SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS

Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.

Necessary Conditions and Situational Variables of Payments for Ecosystem Services

Clarifying the necessary conditions for the emergence of payments for ecosystem services (PES) and the situational variables that affect PES is the basis for their interpretation, prediction, and selection. This article proposes an analytical framework for the emergence of PES and argues that the key to determining whether PES can occur and whether a selected PES program is appropriate is to evaluate the net gain. When payers anticipate that a PES program will provide a satisfactory number of ES and a net gain over the opportunity cost and will cover all costs, it is assumed that the program will be implemented. When it is difficult to accurately evaluate the net gain of PES, the situational variables that affect the costs and benefits need to be examined. The group characteristics, ES characteristics, spatial and temporal contacts between the suppliers and demanders, correlation with private goods and additionality are important situational variables that affect the emergence and choice of PES.

ON NECESSARY CONDITIONS FOR RESONANCE IN TURNING POINT PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

In this paper, some misunderstam igs concerning the necessary conditions for resonance for ordinary differential equations with turning point have been corrected, and a recursive process for finding the sequence of necessary conditions for resonance has beenoffered.

Pointwise Second-Order Necessary Conditions for Stochastic Optimal Control with Jump Diffusions

In this paper,we establish a second-order necessary conditions for stochastic optimal control for jump diffusions.The controlled system is described by a stochastic differential systems driven by Poisson random measure and an independent Brownian motion.The control domain is assumed to be convex.Pointwise second-order maximum principle for controlled jump diffusion in terms of the martingale with respect to the time variable is proved.The proof of the main result is based on variational approach using the stochastic calculus of jump diffusions and some estimates on the state processes.

KUHN-TUCKER NECESSARY CONDITIONS FOR (h, ψ)-MULTIOBJECTIVE OPTIMIZATION PROBLEMS

Five kinds of cones are introduced, which are used to establish the constraints qualifications, under which the generalized Kuhn-Tucker necessary conditions are developed for a class of generalized (h,φ)-differentiable single-objective and multiobjective programming problems by using Motzkin's alternative theorem and Ben-Tal generalized algebraic operations.

Necessary and Sufficient Conditions in Smooth Programming with Generalized Convexity

Hanson and Mond have grven sets of necessary and sufficient conditions for optimality in constrained optimization by introducing classes of generalized functions, called type Ⅰ functions. Recently, Bector definded univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, additional conditions are attached to the Kuhn Tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization, under appropriate constraint qualifications.

NECESSARY CONDITIONS FOR EFFICIENT SOLUTION OF VECTOR OPTIMIZATION PROBLEMS

In this paper, by using Ljusternik＇s theorem and the open mapping theorem of convex process, the author gives necessary conditions for the efficient solution to the vector optimization problems without requiring that the ordering cone in the objective space has a nonempty interior.

The First-Order Necessary Conditions for Sparsity Constrained Optimization

In this paper,we study optimization problems with the sparsity constraints.Firstly we give the expressions of the Mordukhovich(the limiting)normal cone of sparsity constraint and its intersection with a polyhedral set,and then based on these expressions we present the first-order necessary conditions for sparsity constrained optimization.

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

Some results on pointwise second-order necessary conditions for stochastic optimal controls

The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms.When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions,the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.

Necessary conditions for Gabor frames

The goal of this paper is to establish a set of necessary conditions for Gabor frames. These conditions are also sufficient for tight frames.

Necessary and Sufficient Conditions for Reactive Azeotropes in Multi-reaction Mixtures in Terms of Elemental Composition

Necessary and sufficient conditions for azeotropy in reactive mixtures are derived in terms of elemental composition, which shows that in the space of elemental compositions, they take the same functional form as the conditions for azeotropy in non-reactive mixtures. The production of methyl tert-butyl ether (MTBE) is taken as an example. It is found that there are a 'pseudo' intermediate-boiling ternary reactive azeotrope at p = 101.325kPa and two 'real' ternary reactive azeotropes at p = 101.325 Pa. The introduced elemental compositions also reduce the dimensionality of the phase diagrams and provide a natural set of variables for visualization of phase behavior.

NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.

Necessary Optimality Conditions for Multi-Objective Semi-Infinite Variational Problem

In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These conditions are responsible for the development of duality theory which is an extremely important feature for any class of problems, but the literature available so far lacks these necessary optimality conditions for the stated problem. A lemma is also proved to find the topological dual of as it is required to prove the desired result.

Solution of the Matrix Second Semi-Tensor Product Equation A ∘ l X ∘ l B=C

In this paper, the solution of the matrix second semi-tensor product equation A∘lX∘lB=Cis studied. Firstly, the solvability of the matrix-vector second semi-tensor product equation is investigated. At the same time, the compatibility conditions, the sufficient and necessary conditions and the specific solution methods for the matrix solution are given. Secondly, we further consider the solvability of the second semi-tensor product equation of the matrix. For each part, several examples are given to illustrate the validity of the results.

OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.

SECOND-ORDER OPTIMALITY CONDITIONS FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY 3-DIMENSIONAL NEVIER-STOKES EQUATIONS

This article is concerned with second-order necessary and sufficient optimality conditions for optimal control problems governed by 3-dimensional Navier-Stokes equations. The periodic state constraint is considered.

The convergence rate and necessary-and-sufficient condition for the consistency of isogeometric collocation method

Although the isogeometric collocation(IGA-C)method has been successfully utilized in practical applications due to its simplicity and efficiency,only a little theoretical results have been established on the numerical analysis of the IGA-C method.In this paper,we deduce the convergence rate of the consistency of the IGA-C method.Moreover,based on the formula of the convergence rate,the necessary and sufficient condition for the consistency of the IGA-C method is developed.These results advance the numerical analysis of the IGA-C method.

A NECESSARY AND SUFFICIENT CONDITION FOR THEOSCILLATION OF SOLUTIONS OF LIENARD TYPE SYSTEM WITH MULTIPLE SINGULAR POINTS

In this paper, a necessary and sufficient condition for teh solution of Lienard typesystem with muliiple singular points to oscillation under the more general assumptionis given.Results of the papers [1-4] are also extended and improved in this paper.

Some Necessary and Sufficient Condition of Multivariate Random Variable Satisfy Normal Distribution

In this paper,we extended some results of article[1],obtain some sufficient and necessary condition which multivariate random variable satisfy normal distribution.