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 ‎$$\delta(A,B)=\sup\{‎d‎(x,y):~x \in A,~y\in B\}.$$

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    $\begingroup$ If its a contraction we have $d_n\rightarrow 0$ but then for $n$ large enough we have $d_{n+k}\geq \alpha^kd_n+\beta^k\delta(A,B)\geq (\alpha^k+\beta^k)d_n\geq d_n$, a contradiction. Did I missunderstood something? $\endgroup$ – user35593 Nov 11 '15 at 19:12

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