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In this paper, we study quasi-periodic CMV matrices with Verblunsky coefficients given by the skew-shift. We prove the positivity of Lyapunov exponents and Anderson localization for most frequencies, which establish the analogous results of one-dimensional Schr\"{o}dinger operators proved by Bourgain, Goldstein and Schlag.

The authors study the Lagrangian stability for the sublinear Duffing equations ẍ + e(t)∣x∣α−1x = p(t) with 0 < α < 1, where e and p are real analytic quasi-periodic functions with frequency ω. It is proved that if the mean value of e is positive and the frequency ω satisfies Diophantine condition, then every solution of the equation is bounded.

In this article, we prove Lyapunov behavior and dynamical localization for analytic quasi-periodic CMV matrices in the regime of positive Lyapunov exponents. The localization results of Wang and Damanik [28] are extended.

In this paper, we consider the Littlewood's problem for the second order differential equation x ¨ + x 2 n + 1 + q ( t ) x l = p ( t ) , l n , where the functions p ( t ) and q ( t ) do not need to be periodic. Using the theory of non-periodic twist mappings which is developed by Kunze and Ortega in [3] , [4] , [5] , [6] , [7] , we obtained longtime closeness results on the bounded and unbounded solutions of the equation.

In this paper, we prove the existence of quasi-periodic solutions and boundeness of all solutions of the general semilinear quasi-periodic differential equation x″ + ax+ − bx− = Gx(x, t) + f(t), where x+ = max{x, 0}, x− = max{ − x, 0}, a and b are two different positive constants, f(t) is $${{\cal C}^{39}}$$ smooth in t, G(x, t) is $${{\cal C}^{35}}$$ smooth in x and t, f (t) and G(x, t) are quasi-periodic in t with the Diophantine frequency ω = (ω1, ω2), and $$D_x^iD_t^jG\left({x,\;t} \right)$$ is bounded for 0 ⩽ i + j ⩽ 35.

In this paper, we study the reducibility problem for a class of analytic almost periodic linear Hamiltonian systems $$\begin{aligned} \frac{dx}{dt} = J[{A}+\varepsilon {Q}(t)]x \end{aligned}$$ where A is a symmetric matrix, J is an anti-symmetric symplectic matrix, Q(t) is an analytic almost periodic symmetric matrix with respect to t, and $$\varepsilon $$ is a sufficiently small parameter. It is also assumed that JA has possible multiple eigenvalues and the basic frequencies of Q satisfy the non-resonance conditions. It is shown that, under some non-resonant conditions, some non-degeneracy conditions and for most sufficiently small $$\varepsilon $$ , the Hamiltonian system can be reduced to a constant coefficients Hamiltonian system by means of an almost periodic symplectic change of variables with the same basic frequencies as Q(t).

We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without the polynomial-like growth condition. As an application, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+\psi(x)=p(t)$ under the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $\psi(x)$ are periodic and $g(x)$ is bounded.

In this paper, we consider the boundedness of solutions for a class of impact oscillators $$ \{{array}{ll} \displaystyle \ddot{x}+x^{2n+1}+\sum_{i=0}^{2n}p_{i}(t)x^{i}=0,& \quad {\rm for}\quad x(t)> 0, x(t)\geq 0,& \dot{x}(t_{0}^{+})=-\dot{x}(t_{0}^{-}),& \quad {\rm if}\quad x(t_{0})=0, {array}. $$ where $n\in\NN^{+}$, $p_{i}(t+1)=p_{i}(t)$ and $p_{i}(t)\in C^5(\RR/\ZZ).$

In this paper we generalize the comparison result of Bostan and Namah (2007) [8] to the second-order parabolic case and prove two properties of pseudo-almost periodic functions; then by using Perron’s method we prove the existence and uniqueness of time pseudo-almost periodic viscosity solutions of second-order parabolic equations under usual hypotheses.

We consider the following boundary value problem with nonhomogeneous three-point boundary condition $$ \begin{array}{l} \left( {\phi _p ( {u^\prime } )} \right)^{\prime\prime} + a( t )f( u ) = 0, \quad t \in ( {0,1} ), \\ u( 0 ) = \xi u( \eta ) + \lambda , \quad u^\prime ( 0 ) = u^\prime( 1 ) = 0 \\ \end{array}$$ . We derive several existence, nonexistence, and multiplicity results for positive solutions in terms of different values of the parameter λ. The uniqueness of positive solutions and the dependence of positive solutions on the parameter λ are also studied.

