I have seen your question on math.SE and I know how it is when you don't find the right place to ask a question. Consider this a one-time present. If you truly want to learn Mathematica, you need to put in some effort in understanding what's going on here.
Basically, your iteration can be written down almost as you have it:
s[x_] := x^Log[x];
t[x_] := 1/81*ArcSin[x - 1] + 1;
α[n_] := (n + 1)/(3 n + 5);
β[n_] := (n + 3)/(5 n + 7);
x[0] = 1.5;
x[n_] := (1 - α[n - 1])*s[x[n - 1]] + α[n - 1]*t[y[n - 1]];
y[n_] := (1 - β[n])*t[x[n]] + β[n]*s[x[n]];
ListLinePlot[Table[x[n], {n, 0, 10}], PlotRange -> All]

However, the highly recursive nature of this makes it extremely slow. When computing x[n]
, the calculation will call the x[n-1]
several times and it will be calculated over and over again.
You can speed this up using memoization, which remembers what you have already calculated. But you have to take care to call ClearAll[x]
when you change the setting of x[0]
.
To use this, replace the definition for x[n_]
with
x[n_] := x[n] = (1 - α[n - 1])*s[x[n - 1]] + α[n - 1]*t[y[n - 1]];
Again, you need to understand what's going on here and carefully look up everything that you don't know. Please check the following posts
and for your next questions, please show what you have tried and what did not work.