I tried it make it solved and plot but im confused, NDsolve, DSolve, ParametricNDSolve didnt work. Please help me, and please write anything.Thank you.`

Clear[fList, KList, \[Phi]List, LList, A, f1, f2, x, t, gList, p] 
fList[s_, 
  a_] = {-0.3 x1[s] . (1/s) . ((Log[e, (t/s)])^(-a)), -0.4 x2[
     s] . (1/s) . ((Log[e, (t/s)])^(-a))}; 
KList[s_, 
  a_] = {{-0.04 . (1/s) . ((Log[e, (t/s)])^(-a)), 
   0.02 . (1/
      s) . ((Log[e, (t/s)])^(-a))}, {-0.02 . (1/
       s) . ((Log[e, (t/s)])^(-a)), -0.03 . (1/
       s) . ((Log[e, (t/s)])^(-a))}};
gList[s_, a_] = {0.04 . (1/s) . ((Log[e, (t/s)])^(-a)), 
   0.01 . (1/s) . ((Log[e, (t/s)])^(-a))};
f1[x_] := 1/((x^2) + 1);
f2[x_] := (1/2) (Abs[x - 1] - Abs[x + 1]);
g[t_] := {f1[x1[t]], f1[x2[t]]};
\[Phi]List[t_] := {x1[t], x2[t]};
A[a_] := 1/Gamma[1 - a];
B[s_, a_] := (1/s) . ((Log[e, (t/s)])^(-a));
Needs["PlotLegends`"]
\[Alpha] = 0.5
intsyseqn = 
  Thread[\[Phi]List[t] == 
     A[1/2] . 
      Integrate[
       gList[s, 1/2] + fList[s, 1/2] + 
        KList[s, 1/2] . f1[\[Phi]List[t]], {s, 1, t}]] /. {Integrate[
      f_ + g_ + h_ + r_, t_] :> 
     Integrate[f, t] + Integrate[g, t] + Integrate[h, t] + 
      Integrate[r, t]};
{tmin, tmax} = {1, 1000};
solx1 = ParametricNDSolveValue[{intsyseqn, x1[tmin] == 0, 
    x2[tmin] == 0}, {t, tmin, tmax}, {x1, x2}];
xsıfırbes = 
 Plot[Evaluate[{x1[t]} /. solx1], {t, 1, 1000}, 
  PlotRange -> {0.0013, 0.0016}, PlotStyle -> {Black}, 
  PlotLegend -> {"\[Alpha]=0.1"}, LegendPosition -> {0.4, -0.2}, 
  LegendShadow -> None, LegendSize -> {0.5, 0.3}]

Attachments:

son1.nb