On measurability of fuzzy-number-valued functions

This paper gives a new definition of the measurability of fuzzy-number-valued functions, and shows the relationship among this measurability and those derived from the corresponding set-valued functions.     

Existence results for Urysohn integral inclusions involving the Henstock integral

This paper presents two existence results for Urysohn integral inclusions in Banach spaces, the set-valued integral involved being the Henstock integral. In particular, the existence of continuous solutions for integral inclusions of Volterra and of Hammerstein type is obtained. Our results extend the existence theorems given in literature in the single- or set-valued case, under Bochner or Pettis integrability hypothesis. The compactness of solutions set is also studied.     

Second order three boundary value problem in Banach spaces via Henstock and Henstock-Kurzweil-Pettis integral

Existence theorems and some properties of solutions set of three boundary value second order differential equations and inclusions in Banach spaces are obtained under Henstock, respectively Henstock-Kurzweil-Pettis integrability assumptions. Our results extend those obtained by Azzam, Castaing and Thibault in the Bochner integrability setting and in the Pettis integrability one. The continuity of the (unique) solution with respect to a parameter in the single-valued case is also studied.     

On integral means of star-like functions

We study univalent holomorphic functions in the unit diskU = {z: |z| < 1} of the formf(z)=z+∑ _n=2 ^∞ a _n z ⁿ that satisfy the condition Re zf’(z)/f(z) > α with α∈ [0, 1) inU. Some integral means of such funcions are estimated.     

R~n中Banach值函数的Mcshane积分(I)(英文)

R．AGordon在［1］中定义了从R1到Banach空间抽象函数的McShane积分，证明了当X不含C0时，如果f在[a，b]上McShanef可积，则在[a，b]上Petits　可积．在这篇文章中，我们定义了从Rn到Banaach空间抽象函数的Mcshane积分，证明了fMcShane可积，则f是Pattis可积．于是我们推广了[1]的结果．     

Solutions to fuzzy integral equations with arbitrary kernels

Using a parametric Riemann integral representation, a numerical algorithm for solving fuzzy Fredholm and Voltera integral equations of the second kind with arbitrary kernel is proposed and illustrated with examples.     

On the functions of one selfadjoint integral operator

We construct a selfadjoint Akhiezer integral operator S on L ₂(?1, 1) with the spectrum {?1; 0; 1} and the property that every function φ(S) that is not a multiple of S fails to be an Akhiezer integral operator.     

A (CF)-mapping of integral functional of locally lipschitz functions

We find a (CF)-mapping of the integral functional of locally Lipschitz functions f _t parametrized by . In the process of obtaining a (CF)-mapping, the hypothesis of upper semicontinuity of the set-valued map is needed, where denotes a convexificator of f _t at x. As a corollary of our result, we get (CF)-mappings which are obtained by Clarke subdifferentials and Michel–Penot subdifferentials, respectively. Finally, the examples specifically deriving a convexificator of the integral functional are provided.      

On the monotone convergence theorem for conormed-seminormed fuzzy integral

[In this paper, under the condition that the t-conorm and t-seminorm are continuous, we prove the monotone convergence theorem of conormed-seminormed fuzzy integral for non-decreasing monotone measurable function sequence; also we show that the result cannot be extended to the case of the non-increasing function sequence.     

On the restudy of fuzzy complex analysis: Part II. The continuity and differentiation of fuzzy complex functions

We consider functions mapping complex numbers into fuzzy complex numbers or mapping fuzzy complex numbers into complex numbers. We study the continuity and differentiation of such functions.     

On the Theory of (Dual) Projection for Fuzzy Stochastic Processes

Abstract

In a theory similar to one of real-valued stochastic processes, in this paper, we investigate the projection and dual projection for fuzzy stochastic processes. First, the related concepts of fuzzy stochastic processes are introduced, such as adaption, measurability, optionality, predictability, etc. Subsequently, we study fuzzy stochastic integral and fuzzy measure generated by increasing fuzzy stochastic processes. Moreover, (dual) projection w.r.t. (increasing) fuzzy stochastic processes are discussed. We prove the existence and uniqueness of (dual) optional (predictable) projection for (increasing) fuzzy stochastic processes.     

向量值函数的Mcshane积分

本文讨论了向量值函数的Ｍｃｓｈａｎｅ积分的性质．例如收敛定理和原函数的性质等．     

On Fourier series for higher order (partial) derivatives of functions

This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term‐by‐term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto‐dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2rth order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.     

关于Hardy-Hilbert型积分不等式

设1/p=1/q≈1：1且P〉1．通过引入一个适当的积分核函数和参数λ（λ〉-1）,创建了一种新型Hardy～Hilbert型积分不等式．证明了其常数因子（p^λ=1＋q^λ＋1）Г（λ＋1）是最佳的,其中Г（x）Г-函数．特别,当p=2时,得到了一种新的Hilbert型积分不等式．作为应用,给出了它的一种等价形式．     

On the integral representation of superbiharmonic functions

We consider a nonnegative superbiharmonic function w satisfying some growth condition near the boundary of the unit disk in the complex plane. We shall find an integral representation formula for w in terms of the biharmonic Green function and a multiple of the Poisson kernel. This generalizes a Riesz-type formula already found by the author for superbihamonic functions w satisfying the condition 0 ⩽ w(z) ⩽ C(1-|z|) in the unit disk. As an application we shall see that the polynomials are dense in weighted Bergman spaces whose weights are superbiharmonic and satisfy the stated growth condition near the boundary. Research supported in part by IPM under the grant number 83310011.     

广义F积分的表示(I)

与Ｓｕｇｅｎｏ的Ｆ积分表示理论相比较,本文探讨了广义Ｆ积分的表示问题,给出了几个表示定理,如中值定理、重排转化定理等     

模糊集值映射(K)可积性的一个等价条件

In this paper,we first discuss the relationship between the McShane integral and Pettis integral for vector-valued functions.Then by using the embedding theorems for the fuzzy number space E~1,we give a new equivalent condition for(K) integrability of a fuzzy set-valued mapping F:[a,b]→E~1.     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.