‹… β€– 𝐘 ) be two Banach spaces. Let 𝐡 𝐢 ( 𝐑 , 𝐗 ) (resp., 𝐡 𝐢 ( 𝐑 Γ— 𝐘 , 𝐗 ) ) denote the collection of all 𝐗 -valued bounded continuous functions (resp., the class of jointly bounded continuous functions 𝐹 ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 ). The space 𝐡 𝐢 ( 𝐑 , 𝐗 ) equipped with the sup norm β€– β‹… β€– ∞ is a Banach space. Furthermore, 𝐢 ( 𝐑 , 𝐘 ) (resp., 𝐢 ( 𝐑 Γ— 𝐘 , 𝐗 ) ) denotes the class of continuous functions from 𝐑 into 𝐘 (resp., the class of jointly continuous functions 𝐹 ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 ).

The notation 𝐿 ( 𝐗 , 𝐘 ) stands for the Banach space of bounded linear operators from 𝐗 into 𝐘 equipped with its natural topology; in particular, this is simply denoted 𝐿 ( 𝐗 ) whenever 𝐗 = 𝐘 .

Definition 2.1 (see [19]). The Bochner transform 𝑓 𝑏 ( 𝑑 , 𝑠 ) , 𝑑 ∈ 𝐑 , 𝑠 ∈ [ 0 , 1 ] of a function 𝑓 ∢ 𝐑 ↦ 𝐗 is defined by 𝑓 𝑏 ( 𝑑 , 𝑠 ) ∢ = 𝑓 ( 𝑑 + 𝑠 ) .

Remark 2.2. (i) A function πœ‘ ( 𝑑 , 𝑠 ) , 𝑑 ∈ 𝐑 , 𝑠 ∈ [ 0 , 1 ] , is the Bochner transform of a certain function 𝑓 , πœ‘ ( 𝑑 , 𝑠 ) = 𝑓 𝑏 ( 𝑑 , 𝑠 ) , if and only if πœ‘ ( 𝑑 + 𝜏 , 𝑠 βˆ’ 𝜏 ) = πœ‘ ( 𝑠 , 𝑑 ) for all 𝑑 ∈ 𝐑 , 𝑠 ∈ [ 0 , 1 ] and 𝜏 ∈ [ 𝑠 βˆ’ 1 , 𝑠 ] .
(ii) Note that if 𝑓 = β„Ž + πœ‘ , then 𝑓 𝑏 = β„Ž 𝑏 + πœ‘ 𝑏 . Moreover, ( πœ† 𝑓 ) 𝑏 = πœ† 𝑓 𝑏 for each scalar πœ† .

Definition 2.3. The Bochner transform 𝐹 𝑏 ( 𝑑 , 𝑠 , 𝑒 ) , 𝑑 ∈ 𝐑 , 𝑠 ∈ [ 0 , 1 ] , 𝑒 ∈ 𝐗 of a function 𝐹 ( 𝑑 , 𝑒 ) on 𝐑 Γ— 𝐗 , with values in 𝐗 , is defined by 𝐹 𝑏 ( 𝑑 , 𝑠 , 𝑒 ) ∢ = 𝐹 ( 𝑑 + 𝑠 , 𝑒 ) for each 𝑒 ∈ 𝐗 .

Definition 2.4. Let 𝑝 ∈ [ 1 , ∞ ) . The space 𝐡 𝑆 𝑝 ( 𝐗 ) of all Stepanov bounded functions, with the exponent 𝑝 , consists of all measurable functions 𝑓 ∢ 𝐑 ↦ 𝐗 such that 𝑓 𝑏 ∈ 𝐿 ∞ ( 𝐑 ; 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . This is a Banach space with the norm β€– 𝑓 β€– 𝑆 𝑝 β€– β€– 𝑓 ∢ = 𝑏 β€– β€– 𝐿 ∞ ( 𝐑 , 𝐿 𝑝 ) = s u p 𝑑 ∈ 𝐑 ξ‚΅ ξ€œ 𝑑 𝑑 + 1 β€– 𝑓 ( 𝜏 ) β€– 𝑝 ξ‚Ά 𝑑 𝜏 1 / 𝑝 . ( 2 . 1 )

2.1. 𝑆 𝑝 -Pseudo Almost Periodicity

Definition 2.5. A function 𝑓 ∈ 𝐢 ( 𝐑 , 𝐗 ) is called (Bohr) almost periodic if for each πœ€ > 0 there exists 𝑙 ( πœ€ ) > 0 such that every interval of length 𝑙 ( πœ€ ) contains a number 𝜏 with the property that β€– 𝑓 ( 𝑑 + 𝜏 ) βˆ’ 𝑓 ( 𝑑 ) β€– < πœ€ f o r e a c h 𝑑 ∈ 𝐑 . ( 2 . 2 )
The number 𝜏 above is called an πœ€ -translation number of 𝑓 , and the collection of all such functions will be denoted 𝐴 𝑃 ( 𝐗 ) .

