On determination of jumps in terms of the Abel-Poisson mean of Fourier series. II

In this paper, we generalize two important results of Bagota and Móricz [1], and generalize our earlier results in [6] from one-variable to two-variable case. As special applications, we prove that the generalized jump of f(x, y) at some point (x ₀, y ₀) can be determined by the higher order mixed partial derivatives of the Abel-Poisson mean of double Fourier series and the higher order mixed partial derivatives of the Abel-Poisson means of the three conjugate double Fourier series.     

More on determination of jumps

We generalize some old and new results on the determination of jumps of a periodic, Lebesgue integrable function f at each point of discontinuity of first kind in terms of the partial sums of the conjugate series to the Fourier series of f in [1], and in terms of the Abel-Poisson means in [2], to some more general linear operators which satisfy some certain conditions. The linear operators in discussion include the Fejér means, de la Vallée-Poussin means, and Bernstein-Rogosinski sums. Research also supported in part by NSF of China under grant number 10471130.     

Stability of Classes of Mappings and Holder Continuity of Higher Derivatives of Elliptic Solutions to Systems of Nonlinear Differential Equations

Nirenberg published the following well-known result in 1954: Let a function z be a twice continuously differentiable solution to a nonlinear second-order elliptic equation. Suppose that the function F defining the equation is continuous and has continuous first-order partial derivatives with respect to all of its arguments (i.e., independent together with z and the symbols of all first- and second-order partial derivatives of z). Then the partial derivatives of z are locally Holder continuous. Simultaneously with Nirenberg, Morrey obtained an analogous result for elliptic systems of second-order nonlinear equations. In this article, we get the same result for the higher derivatives of elliptic solutions to systems of nonlinear partial differential equations of arbitrary order and a rather general shape. The proof is based on the results of the author's recent research on the study of the stability phenomena in the C ^l-norm of classes of mappings.     

Determination of jumps in terms of Abel–Poisson means

A theorem of Ferenc Lukács determines the jumps of a periodic, Lebesgue integrable function f at each point of discontinuity of first kind in terms of the partial sums of the conjugate series to the Fourier series of f. The aim of this note is to prove an analogous theorem in terms of the Abel-Poisson means. This revised version was published online in June 2006 with corrections to the Cover Date.     

Brownian motion and exposed solutions of differential inclusions

We present a research program designed by A. Bressan and some partial results related to it. First, we construct a probability measure supported on the space of solutions to a planar differential inclusion, where the right-hand side is a Lipschitz continuous segment. Such measure assigns probability one to solutions having derivatives a.e. equal to one of the endpoints of the segment. Second, for a class of planar differential inclusions with Hölder continuous right-hand side F, we prove existence of solutions whose derivatives are exposed points of F. Finally, we complete the research program if the right-hand side of the differential inclusion does not depend on the state and prove a result on the Lipschitz continuity of an auxiliary map. The proofs rely on basic properties of Brownian motion.     

Chaos via Furstenberg family couple

In this paper we define (F₁,F₂)-chaos via Furstenberg family couple F₁ and F₂. It turns out that the Li-Yorke chaos and distributional chaos can be treated as chaos in Furstenberg families sense. Some sufficient conditions such that a system is the (F₁,F₂)-chaotic (Theorems 4.2 and 4.4) are given. In addition, we construct an example as an application. It is showed that the second type of distributional chaos cannot imply the first type of distributional chaos even though the scrambled set is uncountable.     

A characterization of finite symplectic polar spaces of odd prime order

A sufficient condition for the representation group for a nonabelian representation (Definition 1.1) of a finite partial linear space to be a finite p-group is given (Theorem 2.9). We characterize finite symplectic polar spaces of rank r at least two and of odd prime order p as the only finite polar spaces of rank at least two and of prime order admitting nonabelian representations. The representation group of such a polar space is an extraspecial p-group of order p1+2r and of exponent p (Theorems 1.5 and 1.6).     

