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Article
Positive periodic solutions for certain kinds of delayed q-difference equations with biological background
This paper specifically focuses on a specific type of q-difference equations that incorporate multiple delays. The main objective is to explore the existence of positive periodic solutions using coincidence degre...
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Chapter and Conference Paper
Weighted Norms In Advanced Volterra Difference Equations
In this research we explore the existence of bounded solutions, and periodic solutions of Advanced type Volterra difference equations of the form $$\b...
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Chapter
Ordinary Dynamical Systems
In this chapter, we analyze boundedness and stability of solutions of ordinary dynamical systems. We will develop general theorems in which Lyapunov functions play an important role. We begin by considering so...
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Chapter
Volterra Integro-Dynamic Equations
This chapter is exclusively devoted to the study of Volterra integro-dynamic equations with or without delay. We will display some exotic Lyapunov functionals to obtain stability and instability of the zero so...
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Chapter
Volterra Integral Dynamic Equations
In this chapter, we apply the concept of resolvent that we developed in Sect. 1.4.1 for vector Volterra integral dynamic equations and show the boundedness of solutions. The resolvent is an abstract term which...
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Chapter
Periodicity Using Shift Periodic Operators
In Chap. 7, we had to require the time scale to be additive in order to show the existence of periodic solutions. We devote this chapter to the study of of functional delay dynamical systems. The periodic de...
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Chapter
Introduction to Stability and Boundedness in Dynamical Systems
In this chapter, we provide a brief introduction to time scale calculus and introduce fundamental concepts that we need throughout this book. In Sect. 1.2 based on the work of Peterson and Tisdell (J Differ Eq...
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Chapter
Functional Dynamical Systems
In this chapter we consider functional dynamical systems on time scales that we apply to Volterra integro-dynamic equations on time scales. Our general theorems will require the construction of suitable Lyap...
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Chapter
Exotic Lyapunov Functionals for Boundedness and Stability
This chapter is devoted primarily to the construction of exotic Lyapunov functionals for the study of boundedness, stability and exponential stability. Necessary and sufficient conditions for scalar and vector...
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Chapter
Periodic Solutions: The Natural Setup
Among time scales, periodic ones deserve a special interest since they enable researchers to develop a theory for the existence of periodic solutions of dynamic equations on time scales (see for example Bi et ...
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Chapter and Conference Paper
Lyapunov Functionals and Stability in Finite Delays Difference Equations
In this research we prove general theorems regarding the stability of the zero solution of a functional difference equation with finite delay. In the analysis we assume the existence of a Lyapunov functional t...
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Article
Hilger-type impulsive differential inequality and its application to impulsive synchronization of delayed complex networks on time scales
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Book
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Chapter
Stability and Boundedness
In this chapter we provide a brief introduction to difference calculus including basic material on Volterra difference equations. Using the z-transform we state some known theorems regarding stability of the zero...
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Chapter
Fixed Point Theory in Stability and Boundedness
In the past hundred and fifty years, Lyapunov functions/functionals have been exclusively and successfully used in the study of stability and existence of periodic and bounded solutions. The author has extensi...
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Chapter
Population Dynamics
This chapter is devoted to the application of Volterra difference equations in population dynamics and epidemics. We begin the chapter by introducing different types of population models including predator-pre...
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Chapter
Functional Difference Equations
In this chapter we consider functional difference equations that we apply to all types of Volterra difference equations. Our general theorems will require the construction of suitable Lyapunov functionals, a ...
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Chapter
Periodic Solutions
This chapter is devoted to the study of periodic solutions of functional difference systems with finite and infinite delay. We will obtain different results concerning Volterra difference equations with fin...
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Chapter
Exponential and l p-Stability in Volterra Equations
This chapter is devoted primarily to the exponential and lp-stability of Volterra difference equations. Lyapunov functionals are the main tools in the analysis. It is pointed out that in the case of exponential s...
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Chapter and Conference Paper
The Case for Large Contraction in Functional Difference Equations
In this note we review some of the latest research on the qualitative analysis of solutions of difference equations using fixed point theory and Lyapunov functionals. It turns out that the use of fixed point t...
