Wavefronts for a cooperative tridiagonal system of differential equations

Consider the infinite system of nonlinear differential equations \(\dot u\) _n =f(_n?1, u_n, u_n+1),nε?, wherefεC ¹,D ₁ f > 0,D ₃ f>0, andf(0, 0, 0) = 0 =f(1, 1, 1). Existence of wavefronts—i.e., solutions of the formu _n (t) = U(n + ct), wherecε?,U(? ∞) = 0,U(+∞) = 1, andU is strictly increasing—is shown for functionsf which satisfy the condition: there existsa, 0<a<1, such thatf(x, x,x)<0 for 0<x<a andf(x, x, x) > 0 fora < x < 1.