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.