We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itôtype equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of inva...

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov-Perron operator. An example is provided to illustrate the result.

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We ...

‎using nehari manifold methods and mountain pass theorem‎, ‎the existence of nontrivial and radially symmetric solutions for a class of $p$-kirchhoff-type system are established.

First we prove a general spectral theorem for the linear NavierStokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in H (l = 0, 1, 2, · · · ). Then we prove the existence of invariant manifolds. We are also interested in a m...

Hyperholomorphic bundles over a hyperkähler manifold. 0. Intruduction. The main object of this paper is the notion of a hyperholomorphic bundle (Definition 2.4) over a hyperkähler manifold M (Definition 1.1). The hyperholomorphic bundle is a direct sum of holomorphic stable holomor-phic bundles. The first Chern class of a hyperholomorphic bundle is of zero degree. Roughly speaking, the hyperhol...