Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations

Abstract

We discuss the existence and approximation of solutions of asymptotic or periodic boundary value problems of mixed functional differential equations. Our approach is via monotone iteration and non-standard ordering in the profile set for asymptotic boundary value problems and via S 1 -degree and equivariant bifurcation theory for periodic boundary value problems. Applications will be given to wave fronts and to slowly oscillatory spatially periodic traveling waves of lattice delay differential equations arising from population genetics, population dynamics, and neural networks.1997 Academic Press

Stability of nonmonotone critical traveling waves forspatially discrete reaction-diffusion equations with time delay

“…Taking Young's inequality (3.17) with η = 1 , we can see that 19) for some small constant ε > 0. Substituting (3.19) into (3.18), one gets…”

Front propagation for a two-dimensional periodic monostable lattice dynamical system

“…[1,27,9]) to the new system of equations satisfied by w i,j to derive the existence of traveling waves. The super-sub-solutions constructed in [19] are useful in applying this method.…”

Existence of traveling wave solutions in m-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interaction

“…For details, see for example, [4,5,23,24]. Taking into account time delay in population dynamics, Wu and Zou [21] considered the delayed lattice differential equations and studied the existence of traveling wave solutions. As mentioned in Weinberger et al [19], the most interesting population model should involve the interactions of different species, and there are also some concrete system of lattice differential equations which are derived in population dynamics.…”