Reflexive periodic solutions of general periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The objective of this paper is to provide four new iterative methods based on the conjugate gradient normal equation error (CGNE), conjugate gradient normal equation residual (CGNR), and least‐squares QR factorization (LSQR) algorithms to find the reflexive periodic solutions (X₁,Y₁,X₂,Y₂,…,X_σ,Y_σ) of the general periodic matrix equations for i = 1,2,…,σ. The iterative methods are guaranteed to converge in a finite number of steps in the absence of round‐off errors. Finally, some numerical results are performed to illustrate the efficiency and feasibility of new methods.     

A new characterization of periodic oscillations in periodic difference equations

In this paper, we characterize periodic solutions of p-periodic difference equations. We classify the periods into multiples of p and nonmultiples of p. We show that the elements of the set of multiples of p follow the well-known Sharkovsky’s ordering multiplied by p. On the other hand, we show that the elements of the set Γ_p of nonmultiples of p are independent in their existence. Moreover, we show the existence of a p-periodic difference equation with infinite Γ_p-set in which the maps are defined on a compact domain and agree exactly on a countable set. Based on the proposed classification, we give a refinement of Sharkovsky’s theorem for periodic difference equations.     

Generalized periodic solutions of quasilinear equations

We study a boundary-value periodic problem for the quasilinear equationu _ff ?u _xx =F[u,u _f u _x ],u(0,t) =u (π,t),u (x, t + π/q) =u(x, t), 0 ≤x ≤π,t ∈ ?,q ∈ ?. We establish conditions under which the theorem on the uniqueness of a smooth solution is true.     

Bifurcation of periodic solutions of periodic evolution equations in a cone

We consider a class of nonlinear periodic evolution equations, leaving invariant a cone in a real Banach space, and satisfying appropriate monotonicity requirements. A necessary and sufficient condition is found for the existence of a (unique, globally attractive) nontrivial periodic solution within the cone. Such a condition is expressed in terms of an associated linear equation. To establish this result, an abstract version of the familiar comparison techniques for parabolic equations is worked out. Applications and examples are also discussed.     

The almost periodic type difference equations

In this paper, we study the existence of almost periodic type solutions of difference equations by means of exponential dichotomy and trichotomy of linear difference equations.     

The existence of periodic solutions of higher order nonlinear periodic difference equations

Sufficient conditions for the existence of at least one periodic solution of two classes of nonlinear higher order periodic difference equations are established, respectively. The results show us that sufficient conditions for the existence of T ? periodic solutions of difference equation are different from those ones for the existence of T ? periodic solutions of differential equation. Copyright © 2012 John Wiley & Sons, Ltd.     

Time periodic solutions of nonlinear wave equations

Existence and regularity of solutions of $$(1)u_{tt} - u_{xx} = \varepsilon K(x,t,u,u_t )0< x< \pi ,0 \leqslant t \leqslant 2\pi $$ together with the periodicity and boundary conditions $$(2)u(x,t + 2\pi ) = u(x,t),u(0,t) = 0 = u(\pi ,t)$$ is studied both with an without the dissipation u_t. A solution is a pair (χ, u). A main feature of interest here is an infinite dimensional biofurcation problem. Under appropriate conditions on K, global existence results are obtained by a combination of analytical and topological methods.     

非线性发展方程新的周期解

In this paper,some new periodic solutions of nonlinear evolution equations and corresponding travelling wave solutions are obtained by using the double function method and Jacobi elliptic functions.     

Generic periodic solutions of functional differential equations

Consider the class of retarded functional differential equations

$\text{x(t) = f(x}{}_{\mathrm{t}}\text{)}$

, (1) where x_t(θ) = x(t + θ), ?1 ? θ ? 0, so x_t?C = C([?1, 0], Rⁿ), and $\text{f∈}\textcolor{red}{}\text{=C}{}^{\mathrm{r}}\text{(C,R}{}^{\mathrm{n}}\text{)}$. Let 2 ? r ? ∞ and give $\text{X}$ the appropriate (Whitney) topology. Then the set of $\text{f∈}\textcolor{red}{}$ such that all fixed points and all periodic solutions of (¹) are hyperbolic is residual in

.     

Almost periodic properties of ordinary differential equations

Sunto Si tratta di alcune proprietà dell'equazioni differenziali ordinarie con coefficienti quasi-periodici.      

Systems of differential equations with periodic coefficients

We consider linear systems of differential equations with periodic coefficients. We prove the solvability of nonhomogeneous systems in the Sobolev space W ₂ ¹ (R) and establish estimates for the solutions. This result implies a perturbation theorem for the exponential dichotomy of systems of differential equations with periodic coefficients.