Is it possible to find a Recurrence sequence that Satisfying the following inequality

$ d_{n+k}\geq \alpha ^k d_n +\beta^k \delta(A,B),$

where $0<\alpha<1, \alpha ^k+\beta^k\geq 1~and~ k\in \Bbb{N}\cup\{ 0\}.$ With the following conditions: Let $A$ and $B$ be nonempty bounded subsets of a metric space $X.$ $T:A\cup B \to A\cup B$ is a cyclic contraction map(i.e $T(A) \subseteq B ~and ~T(B) \subseteq A.$). Let $x_0 \in A,$ define $x_{n+1}=Tx_n$ for every $n\in \Bbb{N}\cup\{ 0\}$ and let $d_n = d(x_n, x_{n+1}),$ and $$\delta(A,B)=\sup\{d(x,y):~x \in A,~y\in B\}.$$