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We prove a Livšic-type theorem for Hölder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $$(f,\mu )$$ is a non-uniformly hyperbolic system and $$A:M \rightarrow GL(d,\mathbb {R}) $$ is an $$\alpha $$-Hölder continuous map satisfying $$ A(f^{n-1}(p))\ldots A(p)=\text {Id}$$ for every $$p\in \text {Fix}(f^n)$$ and $$n\in \mathbb {N}$$, there exists a measurable map $$P:M\rightarrow GL(d,\mathbb {R})$$ satisfying $$A(x)=P(f(x))P(x)^{-1}$$ for $$\mu $$-almost every $$x\in M$$. Moreover, we prove that whenever the measure $$\mu $$ has local product structure the transfer map P is $$\alpha $$-Hölder continuous in sets with arbitrary large measure.

This article studies the Stochastic Degasperis-Procesi equation on $$ \mathbb {R}$$ with an additive noise. Applying the kinetic theory, and considering the initial conditions in $$L^2(\mathbb {R})\cap L^{2+\delta }( \mathbb {R})$$ , for arbitrary small $$\delta >0$$ , we establish the existence of a global pathwise solution. Restricting to the particular case of zero noise, our result improves the deterministic solvability results that exist in the literature.

In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution which is unique in the class of functions with polynomial growth. In our study, the main tool is the associated of reflected backward stochastic differential equations with jumps with interconnected obstacles for which we show the existence of a unique Markovian solution.

Functional differential equations of mixed type (MFDE) are introduced; in these equations of functional type, the time derivative may depend both on past and future values of the variables. Here the linear autonomous case is considered. We study the spectrum of the (unbounded) operator, and construct continuous semigroups on the stable, center, and unstable subspaces.

A study is made of systems of weakly coupled, semilinear, parabolic equations, namely reaction-diffusion systems, subject to the homogeneous Neumann boundary conditions in parametrized nonconvex domains inR 2. It is assumed that the domain approaches a union of two disjoint domains as the parameter varies. Under some conditions the long-time behavior of bounded solutions is discussed and the existence of a finite-dimensional invariant manifold is shown, together with its attractivity. This manifold is represented by a graph of some function defined in a possibly large bounded region of the phase space, and the original system is reduced to an ODE system on it. Since an explicit form of the reduced ODE system is given, its dynamics can be studied in detail, which in turn reveals the global dynamics of the original reaction-diffusion system. One can thereby prove, among other things, the existence of asymptotically stable equilibrium solutions of the original system having large spatial inhomogeneity. The existence and stability of a spatially inhomogeneous periodic solution of large amplitude are also discussed.

We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of some finite delays are established. In the first situation, the effect of the delays is dominated by non-delayed diagonal negative feedback terms, and sufficient conditions for both the asymptotic and the exponential asymptotic stability of the system are given. In the second case, the stability depends on the size of some bounded diagonal delays and coefficients, although terms with unbounded delay may co-exist. Our results encompass DDEs with discrete and distributed delays, and enhance some recent achievements in the literature.

Trong một bài báo trước đây, chúng tôi đã nghiên cứu các hệ thống tự trị tham số và đưa ra một tiêu chuẩn có thể tính toán được để một quỹ đạo gần đúng nối kết các điểm cân bằng hyperbolic được bóng bởi một quỹ đạo kết nối thực sự. Tiêu chuẩn này đã được sử dụng để cung cấp các ví dụ được xác minh một cách nghiêm ngặt về các quỹ đạo homoclinic saddle-focus Shilnikov trong ba chiều. Điều này bao gồm việc xác minh một điều kiện về giá trị riêng của phép tuyến tính hóa tại điểm cân bằng. Trong các chiều lớn hơn ba, có ba điều kiện bổ sung cần phải được thiết lập: vị trí tổng quát, tiếp xúc tiệm cận và điều kiện xuyên tính. Trong bài báo này, chúng tôi đưa ra các tiêu chuẩn có thể tính toán để xác minh ba điều kiện này. Một ví dụ trong bốn chiều, trong đó các phép toán nghiêm ngặt được thực hiện chi tiết, được đưa ra.

We prove the existence in the sense of sequences of solutions for some system of integro-differential type equations in two dimensions containing the normal diffusion in one direction and the anomalous diffusion in the other direction in $$H^{2}({\mathbb R}^{2}, {{\mathbb {R}}}^{N})$$ using the fixed point technique. The system of elliptic equations contains second order differential operators without the Fredholm property. It is established that, under the reasonable technical assumptions, the convergence in $$L^{1}({{\mathbb {R}}}^{2})$$ of the integral kernels yields the existence and convergence in $$H^{2}({{\mathbb {R}}}^{2}, {\mathbb R}^{N})$$ of the solutions. We emphasize that the study of the systems is more difficult than of the scalar case and requires to overcome more cumbersome technicalities.