u = 2;
c = 3000;
it = 3;
Array[h, {u, c}];
Do[Do[h[i, k] = 0, {i, u}], {k, c}];
F[x_] := Sin[x];
AA[x_] := 1/(x^2 + 1);
w[x_] := F[x] + AA[x];
v[x_] := Sum[w[x + i*2 Pi], {i, -it, it}];

Do[n = 1;
  If[j == 1,
   While[n <= c,
     h[j, n] =
      NDSolveValue[{y''[x] == Cos[y[x]], y'[0] == 0, y[0] == n},
       y, {x, 0, 1},
       "Method" -> {"EquationSimplification" -> {Automatic,
           "TimeConstraint" -> Infinity}}, AccuracyGoal -> 3]; n++];,
   While[n <= c, h[j, n] = NDSolveValue[{y''[x] == Cos[y[x]]
          + Sum[v[h[j - 1, n][x] - h[j - 1, i][x]], {i, 1, c}],
        y'[0] == 0, y[0] == n}, y, {x, 0, 1},
       "Method" -> {"EquationSimplification" -> {Automatic,
           "TimeConstraint" -> Infinity}}, AccuracyGoal -> 3.];
     n++];];, {j, u}];