Using this: Answer form user - Daniel Lichtblau we have:
$$f(x)=\frac{r \left(\frac{1}{\alpha }-1\right)}{\text{nn}-1}-\frac{r (1-p) \exp \left(n \int_0^x f(t) \, dt\right)}{(\text{nn}-1) p}$$
a = 0;(*Assuming constants*)
b = 1;
r = 5;
p = 1/3;
nn = 5;
α = 10;
term = 8;
kernel[x_] := r/(nn - 1)*(1/α - 1)
func[x_, 0] := kernel[x]
ifunc[0][x_] := kernel[x]
func[x_?NumericQ, n_Integer] := kernel[x] - r/(nn - 1)*(1 - p)/p*
Exp[nn*NIntegrate[ifunc[n - 1][y], {y, 0, x}, MinRecursion -> 2]]
ifunc[j_Integer /; j >= 1] := ifunc[j] =
Module[{vals}, vals = Table[{x, func[x, j]}, {x, a, b, 0.002}];
Interpolation[vals]]
Plot[Evaluate[Table[ifunc[j][x], {j, 0, term}]], {x, a, b},
PlotStyle -> {Blue, Red, Green, Purple, Black}]
Plot[Evaluate[Table[ifunc[j][x], {j, 0, term}]][[-1]], {x, a, b},
PlotRange -> {Automatic, {-2, -1}}]
I did eighth iterations to this process.

A numeric check with random numbers:
g[x_, n_] := (-ifunc[n][x] + r/(nn - 1) (1/α - 1) - r/(nn - 1)*(1 - p)/p*
Exp[nn*NIntegrate[ifunc[n][y], {y, 0, x}, MinRecursion -> 2]]);
Table[g[RandomReal[{0, 1}, 10][[k]], term], {k, 1, 10}]
(* {-4.78826*10^-6, -7.19387*10^-7, -5.44634*10^-6, -1.1502*10^-7,
-3.94742*10^-6, -3.83767*10^-6, -6.26247*10^-6, -1.42423*10^-6,
-1.89644*10^-6, -3.32356*10^-7}*)
DSolve
doesn't do integro-differential equations. $\endgroup$ – march Sep 21 '17 at 20:07DSolve
can't solve. $\endgroup$ – Mariusz Iwaniuk Sep 21 '17 at 20:51