proof using (fixed point theorem) I am seeking to solve for a Nash equilibrium in pure strategies $(d_2,d_2)$ involving two players, $1$ and $2$. Given that $h'(.)$ is s strictly decreasing and continuous function, $\Phi(d_1-d_2)$ denoting a convolution function, and $F(.)$ denoting a CDF, I want to prove for existence and uniqueness of equilibrium. My guess is that we we use a fixed point theorem to prove existence. The following is the first order condition for maximization. 
$$g_1(d_1) \equiv h'(d_1)-\gamma\Phi(d_1-d_2)-\eta(1-F(m-d_1))=0 \\
g_2(d_2)≡ h'(d_2)-\gamma[1-\Phi(d_1-d_2)]-\eta(1-F(m-d_2))=0 $$
Note that the parameters are all positive and $d_1$ & $d_2$ are continuous and $m$ is a constant.  I highly appreciate any suggestion towards the proof. 
 A: I don't know the game you're discussing as you only give some FOCs but the general approach I would try first is using Berge+ Kakutani.
On the wikipedia page for Berge, think of $f$ as payoff, $x$ as strategies for $i$, $\theta$ as strategies for the opponent, and $C^*$ as the best-response correspondence.  
I believe you would need to verify then that (conditions for Berges Theorem):  $u_i(d_1,d_2)$ is continuous in both arguments for both players, that the strategy space for both agents is compact conditional upon each choice of strategy by the opposing player, and that the correspondence relating choice of strategy for $-i$ to the permissible strategies of player $i$ is continuous (usually in games the joint strategy space is a product space so this holds trivially).  
By Berge, this then suffices for $C^*$, i.e. the players' best response correspondences to be upper-hemicontinuous (assuming the joint strategy space is a product space).   You would also want to verify that $C^*$ is convex valued.  Sufficient conditions for this are $f$ is quasiconcave and $C$ is convex-valued.  Lastly your basic strategy spaces should be convex/compact.
Then you have best response functions that map convex, compact strategy spaces into themselves, are upper-hemicontinuous and convex-valued, thus by Kakutani a fixed point exists.
Hope this is helpful!
