Chapter 6, Section 6, Problem 5 (page 118)

a = 1;
b = 0.09;
y0 = 0;

sMap[
          y_] := b + a x (y^2) + 
          a Integrate[(((y /. {x -> s}))^2), {s, 1, 
                x}];

    tMap[t_] := b + a (t^2);

nIters = 3;     
theSIters = 
      FixedPointList[sMap, y0, nIters];
theTIters = FixedPointList[tMap, 0, nIters];

Plot[
        Evaluate[Drop[theSIters, 1]], {x, 0, 1}];

tItersErr = 
        Abs[Drop[theTIters, -1] - 
            Drop[theTIters, 1]];
  
sItersErrX = 
        Drop[theSIters, -1] - 
          Drop[theSIters, 1];
  
nSamps = 100; 
sItersErr = 
        Map[Max, 
          Abs[Transpose[
              Table[sItersErrX, {x, 0, 1, N[1/nSamps]}]]]];

lp1 = 
        ListPlot[Log[tItersErr], PlotJoined -> True, 
          PlotStyle -> RGBColor[0, 1, 0], 
          DisplayFunction -> Identity];
lp2 = 
        ListPlot[Log[sItersErr], PlotJoined -> True, 
          PlotStyle -> RGBColor[1, 0, 0], 
          DisplayFunction -> Identity];
Show[{lp1, lp2}, 
        DisplayFunction -> $DisplayFunction];