We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u\left(t\right)=U(t,s)u\left(s\right)+{\int }_s^{t}U(t,\xi )f(\xi ,u\left(\xi \right))d\xi$ when the evolution family ${\left(U(t,s)\right)}_{t\ge s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(t) belongs to some classes of admissible function spaces. These manifolds are formed by trajectories of the solutions belonging...

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$\dot{x}\left(t\right)=ℒ\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\in ℝ\left(\mathrm{P}\right)$$ where $\left\{ℒ\right(t):t\in ℝ\}$ is a family of linear operators from a Banach space $E$ into itself and $f:ℝ\times E\to E$. By $L\left(E\right)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a<b$ and $d>0$, we let $C\left(\right[-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d:[a,b]\times C([-d,0],E)\to E$. Let $\widehat{ℒ}:[a,b]\to L\left(E\right)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define ${\tau }_{t}x\left(s\right)=x(t+s)$ for each $s\in [-d,0]$. We prove that, under certain conditions,...

We develop a difference equations analogue of recent results by F. Gesztesy, K. A. Makarov, and the second author relating the Evans function and Fredholm determinants of operators with semi-separable kernels.

The unstable properties of the linear nonautonomous delay system $x^{\text{'}}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)x(t-r\left(t\right))$, with nonconstant delay $r\left(t\right)$, are studied. It is assumed that the linear system $y^{\text{'}}\left(t\right)=(A\left(t\right)+B\left(t\right))y\left(t\right)$ is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay $r\left(t\right)$ is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function $r\left(t\right)$ and the results depending on the asymptotic properties of the...

We study dichotomous behavior of solutions to a non-autonomous linear difference equation in a Hilbert space. The evolution operator of this equation is not continuously invertible and the corresponding unstable subspace is of infinite dimension in general. We formulate a condition ensuring the dichotomy in terms of a sequence of indefinite metrics in the Hilbert space. We also construct an example of a difference equation in which dichotomous behavior of solutions is not compatible with the signature...

We propose a new rigorous numerical technique to prove the existence of symmetric homoclinic orbits in reversible dynamical systems. The essential idea is to calculate Melnikov functions by the exponential dichotomy and the rigorous numerics. The algorithm of our method is explained in detail by dividing into four steps. An application to a two dimensional reversible system is also treated and the existence of a symmetric homoclinic orbit is rigorously verified as an example.