一类多项式型迭代函数方程在共振点附近的解析解

In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied. As well as in previous work, we reduce this problem with the SchrSder transformation to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous work the constant α given in the Schroder transformation, i.e., the eigenvalue of the linearized f at its fixed point O, is required to fulfill that α is off the unit circle S^1 or lies on the circle with the Diophantine condition. In this paper, we obtain results of analytic solutions in the case of α at resonance, i.e., at a root of the unity and the case of α near resonance under the Brjuno condition.     

Analytic solutions of a second-order nonautonomous iterative functional differential equation

In this paper a second-order nonautonomous iterative functional differential equation is considered. By reducing the equation with the Schröder transformation to another functional differential equation without iteration of the unknown function, we give existence of its local analytic solutions. We first discuss the case that the constant α given in the Schröder transformation does not lie on the unit circle in C and the case that the constant lies on the circle but fulfills the Diophantine condition. Then we further study the case that the constant is a unit root in C but the Diophantine condition is offended. Finally, we investigate analytic solutions of the form of power functions.     

KAM-tori near an analytic elliptic fixed point

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector ω ₀ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω ₀ has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω ₀. This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.     

On the period function of x″+f(x)x′+g(x)=0

We study the period function T of a center O of the title's equation. A sufficient condition for the monotonicity of T, or for the isochronicity of O, is given. Such a condition is also necessary, when f and g are odd and analytic. In this case a characterization of isochronous centers is given. Some classes of plane systems equivalent to such equation are considered, including some Kukles’ systems.     

Kolmogorov Theorem revisited

Kolmogorov Theorem on the persistence of invariant tori of real analytic Hamiltonian systems is revisited. In this paper we are mainly concerned with the lower bound on the constant of the Diophantine condition required by the theorem. From the existing proofs in the literature, this lower bound turns to be of O(ε1/4), where ε is the size of the perturbation. In this paper, by means of careful estimates on Kolmogorov's method, we show that this lower bound can be weakened to be of O(ε1/2). This condition coincides with the optimal one of KAM Theorem. Moreover, we also obtain optimal estimates for the distance between the actions of the perturbed and unperturbed tori. We believe that some ideas contained in this paper may be used for improving several estimates in the general KAM context.     

Classes of analytic functions subordinate to convex functions and extreme points

In this paper we prove two sufficient conditions for an analytic function f to be an extreme point of the set of functions subordinate to a given convex mapping F when the image of the unit disk under F is a convex domain other than a half-plane, a strip or an infinite wedge.     

LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION

We study the local analytic solutions f of the functional equation f(ψ(zf(z)))=(f(z)) for z in some neighborhood of the origin.Whether the solution f vanishes at z=0 or not plays a critical role for local analytic solutions of this equation.In this paper,we obtain results of analytic solutions not only in the case f(0)=0 but also for f(0)≠0.When assuming f(0) =0,for technical reasons,we just get the result for f’(0)≠0.Then when assuming f(0)=ω0≠0,ψ’(0)=s≠0,ψ(z) is analytic at z=0 and(z)is analytic at z=ω0,we give the existence of local analytic solutions f in the case of 0<|sω0|<1 and the case of |sω0|=1 with the Brjuno condition.     

Quasi-periodic solutions with prescribed frequency in a nonlinear Schrdinger equation

In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 σ 1, and each given integer b 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.     

The holomorphic solutions of the generalized Dhombres functional equation

We study holomorphic solutions f of the generalized Dhombres equation f(zf(z))=φ(f(z)), z∈C, where φ is in the class E of entire functions. We show, that there is a nowhere dense set E₀⊂E such that for every φ∈E?E₀, any solution f vanishes at 0 and hence, satisfies the conditions for local analytic solutions with fixed point 0 from our recent paper. Consequently, we are able to provide a characterization of solutions in the typical case where φ∈E?E₀. We also show that for polynomial φ any holomorphic solution on C?{0} can be extended to the whole of C. Using this, in special cases like φ(z)=zk+1, k∈N, we can provide a characterization of the analytic solutions in C.     

On polyharmonic interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C^∞ or analytic functions in the ball BR. We prove two main results on the interpolation of C^∞ or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.     

