Comparative Statics on an infinite System

Hello, For a research project I am encountering the following (infinite) system of equations:

$S=π_1\cdot\left(\frac{(e_{u}+θ (c)\cdot(S_{u}+δ_{u}))^{-γ}(S_{u}+δ_{u})}{c^{-γ}} \right) + π_2\cdot\left(\frac{(e_{d}+θ (c)\cdot(S_{u}+δ_{u}))^{-γ}(S_{u}+δ_{u})}{c^{-γ}} \right)+(1-π_1-π_2)\cdot\left(\frac{(e_{d}+θ (c)\cdot(S_{d}+δ_{d}))^{-γ}(S_{d}+δ_{d})}{c^{-γ}} \right);\forall c$

$\int λ(c) θ(c) dc=0$

c is defined on a compact interval on R+ (positive reals)

Note that the first line represents infinite equations since it holds for all $c$. The second line represents one equation only.

All variables except $S$ and $θ (c)$ are scalars and known, with the exception of $λ(c)$ which is a known (density) function.

------> I would like to:

1) Ideally find an expression for $θ (c)$ (or an approximate expression)

2) Ideally find an expression for $S$ (or an approximate expression)

3) Alternatively perform comparative statics on $S$: I.E. find $\nabla S$

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