Solving delayed differential equations with Mathematica

Question

Attempting to solve a delayed differential equation with the given model below, which has the structure of h[x[t - 1]] - .88*x[t]. We have defined h[x_,t_] as a piecewise below, that gives the value 3.52 if x[t-1] is between 1 and 2, but gives 0 when not in that interval. When we run the entire code, it gives us error code "DSolve::bvnul: For some branches of the general solution, the given boundary conditions lead to an empty solution." We are trying to model this for t between [0,20] When we run the same code but with the structure x[t-1]-.88*x[t] we do not receive an error code. Please help.

h[x_, t_] := 
  Piecewise[{{3.52, 1 <= x[t - 1] <= 2}, {0, x[t - 1] > 2}, {0, 
 x[t - 1] < 1}}];
model = h[x[t - 1]] - .88*x[t];
g[t_] := 2;
tMin = 0;
tMax = 20;
DiscontinuityTree[t0_, Tend_, delays_] := 
 Module[{dt, next, ord}, ord[t_] := Infinity; ord[t0] = 0; 
  next[b_, order_, del_] := Map[dt[b, #, order, del] &, del]; 
  dt[t_, {d_, nq_}, order_, del_] := 
   Module[{b = t + d}, 
    If[b <= Tend, o = order + Boole[! nq]; ord[b] = Min[ord[b], o]; 
     Sow[{t -> b, d}]; next[b, o, del]]]; 
  rules = Reap[next[t0, 0, delays]][[2, 1]]; 
  rules = Tally[rules][[All, 1]]; f[x_?NumericQ] := {x, ord[x]}; 
  f[a_ -> b_] := f[a] -> f[b]; 
  rules[[All, 1]] = Map[f, rules[[All, 1]]]; rules]
tree = Tally[
   DiscontinuityTree[0, 8, {{1, True}, {\[Pi], False}}]][[All, 1]]
IntegrateSmooth[rhs_, history_, delayvars_, pfun_, 
  dvars_, {t_, t0_, t1_}] := 
 Module[{delayvals, dvt, tau, hrule, dvrule, dvrules, oderhs, ode, 
   init, sol}, dvt[tau_] = Map[#[tau] &, dvars]; 
  hrule[pos_] := 
   Thread[dvars -> Map[Function[Evaluate[{t}], #] &, history[[pos]]]];
   dvrule[(dv_)[z_]] := 
   Module[{delay, pos}, delay = t - z; pos = pfun[t0 - delay]; 
    dv[z] -> (dv[z] /. hrule[pos])]; dvrules = Map[dvrule, delayvars];
   oderhs = rhs /. dvrules; ode = Thread[D[dvt[t], t] == oderhs]; 
  init = Thread[dvt[t0] == (dvt[t0] /. hrule[-1])]; 
  sol = DSolve[{ode, init}, dvars, t]; 
  If[Head[sol] === DSolve || Length[sol] == 0, 
   Message[DDESteps::stuck, ode, init]; Throw[$Failed]]; 
  dvt[t] /. First[sol]]; DDESteps::stuck = "Not Working";
sol = DDESteps[h[x[t - 1]] - .88*x[t], 2, x, {t, 0, 3}]