Using a variant of the mountain pass theorem, the authors prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbour interaction. The problem is to solve a second-order forward-backward differential-difference equation in the form \[ c^2u''= V' (u(t+1)-u(t))- V' (u(t)-u(t-1)). \tag{1} \] The following results are proven in the paper: 1. Under appropriate assumptions if \(u\) is a critical point of \[ \int_{R} {{c^2}\over 2}(u'(t))^2-V(u(t+1)-u(t)), \] then \(u\) is a solution of (1). 2. Under appropriate assumptions for every sufficiently large \(c\) the equation (1) has non-trivial non-decreasing solutions from the space \(\{u\in H^1_{loc} (R) : u' \in L^2(R),u(0) = 0\}\). 3. Under appropriate assumptions for every sufficiently large \(c\) the equation (1) has non-trivial non-increasing solutions from the space \(\{u\in H^1_{loc} (R) : u' \in L^2(R),u(0) = 0\}\).