RSolve conditional recurrence expression

Question

I have trouble in expressing condition recurrence in Mathematica.

I have a conditional recurrence relational, that is conditioned on a function value (defined on sequence index).

where $a_1, a_2, c, b$ are constants.

I tried using the 'Min' function in Mathematica for solving the recurrence relation but it doesn't work. Any idea as to how I can express this such that it can be solved by Mathematica.

Answer 1

Manipulate[
 Module[{soln},
  soln = RSolve[{x[k + 1] == x[k] +
        Piecewise[{
          {a1 b^k, x[k] >= c b^k}},
         a2 b^k],
      x[0] == x0}, x[k], k][[1]];
  If[Length[soln] == 1,
   DiscretePlot[x[k] /. soln, {k, 0, 6},
    PlotLabel -> Style[soln[[1]] // TraditionalForm, 14]],
   $Failed]],
 {{x0, 0, Subscript[x, 0]}, Range[10],
  ControlType -> SetterBar},
 Row[{
   Control[{{a1, 1, Subscript[a, 1]}, Range[5]}],
   Spacer[10],
   Control[{{a2, 1, Subscript[a, 2]}, Range[5]}],
   Spacer[10],
   Control[{{b, 1}, Range[5]}],
   Spacer[10],
   Control[{{c, 1}, Range[5]}]}]]