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2018 | Buch

Stability and Boundedness Kapitel lesen Erstes Kapitel lesen

Qualitative Theory of Volterra Difference Equations

verfasst von: Youssef N. Raffoul

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book provides a comprehensive and systematic approach to the study of the qualitative theory of boundedness, periodicity, and stability of Volterra difference equations. The book bridges together the theoretical aspects of Volterra difference equations with its applications to population dynamics. Applications to real-world problems and open-ended problems are included throughout.

This book will be of use as a primary reference to researchers and graduate students who are interested in the study of boundedness of solutions, the stability of the zero solution, or in the existence of periodic solutions using Lyapunov functionals and the notion of fixed point theory.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Stability and Boundedness
Abstract
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 solution of Volterra difference equations of convolution types. We move on to introducing Lyapunov functions for autonomous difference equations and state some known results concerning stability and boundedness. In Section 1.3 we introduce the concept of total stability and its correlation with uniform asymptotic stability for perturbed Volterra difference equations.
Youssef N. Raffoul
Chapter 2. Functional Difference Equations
Abstract
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 task that is difficult but possible. As we have seen in Chapter 1, the concept of resolvent can only apply to linear Volterra difference systems. The theorems on functional difference equations will enable us to qualitatively analyze the theory of boundedness, uniform ultimate boundedness, and stability of solutions of vectors and scalars Volterra difference equations. We extend and prove parallel theorems regarding functional difference equations with finite or infinite delay, and provide many applications. In addition, we will point out the need of more research in delay difference equations. In the second part of the chapter, we state and prove theorems that guide us on how to systematically construct suitable Lyapunov functionals for a specific nonlinear Volterra difference equation. We end the chapter with open problems. Most of the results of this chapter can be found in [37, 38, 128, 133, 135, 141, 147, 181], and [182].
Youssef N. Raffoul
Chapter 3. Fixed Point Theory in Stability and Boundedness
Abstract
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 extensively used Lyapunov functions/functionals for the purpose of analyzing solutions of functional equations, and each time the suitable Lyapunov functional presented us with unique difficulties, that could only overcome by the imposition of severe conditions on the given coefficients.
Youssef N. Raffoul
Chapter 4. Periodic Solutions
Abstract
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 finite and infinite delays, using fixed point theory. Fixed point theory will enable us to obtain results concerning stability, classification of solutions, existence of positive solutions, and the existence of periodic solutions and positive periodic solutions. In the analysis, we make use of Schaefer fixed point theorem, [159], Krasnoselskii’s fixed point theorem, [97], and Schauder fixed point theorem.
Youssef N. Raffoul
Chapter 5. Population Dynamics
Abstract
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-prey models. Most commonly studied version of population models are described by continuous-time dynamics, whereas in real ecosystem the changes in populations of each species due to competitive interaction cannot occur continuously. Hence, discrete-time dynamical systems are often more suitable tool for modeling the dynamics in competing species. Cone theory is introduced and utilized to prove the existence of positive periodic solutions for functional difference equations. We introduce an infinite delay population model which governs the growth of population N(n) of a single species whose members compete among themselves for the limited amount of food that is available to sustain the population, and use the results on cone theory to obtain the existence of a positive periodic solution. Moreover, from a biologist’s point of view, the idea of permanence plays a central role in any competing species.
Youssef N. Raffoul
Chapter 6. Exponential and l p-Stability in Volterra Equations
Abstract
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 stability, Lyapunov functionals are hard to extend to vector Volterra difference equations or to Volterra difference equations with infinite delay. In addition, we use nonstandard discretization scheme due to Mickens [122] and apply them to continuous Volterra integro-differential equations. We will show that under the discretization scheme the stability of the zero solution of the continuous dynamical system is preserved. Also, under the same discretization, using a combination of Lyapunov functionals, Laplace transforms, and z-transforms, we show that the boundedness of solutions of the continuous dynamical system is preserved.
Youssef N. Raffoul
Backmatter
Metadaten
Titel
Qualitative Theory of Volterra Difference Equations
verfasst von
Youssef N. Raffoul
Copyright-Jahr
2018
Electronic ISBN
978-3-319-97190-2
Print ISBN
978-3-319-97189-6
DOI
https://doi.org/10.1007/978-3-319-97190-2

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