Definition 2.6. A function 𝐹 ∈ 𝐢 ( 𝐑 Γ— 𝐘 , 𝐗 ) is called (Bohr) almost periodic in 𝑑 ∈ 𝐑 uniformly in 𝑦 ∈ 𝐾 where 𝐾 βŠ‚ 𝐘 is any compact subset 𝐾 βŠ‚ 𝐘 if for each πœ€ > 0 there exists 𝑙 ( πœ€ ) such that every interval of length 𝑙 ( πœ€ ) contains a number 𝜏 with the property that β€– 𝐹 ( 𝑑 + 𝜏 , 𝑦 ) βˆ’ 𝐹 ( 𝑑 , 𝑦 ) β€– < πœ€ f o r e a c h 𝑑 ∈ 𝐑 , 𝑦 ∈ 𝐾 . ( 2 . 3 ) The collection of those functions is denoted by 𝐴 𝑃 ( 𝐑 Γ— 𝐘 ) .

Define the classes of functions 𝑃 𝐴 𝑃 0 ( 𝐗 ) and 𝑃 𝐴 𝑃 0 ( 𝐑 Γ— 𝐗 ) , respectively, as follows: 𝑃 𝐴 𝑃 0 ξ‚» ( 𝐗 ) ∢ = 𝑒 ∈ 𝐡 𝐢 ( 𝐑 , 𝐗 ) ∢ l i m 𝑇 β†’ ∞ 1 ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 ξ‚Ό , β€– 𝑒 ( 𝑠 ) β€– 𝑑 𝑠 = 0 ( 2 . 4 ) and 𝑃 𝐴 𝑃 0 ( 𝐑 Γ— 𝐘 ) is the collection of all functions 𝐹 ∈ 𝐡 𝐢 ( 𝐑 Γ— 𝐘 , 𝐗 ) such that l i m 𝑇 β†’ ∞ 1 ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 β€– 𝐹 ( 𝑑 , 𝑒 ) β€– 𝑑 𝑑 = 0 ( 2 . 5 ) uniformly in 𝑒 ∈ 𝐘 .

Definition 2.7 (see [13]). A function 𝑓 ∈ 𝐡 𝐢 ( 𝐑 , 𝐗 ) is called pseudo almost periodic if it can be expressed as 𝑓 = β„Ž + πœ‘ , where β„Ž ∈ 𝐴 𝑃 ( 𝐗 ) and πœ‘ ∈ 𝑃 𝐴 𝑃 0 ( 𝐗 ) . The collection of such functions will be denoted by 𝑃 𝐴 𝑃 ( 𝐗 ) .

Definition 2.8 (see [13]). A function 𝐹 ∈ 𝐢 ( 𝐑 Γ— 𝐘 , 𝐗 ) is said to be pseudo almost periodic if it can be expressed as 𝐹 = 𝐺 + Ξ¦ , where 𝐺 ∈ 𝐴 𝑃 ( 𝐑 Γ— 𝐘 ) and πœ™ ∈ 𝑃 𝐴 𝑃 0 ( 𝐑 Γ— 𝐘 ) . The collection of such functions will be denoted by 𝑃 𝐴 𝑃 ( 𝐑 Γ— 𝐘 ) .

Define 𝐴 𝐴 0 ( 𝐑 Γ— 𝐘 ) as the collection of all functions 𝐹 ∈ 𝐡 𝐢 ( 𝐑 Γ— 𝐘 , 𝐗 ) such that l i m 𝑇 β†’ ∞ 1 ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 β€– 𝐹 ( 𝑑 , 𝑒 ) β€– 𝑑 𝑑 = 0 ( 2 . 6 ) uniformly in 𝑒 ∈ 𝐾 , where 𝐾 βŠ‚ 𝐘 is any bounded subset.

Obviously, 𝑃 𝐴 𝑃 0 ( 𝐑 Γ— 𝐘 ) βŠ‚ 𝐴 𝐴 0 ( 𝐑 Γ— 𝐘 ) . ( 2 . 7 )

A weaker version of Definition 2.8 is the following.

Definition 2.9. A function 𝐹 ∈ 𝐢 ( 𝐑 Γ— 𝐘 , 𝐗 ) is said to be B-pseudo almost periodic if it can be expressed as 𝐹 = 𝐺 + Ξ¦ , where 𝐺 ∈ 𝐴 𝑃 ( 𝐑 Γ— 𝐘 ) and πœ™ ∈ 𝐴 𝐴 0 ( 𝐑 Γ— 𝐘 ) . The collection of such functions will be denoted by 𝐡 𝑃 𝐴 𝑃 ( 𝐑 Γ— 𝐘 ) .

