Positive Periodic Solution for Second-Order Nonlinear Differential Equations with Variable Coefficients and Mixed...

Area of Science:

• Differential Equations
• Nonlinear Analysis
• Mathematical Physics

Background:

• Second-order nonlinear differential equations with variable coefficients and mixed delays are crucial in modeling complex dynamical systems.
• Establishing the existence of periodic solutions is fundamental for understanding system stability and behavior.
• Existing models often lack the generality to encompass a wider range of phenomena.

Purpose of the Study:

• To investigate the existence of positive periodic solutions for a specific class of second-order nonlinear differential equations.
• To extend existing theoretical frameworks by considering variable coefficients and mixed delays.
• To develop a more broadly applicable mathematical model for dynamical systems.

Main Methods:

• Application of Krasnoselskii's fixed point theorem.
• Analysis of second-order nonlinear differential equations with variable coefficients.
• Incorporation of mixed delay terms into the differential equations.

Main Results:

• Existence of at least one positive periodic solution is proven.
• The established results are applicable to a more general class of equations than previously studied.
• The findings provide a theoretical foundation for analyzing systems with complex dynamics.

Conclusions:

• The study successfully demonstrates the existence of positive periodic solutions for the investigated equations.
• The generalized approach enhances the applicability of the findings to diverse scientific and engineering fields.
• This work contributes to the advancement of nonlinear analysis and the theory of differential equations.