Nondifferentiable Multiobjective Programming under Generalized d-ρ-(η,θ)-Type I Univexity

In this paper,on the basis of notions of d-ρ-(η,θ)-invex function,type I function and univex function,we present new classes of generalized d-ρ-(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.     

模糊多目标半定规划

This paper first applies the fuzzy set theory to multi-objective semi-definite program-ming （MSDP）, and proposes the fuzzy multi-objective semi-definite programming （FMSDP） model whose optimal efficient solution is defined for the first time, too. By constructing a membership function, the FMSDP is translated to the MSDP. Then we prove that the optimal efficient solution of FMSDP is consistent with the efficient solution of MSDP and present the optimality condition about these programming. At last, we give an algorithm for FMSDP by introducing a new membership function and a series of transformation.     

Best Proximity Point Theorems for p-Proximalα-η-β-Quasi Contractions in Metric Spaces with w0-Distance

In this paper,we propose a new class of non-self mappings called p-proximalα-η-β-quasi contraction,and introduce the concepts ofα-proximal admissible mapping with respect toηand(α,d)regular mapping with respect toη.Based on these new notions,we study the existence and uniqueness of best proximity point for this kind of new contractions in metric spaces with w;-distance and obtain a new theorem,which generalize and complement the results in[Ayari,M.I.et al.Fixed Point Theory Appl.,2017,2017:16]and[Ayari,M.I.et al.Fixed Point Theory Appl.,2019,2019:7].We give an example to show the validity of our main result.Moreover,we obtain several consequences concerning about best proximity point and common fixed point results for two mappings,and we present an application of a corollary to discuss the solutions to a class of systems of Volterra type integral equations.     

一类奇异三阶三点边值问题的正解

In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly increasing.Finally we give an example.     

The Singular Upper Triangle Type Solutions with Spin 1/2 of Quantum Yang-Baxter Equation （I）

In these two papers （I） and （II）, the singular upper triangle type solutions with spin 1/2 of quantum Yang-Baxter equation are given. In the Paper （I）, we give the Yang-Baxter equation and give the general solutions of some function equations for the next paper.     

Banach空间中关于(H,η)-增生算子的一类新变分包含组问题(英文)

In this paper,we first introduce a new class of generalized accretive operators named(H,η)-accretive in Banach space.By studying the properties of(H,η)-accretive,we extend the concept of resolvent operators associated with m-accretive operators to the new (H,η)-accretive operators.In terms of the new resolvent operator technique,we prove the existence and uniqueness of solutions for this new system of variational inclusions.We also construct a new algorithm for approximating the solution of this system a...     

线性互补问题的一种新Lagrange乘子法

A new multiplier method for solving the linear complementarity problem LCP(q, M) is proposed. Based on the Lagrangian of LCP(q,M) introduced here, we construct a new differentiable merit function θ(x,λ) which containing a multiplier vector λ and satisfying θ(x,λ) ≥ 0 and θ(x,λ) = 0 if and if only x solves LCP(q,M). A simple damped Newton-type algorithm which based on the merit function θ(x,λ) is presented. The main feature of the method is that the multiplier self-adjusting step accelerates the local convergence rate without losing global convergence. When M is the P-matrix, the sequence {θ(x^k,λ^k)}where {(x^k,λ^k)} generated by the algorithm is globally linearly convergent to zero and convergent in finite number of iterations if the solution is nondegenerate. Numerical results suggest that the method is high efficient and promising.     

Duality for Dini-Invex Nonsmooth Multiobjective Programming

In this paper we generalize the concept of a Dini-convex function with Dini derivative and introduce a new concept - Dini-invexity. Some properties of Dini invex functions are discussed. On the base of this, we study the Wolfe type duality and Mond-Weir type duality for Dini-invex nonsmooth multiobjective programmings and obtain corresponding duality theorems.     

Generalized Logan's Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson's Inequality in L_2(R~3)

We study Jackson's inequality between the best approximation of a function f ∈ L_2(R~3) by entire functions of exponential spherical type and its generalized modulus of continuity.We prove Jackson's inequality with the exact constant and the optimal argument in the modulus of continuity.In particular,Jackson's inequality with the optimal parameters is obtained for classical modulus of continuity of order r and Thue–Morse modulus of continuity of order r ∈ N.These results are based on the solution of the generalized Logan problem for entire functions of exponential type.For it we construct a new quadrature formulas for entire functions of exponential type.     

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.     

Second order nondifferentiable multiobjective fractional programming under generalized univex functions

In this paper, a new class of second order $(d,\rho,\eta,\theta)$-type 1 univex function is introduced. The Wolfe type second order dual problem (SFD) of the nondifferentiable multiobjective fractional programming problem (MFP) is considered, where the objective and constraint functions involved are directionally differentiable. Also the duality results under second order$(d,\rho,\eta,\theta)$-type 1 univex functions are established.     

