Generalized Variational Relation Problems with Applications

In this paper, we first obtain an existence theorem of the solutions for a variational relation problem. An existence theorem for a variational inclusion problem, a KKM theorem and an extension of the well know Ky Fan inequality will be established, as particular cases. Some applications concerning a saddle point problem with constraints, existence of a common fixed point for two mappings and an optimization problem with constraints, will be given in the last section of the paper.     

Existence Results for Functional Differential Inclusions with Infinite Delay

The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the setcontraction theory.     

Existence Theorems for Periodic Differential Inclusions in IR^N

We study the periodic problem for differential inclusions in R~N.First we look for extremal periodicsolutions.Using techniques from multivalued analysis and a fixed point argument we establish an existencetheorem under some general hypotheses.We also consider the“nonconvex periodic problem”under lowersemicontinuity hypotheses,and the“convex periodic problem”under general upper semicontinuity hypotheseson the multivalued vector field.For both problems,we prove existence theorems under very general hypotheses.Our approach extends existing results in the literature and appear to be the most general results on the nonconvexperiodic problem.     

Existence and Stability of Solutions for Generalized Ky Fan Inequality Problems with Trifunctions

In this paper, an existence theorem for solutions to the generalized Ky Fan Inequality problem is obtained by means of the Kakutani-Fan-Glicksberg fixed-point theorem without imposing the condition that the dual of the ordering cone has a weak* compact base. In addition, the stability of the solution set is shown.     

Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems with Discontinuous Mappings

Let X, Y be two finite-dimensional topological vector spaces, Z a Hausdorff topological vector space, K C X and D C Z be two nonempty sets, C be a pointed, closed, and convex cone in Y with int C ≠θ Let S ： K → 2^K and T ： K → 2^D be two multivalued mappings, and φ ： K × D × K → Y be a trifunction. In this paper, we consider the generalized vector quasi-equilibrium problem, which is formulated by finding X∈ K and y∈ T（x） such that x∈ E S（x） and φ（x,y, u） （∈/） -int C for all u ∈ S（x）. We establish an existence result in which T is not supposed to have any continuity property. Our results extend and improve the corresponding results of Cubiotti, Yao and Guo.     

New Generalized R-KKM Type Theorems in General Topological Spaces and Applications

In this paper, some new generalized R-KKM type theorems for generalized R-KKM mappings with finitely closed values and with finitely open values are established in noncompact topological spaces without any convexity structure under much weaker assumptions. As applications, some new minimax inequalities, saddle point theorem and equilibrium existence theorem for equilibrium problems with lower and upper bounds are established in general noncompact topological spaces. These theorems unify and generalize many known results in the literature.     

General Maximum Principles for Partially Observed Risk-Sensitive Optimal Control Problems and Applications to Finance

This paper is concerned with partially observed risk-sensitive optimal control problems. Combining Girsanov’s theorem with a standard spike variational technique, we obtain some general maximum principles for the aforementioned problems. One of the distinctive differences between our results and the standard risk-neutral case is that the adjoint equations and variational inequalities strongly depend on a risk-sensitive parameter γ. Two examples are given to illustrate the applications of the theoretical results obtained in this paper. As a natural deduction, a general maximum principle is also obtained for a fully observed risk-sensitive case. At last, this result is applied to study a risk-sensitive optimal portfolio problem. An explicit optimal investment strategy and a cost functional are obtained. A numerical simulation result shows the influence of a risk-sensitive parameter on an optimal investment proportion; this coincides with its economic meaning and theoretical results. This work was partially supported by the National Natural Science Foundation (10671112), the National Basic Research Program of China (973 Program, No. 2007CB814904), the Natural Science Foundation of Shandong Province (Z2006A01) and the Doctoral Fund of the Education Ministry of China.     

The Generalized KKM Map and Its Applications to Variational Inequalities

In this paper,applying the concept of generalized KKM map,we study problems ofvariational inequalities.We weaken convexity(concavity)conditions for a functional of two variables■(x,y)in the general variational inequalities.Last,we show a proof of non-topological degree meth-od of acute angle principle about monotone operator as an application of these results.     

Combinatorial Integer Labeling Theorems on Finite Sets with Applications

Tucker’s well-known combinatorial lemma states that, for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1,±2,…,±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper, we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {±1,±2,…,±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state that, under some mild conditions, there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set {0,1} ⁿ +q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided.     

Existence Theorems of Hartman–Stampacchia Type for Hemivariational Inequalities and Applications

We give some versions of theorems of Hartman-Stampacchia's type for the case of Hemivariational Inequalities on compact or on closed and convex subsets in infinite and finite dimensional Banach spaces. Several problems from Nonsmooth Mechanics are solved with these abstract results.     

