We study the initial value problem for the Ishimori system. Elliptic-hyperbolic and hyperbolic-elliptic cases were treated by inverse scattering techniques [6, 7], [9]. Elliptic-elliptic and hyperbolic-elliptic cases were studied in [8] without using inverse scattering techniques. In the previous paper [4] we proved the global existence of small solutions to the Ishimori and the Davey-Stewartson systems in the elliptic-hyperbolic and hyperbolic-hyperbolic cases under exponential decay on the data. Our purpose in this paper is to remove that condition in the case of the Ishimori system.

We prove a comparison theorem for viscosity solutions of degenerate parabolic equations with capillary boundary condition on a half space. We apply our theorem to construct a global generalized evolution for the capillary problem.

We develop results on (a) almost periodicity properties and (b) (exponential) asymptotic stability of solutions to the functional differential equations with infinite delay (FDE) $ \dot{x}(t) + \alpha x(t) + Bx(t) \ni F(x_t) , $ $ t \geq 0,$ $x |_{\mathbb{R}^-} = \varphi \in E $ in a context which allows for a (generally) nonlinear and multivalued accretive state-responsive operator $ B \subset X \times X $ in a Banach state space $ X $ and a locally defined and locally Lipschitz continuous history-responsive function $ F : D(F) \subset E \to X, $ in an appropriate initial history space $ E $ of continuous functions from $ \mathbb{R}^- $ into $ X. $ Applications to models from population dynamics (delay logistic equation) and biology (Goodwin oscillator) are presented.

Let $\Lambda$ be the Laplace--Beltrami operator on $S^{n-1}$. The aim of this paper is to prove that any continuum of nontrivial solutions of the equation $-\Lambda u = f(u,\lambda),$ which bifurcate from the set of trivial solutions, is unbounded in $H^1(S^{n-1}) \times R$. As the main tool we use degree theory for $S^1$--equivariant, gradient operators defined in [15].

We obtain some general comparison results for entropy solutions of weakly coupled systems of conservation laws with source. For some particular couplings, by using these comparison methods, we show the global existence of the solutions to the Cauchy problem.

Sharp existence results are given for positive solutions to the boundary value problem $v''+g(t,v) =0$, $v(0)= v(t_0)=0$, where $g(t,v)$ is allowed to be singular at both $t=0$ and $t=t_0$. These results are applied to radial solutions of the semilinear elliptic problem $\Delta u +f(r,u)= 0$ on a ball. Examples and corollaries illustrate the wide class of equations to which the results apply. The proofs use the Mountain Pass Lemma in conjunction with some delicate estimates, which allow the treatment of singular equations.

In this paper we obtain a variation of constants formula for the Cauchy problem associated to the nonhomogeneous retarded functional differential equation $$ x'(t)=L(x_t)+f(t), \quad t\geq 0 , \tag{*} $$ where $L:C([-r,0];E)\longrightarrow E$ is a bounded linear operator and $E$ is a Banach space. Once the existence and uniqueness theorem is proved, we define a fundamental solution associated with this problem and we use it to derive a variation of constants formula. The known results on the decomposition of the space $C([-r,0];E)$ with respect to the homogeneous equation $x'(t)=L(x_t)$ allow us to obtain a similar decomposition for the solutions of $(*)$. Moreover, the projection of solutions on a certain invariant finite-dimensional subspace is characterized in terms of the semigroup defined by the homogeneous problem.

The so-called Butler-McGehee lemma was first stated and proposed by Freedman and Waltman [11] to study persistence in three interacting predator-prey population models. Roughly speaking, the lemma says that if a trajectory, not on the stable manifold of a given isolated hyperbolic equilibrium $P,$ has that equilibrium in its $\omega$-limit set, then its $\omega$-limit set also contains points on the stable and unstable manifolds of the equilibrium different from $P.$ The lemma has been extended to different forms. The main purpose of this paper is to generalize one of the various formats of the Butler-McGehee lemma (Butler and Waltman [4]) in such a way as to encompass orbits from a set $G$ rather than from a single point. An application to the uniform persistence of a class of dynamical systems which are not necessarily point dissipative is given.