Asymptotic constancy for pseudo monotone dynamical systems on function spaces

Convergence for pseudo monotone semiflows on product ordered topological spaces

“…Even though the convergence principle in [7] has been successfully applied to neutral functional differential equations and semilinear parabolic partial differential equations with Neumamn boundary condition, its requirements on the phase space, the set of equilibria and even the monotonicity properties are still too restrictive and therefore, its limitations seem natural. In fact, the convergence principle in [7] cannot be applied to some important examples like the following scalar delay differential equation:…”

“…We know that very little has been accomplished in this direction. For instance, Haddock et al [7] recently introduced a class of eventually strongly pseudo monotone semiflows defined on a function subspace X ⊆ C(M, R 1 ) which has a topology making its inclusion into C(M, R 1 ) continuous, where M is a compact topological space and R 1 denotes the set of all real numbers, and proved that each precompact orbit tends to a constant function whenever each constant function is an equilibrium point for such semiflows.…”

“…In fact, the convergence principle in [7] cannot be applied to some important examples like the following scalar delay differential equation:…”

Translation-invariant monotone systems II: Almost periodic/automorphic case

“…A series of conditions have been found by different researchers to guarantee global convergence for monotone dynamical systems. For monotone semiflows or mappings, these conditions are orbital stability/fixed point stability [1,2,3,42,10,20,25,26,27], possessing a first integral or invariant function with positive gradient [32,28,22,34,7,8,18,19,45,46], sublinearity [16,17,40,44,43], minimal equilibria [47,12], respectively.…”

On the Asymptotic Periodic Solutions of Abstract Functional Differential Equations

“…This problem has a long history and has been considered in part by many authors, see e.g. [4,5,11,13,16,29,30,35,39] and the references therein. On the other hand, it arises naturally from recent studies on the existence of (almost) periodic solutions of evolution equations (see e.g.…”