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Nonautonomous Young Differential Equations Revisited.

Nguyen Dinh Cong¹, Luu Hoang Duc^1,2, Phan Thanh Hong³

¹1Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam.

Journal of Dynamics and Differential Equations

|November 22, 2018

Summary

This study proves that nonautonomous Young differential equations have a unique, continuously dependent solution under mild conditions. The research utilizes advanced mathematical techniques for robust findings in differential equations.

Keywords:

Fractional Brownian motion (fBm)Stochastic differential equations (SDE)Young integral p-variation

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Area of Science:

Differential Equations
Mathematical Analysis

Background:

Nonautonomous differential equations present unique challenges in solution analysis.
Understanding solution behavior concerning initial conditions is crucial for applications.

Purpose of the Study:

To establish the existence and uniqueness of solutions for nonautonomous Young differential equations.
To demonstrate the continuous dependence of these solutions on initial conditions.

Main Methods:

Utilizing estimates in p-variation norms.
Constructing a greedy sequence of times.
Applying a Gronwall-type lemma and Schauder's fixed-point theorem.

Main Results:

A unique solution is proven to exist for the nonautonomous Young differential equation.
The solution exhibits continuous dependence on the initial conditions.

Conclusions:

The findings confirm the well-posedness of nonautonomous Young differential equations under specified conditions.
The employed mathematical framework provides a rigorous basis for analyzing such equations.