Difference Equations

Abstract

Nonlinear difference equations lead to special classes of finite-dimensional systems to which Brouwer degree has been successfully applied in recent years. For first order equations, special attention is paid on their periodic solutions, by analogy with the case of ordinary differential equations. The method of lower and upper solutions is developed and presents some interesting distinct features from the corresponding differential case. Combined with the Brouwer degree, it provides multiplicity results of the Ambrosetti-Prodi type for problems depending upon a parameter which can have zero, at least one or at least two solutions depending upon the value of the parameter. Such results have applications in population dynamics, and in particular to difference equations of the Verhulst or Lotka-Volterra type. For second order difference equations, a typical question is the Dirichlet problem, whose spectral theory in the linear case is more complicated than for the periodic problem in the first order case. Bifurcation theory is applied to the obtention of nontrivial solution, and bifurcation from infinity is combined to the Brouwer degree to provide multiplicity results when the spectral parameter crosses an eigenvalue and the nonlinearity satisfies a Landesman-Lazer condition, also present in the study of equations with nonlinearities bounded from above or from below. A method of lower and upper solutions is also developed for second order Dirichlet problems, and provides again multiplicity results of the Ambrosetti-Prodi type.