On a theorem of Choquet and Dolecki

The results of this paper clarify and extend slightly the previous work of Dolecki and Lechicki (C. R. Acad. Sci. Paris, 293 1981, 219–221; J. Math. Anal. Appl., 88 1982, 547–584) and Hansell, Jayne, Rogers and the author (Math. Z., 189 1985, 297–318). Let X, Y be Hausdorff spaces and F: X → Y an upper semicontinuous set-valued map. A subset K of F(x) is said to be a peak of F at x, if, for every open set V containing K, there exists a neighbourhood U of x such that F(U)/F(x)/t(V). Criteria (“Choquet-Dolecki Theorems”) are given in order that F has the smallest possible peak. It turns out that in unexpectedly general situations an upper semicontinuous map F has, for every x in X, a peak which is the smallest possible at x and moreover compact.     

Stability of Classes of Mappings and Holder Continuity of Higher Derivatives of Elliptic Solutions to Systems of Nonlinear Differential Equations

Nirenberg published the following well-known result in 1954: Let a function z be a twice continuously differentiable solution to a nonlinear second-order elliptic equation. Suppose that the function F defining the equation is continuous and has continuous first-order partial derivatives with respect to all of its arguments (i.e., independent together with z and the symbols of all first- and second-order partial derivatives of z). Then the partial derivatives of z are locally Holder continuous. Simultaneously with Nirenberg, Morrey obtained an analogous result for elliptic systems of second-order nonlinear equations. In this article, we get the same result for the higher derivatives of elliptic solutions to systems of nonlinear partial differential equations of arbitrary order and a rather general shape. The proof is based on the results of the author's recent research on the study of the stability phenomena in the C ^l-norm of classes of mappings.     

Darstellung durch Spinorgeschlechter ternärer quadratischer formen

Let V be a regular ternary quadratic space over the algebraic number field F, L a lattice on V over the maximal order o of F. A number a, which is represented by all completions L_p, is not necessarily represented by L itself, but only by a lattice in the genus of L. We determine in which cases such a number is not represented by all spinor genera in the genus. Theorem 1 repeats the known [8,3] necessary conditions for this, which show that this behaviour is exceptional. They are sharpened in Theorem 2 to a necessary and sufficient condition in terms of certain groups Θ(L_p, a) (Definition 1). These groups are computed in Theorems 3 and 4 for nondyadic and 2-adip p. Some applications are given in the last section: We give new proofs (and in one case a correction) of results from [5] on the numbers represented by some genera of positive definite ternaries.     

Fejér type theorems for Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2], then the nth partial sum of its formally differentiated Fourier series divided by n converges to ^-1 [F(x+0) - F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of nonperiodic functions of bounded variation is also known. These theorems can be interpreted in such a way that the terms of the Fourier-Stieltjes (or Fourier) series of F determine the atoms of the finite Borel measure on the torus T:= [0, 2) induced by an appropriate extension of F (or by F itself in the periodic case). The aim of the present paper is to extend all of these results to the Cesàro as well as Abel-Poisson means of Fourier-Stieltjes (or Fourier) series of a nonperiodic (or periodic) function F of bounded variation. At the end, we sketch a possible extension of these results to linear means defined by more general kernels.     

Stability of a class of stochastic distributed parameter systems with random boundary conditions

In this article we consider the question of stability of a class of stochastic systems governed by elliptic and parabolic second order partial differential equations with Neumann boundary conditions. Results on the “stability in the mean” are given in Theorems 1 and 2, and those on “almost sure stability” are presented in Theorems 3 and 4. These results are proved under the assumption that the perturbing forces are measurable stochastic processes defined on $\text{I × Ω}$. In Theorem 5 it is shown that the proofs require only minor modification to admit progressively measurable (predictable or optional) processes.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications to differential equations