Definition 2.10 (see [20, 21]). A function 𝑓 ∈ 𝐡 𝑆 𝑝 ( 𝐗 ) is called 𝑆 𝑝 -pseudo almost periodic (or Stepanov-like pseudo almost periodic) if it can be expressed as 𝑓 = β„Ž + πœ‘ , where β„Ž 𝑏 ∈ 𝐴 𝑃 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and πœ‘ 𝑏 ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . The collection of such functions will be denoted by 𝑃 𝐴 𝑃 𝑝 ( 𝐗 ) .

In other words, a function 𝑓 ∈ 𝐿 𝑝 l o c ( 𝐑 , 𝐗 ) is said to be 𝑆 𝑝 -pseudo almost periodic if its Bochner transform 𝑓 𝑏 ∢ 𝐑 β†’ 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) is pseudo almost periodic in the sense that there exist two functions β„Ž , πœ‘ ∢ 𝐑 ↦ 𝐗 such that 𝑓 = β„Ž + πœ‘ , where β„Ž 𝑏 ∈ 𝐴 𝑃 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and πœ‘ 𝑏 ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) .

To define the notion of 𝑆 𝑝 -pseudo almost automorphy for functions of the form 𝐹 ∢ 𝐑 Γ— 𝐘 ↦ 𝐘 , we need to define the 𝑆 𝑝 -pseudo almost periodicity for these functions as follows.

Definition 2.11. A function 𝐹 ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 , ( 𝑑 , 𝑒 ) ↦ 𝐹 ( 𝑑 , 𝑒 ) with 𝐹 ( β‹… , 𝑒 ) ∈ 𝐿 𝑝 l o c ( 𝐑 , 𝐗 ) for each 𝑒 ∈ 𝐗 , is said to be 𝑆 𝑝 -pseudo almost periodic if there exist two functions 𝐻 , Ξ¦ ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 such that 𝐹 = 𝐻 + Ξ¦ , where 𝐻 𝑏 ∈ 𝐴 𝑃 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and Ξ¦ 𝑏 ∈ 𝐴 𝐴 0 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) .

The collection of those 𝑆 𝑝 -pseudo almost periodic functions 𝐹 ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 will be denoted 𝑃 𝐴 𝑃 𝑝 ( 𝐑 Γ— 𝐘 ) .

2.2. 𝑆 𝑝 -Almost Automorphy

The notion of S 𝑝 -almost automorphy is a new notion due to N'Guérékata and Pankov [22].

Definition 2.12 (Bochner). A function 𝑓 ∈ 𝐢 ( 𝐑 , 𝐗 ) is said to be almost automorphic if for every sequence of real numbers ( 𝑠 ξ…ž 𝑛 ) 𝑛 ∈ β„• there exists a subsequence ( 𝑠 𝑛 ) 𝑛 ∈ β„• such that 𝑔 ( 𝑑 ) ∢ = l i m 𝑛 β†’ ∞ 𝑓 ξ€· 𝑑 + 𝑠 𝑛 ξ€Έ ( 2 . 8 ) is well defined for each 𝑑 ∈ 𝐑 , and l i m 𝑛 β†’ ∞ 𝑔 ξ€· 𝑑 βˆ’ 𝑠 𝑛 ξ€Έ = 𝑓 ( 𝑑 ) ( 2 . 9 ) for each 𝑑 ∈ 𝐑 .

Remark 2.13. The function 𝑔 in Definition 2.12 is measurable but not necessarily continuous. Moreover, if 𝑔 is continuous, then 𝑓 is uniformly continuous. If the convergence above is uniform in 𝑑 ∈ 𝐑 , then 𝑓 is almost periodic. Denote by 𝐴 𝐴 ( 𝐗 ) the collection of all almost automorphic functions 𝐑 β†’ 𝐗 . Note that 𝐴 𝐴 ( 𝐗 ) equipped with the sup norm, β€– β‹… β€– ∞ , turns out to be a Banach space.

We will denote by 𝐴 𝐴 𝑒 ( 𝐗 ) the closed subspace of all functions 𝑓 ∈ 𝐴 𝐴 ( 𝐗 ) with 𝑔 ∈ 𝐢 ( 𝐑 , 𝐗 ) . Equivalently, 𝑓 ∈ 𝐴 𝐴 𝑒 ( 𝐗 ) if and only if 𝑓 is almost automorphic, and the convergences in Definition 2.12 are uniform on compact intervals, that is, in the Fréchet space 𝐢 ( 𝐑 , 𝐗 ) . Indeed, if 𝑓 is almost automorphic, then its range is relatively compact. Obviously, the following inclusions hold: 𝐴 𝑃 ( 𝐗 ) βŠ‚ 𝐴 𝐴 𝑒 ( 𝐗 ) βŠ‚ 𝐴 𝐴 ( 𝐗 ) βŠ‚ 𝐡 𝐢 ( 𝐗 ) . ( 2 . 1 0 )

