广义 $d$-$\rho$-$(\eta,\theta)$-type I 一致凸非可微多目标规划

In this puper, on the basis of notions of d-p-（η, θ）-invex function, type I function and univex function, we present new classes of generalized d-p-（η, θ）-type I univex functions. By using these new concepts, we obtain several sufficient optimality conditions for a feasible solution to be an efficient solution, and derive some Mond-Weir type duality results.     

Higher order duality for a new class of nonconvex semi-infinite multiobjective fractional programming with support functions

In the paper, a new class of semi-infinite multiobjective fractional programming problems with support functions in the objective and constraint functions is considered. For such vector optimization problems, higher order dual problems in the sense of Mond-Weir and Schaible are defined. Then, various duality results between the considered multiobjective fractional semi-infinite programming problem and its higher order dual problems mentioned above are established under assumptions that the involved functions are higher order $\left(\Phi,\rho,\sigma^{\alpha}\right)$-type I functions. The results established in the paper generalize several similar results previously established in the literature.     

二阶($F, \alpha,\rho, d,p$) 一致凸性与极小极大分式规划的对偶

In this paper,we introduce a class of generalized second order(F,α,ρ,d,p)-univex functions.Two types of second order dual models are considered for a minimax fractional programming problem and the duality results are established by using the assumptions on the functions involved.     

角域内代数体函数的唯一性定理

Let W (z) and M(z) be v-valued and k-valued algebroidal functions respectively,(θ) be a b-cluster line of order ∞ (or ρ(r)) of W (z) (or M(z)).It is shown that W (z) ≡ M(z) provided E(a j ,W (z)) = E(a j ,M(z)) (j = 1,...,2v + 2k + 1) holds in the angular domain Ω(θ- δ,θ + δ),where b,a j (j = 1,...,2v + 2k + 1) are complex constants.The same results are obtained for the case that (θ) is a Borel direction of order ∞ (or ρ(r)) of W (z) (or M(z)).     

THE BEHAVIOR OF SOLUTIONS IN THE VICINITY OF A BOUNDED SOLUTION TO AUTONOMOUS DIFFERENTIAL EQUATIONS

By using the exponential dichotomy,this paper investigates the behavior of solutionsin the vicinity of a bounded solution to the autonomous differential systemdx/dt=f(x).(1)Suppose x=u(t)is a nontrivial bounded solution of system(1).By discussing theequivalent equations of system(1)dθ/dt=1 (p,θ)dp/dt=A(θ)p (p,θ)(2)with respect to the moving orthonormal transformationx=u(θ) s(θ)p,the author proves that if linear system corresponding to(2)admits exponential dichotomy,then the given bounded solution x=u(t)should be periodic.The author also discusses thestadility of the obtained periodic solution.Finally,this paper discusses perturbation of thebounded solution of autonomous system(1).     

Higher-Order (F,α,β,ρ,d,E)-Convexity in Fractional Programming

In this paper we define higher order (F,α,β,ρ,d,E)-convex function with respect to E-differentiable function K and obtain optimality conditions for nonlinear programming problem (NP) from the concept of higher order (F,α,β,ρ,d)-convexity. Here, we establish Mond-Weir and Wolfe duality for (NP) and utilize these duality in nonlinear fractional programming problem.     

Some Characterizations of Integral Operators Associated with Certain Classes of <Emphasis Type="Italic">p</Emphasis>-Valent Functions Defined by the Srivastava–Saigo–Owa Fractional Differintegral Operator

The purpose of this paper is to introduce new integral operators associated with Srivastava–Saigo–Owa fractional differintegral operator. We investigate some properties for the integral operators \({\mathcal {F}}_{p,\eta ,\mu }^{\lambda ,\delta }(z)\) and \({\mathcal {G}}_{p,\eta ,\mu }^{\lambda ,\delta }(z)\) to be in the classes \({\mathcal {R}}_{k}^{\zeta }\left( p,\rho \right) \) and \({\mathcal {V}}_{k}^{\zeta }\left( p,\rho \right) \).     

一类三点边值问题正解的存在性

在与线性问题第一特征值相关的条件下,通过应用不动点指数理论讨论了三点边值问题u″ 9(t)f(u)=0,t∈(0,1),u′(0)=0,u(1)=αu(η)正解的存在性,这里η∈(0,1),α∈R且0<α<1.本文结果推广和改进了文献[1]的主要结论.     

