Analytic solutions of an iterative differential equation under Brjuno condition

In this paper, the differential equation involving iterates of the unknown function,
x＇（z）=[a^2-x^2（z）]x^[m]（z）
with a complex parameter a, is investigated in the complex field C for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the SchrSder transformation x（z） = y（αy^-1（z）） to the following another functional differential equation without iteration of the unknown function
αy＇（αz）=[a^2-y^2（αz）]y＇（z）y（α^mz）,
which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y（αy^-1 （z）） for the original iterative differential equation are obtained. We discuss not only these α given in SchrSder transformation in the hyperbolic case 0 〈 ｜α｜ 〈 1 and resonance, i.e., at a root of the unity, but also those α near resonance （i.e., near a root of the unity） under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.     

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.     

Analytic invariant curves for a planar mapping

This paper is concerned with the existence of analytic invariant curves for a planar mapping. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. In this paper, we discuss not only the general case, but also the critical cases as well, in particular, the case where β is a unit root is discussed.     

All General Solutions of Post Equations

In a previous paper, we have described all reproductive general solutions of a Post equation, supposing that a general solution is known. In this paper we describe all general solutions of Post equation, supposing that a general solution of this equation is known （Theorem 6）. As a special case we get the previous characterization of reproductive solutions and a similar result for Boolean equations （Theorem 9）.     

Partial Regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set S of suitable weak solutions and the parameter α in the nonlinear term in the following parabolic equation■ It is shown that when 5/3 ≤ α < 7/3, the ■-dimensional parabolic Hausdorff measure of S is zero,which generalizes the recent corresponding work of Ozánski and Robinson in [SIAM J. Math. Anal.,51, 228–255(2019)] for α = 2 and f = 0. The same result is valid for a 3...     

一类系数为间隙级数的高阶线性微分方程解的增长性

In this paper, we investigate the growth of solutions of a class of higher order linear differential equations with coefficients being gap series. In this case, we remove the condition that the order of coefficients in equations is less than 1/2, and obtain some results which improve the previous results.     

CONTINUITY IN FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper, we investigate the continuous dependence of solutions of the functional dif-ferential equation with infinite delay x'(t) = f(t, x_t) on initial functions. Endowing the phasespare a g-norm as well as a supremum norm. we show that if the equation satifies a mild fadingmemory dondition, then the continuity of f in respect to the topology induced by the supremumnorm can yield the continuity of solutions of the equation in respect to the topology induced bythe g-norm which is stronger than the ahead one.     

ON THE GROWTH OF THE SOLUTIONS FOR 1ST KIND OF HIGHER ORDER DIFFERENTIAL EQUATIONS IN THE UNIT DISC

In this paper, we study the growth of the solutions for 1st kind of differential equations of higher order in the unit disc. We give a sufficient condition for all solutions of second order linear differential equation to be inadmissible and a sufficient condition for all solutions of higher order linear differential equation to be of infinite order.     

STABILITY OF THE SOLUTIONS TO STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper,we obtain the stability of solutions to stochastic functional differential equations with infinite delay at phase space BC((-∞,0];Rd),under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition by means of the corollary of Bihari inequality.     

Existence and Blowup of Solutions for Neutral Partial Integro-differential Equations with State-dependent Delay

In this paper, we study the existence and blowup of solutions for a neutral partial functional integro-differential equation with state-dependent delay in Banach space. The mild solutions are obtained by Sadovskii fixed point theorem under compactness condition for the resolvent operator, the theory of fractional power and α-norm are also used in the discussion since the nonlinear terms of the system involve spacial derivatives. The strong solutions are obtained under the lipschitz condition. In addition, based on the local existence result and a piecewise extended method, we achieve a blowup alternative result as well for the considered equation. Finally, an example is provided to illustrate the application of the obtained results.     

在共振点附近的一类二阶泛函微分方程的解析解

在复域C内研究一类包含未知函数迭代的二阶微分方程x″(z)=G(z,x(z),x~2(z),…,x~m(z))解析解的存在性.通过Schr(?)der变换,即x(z)=y(αy~(-1)(z)),把这类方程转化为一种不含未知函数迭代的泛函微分方程α~2y″(αz)y″(z)-αy′(αz)y″(z)= (y′(z))~3G(y(z),y(αz),…,y(α~mz)),并给出它的局部可逆解析解.本文不仅讨论了双曲型情形0<|α|<1和共振的情形(α是一个单位根),而且还在Brjuno条件下讨论了共振点附近的情形(即单位根附近).     

复指数函数系的极小性

A sufficient condition is obtained for the minimality of the complex exponential system E（A, M） = {z^le^λnz： l = 0, 1,,.., mn - 1; n = 1, 2,...} in the Banaeh space La^p consisting of all functions f such that f^-a ∈ LP（N）. Moreover, if the incompleteness holds, each function in the closure of the linear span of exponential system E（A, M） can be extended to an analytic function represented by a Taylor-Dirichlet series.     