Definition 2.14 (see [22]). The space 𝐴 𝑆 𝑝 ( 𝐗 ) of Stepanov-like almost automorphic functions (or 𝑆 𝑝 -almost automorphic) consists of all 𝑓 ∈ 𝐡 𝑆 𝑝 ( 𝐗 ) such that 𝑓 𝑏 ∈ 𝐴 𝐴 ( 𝐿 𝑝 ( 0 , 1 ; 𝐗 ) ) . That is, a function 𝑓 ∈ 𝐿 𝑝 l o c ( 𝐑 ; 𝐗 ) is said to be S 𝑝 -almost automorphic if its Bochner transform 𝑓 𝑏 ∢ 𝐑 β†’ 𝐿 𝑝 ( 0 , 1 ; 𝐗 ) is almost automorphic in the sense that for every sequence of real numbers ( 𝑠 ξ…ž 𝑛 ) 𝑛 ∈ β„• there exists a subsequence ( 𝑠 𝑛 ) 𝑛 ∈ β„• and a function 𝑔 ∈ 𝐿 𝑝 l o c ( 𝐑 ; 𝐗 ) such that ξ‚Έ ξ€œ 𝑑 𝑑 + 1 β€– β€– 𝑓 ξ€· 𝑠 𝑛 ξ€Έ β€– β€– + 𝑠 βˆ’ 𝑔 ( 𝑠 ) 𝑝 ξ‚Ή 𝑑 𝑠 1 / 𝑝 ξ‚Έ ξ€œ ⟢ 0 , 𝑑 𝑑 + 1 β€– β€– 𝑔 ξ€· 𝑠 βˆ’ 𝑠 𝑛 ξ€Έ β€– β€– βˆ’ 𝑓 ( 𝑠 ) 𝑝 ξ‚Ή 𝑑 𝑠 1 / 𝑝 ⟢ 0 ( 2 . 1 1 ) as 𝑛 β†’ ∞ pointwise on 𝐑 .

Remark 2.15. It is clear that if 1 ≀ 𝑝 < π‘ž < ∞ and 𝑓 ∈ 𝐿 π‘ž l o c ( 𝐑 ; 𝐗 ) is 𝑆 π‘ž -almost automorphic, then 𝑓 is 𝑆 𝑝 -almost automorphic. Also if 𝑓 ∈ 𝐴 𝐴 ( 𝐗 ) , then 𝑓 is 𝑆 𝑝 -almost automorphic for any 1 ≀ 𝑝 < ∞ . Moreover, it is clear that 𝑓 ∈ 𝐴 𝐴 𝑒 ( 𝐗 ) if and only if 𝑓 𝑏 ∈ 𝐴 𝐴 ( 𝐿 ∞ ( 0 , 1 ; 𝐗 ) ) . Thus, 𝐴 𝐴 𝑒 ( 𝐗 ) can be considered as 𝐴 𝑆 ∞ ( 𝐗 ) .

Definition 2.16. A function 𝐹 ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 , ( 𝑑 , 𝑒 ) ↦ 𝐹 ( 𝑑 , 𝑒 ) with 𝐹 ( β‹… , 𝑒 ) ∈ 𝐿 𝑝 l o c ( 𝐑 ; 𝐗 ) for each 𝑒 ∈ 𝐘 , is said to be S 𝑝 -almost automorphic in 𝑑 ∈ 𝐑 uniformly in 𝑒 ∈ 𝐘 if 𝑑 ↦ 𝐹 ( 𝑑 , 𝑒 ) is 𝑆 𝑝 -almost automorphic for each 𝑒 ∈ 𝐘 ; that is, for every sequence of real numbers ( 𝑠 ξ…ž 𝑛 ) 𝑛 ∈ β„• , there exists a subsequence ( 𝑠 𝑛 ) 𝑛 ∈ β„• and a function 𝐺 ( β‹… , 𝑒 ) ∈ 𝐿 𝑝 l o c ( 𝐑 ; 𝐗 ) such that ξ‚Έ ξ€œ 𝑑 𝑑 + 1 β€– β€– 𝐹 ( 𝑠 𝑛 β€– β€– + 𝑠 , 𝑒 ) βˆ’ 𝐺 ( 𝑠 , 𝑒 ) 𝑝 ξ‚Ή 𝑑 𝑠 1 / 𝑝 ξ‚Έ ξ€œ ⟢ 0 , 𝑑 𝑑 + 1 β€– β€– 𝐺 ( 𝑠 βˆ’ 𝑠 𝑛 β€– β€– , 𝑒 ) βˆ’ 𝐹 ( 𝑠 , 𝑒 ) 𝑝 ξ‚Ή 𝑑 𝑠 1 / 𝑝 ⟢ 0 ( 2 . 1 2 ) as 𝑛 β†’ ∞ pointwise on 𝐑 for each 𝑒 ∈ 𝐘 .

