迭代函数方程$G(x,f(x),\ldots,f^n(x))=F(x)$的$C^1$解

In this paper we consider the iterative equation G(x,f(x),...,f n(x)) = F(x) on R,and give the existence of C 1 solutions near the fixed point of F,which generalize some results on the leading coefficient problem from the form of the polynomial-like iterative equations to the general form.     

含k-次增生算子方程（1–k）x＋Tx=f解的Ishikawa迭代逼近

在一致光滑的实Banach空间中,研究当T为k-次增生算子时,非线性方程（1-k）x＋Tx=f具混合误差的Ishikawa迭代解,给出了强收敛定理,并对Ishikawa迭代程序关于含k-次增生算子方程（1-k）x＋Tx=f的稳定性进行了讨论,推广和改进了近期一些文献的相关结果.     

On the equationx″+f(x)h(x′)x′+g(x)=e(t)

Summary We consider the equation x″+f(x)h(x′)x′+g(x)=e(t) in which f, g, and h are continuous, e is sectionally continuous and absolutely integrable, h(u)>0, xg(x)>0 if x ≠ 0, and f(x)≥0. Necessary and sufficient conditions are given for boundedness of all solutions and their derivatives. When f(0)>(0) we give necessary and sufficient conditions for all solutions and their derivatives to converge to zero. Entrata in Redazione il 14 giugno 1969.     

On the functional equation $$f(x+y)=f(x)+f(y)$$f(x+y)=f(x)+f(y)

Monatshefte für Mathematik - An extension of the linearity of the continuous solutions of the functional equation $$f(x+y)=f(x)+f(y)$$ to measurable functions is presented.     

关于函数y=f(x)与y=|f(x)|

本文对函数 y=f ( x)与 y=|f ( x) |的部分性质进行了比较和分析 ,指出了它们性质上的区别与联系 .     

A note about the cyclotomic polynomial Φpq(x) and some related results

The expression of cyclotomic polynomial Φpq(x) is concerned for a long time. A simple and explicit expression of Φpq(x) in Z[x] has been showed. The form of the factors of Φpq(x) over F2 and the upper, lower bounds of their Hamming weight are provided.     

一种预测—校正式单点迭代方法

本文以Ｎｅｗｔｏｎ迭代法（ｘｎ＋１＝ｘｎ－ｆ（ｘｎ）／ｆ′（ｘｎ），收敛阶为２）为基础，给出了一种新的实用的预测—校正式单点迭代方法（ｘｎ＋１＝ｘｎ－ｕ（ｘｎ）ｆ（ｘｎ）＋１２ｆ（ｘｎ－ｕ（ｘｎ））ｆ（ｘｎ）－１２ｆ（ｘｎ－ｕ（ｘｎ））收敛阶为４）．该方法不仅公式简洁，计算方便，计算量小，而且收敛阶高，收敛速度快     

Classification of permutation polynomials of the form $$x^3g(x^{q-1})$$ x 3 g ( x q - 1 ) of $${\mathbb F}_{q^2}$$ F q 2 where $$g(x)=x^3+bx+c$$ g ( x ) = x 3 + b x + c and $$b,c \in {\mathbb F}_q^*$$ b,c ∈ F q ∗

Designs, Codes and Cryptography - We classify all permutation polynomials of the form $$x^3g(x^{q-1})$$ of $${\mathbb F}_{q^2}$$ where $$g(x)=x^3+bx+c$$ and $$b,c \in {\mathbb F}_q^*$$ . Moreover...     

Generalized p(x)-elliptic system with nonlinear physical data

This paper considers the following Dirichlet problem of the form \[-\text{div}\,\big(\Phi(Du-\Theta(u)\big)=v(x)+f(x,u)+\text{div}\,\big(g(x,u)\big),\] which corresponds to a diffusion problem with a source $v$ in moving and dissolving substance, the motion is described by $g$ and the dissolution by $f$. By the theory of Young measure we will prove the existence result in variable exponent Sobolev spaces $W^{1,p(x)}_0(\Omega;\R^m)$.     

