Weighted Estimates for a Class of Global Maximal Operators Associated with Dispersive Equation

For a function?satisfying some suitable growth conditions,consider the following general dispersive equation defined by■where■is a pseudo-differential operator with symbol?(|ξ|).In the present paper,when the initial data f belongs to Sobolev space,we give the local and global weighted L~q estimate for the global maximal operator S_?^**defined by S_?^**f (x)=■where■is a formal solution of the equation (*).     

STABILITY OF THE SOLUTION OF THE ITERATED EQUATION sum from i=1 to n λ_if(x)=F(x)

This note is a further work of discussion on the iterated equation λ_1f(x)+λ_2f~2(x)+…λ_nf~n(x)-P(x), following the author's conclusions of the existence and uniqueness of this equation. In this note we prove that the sufficient condition of the stability of the solution is the same one as the uniqueness of the solution (see [4]).     

SIGN-CHANGING SOLUTIONS FOR SCHRODINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS

In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation
-△u ＋ λV（x）u = K（x）｜u｜^p-2u x∈R^N, u→0 as ｜x｜→ ＋∞,
2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V（x） and K（x） satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].     

一类高阶线性微分方程解的复振荡

In this paper,we shall use Nevanlinna theory of meromorphic functions to investigate the complex oscillation theory of solutions of some higher order linear differential equation.Suppose that A is a transcendental entire function with ρ(A)<1/2.Suppose that k≥2 and f(k)+A(z)f=0 has a solution f with λ(f)<ρ(A),and suppose that A1=A+h,where h≡0 is an entire function with ρ(h)<ρ(A).Then g(k)+A1(z)g=0 does not have a solution g with λ(g)<∞.     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

Growth,Zeros and Fixed points of Differences of Meromorphic Solutions of Difference Equations

In this paper,we study the di erence equation a1(z)f(z+1)+a0(z)f(z)=0;where a1(z)and a0(z)are entire functions of nite order.Under some conditions,we obtain some properties,such as xed points,zeros etc.,of the di erences and forward di erences of meromorphic solutions of the above equation.     

SIGN-CHANGING SOLUTIONS FOR SCHRDINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS

In this article, we study the existence of sign-changing solutions for the following Schrdinger equation-△u + λV(x)u = K(x)|u|p-2u x ∈ RN, u → 0 as |x| → +∞,where N ≥ 3, λ 0 is a parameter, 2 p 2N N-2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow,we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].     

On an Asymptotic Behavior of Exponential Functional Equation

The stability problems of the exponential （functional） equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved： Let X be a real （or complex） normed space. A mapping f ： X → C is exponential if and only if f（x ＋ y） - f（x）f（y） → 0 as ｜｜x｜｜ ＋ ｜｜y｜｜ → ∞ under some suitable conditions.     

Unb ounded Motions in Asymmetric Oscillators Dep ending on Derivatives

In this paper, we consider the unboundedness of solutions for the asymmetric equation x00+ax+?bx?+?(x)ψ(x0)+f(x)+g(x0)=p(t), where x+ = max{x, 0}, x? = max{?x, 0}, a and b are two different positive constants, f (x) is locally Lipschitz continuous and bounded,?(x), ψ(x), g(x) and p(t) are continuous functions, p(t) is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case √1a+ √1b ∈Q and the nonresonance case√1a + √1b /∈Q.     

