Likely Limit Sets of Feigenbaum''''s Maps and Their Hausdorf f Dimensions

A continuous map from a closed interval into itself is called a Feigenbaum's map if it is a solution of the functional equation f2(λx)=λf(x).In this paper, the likely limit sets of a type of Feigenbaum's maps are studied and their Hausdorff dimensions are estimated.As an application, we prove that for any 0相似文献   

DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL HALF SPACE

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n + : u tt u + u t + divf (u) = 0, t > 0, x = (x 1 , x ′ ) ∈ R n + := R + × R n 1 , u(0, x) = u 0 (x) → u + , as x 1 → + ∞ , u t (0, x) = u 1 (x), u(t, 0, x ′ ) = u b , x ′ = (x 2 , x 3 , ··· , x n ) ∈ R n 1 . (I) For the non-degenerate case f ′ 1 (u + ) < 0, it was shown in [10] that the above initialboundary value problem (I) admits a unique global solution u(t, x) which converges to the corresponding planar stationary wave φ(x 1 ) uniformly in x 1 ∈ R + as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u(t, x) for t → + ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ (u) < 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 (u b ) ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.     

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].     

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].     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

DISCUSSION ON THE CONTINUOUS SOLUTIONS OF A FUNCTIONAL EQUATION

1 IntroductionIn an issue of the American Mathematical Monthly [1] the following problemwas raised:"Determine all homeomorphisms f from [0,1] onto [0,1] that are solutionsof the functional equationf(Zx -- f(x)) = x for all x E [0,IJ.',Ratti and Lin [2] generalized the problem in two different directions. In thispapert we consider the following functional equationf(cr(x) df(x)) = F(x), (1.1)where c < 0,d > 0, c d = 1. We shall give the sufficient conditions ofthe existence, uniqueness a…     

ABOUT SOME NOTES IN THE VARIATIONAL PRINCIPLE

The bilinear from a(·,·)on the real Hilbert space is said to be coercive if there is a α>0) such that α(v,v)≥α‖v‖_H v∈H. (0.1)Assume that α(·,·)is a bilinear continuous and coercive form on H.Then, giving a f∈H′arbitrarily,There exists a unique u∈H such that α(u,v) = f(v). (0.2)Furthermore, u depends continuously on f. This is the Lax-Milgrarm Theorem obtained in 1954.In 1972, A. K. Aziz[1] improved the condition (0.1) and obtained the sufficient condition of Eq. (0.2) being Well-posed.In this paper,by different method,we proved, more simply, this sufficient condition in §1. Furthermore, we proved that this sufficient condition is also the necessary one. In §2 we improved the error estimate of approximate solution of (0.2). which was obtained by Céa[3] in 1964. In §3 we discussed the Well-posed problem of Eq.(2), when a(·, ·) is monotonic. This result may be considered as a generalization of the dffinition of coercive,and the Lax-Milgram Theorem as a special example. In §4 we exte     

Quenching Phenomenon for a Parabolic MEMS Equation

This paper deals with the electrostatic MEMS-device parabolic equation u_t-?u =λf(x)/(1-u)~p in a bounded domain ? of R~N,with Dirichlet boundary condition,an initial condition u0(x) ∈ [0,1) and a nonnegative profile f,where λ 0,p 1.The study is motivated by a simplified micro-electromechanical system(MEMS for short) device model.In this paper,the author first gives an asymptotic behavior of the quenching time T*for the solution u to the parabolic problem with zero initial data.Secondly,the author investigates when the solution u will quench,with general λ,u0(x).Finally,a global existence in the MEMS modeling is shown.     

On 2n—th—Order Eigenvalue Problem

In [1], Lu discussed the existence of positive solution for the BVPwhere f: [0, 1] × [0, ∞) → R is continuous and satisfies the following conditions(H1) There exists an M > 0 such that(H2) limx→∞ f(t, x)/x =∞uniformly for t [α,β] (0, 1) for some α < β and λ > 0 is a parameter.In this paper, the result of [1] is generalized to the following 2n-th-order BVPThe result of this paper isTheorem 1 Assume (H1) and (H2) hold. Then the BVP (3)-(4) has a positive solutionifλ > 0 is sm…     

A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRDINGER EQUATION

We use the Fokas method to analyze the derivative nonlinear Schrdinger(DNLS)equation iqt(x, t) =-qxx(x, t)+(rq2)x on the interval [0, L]. Assuming that the solution q(x, t)exists, we show that it can be represented in terms of the solution of a matrix RiemannHilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit(x, t) dependence, and it has jumps across {ξ∈ C|Imξ4= 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a(ξ), b(ξ)}, {A(ξ), B(ξ)}, and{A(ξ), B(ξ)}, which in turn are defined in terms of the initial data q0(x) = q(x, 0), the boundary data g0(t) = q(0, t), g1(t) = qx(0, t), and another boundary values f0(t) = q(L, t), f1(t) =qx(L, t). The spectral functions are not independent, but related by a compatibility condition,the so-called global relation.     

Essential and discrete spectra of partially integral operators

Let Ω₁, Ω₂ ⊂ ℝ^ν be compact sets. In the Hilbert space L ₂(Ω₁ × Ω₂), we study the spectral properties of selfadjoint partially integral operators T ₁, T ₂, and T ₁ + T ₂, with

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

On super vertex-graceful unicyclic graphs

A graph G with p vertices and q edges, vertex set V(G) and edge set E(G), is said to be super vertex-graceful (in short SVG), if there exists a function pair (f, f ⁺) where f is a bijection from V(G) onto P, f ⁺ is a bijection from E(G) onto Q, f ⁺((u, v)) = f(u) + f(v) for any (u, v) ∈ E(G),

Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

Three symmetric positive solutions for second-order nonlocal boundary value problems

Using the Leggett-Williams fixed point theorem,we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form u(t)+g(t)f(t,u(t))=0,0相似文献   

Numerical Approximation of a Reaction-Diffusion System with Fast Reversible Reaction

The authors consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction.It is deduced from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the kinetic rate.It follows that the solution converges to the solution of a nonlinear diffusion problem,as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.     

Description of the lipschitz and zygmund classes in terms of the modulus of continuity

Let C be the space of 2π-periodic continuous real-valued functions, let

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

Duals of holomorphic Besov spaces on the polydisks and diagonal mappings

Let U ⁿ be the unit polydisk in C ⁿ and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω ₁, ..., ω _n ), ω _j ∈ S(1 ≤ j ≤ n) and f ∈ H(U ⁿ ). The function f is said to be in holomorphic Besov space B _p (ω) if