Thank you for the text of your problem. That was very helpful.
I believe there were slight differences between your text and your image. I assumed the text was supposed to match the image and made changes to accomplish that.
a=1.1; b=0.02; c=0.035; d=1.0; k=0.3; e=0.5; f=0.2; α=0.98; h=1; n=1+t;U[0]=0.3; V[0]=0.2; W[0]=0.5;
S1[t_]:=(t-1)/α(n+1-(t+1))^α-(t+1)/α(n+1-(t+1))^α-1/(α(α+1))(n+1-(t+1))^(α-1)-((t-1)/α(n+1-t)^α-
(t)/α(n+1-t)^α-1/(α(α+1))(n+1-t)^(α-1));
S2[t_]:=t/α (n+1-(t+1))^α-(t+1)/α(n+1-(t+1))^α-1/(α(α+1))(n+1-(t +1))^(α-1)+1/(α(α+1))*(n+1-t)^(α-1);
Ut=U[t]==U[0]+1/Gamma[α]((U[t] c d (V[t])^a-f c V[t](U[t])^a)/h S1[t]-(b(U[t-1]c d V[t-1]^a-f c V[t-1]*
(U[t-1])^a))/h S2[t]);
Vt=V[t]==V[0]+1/Gamma[α]((c f(1-2 V[t]-W[t])(U[t])^a+d c (V[t])^a*U[t])/h S1[t]-(c f(1-2 V[t-1]-W[t-1])*
(U[t-1])^a+d c(V[t-1])^a U[t-1])/h S2[t]);
Wt=W[t]==W[0]+1/Gamma[α]((d f(1-V[t]-W[t])(U[t])^a)/h S1[t]-(d f(1-V[t-1]-W[t-1])(U[t-1])^a)/h S2[t]);
{Us,Vs,Ws}={U0,V0,W0};
Do[{Us,Vs,Ws}={U[t],V[t],W[t]}/.Last[NSolve[{Ut,Vt,Wt}/.
Thread[{U[t-1],V[t-1],W[t-1]}->{Us,Vs,Ws}],{U[t],V[t],W[t]}]];
sol[t]={Us,Vs,Ws};,{t,1,10}];
Table[sol[i],{i,1,10}]
I believe your first NSolve will be of approximately the form
NSolve[{
U[1]==U0+0.9882*(0.00002909-0.03519*(-0.007*U[1]^1.1*V[1]+0.035*U[1]*V[1]^1.1)),
V[1]==V0+0.9882*(0.002028-0.03519*(0.035*U[1]*V[1]^1.1+0.007*U[1]^1.1*(1-2*V[1]-W[1]))),
W[1]==W0+0.9882*(0.01640-0.007038*U[1]^1.1*(1-V[1]-W[1]))},
{U[1],V[1],W[1]}]
and, since NSolve is a numerical solver, I believe that it will not know how to deal with the unknowns U0,V0,W0.
I assume U0,V0,W0 where not present when you were able to successfully solve the problem without iteration and that was how it was able to solve that problem.
If you have numeric constants for the values of U0,V0,W0 then it should be fairly easy to modify the code to make it work. If you don't have numeric values for those then a different method of solving for the subsequent U[t],V[t],W[t] may be needed.