依赖于导数项的非对称振子解的无界性（英文）

In this paper, we consider the unboundedness of solutions for the asymmetric equation x'+ax~+-bx~-+(x)ψ(x')+f(x)+g(x')=p(t),where x~+= max{x, 0}, x~-= max{-x, 0}, a and b are two different positive constants,f(x) is locally Lipschitz continuous and bounded, (x), ψ(x), g(x) and p(t) are continuous functions, p(t) is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case 1/a~(1/2)+1/b~(1/2)∈Q and the nonresonance case 1/a~(1/2)+1/b~(1/2)?Q     

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRDINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS

In this paper,we study the following generalized quasilinear Schrdinger equations with critical or supercritical growths-div(g~2(u)▽u) + g(u)g′(u)|▽u|~2+ V(x)u = f(x,u) + λ|u|~(p-2)u,x∈R~N,where λ0,N≥3,g:R → R~+ is a C~1 even function,g(0) = 1,g′(s) ≥ 0 for all s ≥ 0,lim_(|s|→+∞)g(s)/|s|~(α-1):= β 0 for some α≥ 1 and(α-1)g(s) g′(s)s for all s 0 and p ≥α2*.Under some suitable conditions,we prove that the equation has a nontrivial solution for smallλ 0 using a change of variables and variational method.     

A Note on a Problem of M. S. Berger

In this paper, we discuss the problem concerning global and local structure of solutions of an operator equation posed by M. S. Berger. Let f : U (?)E→ F be a C1 map, where E and F are Banach spaces and U is open in E. We show that the solution set of the equation f(x)=y for a fixed generalized regular value y of f is represented as a union of disjoint connected C1 Banach submanifolds of U, each of which has a dimension and its tangent space is given. In particular, a characterization of the isolated solutions of the equation f(x) = y is obtained.     

有关M.S.Berger问题的注记

In this paper, we discuss the problem concerning global and local structure of solutions of an operator equation posed by M. S. Berger. Let f : U 真包含 E → F be a C1 map, where E and F are Banach spaces and U is open in E. We show that the solution set of the equation f(x) = y for a fixed generalized regular value y of f is represented as a union of disjoint connected C1 Banach submanifolds of U, each of which has a dimension and its tangent space is given. In particular, a characterization of the isolated solutions of the equation f(x) = y is obtained.     

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

THE EXISTENCE AND UNIQUENESS OF ALMOST PERIODIC SOLUTION OF x" + cx'''' + g(x) =P(t,x)

This paper studies equation x" + cx' + g(x) = P(t,x). Under some suitable conditions the existence and uniqueness of almost periodic solution of this equation are given.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

Existence and Multiplicity Results for a Degenerate Elliptic Equation

The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f（x, s）, we show the following problem：-△pu=λu^α-a（x）u^q in Ω,u│δΩ=0 possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N ≥ 2, with C^2 boundary, λ is a positive parameter, Ap is the p-Laplacian operator with p 〉 1, α, q are given constants such that p - 1 〈α 〈 q, and a（x） is a continuous positive function in Ω^-.     

SINGULAR PERTURBATIONS OF BOUNDARY VALUE PROBLEMS FOR THIRD ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

In this paper we obtain some results about the convergence of aolutions of the boundary value problems of the third order nonlinear ordinary differential equation with a small parameter ε>0: (i=0, 1, 2) to a solution of their reduced problem as ε→0, hero z=ψ(t, x, y) is a root of the equation f(t, x, y, z, 0)=0, and about the existence of solutions of the reduced problem. In addition, under certain conditions we prove the existence of solutions of the boundary value problems (1), (2_i) (i=1, 2), and give their asymptotic estimations.     

OSCILLATION OF A CLASS OF HILLE EQUATION

This paper studies a class of Hille equation. A formula for solutions of a class of Hille equation is given. Under some suitable conditions the oscillation and nonoscillation of a class of Hille equation are established. Our results generalize the known Hille's ones.     

On approximation properties of non-convolution type nonlinear integral operators

In the present paper we state some approximation theorems concerning pointwise convergence and its rate for a class of non-convolution type nonlinear integral operators of the form:Tλ (f;x) = B A Kλ (t,x, f (t))dt , x ∈< a,b >, λ∈Λ. In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 of f as (x,λ ) → (x0,λ0) in L1 < A,B >, where < a,b > and < A,B > are is an arbitrary intervals in R, Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology. The results of the present paper generalize several ones obtained previously in the papers [19]-[23].