ON THE PROBLEMS OF THE INFINITESIMAL DEFORMATION OF THE SURFACES OF REVOLUTION WITH MIXED CURVATURE

In this paper the problem of the infinitesimal deformation of the surfaces of revolution with mixed Ganss curvature is studied. In connection with this problem a differential equation of mixed type, which belongs to the second degenerate type, in the form , \[k(\rho ){w_{\rho \rho }} + {w_{\theta \theta }} + \rho {w_\rho } = 0({\rho _1} < \rho < {\rho _2},0 \leqslant \theta \leqslant 2\pi )\] is obtained, where w is the component of the displacement vector of the infinitesimal deformation in the direction of the rotation axis, and\[k(\rho ) = \rho {z^'}(\rho )/{z^{'}}(\rho ),z(\rho )\] being the meridian curve of the surface of revolution. Suppose a surface of revolution S has two holes \[{L_1}(\rho = {\rho _1})\] and \[{L_2}(\rho = {\rho _2})\], then the meridian of the surface satisfies the condition \[{z^'}(\rho ) = 0\] on \[\rho = {\rho _0}({\rho _1} < {\rho _0} < {\rho _2})\]. If the Gauss curvature K of the surface is a strictly monotone increasing function of \[\rho \], \[{K^'}(\rho ) > 0,{\rho _1} < \rho < {\rho _2}\],then the surface S does not permit of the non-trivial sliding on the plane containing the boundary L2 of the surface. The rigidity of the surface is proved by the energy integral method. Moreover, the uniqueness of the Tricomi problem, the generalized Tricomi problem, the degenerate Tricomi problem and the Frankl's problem for a piece of surface with mixed curvature are studied.     

依赖于一阶导数的二阶脉冲微分方程边值问题的正解(英)

本文研究一类二阶脉冲微分方程：■的正解存在性．其中,0＜η＜1,0＜α＜1,f：[0,1]×[0,∞)×R→[0,∞),I_i：[0,∞)×R→R,J_i：[0,∞)×R→R,(i=1,2,…,k)均为连续函数．本文所用方法是文献[5]推广的Krasnoselskii不动点定理,此定理为解决依赖于一阶导数的边值问题提供了理论依据．基于此定理,获得了问题正解存在性定理．特别地,我们获得此类问题的Green函数,使问题的解决更直观和简单．     

Efficient and robust tests for semiparametric models

In this paper, we investigate a hypothesis testing problem in regular semiparametric models using the Hellinger distance approach. Specifically, given a sample from a semiparametric family of \(\nu \)-densities of the form \(\{f_{\theta ,\eta }:\theta \in \Theta ,\eta \in \Gamma \},\) we consider the problem of testing a null hypothesis \(H_{0}:\theta \in \Theta _{0}\) against an alternative hypothesis \(H_{1}:\theta \in \Theta _{1},\) where \(\eta \) is a nuisance parameter (possibly of infinite dimensional), \(\nu \) is a \(\sigma \)-finite measure, \(\Theta \) is a bounded open subset of \(\mathbb {R}^{p}\), and \(\Gamma \) is a subset of some Banach or Hilbert space. We employ the Hellinger distance to construct a test statistic. The proposed method results in an explicit form of the test statistic. We show that the proposed test is asymptotically optimal (i.e., locally uniformly most powerful) and has some desirable robustness properties, such as resistance to deviations from the postulated model and in the presence of outliers.     

The Starlikeness of Analytic Functions of Koebe Type with Complex Order

For α 0, λ 0 and β,η∈R, we consider the M(α,λ)_b of normalized analyticα—λ convex functions defined in the open unit disc U. In this paper we investigate the class M(α, λ,β,η)_b,with f_b := z/(1-z~n)~b being Koebe type. By making use of Jack's Lemma as well as several differential and other inequalities, the authors derive sufficient conditions for starlikeness of the class M(α, λ, β, η)_b of n-fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in earlier works are also indicated.     

