Let $(\alpha,\gamma,\beta)$ real number such that $\alpha+\gamma+\beta=1$, let $(e_n)_{n\in\mathbb{N}}$ be a summable non-negative sequence and let $(u_n)$ be a non-negative sequence such that

$\forall n\in\mathbb{N}$, $u_{n+3}\leq \alpha u_{n+2}+\beta u_{n+1} +\gamma u_n+e_n$.

I'm looking for sufficient conditions on $(\alpha,\beta,\gamma)$ to obtain the convergence of $(u_n)$.

Suppose that now $\forall n\in\mathbb{N}$, $u_{n+3}\leq \alpha_n u_{n+2}+\beta_n u_{n+1} +\gamma_n u_n+e_n$. Can we obtain sufficient conditions on the sequences $(\alpha_n)$,$(\beta_n)$, and $(\gamma_n)$ ?

For example, can we prove the convergence of the sequence $(u_n)$ such that $\forall n\in\mathbb{N}$, $u_{n+3}\leq \frac{3}{2} u_{n+2} -u_{n+1} +\frac{1}{2} u_n+e_n$.

Thank you !


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