The collection of those 𝑆 𝑝 -almost automorphic functions 𝐹 ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 will be denoted by 𝐴 𝑆 𝑝 ( 𝐑 Γ— 𝐘 ) .

2.3. Pseudo Almost Automorphy

The notion of pseudo almost automorphy is a new notion due to Liang et al. [2, 9].

Definition 2.17. A function 𝑓 ∈ 𝐢 ( 𝐑 , 𝐗 ) is called pseudo almost automorphic if it can be expressed as 𝑓 = β„Ž + πœ‘ , where β„Ž ∈ 𝐴 𝐴 ( 𝐗 ) and πœ‘ ∈ 𝑃 𝐴 𝑃 0 ( 𝐗 ) . The collection of such functions will be denoted by 𝑃 𝐴 𝐴 ( 𝐗 ) .

Obviously, the following inclusions hold: 𝐴 𝑃 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝑃 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝐴 ( 𝐗 ) , 𝐴 𝑃 ( 𝐗 ) βŠ‚ 𝐴 𝐴 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝐴 ( 𝐗 ) . ( 2 . 1 3 )

Definition 2.18. A function 𝐹 ∈ 𝐢 ( 𝐑 Γ— 𝐘 , 𝐗 ) is said to be pseudo almost automorphic if it can be expressed as 𝐹 = 𝐺 + Ξ¦ , where 𝐺 ∈ 𝐴 𝐴 ( 𝐑 Γ— 𝐘 ) and πœ‘ ∈ 𝐴 𝐴 0 ( 𝐑 Γ— 𝐘 ) . The collection of such functions will be denoted by 𝑃 𝐴 𝐴 ( 𝐑 Γ— 𝐘 ) .

A substantial result is the next theorem, which is due to Liang et al. [2].

Theorem 2.19 (see [2]). The space 𝑃 𝐴 𝐴 ( 𝐗 ) equipped with the sup norm β€– β‹… β€– ∞ is a Banach space.

We also have the following composition result.

Theorem 2.20 (see [2]). If 𝑓 ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 belongs to 𝑃 𝐴 𝐴 ( 𝐑 Γ— 𝐘 ) and if π‘₯ ↦ 𝑓 ( 𝑑 , π‘₯ ) is uniformly continuous on any bounded subset 𝐾 of 𝐘 for each 𝑑 ∈ 𝐑 , then the function defined by β„Ž ( 𝑑 ) = 𝑓 ( 𝑑 , πœ‘ ( 𝑑 ) ) belongs to 𝑃 𝐴 𝐴 ( 𝐗 ) provided πœ‘ ∈ 𝑃 𝐴 𝐴 ( 𝐘 ) .

3. 𝑆 𝑝 -Pseudo Almost Automorphy

This section is devoted to the notion of 𝑆 𝑝 -pseudo almost automorphy. Such a concept is completely new and is due to Diagana [7].

Definition 3.1 (see [7]). A function 𝑓 ∈ 𝐡 𝑆 𝑝 ( 𝐗 ) is called 𝑆 𝑝 -pseudo almost automorphic (or Stepanov-like pseudo almost automorphic) if it can be expressed as 𝑓 = β„Ž + πœ‘ , ( 3 . 1 ) where β„Ž 𝑏 ∈ 𝐴 𝐴 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and πœ‘ 𝑏 ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . The collection of such functions will be denoted by 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) .

Clearly, a function 𝑓 ∈ 𝐿 𝑝 l o c ( 𝐑 , 𝐗 ) is said to be 𝑆 𝑝 -pseudo almost automorphic if its Bochner transform 𝑓 𝑏 ∢ 𝐑 β†’ 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) is pseudo almost automorphic in the sense that there exist two functions β„Ž , πœ‘ ∢ 𝐑 ↦ 𝐗 such that 𝑓 = β„Ž + πœ‘ , where β„Ž 𝑏 ∈ 𝐴 𝐴 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and πœ‘ 𝑏 ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) .

Theorem 3.2 (see [7]). If 𝑓 ∈ 𝑃 𝐴 𝐴 ( 𝐗 ) , then 𝑓 ∈ 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) for each 1 ≀ 𝑝 < ∞ . In other words, 𝑃 𝐴 𝐴 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) .

Obviously, the following inclusions hold: 𝐴 𝑃 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝑃 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝐴 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) , 𝐴 𝑃 ( 𝐗 ) βŠ‚ 𝐴 𝐴 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝐴 ( 𝐗 ) βŠ‚ 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) . ( 3 . 2 )

Theorem 3.3 (see [7]). The space 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) equipped with the norm β€– β‹… β€– 𝐒 𝑝 is a Banach space.