Hybrid Moreau’s Proximal Algorithms and Convergence Theorems for Minimization Problems in Hilbert Spaces with Applications

Abstract

In this paper, motivated by Moreau’s proximal algorithm, we give several algorithms and related weak and strong convergence theorems for minimization problems under suitable conditions. These algorithms and convergence theorems are different from the results in the literatures. Besides, we also study algorithms and convergence theorems for the split feasibility problem in real Hilbert spaces. Finally, we give numerical results for our main results.     

Coincidence Theorems with Applications to Minimax Inequalities, Section Theorem, Best Approximation and Multiobjective Games in Topological Spaces

Some new coincidence theorems involving admissible set-valued mappings are proved in general noncompact topological spaces. As applications, some new minimax inequalities, section theorem, best approximation theorem, existence theorems of weighted Nash equilibria and Pareto equilibria for multiobjective games are given in general topological spaces.     

Existence’s Results for Parabolic Problems Related to Fully Non Linear Operators Degenerate or Singular

In this paper we prove some existence and regularity results concerning parabolic equations

$ u_t = F(x, \nabla u, D^2 u) + f(x,t) $

with some boundary conditions, on Ω×]0, T[, where Ω is some bounded domain which possesses the exterior cone property and F is some fully nonlinear elliptic operator, singular or degenerate.

     

Multiplicity Results for Second-order Generalized Sturm-Liouville Boundary Value Problems with Dependence on the First Order Derivative

By using fixed point theorems,we consider multiplicity of positive solutions for second-order generalized Sturm-Liouville boundary value problem,where the first order derivative is involved in the nonlinear term explicitly.We show the existence of multiple positive solutions for the problems.Example is given to illustrate the main results of the article.     

Hybrid Steepest Descent Methods for Zeros of Nonlinear Operators with Applications to Variational Inequalities

In this paper, the hybrid steepest descent methods are extended to develop new iterative schemes for finding the zeros of bounded, demicontinuous and φ-strongly accretive mappings in uniformly smooth Banach spaces. Two iterative schemes are proposed. Strong convergence results are established and applications to variational inequalities are given. In this research, the first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality (075105118). The third author was partially supported by Grant NSC 96-2628-E-110-014-MY3.     

A Characterization of the Existence of Solutions for Hamilton—Jacobi Equations in Ergodic Control Problems with Applications

We give a characterization of the existence of bounded solutions for Hamilton—Jacobi equations in ergodic control problems with state-constraint. This result is applied to the reexamination of the counterexample given in [5] concerning the existence of solutions for ergodic control problems in infinite-dimensional Hilbert spaces and also establishing results on effective Hamiltonians in periodic homogenization of Hamilton—Jacobi equations. Accepted 1 December 1999     

Existence and Uniquenes Theorems of Solutions for a Class of 2nth-order Boundary Value Problems

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Continuous Selections and Fixed Points for $ϕ$-Maps with Their Applications to Section Problems

The concept of finitely continuous topological space is introduced and the basic properties of the space are given. Several continuous selection theorems and fixed point theorems for $ϕ$-maps are established, and as applications of the above fixed point theorems, some section problems are discussed. The results generalize and improve many corresponding conclusions.     

Generalized hyper-heuristics for solving 2D Regular and Irregular Packing Problems

The idea behind hyper-heuristics is to discover some combination of straightforward heuristics to solve a wide range of problems. To be worthwhile, such a combination should outperform the single heuristics. This article presents a GA-based method that produces general hyper-heuristics that solve two-dimensional regular (rectangular) and irregular (convex polygonal) bin-packing problems. A hyper-heuristic is used to define a high-level heuristic that controls low-level heuristics. The hyper-heuristic should decide when and where to apply each single low-level heuristic, depending on the given problem state. In this investigation two kinds of heuristics were considered: for selecting the figures (pieces) and objects (bins), and for placing the figures into the objects. Some of the heuristics were taken from the literature, others were adapted, and some other variations developed by us. We chose the most representative heuristics of their type, considering their individual performance in various studies and also in an initial experimentation on a collection of benchmark problems. The GA included in the proposed model uses a variable-length representation, which evolves combinations of condition-action rules producing hyper-heuristics after going through a learning process which includes training and testing phases. Such hyper-heuristics, when tested with a large set of benchmark problems, produce outstanding results for most of the cases. The testbed is composed of problems used in other similar studies in the literature. Some additional instances for the testbed were randomly generated.     

Levitin–Polyak Well-Posedness for Optimization Problems with Generalized Equilibrium Constraints

In this paper, we consider Levitin–Polyak well-posedness of parametric generalized equilibrium problems and optimization problems with generalized equilibrium constraints. Some criteria for these types of well-posedness are derived. In particular, under certain conditions, we show that generalized Levitin–Polyak well-posedness of a parametric generalized equilibrium problem is equivalent to the nonemptiness and compactness of its solution set. Finally, for an optimization problem with generalized equilibrium constraints, we also obtain that, under certain conditions, Levitin–Polyak well-posedness in the generalized sense is equivalent to the nonemptiness and compactness of its solution set.