Bounded solutions and wavefronts for discrete dynamics

This paper deals with the second-order nonlinear difference equation Delta(r(k)Deltau(k)) + q(k)g(u(k+1)) = 0 where {r(k)} and {q(k)} are positive real sequences defined on N, and the nonlinearity g : R --> R is nonnegative and nontrivial. Sufficient and necessary conditions are given, for the existence of bounded solutions starting from a fixed initial condition u(0). The same dynamic, with f instead of g such that uf (u) > 0 for u not equal 0, was recently extensively investigated. On the contrary, our nonlinearity g is of a small appearance in the discrete case. Its introduction is motivated by the analysis of wavefront profiles in biological and chemical models. The paper emphasizes the many different dynamical behaviors caused by such a g with respect to the equation involving function f. Some applications in the study of wavefronts complete this work.