Definition 3.4. A function 𝐹 ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 , ( 𝑑 , 𝑒 ) ↦ 𝐹 ( 𝑑 , 𝑒 ) with 𝐹 ( β‹… , 𝑒 ) ∈ 𝐿 𝑝 ( 𝐑 , 𝐗 ) for each 𝑒 ∈ 𝐘 , is said to be 𝑆 𝑝 -pseudo almost automorphic if there exists two functions 𝐻 , Ξ¦ ∢ 𝐑 Γ— 𝐘 ↦ 𝐗 such that 𝐹 = 𝐻 + Ξ¦ , ( 3 . 3 ) where 𝐻 𝑏 ∈ 𝐴 𝐴 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and Ξ¦ 𝑏 ∈ 𝐴 𝐴 0 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . The collection of those 𝑆 𝑝 -pseudo almost automorphic functions will be denoted by 𝑃 𝐴 𝐴 𝑝 ( 𝐑 Γ— 𝐘 ) .

We have the following composition theorems.

Theorem 3.5. Let 𝐹 ∢ 𝐑 Γ— 𝐗 ↦ 𝐗 be a 𝑆 𝑝 -pseudo almost automorphic function. Suppose that 𝐹 ( 𝑑 , 𝑒 ) is Lipschitzian in 𝑒 ∈ 𝐗 uniformly in 𝑑 ∈ 𝐑 ; that is there exists 𝐿 > 0 such β€– 𝐹 ( 𝑑 , 𝑒 ) βˆ’ 𝐹 ( 𝑑 , 𝑣 ) β€– ≀ 𝐿 β‹… β€– 𝑒 βˆ’ 𝑣 β€– ( 3 . 4 ) for all 𝑑 ∈ 𝐑 , ( 𝑒 , 𝑣 ) ∈ 𝐗 Γ— 𝐗 .
If πœ™ ∈ 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) , then Ξ“ ∢ 𝐑 β†’ 𝐗 defined by Ξ“ ( β‹… ) ∢ = 𝐹 ( β‹… , πœ™ ( β‹… ) ) belongs to 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) .

Proof. Let 𝐹 = 𝐻 + Ξ¦ , where 𝐻 𝑏 ∈ 𝐴 𝐴 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and Ξ¦ 𝑏 ∈ 𝐴 𝐴 0 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . Similarly, let πœ™ = πœ™ 1 + πœ™ 2 , where πœ™ 𝑏 1 ∈ 𝐴 𝐴 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and πœ™ 𝑏 2 ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) , that is, l i m 𝑇 β†’ ∞ 1 ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 ξ‚΅ ξ€œ 𝑑 𝑑 + 1 β€– β€– πœ‘ 2 β€– β€– ( 𝜎 ) 𝑝 ξ‚Ά 𝑑 𝜎 1 / 𝑝 𝑑 𝑑 = 0 ( 3 . 5 ) for all 𝑑 ∈ 𝐑 .
It is obvious to see that 𝐹 𝑏 ( β‹… , πœ™ ( β‹… ) ) ∢ 𝐑 ↦ 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) . Now decompose 𝐹 𝑏 as follows: 𝐹 𝑏 ( β‹… , πœ™ ( β‹… ) ) = 𝐻 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ ( β‹… ) + 𝐹 𝑏 ( β‹… , πœ™ ( β‹… ) ) βˆ’ 𝐻 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ ( β‹… ) = 𝐻 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ ( β‹… ) + 𝐹 𝑏 ( β‹… , πœ™ ( β‹… ) ) βˆ’ 𝐹 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ ( β‹… ) + Ξ¦ 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ . ( β‹… ) ( 3 . 6 )
Using the theorem of composition of almost automorphic functions, it is easy to see that 𝐻 𝑏 ( β‹… , πœ™ 1 ( β‹… ) ) ∈ 𝐴 𝐴 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . Now, set 𝐺 𝑏 ( β‹… ) ∢ = 𝐹 𝑏 ( β‹… , πœ™ ( β‹… ) ) βˆ’ 𝐹 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ . ( β‹… ) ( 3 . 7 )
Clearly, 𝐺 𝑏 ( β‹… ) ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . Indeed, we have ξ€œ 𝑑 𝑑 + 1 β€– 𝐺 ( 𝜎 ) β€– 𝑝 ξ€œ 𝑑 𝜎 = 𝑑 𝑑 + 1 β€– β€– 𝐹 ( 𝜎 , πœ™ ( 𝜎 ) ) βˆ’ 𝐹 ( 𝜎 , πœ™ 1 β€– β€– ( 𝜎 ) ) 𝑝 𝑑 𝜎 ≀ 𝐿 𝑝 ξ€œ 𝑑 𝑑 + 1 β€– β€– πœ™ ( 𝜎 ) βˆ’ πœ™ 1 β€– β€– ( 𝜎 ) 𝑝 𝑑 𝜎 = 𝐿 𝑝 ξ€œ 𝑑 𝑑 + 1 β€– β€– πœ™ 2 β€– β€– ( 𝜎 ) 𝑝 𝑑 𝜎 , ( 3 . 8 ) and hence for 𝑇 > 0 , 1 ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 ξ‚΅ ξ€œ 𝑑 𝑑 + 1 β€– 𝐺 ( 𝜎 ) β€– 𝑝 ξ‚Ά 𝑑 𝜎 1 / 𝑝 𝐿 𝑑 𝑑 ≀ ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 ξ‚΅ ξ€œ 𝑑 𝑑 + 1 β€– β€– πœ™ 2 β€– β€– ( 𝜎 ) 𝑝 ξ‚Ά 𝑑 𝜎 1 / 𝑝 𝑑 𝑑 . ( 3 . 9 )
Now using (3.5), it follows that l i m 𝑇 β†’ ∞ 1 ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 ξ‚΅ ξ€œ 𝑑 𝑑 + 1 β€– 𝐺 ( 𝜎 ) β€– 𝑝 ξ‚Ά 𝑑 𝜎 1 / 𝑝 𝑑 𝑑 = 0 . ( 3 . 1 0 )
Using the theorem of composition of functions of 𝑃 𝐴 𝑃 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) (see [13]) it is easy to see that Ξ¦ 𝑏 ( β‹… , πœ™ 1 ( β‹… ) ) ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) .

