Existence and Uniqueness Theorem of Higher-Order Uncertain Differential Equation with Local Lipschitz Condition

As a type of differential equation driven by uncertain processes, uncertain differential equations are fundamental to modeling dynamic systems under uncertainty. While significant research attention has been directed towards this field, the theoretical foundations for their more complex forms, specifically multi-dimensional and higher-order uncertain differential equations, require further rigorous development. This paper addresses this gap by establishing comprehensive existence and uniqueness theorems for solutions to both multi-dimensional and higher-order uncertain differential equations under the condition of local Lipschitz continuity. We provide rigorous proofs for these theorems, thereby solidifying the mathematical underpinnings necessary for the application of these models.