We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation and using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.

In this paper we consider the problem of the boundedness of all solutions for the reversible system x ″ + ∑ j = 0 l b j ( t ) x 2 j + 1 x ′ + x 2 n + 1 + ∑ i = 0 n − 1 a i ( t ) x 2 i + 1 = 0 . It is shown that all the solutions are bounded provided that the a i ( t ) ( 0 ⩽ i ⩽ [ ( n − 1 ) / 2 ] ) are of bounded variation in [ 0 , 1 ] and the derivatives of b j ( t ) and a i ( t ) ( [ ( n − 1 ) / 2 ] + 1 ⩽ i ⩽ n − 1 , 0 ⩽ j ⩽ l ) are Lipschitzian. It is also shown that there exist a i 's being discontinuous everywhere such that all solutions of the equation are bounded. This implies that the continuity of a i 's is not necessary for the boundedness of solutions of the equation.

In this paper we investigate the existence and uniqueness of almost periodic and pseudo almost periodic solution for nonlinear differential equation with reflection of argument. For the case of almost periodic forced term, we consider the frequency modules of the solutions.

In this paper we investigate the existence and uniqueness of periodic and almost periodic solution of the differential equation with reflection of argument. The relationship between modules of forced term and solution of the equation is considered.

In this paper, we give a definition of pseudo almost periodic sequence and study the existence of pseudo almost periodic sequence to difference equation. Based on these, we investigate the existence of pseudo almost periodic solutions to differential equations with piecewise constant arguments which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems.

The authors discuss the existence of pseudo almost periodic solutions of differential equations with piecewise constant argument by means of introducing new concept, pseudo almost periodic sequence.

In this paper, we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian and an oscillating term $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)=G_x(x,t)+f(t)$, where$x^+=\max (x,0)$,$x^- =\max(-x,0)$,$\phi_p(s)=|s|^{p-2}s$,$p\geq2$, $a $ and $b$ are positive constants $(a\not=b)$, the perturbation $f(t)\in {\cal C}^{23}(\RR/2\pi_p \ZZ)$, the oscillating term $G\in {\cal C}^{21}(\RR\times\RR/2\pi_p \ZZ)$,where $\pi_p=\frac{2\pi(p-1)^{\frac{1}{p}}}{p\sin\frac{\pi}{p}},$ and $G(x,t)$ satisfies $\label{G} |D_x^iD_t^jG(x,t)|\le C,\quad 0\le i+j\le 21,$ and $\label{hatG} |D_t^j\hat{G}|\le C,\quad 0\le j\le 21$ for some $C>0$, where $\hat{G}$ is some function satisfying $\frac{\pa \hat{G}}{\pa x}=G$.

By means of the fixed point methods and the properties of the weighted pseudo almost periodic functions, the existence and uniqueness of weighted pseudo almost periodic solutions are obtained for differential equations involving reflection of the argument. Key words: Almost periodic functions, weighted pseudo almost periodic functions, fixed point methods, differential equations, reflection of argument.

The existence of solutions for stochastic differential equations under G-Brownian motion (G-SDEs) of the typein the presence of a lower solution and an upper solution in a reverse order is established. By the method of upper and lower solutions in the reverse order, it is shown that the G-SDEs have more than one solution if the drift coefficients are discontinuous functions. Key words: Upper and lower solutions in reverse order, stochastic differential equations, G-Brownian motion, discontinuous drift coefficient, existence.

We study some properties of the remotely almost periodic functions. This paper studies viscosity solutions of general Hamilton-Jacobi equations in the time remotely almost periodic case. Existence and uniqueness results are presented under usual hypotheses.

In this paper we investigate the existence and uniqueness of pseudo almost periodic solutions and unboundedness of other solutions for the systems of differential equations with piecewise constant argument [t + 1/2] by means of new notion of pseudo almost periodic vector sequences. The case in which the characteristic equation has multiple roots is considered.

In this paper, we prove the boundedness of all solutions for the periodic equation x ″ + ω 2 x + ϕ ( x ) = G x ( x , t ) + p ( t ) , where ω satisfies the Diophantine condition, ϕ ( x ) ∈ C 19 is bounded, p ( t ) ∈ C 23 and D x i D t j G ( x , t ) is bounded for 0 ⩽ i + j ⩽ 21 .