Theorem 3.6. Let 𝐹 = 𝐻 + Ξ¦ ∢ 𝐑 Γ— 𝐗 ↦ 𝐗 be an 𝑆 𝑝 -pseudo almost automorphic function, where 𝐻 𝑏 ∈ 𝐴 𝐴 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and Ξ¦ 𝑏 ∈ 𝐴 𝐴 0 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . Suppose that 𝐹 ( 𝑑 , 𝑒 ) and Ξ¦ ( 𝑑 , π‘₯ ) are uniformly continuous in every bounded subset 𝐾 βŠ‚ 𝐗 uniformly for 𝑑 ∈ 𝐑 . If 𝑔 ∈ 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) , then Ξ“ ∢ 𝐑 β†’ 𝐗 defined by Ξ“ ( β‹… ) ∢ = 𝐹 ( β‹… , 𝑔 ( β‹… ) ) belongs to 𝑃 𝐴 𝐴 𝑝 ( 𝐗 ) .

Proof. Let 𝐹 = 𝐻 + Ξ¦ , where 𝐻 𝑏 ∈ 𝐴 𝐴 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and Ξ¦ 𝑏 ∈ 𝐴 𝐴 0 ( 𝐑 Γ— 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . Similarly, let 𝑔 = πœ™ 1 + πœ™ 2 , where πœ™ 𝑏 1 ∈ 𝐴 𝐴 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) and πœ™ 𝑏 2 ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) .
It is obvious to see that 𝐹 𝑏 ( β‹… , 𝑔 ( β‹… ) ) ∢ 𝐑 ↦ 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) . Now decompose 𝐹 𝑏 as follows: 𝐹 𝑏 ( β‹… , 𝑔 ( β‹… ) ) = 𝐻 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ ( β‹… ) + 𝐹 𝑏 ( β‹… , 𝑔 ( β‹… ) ) βˆ’ 𝐻 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ ( β‹… ) = 𝐻 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ ( β‹… ) + 𝐹 𝑏 ( β‹… , 𝑔 ( β‹… ) ) βˆ’ 𝐹 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ ( β‹… ) + Ξ¦ 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ . ( β‹… ) ( 3 . 1 1 )
Using the theorem of composition of almost automorphic functions, it is easy to see that 𝐻 𝑏 ( β‹… , πœ™ 1 ( β‹… ) ) ∈ 𝐴 𝐴 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . Now, set 𝐺 𝑏 ( β‹… ) ∢ = 𝐹 𝑏 ( β‹… , 𝑔 ( β‹… ) ) βˆ’ 𝐹 𝑏 ξ€· β‹… , πœ™ 1 ξ€Έ . ( β‹… ) ( 3 . 1 2 )
We claim that 𝐺 𝑏 ( β‹… ) ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) . First of all, note that the uniformly continuity of 𝐹 on bounded subsets 𝐾 βŠ‚ 𝐗 yields the uniform continuity of its Bohr transform 𝐹 𝑏 on bounded subsets of 𝐗 . Since both 𝑔 , πœ™ 1 are bounded functions, it follows that there exists 𝐾 βŠ‚ 𝐗 a bounded subset such that 𝑔 ( 𝜎 ) , πœ™ 1 ( 𝜎 ) ∈ 𝐾 for each 𝜎 ∈ 𝐑 . Now from the uniform continuity of 𝐹 𝑏 on bounded subsets of 𝐗 , it obviously follows that 𝐹 𝑏 is uniformly continuous on 𝐾 uniformly for each 𝑑 ∈ 𝐑 . Therefore for every πœ€ > 0 there exists 𝛿 > 0 such that for all 𝑋 , π‘Œ ∈ 𝐾 with β€– 𝑋 βˆ’ π‘Œ β€– < 𝛿 yield β€– β€– 𝐹 𝑏 ( 𝜎 , 𝑋 ) βˆ’ 𝐹 𝑏 β€– β€– ( 𝜎 , 𝑋 ) < πœ€ βˆ€ 𝜎 ∈ 𝐑 . ( 3 . 1 3 ) Using the proof of the composition theorem [2, Theorem 2.4], (applied to 𝐹 𝑏 ) it follows l i m 𝑇 β†’ ∞ 1 ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 ξ‚΅ ξ€œ 𝑑 𝑑 + 1 β€– 𝐺 ( 𝜎 ) β€– 𝑝 ξ‚Ά 𝑑 𝜎 1 / 𝑝 𝑑 𝑑 = 0 . ( 3 . 1 4 ) Using the theorem of composition [2, Theorem 2.4] for functions of 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) it is easy to see that Ξ¦ 𝑏 ( β‹… , πœ™ 1 ( β‹… ) ) ∈ 𝑃 𝐴 𝑃 0 ( 𝐿 𝑝 ( ( 0 , 1 ) , 𝐗 ) ) .

