I want to solve numerically for a variable, let’s say x. The equation that defines x has as argument a piecewise function that has its pieces depending on x itself. Moreover, some parts of this piecewise function are non-linear solutions of x. Let me try to better explain with an example.

The easiest example I could come up with is given below:

    a = 1; b = 0.1; y = 1 - x[p] - (b + a - 1); FG[p_?NumericQ] =    FindRoot[Y - x + a*Log[1 + x] - x[p] == 0, {x, 0.5}][[1, 2]]; V[p_?NumericQ] :=   Piecewise[{{p*x[p] + (1 - p)*y,      x[p] < 1 + a*Log[b + a] - (b + a - 1) &&       b*Log[b + a] < y < 1 + b*Log[b + a] - (b + a - 1)}, {p*       x[p] + (1 - p)*y,      x[p] < 1 + a*Log[b + a] - (b + a - 1) &&       y < b*Log[b + a]}, {p*x[p] + (1 - p)*y,      x[p] < 1 + a*Log[b + a] - (b + a - 1) &&       y >= 1 +         b*Log[b + a] - (b + a - 1)}, {p*(1 - FG[p] +          a*Log[1 + FG[p]]) + (1 - p)*y,      x[p] >= 1 + a*Log[b + a] - (b + a - 1) &&       b*Log[b + a] < y < 1 + b*Log[b + a] - (b + a - 1)}}, 0] 

I want to solve:

p = 1;  FindRoot[x + a*Log[b + a] +    V[p] - (1 + 0.56*(1 - p) + 0.5*p), {x, -.5}] 

This is not working and I can’t figure out way. I have tried to inactivate FG[p] and or V[p] but it doesn’t help. I am not sure what exactly I am doing wrong. Anyone has any idea?