The Asymptotic and Oscillatory Behaviour of Difference and Differential Equations

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.