Almost automorphic delayed differential equations and Lasota-Wazewska model

Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and uniqueness of almost automorphic solutions are given.Comment: 16 page

Qualitative analysis of dynamic equations on time scales

In this article, we establish the Picard-Lindelof theorem and approximating results for dynamic equations on time scale. We present a simple proof for the existence and uniqueness of the solution. The proof is produced by using convergence and Weierstrass M-test. Furthermore, we show that the Lispchitz condition is not necessary for uniqueness. The existence of epsilon-approximate solution is established under suitable assumptions. Moreover, we study the approximate solution of the dynamic equation with delay by studying the solution of the corresponding dynamic equation with piecewise constant argument. We show that the exponential stability is preserved in such approximations.Comment: 13 page

On the reduction principle for differential equations with piecewise constant argument of generalized type

In this paper we introduce a new type of differential equations with piecewise constant argument (EPCAG), more general than EPCA. The Reduction Principle is proved for EPCAG. The structure of the set of solutions is specified. We establish also the existence of global integral manifolds of quasilinear EPCAG in the so called critical case and investigate the stability of the zero solution

Convergence to equilibrium in degenerate parabolic equations with delay

© 2012 Elsevier Ltd In [11], Busenberg & Huang (1996) showed that small positive equilibria can undergo supercritical Hopf bifurcation in a delay-logistic reaction–diffusion equation with Dirichlet boundary conditions. Consequently, stable spatially inhomogeneous time-periodic solutions exist. Previously in [12] Badii, Diaz & Tesei (1987) considered a similar logistic-type delay-diffusion equation, but differing in two important respects: firstly by the inclusion of nonlinear degenerate diffusion of so-called porous medium type, and secondly by the inclusion of an additional ‘dominating instantaneous negative feedback’ (where terms local in time majorize the delay terms, in some sense). Sufficient conditions were given ensuring convergence of non-negative solutions to a unique positive equilibrium. A natural question to ask, and one which motivated the present work, is: can one still ensure convergence to equilibrium in delay-logistic diffusion equations in the presence of nonlinear degenerate diffusion, but in the absence of dominating instantaneous negative feedback? The present paper considers this question and provides sufficient conditions to answer in the affirmative. In fact the results are much stronger, establishing global convergence for a much wider class of problems which generalize the porous medium diffusion and delay-logistic terms to larger classes of nonlinearities. Furthermore the results obtained are independent of the size of the delay