Uniqueness of the Solutions of Differential Equations: An Alternative to the Lipschitz Condition

Summary of Research Project: Uniqueness of the solutions of differential equations: An alternative to the Lipschitz condition Author: Khayelihle Dube 1. Objective. The general setting of the project is the differential equation dy dx =f(x, y), where f is defined and continuous on an open set D⊆R2. The local existence of a solution through any point (x0, y0)∈Dfollows from the assumed continuity of f. The project focuses on the uniqueness theory, namely conditions providing a unique solution given an initial condition y(x0) = y0, where (x0, y0)∈D. The goal is to establish uniqueness using a geometric condition—strict monotonicity of falong a conical region around the tangent direction—rather than the traditional Lipschitz condition. 2. Motivation. The project provides a thorough review of the existing uniqueness theory, which is based on the Lipschitz condition and its various generalizations, namely one-sided Lipschitz, Osgood, and FitzHugh–Nagumo conditions. All these conditions are sufficient, but not necessary. Therefore, the goal of establishing a new type of condition, which is not based on limiting the growth in yis relevant. This is also illustrated by examples, where the mentioned conditions are violated, but nevertheless the uniqueness property is satisfied. 3. Main Results. The key idea is that the strict monotonicity of fin the direction of the tangent vector at a given point within a narrow cone can prevent two distinct solution curves emerging from this point. This geometric insight replaces quantitative bounds with a qualitative, order-preserving property. Theorem 1 (Special case). Let (0,0) ∈Dand let f(0,0) = 0. If there exists δ > 0, α > 0 such that U={(x, y) : x∈(0, δ),|y|< αx} ⊆ Dand the function f(x, y) is a strictly monotone (either increasing or decreasing) function on Uwith respect to xand f(x, y)= 0, (x, y)∈U, then there exists γ > 0 so that the initial value problem dy dx =f(x, y), y(0) = 0 has at most one solution on the interval [0, γ). Analogous result (Theorem 2) holds for backward uniqueness (x≤0). The results are generalized to arbitrary initial points and non-horizontal tangents as follows. Theorem 4. For any point (x0, y0)∈D, uniqueness in the forward direction holds if fis strictly monotone along the tangent vector 1 f(x0, y0)and nonzero in a slanted cone: U={(x, y) : x∈(x0, x0+δ),|y−y0−f(x0, y0)(x−x0)|< α(x−x0)} ⊆ D Theorem 5. (Two-sided uniqueness): Combines forward and backward cones, allowing different monotonicity types in each direction. The application of these results is illustrated on several differential equations dy dx =x+|y|1/2,dy dx = 1 + |y|1/2dy dx =x± |y|1/3. In these examples, fis not Lipschitz around any point on the x-axis. However, the stated theorems are applicable and yield uniqueness of the solution. For instance, the function f(x, y) = x+|y|1/2is strictly increasing in xand non-zero on the forward cone Uin Theorem 1 and thus, ensuring forward uniqueness for the initial condition y(0) = 0. Contradiction: two solutions ϕ1,ϕ2vs. fincreasing in x. 4. Proof techniques. Geometrically, in the setting of Theorem 1, if two distinct solutions existed, the monotonicity of fwould be violated, leading to a contradiction, see the figure. Inverse function theorems are used to rigorously derive the proof. Theorems 4 and 5 are derived from Theorem 1 via coordinate transformation. 5. Contribution. This work introduces a monotonicity-based uniqueness criterion that is both geometrically natural and widely applicable. Similarly to the Lipschitz condition, it is only a sufficient condition. However, it can be applied at points where the Lipschitz condition fails. In general, it is not a replacement for the Lipschitz condition, but rather, in joint application with the Lipschitz condition, it can expand the class of differential equations for which uniqueness can be rigorously established.