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Combined Measure and Shift Invariance Theory of Time Scales and Applications

Part of the Developments in Mathematics Book Series (volume 77)

DOI: 10.1007/978-3-031-11619-3

ISBN: 978-3-031-11618-6

Publisher: Springer International Publishing - Springer Journals

Jun 28, 2022

1 - 9 of 9 Chapters

LitStream

Chapter 1Riemann Integration, Stochastic Calculus, and Shift Operators on Time Scales

Wang, Chao; Agarwal, Ravi P.

Pages: 1 –115

[This chapter mainly introduces basic knowledge of calculus on time scales. In Sect. 1.1, concepts and fundamental properties of Riemann delta and nabla integration on time scales are introduced including some basic results of Riemann integral and fundamental theorems of calculus. In Sect. 1.2,...
LitStream

Chapter 2♢α-Measurability and Combined Measure Theory on Time Scales

Wang, Chao; Agarwal, Ravi P.

Pages: 117 –195

[In this chapter, combined measure theory on time scales is established including ♢α-measurability, ♢α-measurable functions, Lebesgue-Stieltjes combined ♢α-measure, and related integrals. In Sects. 2.1–2.3, concept of ♢α-measurability is introduced and combined measure theory on time scales is...
LitStream

Chapter 3Shift Invariance and Matched Spaces of Time Scales

Wang, Chao; Agarwal, Ravi P.

Pages: 197 –226

[In this chapter, the concept of matched spaces of time scales is introduced and their basic properties are established. Based on this, the fundamental theory of shift invariance of time scales is addressed. In Sect. 3.1, the knowledge of periodic time scales with shift operators is introduced,...
LitStream

Chapter 4Almost Periodic Functions Under Matched Spaces of Time Scales

Wang, Chao; Agarwal, Ravi P.

Pages: 227 –276

[This chapter is mainly devoted to establish a theory of almost periodic functions through the theory of matched spaces of time scales. In Sect. 4.1, we present some basic notions and properties of periodic functions under the complete-closed time scales in shifts. In Sect. 4.2, a theory of...
LitStream

Chapter 5Almost Automorphic Functions Under Matched Spaces of Time Scales

Wang, Chao; Agarwal, Ravi P.

Pages: 277 –322

[In this chapter, based on the theory of matched space of time scales, we develop a theory of almost automorphic functions. In Sect. 5.1, a notion of weighted pseudo δ-almost automorphic functions under matched spaces is introduced and studied. In Sect. 5.2, a new generalized type of almost...
LitStream

Chapter 6C0-Semigroup and Stepanov-Like Almost Automorphic Functions on Hybrid Time Scales

Wang, Chao; Agarwal, Ravi P.

Pages: 323 –345

[In this chapter, some basic notions of C0-semigroup and Stepanov-like almost automorphic functions in matched spaces of time scales are introduced and studied. In Sect. 6.1, the concept of C0-semigroup on a quantum time scale is proposed and some basic properties are established. In Sects 6.2...
LitStream

Chapter 7Almost Periodic Dynamic Equations Under Matched Spaces

Wang, Chao; Agarwal, Ravi P.

Pages: 347 –367

[In this chapter, based on the theory of δ-almost periodic functions and n0-order Δ-almost periodic functions developed in Chap. 4, some fundamental results of the δ-almost periodic solutions and n0-order Δ-almost periodic solutions of a general dynamic equations are established under matched...
LitStream

Chapter 8Almost Automorphic Dynamic Equations Under Matched Spaces

Wang, Chao; Agarwal, Ravi P.

Pages: 369 –395

[In this chapter, based on the theory established in Chap. 5, two types of almost automorphic solutions of dynamic equations under the matched spaces are discussed systematically. In Sects. 8.1–8.2, the weighted pseudo δ-almost automorphic solutions and the n0-order weighted pseudo Δ-almost...
LitStream

Chapter 9Applications to Dynamics Models Under Matched Spaces

Wang, Chao; Agarwal, Ravi P.

Pages: 397 –421

[In this chapter, based on the theory of almost periodic functions with shift operators proposed in the previous chapters, we will introduce some new types of neutral impulsive stochastic dynamical models on irregular time scales. In Sect. 9.1, by using almost periodic results of stochastic...

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