函数f(x)关于｜x｜的单调性定理

本文提出了函数ｆ（ｘ）在对称区间上关于｜ｘ｜的单调性定理,并讨论了它的应用     

含有k-次增生算子T的方程x+Tx=f的Ishikawa迭代解

建立起了强收敛于方程x Ty=f的解的Ishikawa迭代过程．其中T是一致光滑Banach空间中的一个在D(T)上的既非有界又非Lipschitz的k-次增生算子，推广了一些已有的结果。     

On the functional equation x + f(y + f(x))=y + f(x + f(y))

For an abelian group (G, + ,0) we consider the functional equation $$f : G \to G, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G), \quad\quad\qquad (1)$$ most times together with the condition $$f(0) = 0.\qquad\qquad\qquad\qquad\qquad (0)$$ Our main question is whether a solution of ${(1) \wedge (0)}$ must be additive, i.e., an endomorphism of G. We shall answer this question in the negative (Example 3.14) Rätz (Aequationes Math 81:300, 2011).     

含有k-次增生算子T的方程x Tx=f的迭代解

在一致光滑的Ｂａｎａｃｈ空间研究了非线性算子方程ｘ＋Ｔｘ＝ｆ的迭代解．其中Ｔ不必是增生的,也不必是Ｌｉｐｓｃｈｉｔｚｉａｎ的．从而推广了一些已知的结果．     

On the existence of periodic solutions for the equation\ddot x + f(x)\dot x + g(x) = 0

We establish in this work sufficient conditions for the existence of periodic solutions for the Liénard equation .     

Banach空间中增生算子T的方程f=x＋Tx的迭代解

在具有一致凸对偶空间的Banach空间中讨论了关于增生算子T的方程f=x Tx的迭代解,其结果推广和改进了Chidume和Zhu的结果。     

Uniform boundedness inL p (p=p(x)) of some families of convolution operators

Suppose that a measurable 2π-periodic essentially bounded function (the kernel) κ_λ =κ_λ(x) is given for any realλ≥1. We consider the following linear convolution operator inL ^p: $$\kappa _\lambda = \kappa _\lambda f = (\kappa _\lambda f)(x) = \int_{ - \pi }^\pi {f(t)} k_\lambda (t - x) dt.$$ Uniform boundedness of the family of operators {Κ_λ}_λ≥1 is studied. Conditions on the variable exponentp=p(x) and on the kernel κ_λ that ensure the uniform boundedness of the operator family {Κ_λ}_λ≥1 inL ^p are obtained. The condition on the exponentp=p(x) is given in its final form.     

APPROXIMATION PROPERTIES OF LAGRANGE INTERPOLATION POLYNOMIAL BASED ON THE ZEROS OF （1- x^2）cosnarccosx

We study some approximation properties of Lagrange interpolation polynomial based on the zeros of （1-x^2）cosnarccosx. By using a decomposition for f（x） ∈ C^τC^τ＋1 we obtain an estimate of ‖f（x） -Ln＋2（f, x）‖ which reflects the influence of the position of the x＇s and ω（f^（r＋1）,δ）j,j = 0, 1,... , s,on the error of approximation.     

Lipschitzian solutions of the implicit Cauchy problemg(x′)=f(t,x),x(0)=0, withf discontinuous inx

In this paper, we study the existence of lipschitzian solutions of the implicit Cauchy problemg(x′)=f(t,x),x(0)=0, without assuming that the functionf is continuous with respect tox. After establishing our main results, we also present several counterexamples to possible improvement of them.     

Solutions and Multiple Solutions for p(x)-Laplacian Equations with Nonlinear Boundary Condition

The authors study the p(x)-Laplacian equations with nonlinear boundary condition.By using the variational method,under appropriate assumptions on the perturbation terms f1(x,u),f2(x,u) and h1(x),h2(x),such that the associated functional satisfies the "mountain pass lemma" and "fountain theorem" respectively,the existence and multiplicity of solutions are obtained.The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.