i would like to give an example of one sided condition of Lipschitz, but its difficult to me and i can't think of any. Besides, there is almost nothing on the internet. Does anyone have an example?
My definition of one-sided Lipschitz is given in the following theorem that you can find in R.P. Agarwal, V. Lakshmikantham (1993) Uniqueness and nonuniqueness criteria for ordinary differential equations, World Scientific:
Let $f(x, y)$ be continuous in $$\bar{S}_{+}=\{(x,y)\in\mathbb{R}^{2}: x_{0}\leq x\leq x_{0}+a, \ y_{0}-b\leq y\leq y_{0}+ b\}$$ and for all $(x, y), (x, \bar{y})$ in $\bar{S}_{+}$ with $\bar{y}\geq y$ it satisfies one - sided Lipschitz condition $$ f(x,\bar{y})-f(x,y)\leq L(\bar{y}-y)$$ Then, the initial value problem has at most one solution in $[x_0 , x_0 + a]$.