On the structure of periodic solutions of differential equations

This paper studies the “internal structure” of the periodic solutions of differential equations with the aim of stating when they are constant functions. Yorke [21] and Lasota and Yorke [10] are the first works which show the existence, uńder certain conditions, of a lower bound for the period of non-constant solutions. As applications of the general results proved in Section 1 we obtain a negative solution to an open problem of Browder, the discovery that the periodic solutions ensured by Vidossich [17, Theorem 3.16], are constant functions, and conditions under which the periodic solutions of hyperbolic and parabolic equations are constant functions. Finally, we note that Li [11] applies the results of Section 1 to differential equations with delay.Various result of this paper point out a strong connection between the existence of periodic solutions of small period of x′ = f(x) and the fact that the origin belongs to the range of f. This situation is explored in [19].     

Algorithms for projecting a point onto a level surface of a continuous function on a compact set

Given an equation f(x) = 0, the problem of finding its solution nearest to a given point is considered. In contrast to the authors’ previous works dealing with this problem, exact algorithms are proposed assuming that the function f is continuous on a compact set. The convergence of the algorithms is proved, and their performance is illustrated with test examples.     

Bifurcation and chaos near sliding homoclinics

We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(α) in such a way that if M(α) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.     

Positive solutions for a system of generalized Lidstone problems

In this paper, we study the existence and multiplicity of positive solutions for the system of the generalized Lidstone problems We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some properties of concave functions, so that the nonlinearities f and g are allowed to grow in distinct manners, with one of them growing superlinearly and the other growing sublinearly.     

Traveling waves for nonlinear cellular neural networks with distributed delays

In this paper, we will establish the existence and nonexistence of traveling waves for nonlinear cellular neural networks with finite or infinite distributed delays. The dynamics of each given cell depends on itself and its nearest m left or l right neighborhood cells where delays exist in self-feedback and left or right neighborhood interactions. Our approach is to use Schauder?s fixed point theorem coupled with upper and lower solutions of the integral equation in a suitable Banach space. Further, we obtain the exponential asymptotic behavior in the negative infinity and the existence of traveling waves for the minimal wave speed by the limiting argument. Our results improve and cover some previous works.     

Regularity of a class of subLaplacians on the 3-dimensional torus

In this paper we present a necessary and sufficient condition for a family of sums of squares operators to be globally hypoelliptic on a torus. This condition says that either a Diophantine condition is satisfied or there exists a point of finite type. Also, we describe the analytic and Gevrey versions of this result. The proof is based on L²-estimates and microlocal analysis.     

Units of compatible nearrings, III

This paper is a continuation of two previous works studying the units of a compatible nearring R satisfying the descending chain condition on right ideals using a faithful compatible module G of R. A crucial point in doing this involves determining 1 +  Ann _R (G/H) where H is a direct sum of isomorphic minimal R-ideals. The high point of this paper is extending this determination from the cases in the previous works to the case where G/H and H contain no isomorphic minimal factors. We also shall further expand our knowledge of when a special type of principal series for G introduced in the second of these previous works called a quasi c-chain exists.     

Positive solutions for a high-order semipositone fractional differential equation with integral boundary conditions

In this paper, we consider the existence of positive solutions for a high-order semipositone fractional differential equation with Riemann-Stieltjes integral boundary conditions. By Krasnoselskii-Zabreiko fixed point theorem and some inequalities associated with Green’s function, two new existence theorems are obtained in the case that the nonlinearity f is allowed to grow both superlinearly and sublinearly. Finally, two examples are given to illustrate the main results.     

Unique solvability of the stationary Fokker-Planck equation in a class of positive functions

We study the stationary Focker-Planck equation Δu ? div(u f) = 0 with a given vector field f of the class C ₀ ^∞ (R ⁿ ) on the basis of a fixed point principle that generalizes the contraction mapping method. Next, we introduce a parameter in the equation and prove the unique solvability of the equation Δu ? div(uγ f) = 0 with the parameter in the class of positive slowly increasing functions. We reveal the analytic dependence of the positive solution u on the parameter γ. Pointwise estimates for positive solutions are proved.     

Univalence criterion and convexity for an integral operator

For analytic functions f and g in the open unit disk U, a new integral operator I₁(f,g)(z) is introduced. The main object of this paper is to obtain a univalence condition and the order of convexity for the integral operator I₁(f,g)(z).