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.     

SOME PROPERTIES OF THE $\[{l_2}\]$-VALUED LONG JAMES BANACH SPACE $\[J(\eta ,{l_2})\]$

The main result of this paper is to show that the bidual $\[J(\eta ,{l_2})\]$ of the long James type $\[{l_2}\]$-valued Banach, space $\[J(\eta ,{l_2})\]$ can be identified with transfinite matrices of scalars $\[{[({b_{a,i}})i \in [0,\omega )]_{a \in [0,\eta )}}\]$ with some conditions and the norm of the element x** in $\[J(\eta ,{l_2})\]$** equals $\[\mathop {\sup }\limits_{\gamma \in [0,\eta )} {\left\| {\sum\limits_{\alpha \in [0,\gamma )} {\sum\limits_{i \in [0,\omega )} {{b_{a,i}}{\phi _{a,i}}} } } \right\|_{J{{(\eta ,l)}^{**}}}}\]$.     

Transverse homoclinic orbit bifurcated from a homoclinic manifold by the higher order melnikov integrals

Consider an autonomous ordinary differential equation in $\mathbb{R}^n$ that has a $d$ dimensional homoclinic solution manifold $W^H$. Suppose the homoclinic manifold can be locally parametrized by $(\alpha,\theta) \in \mathbb{R}^{d-1}\times \mathbb{R}$. We study the bifurcation of the homoclinic solution manifold $W^H$ under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed $(\alpha_0,\theta_0)$ on $W^H$, if the Melnikov function have a simple zeros, then the perturbed system can have transverse homoclinic solutions near $W^H$.     

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.     

Remarks on Risk-Sensitive Control Problems

The main purpose of this paper is to investigate the asymptotic behavior of the discounted risk-sensitive control problem for periodic diffusion processes when the discount factor $\alpha$ goes to zero. If $u_\alpha(\theta,x)$ denotes the optimal cost function, $\theta$ being the risk factor, then it is shown that $\lim_{\alpha\to 0}\alpha u_\alpha(\theta,x)=\xi(\theta)$ where $\xi(\theta)$ is the average on $]0,\theta[$ of the optimal cost of the (usual) infinite horizon risk-sensitive control problem.     

ON SMOOTHNESS AND DIFFERENTIABILITY OF FUNCTIONS

Let \(f(x)\) be a bounded real function on [-1,1],we define the modulus of continuity of f as \[\omega (f,\delta ) = \mathop {\sup }\limits_{x,y \in [ - 1,1],\left| {x - y} \right| \le \delta } \left| {f(x) - f(y)} \right|\] and the modulus of smoothness of f as \[{\omega _2}(f,\delta ) = \mathop {\sup }\limits_{x \pm h \in [ - 1,1],\left| h \right| \le \delta } \left| {f(x + h) + f(x - h) - 2f(x)} \right|\] Functions \(f(x)\), continuous on [-1,1] and \({\omega _2}(f,\delta ) = o(\delta )\) ,are called uniformly smooth functions. It is well known that there is a uniformly smooth functions whose derivative exisits on a null-set only. It would is of interest to discuss what condition should be added on the nonnegative function \(\varphi (\delta )\), \(\left( {0 \le \delta \le \frac{1}{2}} \right)\),in order that every bounded function f satisfying\[{\omega _2}(f,\delta ) = O(\varphi (\delta ))\] possess continous (or finite) derivative. the main result of this paper are the following two theorems. Theorem 1 let \(\varphi (\delta )\),\(\left( {0 \le \delta \le \frac{1}{2}} \right)\) ,be a nonnegative function, then, in order that every bounded function \(f(x)\) satisfying condition \[{\omega _2}(f,\delta ) = O(\varphi (\delta ))\] possess continous (or finite) derivative \(f'(x)\) on [-1,1],it is necessary and sufficient that the following condition hold \[\int_0^{\frac{1}{2}} {\frac{{\tilde \varphi (t)}}{t}} dt < \infty \] where \[\tilde \varphi (\delta ) = {\delta ^2}\mathop {\inf }\limits_{0 \le \eta \le \delta } \left\{ {{\eta ^{ - 2}}\mathop {\inf }\limits_{\eta \le \xi \le 1/2} \varphi (\xi )} \right\}\] Theorm 2 Let \(f(x)\) be a bounded function with \[\int_0^{\frac{1}{2}} {\frac{{{\omega _2}(f,t)}}{{{t^2}}}} dt < \infty \] then \(f'(x)\) is a continous function and \[{\omega _2}(f',\delta ) = O\left\{ {\int_0^\delta {\frac{{{\omega _2}(f,t)}}{{{t^2}}}} dt} \right\}\].     

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).