Let X, Y be real Banach spaces, T: X → YA-proper, and C: X → Y compact. Section 1 of this paper is devoted to the study of bifurcation and asymptotic bifurcation problems for Eq. $\text{(1): Tx ? λCx = 0}$. In Theorem 1 it is shown that if T(0) = C(0) = 0 and T and C have F-derivatives T₀ and C₀ at 0 with T₀A-proper and injective, then each eigenvalue of T₀x ? λC₀x = 0 of odd multiplicity is a bifurcation point for Eq. (1). Theorem 2 shows that if T and C have asymptotic derivatives T_∞ and C_∞, then each eigenvalue of T_∞x ? λC_∞x = 0 of odd multiplicity is an asymptotic bifurcation point for Eq. (1). Special cases are treated when Y = X and T = I ? F with Fk-ball-contractive or when Y ≠ X and T is either of type (S) or of strongly accretive type. Section 2 is devoted to applications of Theorems 1 and 2 to bifurcation problems involving elliptic operators. The usefulness of Theorems 1 and 2 stems from the fact that they are directly applicable to differential eigenvalue problems without the preliminary reduction of Eq. (1) to equivalent problems involving compact operators. Moreover, in some cases they are applicable in situations to which the known bifurcation results are not applicable.     

Extinction, persistence and global stability in models of population growth

First, we systematize earlier results on the global stability of discrete model An+1=λAn+F(An−m) of population growth. Second, we invent the effect of delay m when F is unimodal. New, deep and strong results are discussed in Section 4, although Theorems 3-5 (Section 3) are still freshly new. This paper may be considered as a discrete version of our earlier work on the model [D.V. Giang, Y. Lenbury, Nonlinear delay differential equations involving population growth, Math. Comput. Modelling 40 (2004) 583-590]. We are mainly using ω-limit set of persistent solution, which is discussed in more general by P. Walters [An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982].     

On the rate of convergence of some difference schemes for second order elliptic equations

The purpose of this paper is to estimate the rate of convergence for some natural difference analogues of Dirichlet's problem for uniformly elliptic differential equations, $$\begin{gathered} \sum\limits_{j,k = 1}^N {\frac{\partial }{{\partial x_j }}} \left( {a_{jk} \frac{{\partial u}}{{\partial x_k }}} \right) = F in R, \hfill \\ u = f on B, \hfill \\ \end{gathered}$$ in aN-dimensional domainR with boundaryB. These schemes will in general not be of positive type, and the analysis will therefore be carried out in discreteL ²-norms rather than in the maximum norm. Since our approximation of the boundary condition is rather crude, we will only arrive at a rate of convergence of first order for smoothF andf. Special emphasis will be put on appraising the dependence of the rate of convergence on the regularity ofF andf.     

Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces

Let X be a non-empty set and F:X×X→X be a given mapping. An element (x,y)∈X×X is said to be a coupled fixed point of the mapping F if F(x,y)=x and F(y,x)=y. In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393].     

Basic matrix theorems for \mathcal{I}-convergence in (?)-groups

Some aspects of the theory of order and (D)-convergence in (?)-groups with respect to ideals are investigated. Moreover some new Basic Matrix Theorems are proved.     

Further results on factorization of meromorphic solutions of partial differential equations

This paper is a continuation of Hu-Yang [2]. Here we extend Malmquist type theorem ofalgebraic differential equations of Steinmetz [3] and Tu [4] to higher order partial differential equations. The results also generalize Theorems 4.2 and 4.3 in [2].     

New higher order methods for solving nonlinear equations with multiple roots

For a nonlinear equation f(x)=0 having a multiple root we consider Steffensen’s transformation, T. Using the transformation, say, F_q(x)=T^qf(x) for integer q≥2, repeatedly, we develop higher order iterative methods which require neither derivatives of f(x) nor the multiplicity of the root. It is proved that the convergence order of the proposed iterative method is 1+2^q−2 for any equation having a multiple root of multiplicity m≥2. The efficiency of the new method is shown by the results for some numerical examples.