Infinitely many bound state solutions of Schrodinger-Poisson equations in $\mathbb{R}^3$

In this paper, we study a system of Schr\"odinger-Poisson equation \[ \left\{ \begin{array}{c} -\Delta u+a(x)u+K(x)\phi u=|u|^{p-2}u,\quad \quad \quad \ \ \ \ \ \ x\in \mathbb{R}^3, \-\Delta \phi=K(x)u^2,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ x\in \mathbb{R}^3, \end{array} \right. \] where $p\in (4,6)$ and $ K\geq (\not\equiv) 0$. Under some suitable decay assumptions but without any symmetry property on $a$ and $K$, we obtain infinitely many solutions of this system.     

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.     

Infinitely many solutions for fractional Schrodinger-Maxwell equations

In this paper using fountain theorems we study the existence of infinitely many solutions for fractional Schr\"{o}dinger-Maxwell equations \[\begin{cases} (-\Delta)^\alpha u+\lambda V(x)u+\phi u=f(x,u)-\mu g(x)|u|^{q-2}u, \text{ in } \mathbb R^3,\(-\Delta)^\alpha \phi=K_\alpha u^2, \text{ in } \mathbb R^3, \end{cases}\] where $\lambda,\mu >0$ are two parameters, $\alpha\in (0,1]$, $K_\alpha=\frac{\pi^{-\alpha}\Gamma(\alpha)}{\pi^{-(3-2\alpha)/2}\Gamma((3-2\alpha)/2)}$ and $(-\Delta)^\alpha$ is the fractional Laplacian. Under appropriate assumptions on $f$ and $g$ we obtain an existence theorem for this system.     

$\alpha$阶星象亚纯函数

令$S(p)$表示单位圆盘$\mathbb{D}$上在$p\in(0,1)$处有一个简单极点的单叶亚纯函数全体.令$\alpha\in[0,1)$,我们用$\Sigma^{*}(p,\omega_{0},\alpha)$表示$f\in S(p)$使得$\hat{\mathbb{C}}\setminus f(\mathbb{D})$是关于不动点$\omega_{0}\neq0$, $\infty$星象的$\alphga$阶区域的函数全体.在本文中,$f\in\Sigma^{*}(p,\omega_{0},\alpha)$的一些解析刻画条件和系数估计被考虑.     

Positive Ground State Solutions for Schrödinger-Poisson System with General Nonlinearity and Critical Exponent

In this paper, we consider the following Schrödinger-Poisson system \begin{equation*}\begin{cases} -\Delta u + \eta\phi u = f(x,u) + u^5,& x\in\Omega,\\ -\Delta\phi=u^2,& x\in\Omega,\\u = \phi =0,& x\in \partial\Omega, \end{cases}\end{equation*} where $\Omega$ is a smooth bounded domain in $R^3$, $\eta=\pm1$ and the continuous function $f$ satisfies some suitable conditions. Based on the Mountain pass theorem, we prove the existence of positive ground state solutions.     

Analytic Invariant Curves for an Iterative Equation Related to Ricker-type Second-order Equation

In this paper, we consider the existence of analytic invariant curves of an iterative equation f(f(x)) = xe~(a-x-f(x)) which arises from Ricker-type second-order equation. By reducing the equation with the Schr?der transformation to an auxiliary equation, the author discusses not only that the parameter at resonance,i.e., at a root of the unity, but also the parameter near resonance under the Brjuno condition.     

Quasi-periodic Solutions for the Derivative Nonlinear SchrÖdinger Equation with Finitely Differentiable Nonlinearities

The authors are concerned with a class of derivative nonlinear Schr¨odinger equation iu_t + u_(xx) + i?f(u, ū, ωt)u_x=0,(t, x) ∈ R × [0, π],subject to Dirichlet boundary condition, where the nonlinearity f(z1, z2, ?) is merely finitely differentiable with respect to all variables rather than analytic and quasi-periodically forced in time. By developing a smoothing and approximation theory, the existence of many quasi-periodic solutions of the above equation is proved.     

Local analytic solutions of a functional differential equation

In this paper, we are concerned with the existence of analytic solution of a functional differential equation αz+βx^′(z)=x(az+bx^″(z)), where are four complex numbers. We first discuss the existence of analytic solutions for some special cases of the above equation. Then, by reducing the equation with the Schröder transformation to the another functional equation with proportional delay, an existence theorem is established for analytic solutions of the original equation. For the constant λ given in the Schröder transformation, we discuss the case 0<|λ|<1 and λ on the unit circle S¹, i.e., |λ|=1. We study λ is at resonance, i.e., at a root of the unity and λ is near resonance under the Brjuno condition.