4. Sectorial Linear Operators

Definition 4.1. A linear operator 𝐴 ∢ 𝐷 ( 𝐴 ) βŠ‚ 𝐗 ↦ 𝐗 (not necessarily densely defined) is said to be sectorial if the following holds: there exist constants πœ” ∈ 𝐑 , πœƒ ∈ ( πœ‹ / 2 , πœ‹ ) , and 𝑀 > 0 such that 𝜌 ( 𝐴 ) βŠƒ 𝑆 πœƒ , πœ” , 𝑆 πœƒ , πœ” ξ€½ | | | | ξ€Ύ , 𝑀 ∢ = πœ† ∈ β„‚ ∢ πœ† β‰  πœ” , a r g ( πœ† βˆ’ πœ” ) < πœƒ β€– 𝑅 ( πœ† , 𝐴 ) β€– ≀ | | | | πœ† βˆ’ πœ” , πœ† ∈ 𝑆 πœƒ , πœ” . ( 4 . 1 )

The class of sectorial operators is very rich and contains most of classical operators encountered in literature.

Example 4.2. Let 𝑝 β‰₯ 1 and let Ξ© βŠ‚ 𝐑 𝑑 be open bounded subset with regular boundary πœ• Ξ© . Let 𝐗 ∢ = ( 𝐋 𝑝 ( Ξ© ) , β€– β‹… β€– 𝑝 ) be the Lebesgue space.
Define the linear operator 𝐴 as follows: 𝐷 ( 𝐴 ) = π‘Š 2 , 𝑝 ( Ξ© ) ∩ π‘Š 0 1 , 𝑝 ( Ξ© ) , 𝐴 ( πœ‘ ) = 𝚫 πœ‘ , βˆ€ πœ‘ ∈ 𝐷 ( 𝐴 ) . ( 4 . 2 )
It can be checked that the operator 𝐴 is sectorial on 𝐿 𝑝 ( Ξ© ) .

It is wellknown that [14] if 𝐴 is sectorial, then it generates an analytic semigroup ( 𝑇 ( 𝑑 ) ) 𝑑 β‰₯ 0 , which maps ( 0 , ∞ ) into 𝐡 ( 𝐗 ) and such that there exist 𝑀 0 , 𝑀 1 > 0 with β€– 𝑇 ( 𝑑 ) β€– ≀ 𝑀 0 𝑒 πœ” 𝑑 , 𝑑 > 0 , ( 4 . 3 ) β€– 𝑑 ( 𝐴 βˆ’ πœ” ) 𝑇 ( 𝑑 ) β€– ≀ 𝑀 1 𝑒 πœ” 𝑑 , 𝑑 > 0 . ( 4 . 4 )

Throughout the rest of the paper, we suppose that the semigroup ( 𝑇 ( 𝑑 ) ) 𝑑 β‰₯ 0 is hyperbolic; that is, there exist a projection 𝑃 and constants 𝑀 , 𝛿 > 0 such that 𝑇 ( 𝑑 ) commutes with 𝑃 , 𝑁 ( 𝑃 ) is invariant with respect to 𝑇 ( 𝑑 ) , 𝑇 ( 𝑑 ) ∢ 𝑅 ( 𝑄 ) ↦ 𝑅 ( 𝑄 ) is invertible, and the following hold: β€– 𝑇 ( 𝑑 ) 𝑃 π‘₯ β€– ≀ 𝑀 𝑒 βˆ’ 𝛿 𝑑 β€– π‘₯ β€– f o r 𝑑 β‰₯