This thesis deals with the asymptotic and oscillatory behaviour of the solutions of certain differential and difference equations.
It mainly consists of three parts. The first part is to study the asymptotic behaviour of certain differential equations. The second part is to look for oscillatory criteria for certain nonlinear neutral differential equations. And the third part is to establish new criteria for a class of nonlinear neutral difference equations of any order with continuous variable and another type of higher even order nonlinear neutral difference equations to be oscillatory.
A functional differential equation is a differential equation involving the values of the unknown functions at present, as well as at past or future time. The word “time” here stands for the independent variable. In the thesis, the concept of a functional differential equation is confined to ordinary differential equations, although it suits partial ones as well. Functional differential equations can be classified into four types according to their deviations: retarded, advanced, neutral and mixed.
A neutral equation is one in which derivative of functionals of the past history and the present state are involved, but no future states occur in the equation. The order of a differential equation is the order of the highest derivative of the unknown function.
A difference equation is a specific type of recurrence relation, which is an equation that defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. On the other hand, difference equations can be thought of as the discrete analogue of the corresponding differential equations.
By analogy with differential equations, difference equations also can be classified into four types: delay, advanced, neutral, and mixed. The order of a difference equation is the difference between the largest and the smallest values of the integer variable explicitly involved in the difference equation.
Inhaltsverzeichnis (Table of Contents)
- INTRODUCTION
- TERMINOLOGY
- PRELIMINARIES
- BACKGROUND AND HISTORY REVIEW
- MOTIVATION FOR THE THESIS
- OUTLINE OF THE THESIS
- STABILITY OF DIFFERENTIAL EQUATIONS
- INTRODUCTION
- MAIN RESULT
- THE PROOF
- CONCLUSION
- OSCILLATION OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS
- INTRODUCTION
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This thesis examines the asymptotic and oscillatory behavior of solutions to certain differential and difference equations. It consists of three main sections: an analysis of the asymptotic behavior of differential equations, an investigation into oscillatory criteria for nonlinear neutral differential equations, and a study of new criteria for oscillatory behavior in nonlinear neutral difference equations of various orders.
- Asymptotic behavior of differential equations
- Oscillatory criteria for nonlinear neutral differential equations
- New criteria for oscillatory behavior in nonlinear neutral difference equations
- Stability of solutions to differential equations
- Bounded oscillation, bounded almost oscillation, and almost oscillation of solutions
Zusammenfassung der Kapitel (Chapter Summaries)
- Introduction: This chapter introduces the terminology, preliminaries, background, motivation, and outline of the thesis. It provides a framework for understanding the research undertaken.
- Stability of Differential Equations: This chapter focuses on the stability of solutions to first-order differential systems. It establishes sufficient conditions for the stability of solutions based on the properties of the functions involved.
- Oscillation of Nonlinear Neutral Differential Equations: This chapter investigates the oscillatory behavior of second-order nonlinear neutral differential equations. It explores different types of oscillation and provides criteria for determining their presence.
Schlüsselwörter (Keywords)
Key terms and concepts in this thesis include: asymptotic behavior, oscillatory behavior, differential equations, difference equations, neutral equations, nonlinear equations, stability, oscillation, bounded oscillation, bounded almost oscillation, almost oscillation, Riccati's technique.
- Quote paper
- Shuhui Wu (Author), 2009, The Asymptotic and Oscillatory Behaviour of Difference and Differential Equations, Munich, GRIN Verlag, https://www.grin.com/document/1176159
