Linear convergence rate

Question

I have a non-negative sequence $\alpha_k$ such that $$ 0 < \alpha_{k+1} \leq \left(\frac{1}{1+a}\right)^{k+1} \alpha_{0} + \sum_{j=0}^{k}{\left(\frac{1}{1+a}\right)^{k-j+1} \beta_j^2}$$ where $a >0$ and the sequence $\{\beta_j\}$ is a non-negative non-increasing sequence such that $\sum_{j=0}^{\infty} \beta_j^2 < \infty$. In the abscense of the second term on the right hand side, I would have been able to conclude that $\{\alpha_k\}$ has a linear convergence rate, but with its presence, not sure if I can still say something about the convergence of the sequence or its rate if it is convergent. I can impose additional assumptions on the sequence $\{\beta_k\}$ if needed.