二阶两点边值问题单调正解的存在性

考察了形如{x″(t)+f(t,x(t))=0,0≤t≤1,x(0)=ξx(1),x′(1)=ηx′(0)的二阶非线性微分方程两点边值问题,这里ξ,η∈(0,1)∪(1,∞)为给定的常数,f:[0,1]×[0,∞)→[0,∞)连续。在某些适当的增长性条件下,应用Avery-Anderson-Krueger不动点定理证明了单调正解的存在性。     

Partial Differential Equations of Higher Order and Symmetric Positive Systems

In this paper we study the relation between symmetric positive systems and equations of higher order. The main result is: Theorem 1. An equation of second order $L\phi =f$ can be transformed into a symmetric positive system by introducing new unknown functions $u_i=\sum\limits_{j=0}^n {\alpha_ij \varphi _j(i=0,1,\cdots,n),\varphi_0=\varphi_2,\varphi_j=\partial \varphi /\partial x_j}$ iff there exists L_1 of order 1 such that $Re(L_1 \varphi \cdot \bar {L\varphi})=\sum\limits_{i=1}^n{\frac{\partial}{\partial x_i}}+B(\varphi,\varphi)$, where P_i(\varphi,\varphi)(i=1,2,\cdots,n),B(\varphi,\varphi) are differential quadarlic forms and B(\varphi,\varphi) is positive definite. This Theorem can be extended into equations of higher order. Some examples of deducing equations of higher order into symmetric positive systems are given. Finally, we give a counter example which shows that a boundary problem of a symmetric positive system deduced from an equation of higher order is admissible, but its corresponding bounbary problem of the original equation is not well-posed.     

Strong convergence theorems for a Mann-type iterative scheme for a family of Lipschitzian mappings

In the paper, we obtain the existence of triple positive solutions for the following second order three-point boundary value problem, $$\left\{\begin{array}{l}(\phi_p(u'))'(t)+q(t)f(u(t),u'(t),(Tu)(t),(Su)(t))=0,\quad 0\leq t\leq1,\\[4pt]u'(0)=\beta u'(\eta),\qquad u(1)=g(u'(1)),\end{array}\right.$$ where $\phi_{p}(s)=|s|^{p-2}s,p>1,\beta\in[0,1),\eta\in(0,\frac{1}{2}]$ , T and S are all linear operators, g(t) is continuous.     

Optimality and duality for multiple-objective optimization under generalized type I univexity

In this paper, we extend the classes of generalized type I vector-valued functions introduced by Aghezzaf and Hachimi in [J. Global Optim. 18 (2000) 91-101] to generalized univex type I vector-valued functions and consider a multiple-objective optimization problem involving generalized type I univex functions. A number of Kuhn-Tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution. The Mond-Weir and general Mond-Weir type duality results are also presented.     

On nonlinear multiple objective fractional programming involving semilocally type-I univex functions

The aim of this paper is to obtain sufficient optimality conditions for a nonlinear multiple objective fractional programming problem involving semilocally type-I univex and related functions. Furthermore, a general dual is formulated and duality results are proved under the assumptions of generalized semilocally type-I univex and related functions. Our results generalize several known results in the literature.     

二阶三点边值问题的正解

该文讨论了二阶三点边值问题$-u'(t)=b(t)f(u(t))$满足$u'(0)=0$, $u(1)={\alpha}u({\eta})$ 正解的存在性与多重性, 其中常数$\alpha, \eta\in(0,1)$, $f\in C ([0,\infty),[0,\infty) )$, $b\in C ([0,1],[0,\infty) )$且存在$t_0\in[0,1]$使$b(t_0)>0$. 利用该问题相应的Green函数, 将其转化为Hammerstein型积分方程, 借助于锥上的不动点指数理论,给出了该问题单个正解和多个正解存在的与其相应线性问题的第一特征值有关的最佳充分性条件.     

A new proof of the transformation law of Jacobi's theta function

We present a new proof, using Residue Calculus, of the transformation law of the Jacobi theta function defined in the upper half plane. Our proof is inspired by Siegel's proof of the transformation law of the Dedekind eta function.

     

On the ε-optimal control of quasilinear integral equations

We consider an optimal control problem for the stochastic quasilinear integro-functional equation , with cost functional . Two controls are introduced: a program control and a feedback control, generated by the optimal control of the problem under consideration when=0. It is proved that both controls are-optimal for the original problem.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 96–100, 1986.