Assume $f$ is convex and $f'$ is Lipschitz continuous. Is the sequence $(x_n)$ defined by $x_0=a$ and $x_{n+1} = x_n+\lambda f'(x_k)$ convergent?

Question

Assume $f:\mathbb R \to \mathbb R$ such that $f$ is convex and $f'$ is $L$-Lipschitz continuous. We fix $\{a, \lambda\} \subseteq \mathbb R$ and define the sequence $(x_n)$ recursively by $x_0=a$ and $x_{n+1} = x_n+\lambda f'(x_k)$.

I would like to find an example $(f,a,\lambda)$ satisfying above conditions in which the sequence $(x_k)$ is not convergent. Thank you so much!

Answer 1

Let $f(x) = x, \lambda =1$ then $x_n = a + n \lambda$.