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Theory of Translation Closedness for Time Scales

Part of the Developments in Mathematics Book Series (volume 62)

DOI: 10.1007/978-3-030-38644-3

ISBN: 978-3-030-38643-6

Publisher: Springer International Publishing - Springer Journals

May 6, 2020

1 - 9 of 9 Chapters

LitStream

Chapter 1Preliminaries and Basic Knowledge on Time Scales

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 1 –50

[A time scale is an arbitrary nonempty closed subset of the real numbers. The theory of time scales calculus is a powerful tool not only to unify continuous and discrete analysis but also to model various natural phenomena that occur on hybrid time domains. In this chapter, some necessary...
LitStream

Chapter 2A Classification of Closedness of Time Scales Under Translations

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 51 –167

[The study of almost periodic and almost automorphic dynamic equations on time scales reveals some interesting dynamical behavior. Before studying almost periodicity and almost automorphy of dynamical behavior of solutions for dynamic equations, the first and important thing is to introduce and...
LitStream

Chapter 3Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 169 –237

[This Chapter mainly deals with almost periodic functions and its generalizations on complete-closed time scales under translations and it is divided into six sections and is organized as follows. In Sect. 3.1, some basic results of almost periodic functions are established which include their...
LitStream

Chapter 4Piecewise Almost Periodic Functions and Generalizations on Translation Time Scales

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 239 –281

[This chapter is mainly concerned with piecewise almost periodic functions and their generalizations on various types of translation time scales and is divided into four sections. Section 4.1 establishes some basic results of piecewise almost periodic functions on complete-closed time scales...
LitStream

Chapter 5Almost Automorphic Functions and Generalizations on Translation Time Scales

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 283 –336

[Based on the theory of translation time scales we develop a theory of almost automorphic functions and their generalizations in this chapter. In Sect. 5.1, the almost automorphic functions on complete-closed time scales are discussed and some basic properties are established. In Sect. 5.2, on...
LitStream

Chapter 6Nonlinear Dynamic Equations on Translation Time Scales

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 337 –387

[Since time scales can be divided into several types depending on their translations, it is very important to study the dynamical behavior of solutions for nonlinear dynamic equations with the properties of corresponding translation functions. In this chapter, different forms of generalized...
LitStream

Chapter 7Impulsive Dynamic Equations on Translation Time Scales

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 389 –476

[Some phenomena in nature are characterized by the fact that their states are subject to sudden changes at certain moments and therefore can be described by impulsive system. In this chapter, we discuss impulsive dynamic equations on translation time scales. In Sect. 7.1, the Cauchy matrix and...
LitStream

Chapter 8Almost Automorphic Dynamic Equations on Translation Time Scales

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 477 –504

[This chapter is mainly concerned with almost automorphic and generalized solutions for dynamic equations of different types on translation time scales and the chapter is divided into four sections. In Sect. 8.1, the existence of weighted pseudo almost automorphic solutions of abstract dynamic...
LitStream

Chapter 9Analysis of Dynamical System Models on Translation Time Scales

Wang, Chao; Agarwal, Ravi P.; O’Regan, Donal; Sakthivel, Rathinasamy

Pages: 505 –561

[This chapter is divided into three sections. In Sect. 9.1, we discuss the exponential dichotomies for impulsive dynamic equations on time scales and exponential type of bounds of solutions for three representative impulsive dynamic equations are derived, some new mean-value criteria for...

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