I only saw the difference now, excuse me. Thank you very much for your help :)

Here's my code , I did like you told me

Module[
 {
  \[Alpha] = 100,
  \[Beta] = 10,
  c = 10,
  b = 0.05,
  e = 100,
  k = 500,
  a = 10000,
  t = 0.01 *Range[5],
  m,
  f
  },
 (*m=(a+ \[Alpha] p[[n-1]]- \[Alpha] c - (Length[t]-n) \
a)/(\[Alpha](b+(Length[t]-n+1)t[n]- Sum[t[[i]],{i,n+1, Length[t]}]));*)


 f[x0_, pList_] := (pList[[1]] - 
      c - (b + t[[1]]) x0) (a - \[Alpha] pList[[1]]) +
   Sum[ (pList[[i]] - 
       c - (b + t[[i]]) x0) (a - \[Beta]  pList[[
         i]] + \[Beta]  pList[[i - 1]] - \[Alpha] pList[[1]]), {i, 2, 
     Length[t]}] - k (e - x0)^2;

 Maximize[
  {
   f[\[FormalX], Array[\[FormalP], Length[t]]],
     And @@ Thread[Array[\[FormalP], Length[t]] > 0] && 
    And @@ Table[
      a - \[Beta] (p[i] + p[i - 1]) - \[Alpha] p[1] > 0, {i, 2, 
       Length[t]}] 
   },
  Flatten[{\[FormalX], Array[\[FormalP], Length[t]]}] 
  ]

 ]

and i have this output :

Maximize[{-500 (100 - \[FormalX])^2 + (10000 - 
      100 \[FormalP][1]) (-10 - 
      0.06 \[FormalX] + \[FormalP][1]) + (10000 - 90 \[FormalP][1] - 
      10 \[FormalP][2]) (-10 - 
      0.07 \[FormalX] + \[FormalP][2]) + (10000 - 100 \[FormalP][1] + 
      10 \[FormalP][2] - 10 \[FormalP][3]) (-10 - 
      0.08 \[FormalX] + \[FormalP][3]) + (10000 - 100 \[FormalP][1] + 
      10 \[FormalP][3] - 10 \[FormalP][4]) (-10 - 
      0.09 \[FormalX] + \[FormalP][4]) + (10000 - 100 \[FormalP][1] + 
      10 \[FormalP][4] - 10 \[FormalP][5]) (-10 - 
      0.1 \[FormalX] + \[FormalP][5]), \[FormalP][1] > 
    0 && \[FormalP][2] > 0 && \[FormalP][3] > 0 && \[FormalP][4] > 
    0 && \[FormalP][5] > 0 && 
   10000 - 100 p[1] - 10 (p[1] + p[2]) > 0 && 
   10000 - 100 p[1] - 10 (p[2] + p[3]) > 0 && 
   10000 - 100 p[1] - 10 (p[3] + p[4]) > 0 && 
   10000 - 100 p[1] - 10 (p[4] + p[5]) > 0}, {\[FormalX], \[FormalP][
   1], \[FormalP][2], \[FormalP][3], \[FormalP][4], \[FormalP][5]}]

What do you mean by "it doesn't work"? does it run but not return an answer, does it throw an error, or does it not run and returns your Maximize[...] expression?

It ran for me (but warned about convergence). This would be a problem in your equations.

Did you change my "p" s to formalP's?

I suggest removing everything from the Module[] for debugging so you can check each expression and then package it up in the Module.

Your other constraint looks like this:

And @@ Thread[Array[p, Length[t]] > 0] && 
 And @@ Table[
   a - \[Beta] (p[i] + p[i - 1]) - \[Alpha] p[1] > 0, {i, 2, 
    Length[t]}]

to get

p[1]>0&&p[2]>0&&p[3]>0&&p[4]>0&&p[5]>0&&10000-100 p[1]-10 (p[1]+p[2])>0&&10000-100 p[1]-10 (p[2]+p[3])>0&&10000-100 p[1]-10 (p[3]+p[4])>0&&10000-100 p[1]-10 (p[4]+p[5])>0

I ran this and it returned an answer:

Maximize[{f[\[FormalX], Array[\[FormalP], Length[t]]], 
  And @@ Thread[Array[\[FormalP], Length[t]] > 0] && 
   And @@ Table[
     a - \[Beta] (\[FormalP][i] + \[FormalP][
           i - 1]) - \[Alpha] \[FormalP][1] > 0, {i, 2, 5}]}, 
 Flatten[{\[FormalX], Array[\[FormalP], Length[t]]}]]

to get:

{1.66918*10^7, {\[FormalX] -> 96.3516, \[FormalP][1] -> 
   5.00841*10^-12, \[FormalP][2] -> 498.529, \[FormalP][3] -> 
   501.471, \[FormalP][4] -> 498.529, \[FormalP][5] -> 501.471}}

Narjes,

You can add the extra constraints on at the end of your constraint list:

Maximize[{f[\[FormalX], Array[\[FormalP], Length[t]]], 
  And @@ Thread[Array[\[FormalP], Length[t]] > 0] && 
   And @@ Table[
     a - \[FormalP][i] + \[FormalP][i - 1] > 0, {i, 2, Length[t]}]}, 
 Flatten[{\[FormalX], Array[\[FormalP], Length[t]]}]]

Constraints now look like this (dropping the formalp). Note the index of the second set must start at 2 because you do p[i-1]

And @@ Thread[Array[p, Length[t]] > 0] && 
 And @@ Table[a - p[i] + p[i - 1] > 0, {i, 2, Length[t]}]

To get

p[1] > 0 && p[2] > 0 && p[3] > 0 && p[4] > 0 && p[5] > 0 && 
 10000 + p[1] - p[2] > 0 && 10000 + p[2] - p[3] > 0 && 
 10000 + p[3] - p[4] > 0 && 10000 + p[4] - p[5] > 0