Ekeland’s principle for vector equilibrium problems

In this paper, the authors deal with bifunctions defined on complete metric spaces and with values in locally convex spaces ordered by closed convex cones. The aim is to provide a vector version of Ekeland’s theorem related to equilibrium problems. To prove this principle, a weak notion of continuity of a vector-valued function is considered, and some of its properties are presented. Via the vector Ekeland’s principle, existence results for vector equilibria are proved in both compact and noncompact domains.     

An Ekeland’s variational principle for set-valued mappings with applications

In this paper, we obtain a general Ekeland’s variational principle for set-valued mappings in complete metric space, which is different from those in [G.Y. Chen, X.X. Huang, Ekeland’s ε-variational principle for set-valued mapping, Mathematical Methods of Operations Research 48 (1998) 181–186; G.Y. Chen, X.X. Huang, S.H. Hou, General Ekeland’s Variational Principle for Set-Valued Mappings, Journal of Optimization Theory and Applications 106 (2000) 151–164; S.J. Li, W.Y. Zhang, On Ekeland’s variational Principle for set-valued mappings, Acta Mathematicae Application Sinica, English Series 23 (2007) 141–148]. By the result, we prove some existence results for a general vector equilibrium problem under nonconvex and compact or noncompact assumptions of its domain, respectively. Moreover, we give some equivalent results to the variational principle.     

An optimal birth control problem for a dynamical population model with size-structure

This work is concerned with an optimal control problem for a size-structured population model, which takes fertility as the control variable. The existence and uniqueness of solutions to the basic state system and the dual system are proven via the Banach fixed point theorem. Necessary optimality conditions of first order are established in the form of an Euler-Lagrange system by the use of tangent-normal cone technique. The existence of a unique optimal controller is established by means of Ekeland’s variational principle. An example and some comments are presented.     

Variational inclusions problems with applications to Ekeland’s variational principle,fixed point and optimization problems

In this paper, we prove the existence theorems of two types of systems of variational inclusions problem. From these existence results, we establish Ekeland’s variational principle on topological vector space, existence theorems of common fixed point, existence theorems for the semi-infinite problems, mathematical programs with fixed points and equilibrium constraints, and vector mathematical programs with variational inclusions constraints.     

Existence and multiplicity of positive solutions for some Hardy-Sobolev critical elliptic equation with boundary singularities

In this paper, by Ekeland’s variational principle and strong maximum principle, we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical Hardy-Sobolev exponents and Hardy terms with boundary singularities.     

Vector Ekeland’s variational principle in an <Emphasis Type="Italic">F</Emphasis>-type topological space

In this paper, we first give a vector-valued version of Brézis and Browder’s scalar general principle. We then apply the vector-valued general principle to study a vector Ekeland’s variational principle in a F-type topological space, which unifies and improves the corresponding vector-valued Ekeland’s variational results in complete metric space. This project was partially supported by the Research Grants Council of Hong Kong (BG771) and National Natural Science Foundation of China (70501015, 70401006).     

Ekeland’s variational principle for vectorial multivalued mappings in a uniform space

In this paper, we establish Ekeland’s variational principle and an equilibrium version of Ekeland’s variational principle for vectorial multivalued mappings in the setting of separated, sequentially complete uniform spaces. Our approaches and results are different from those in Chen et al. (2008), Hamel (2005), and Lin and Chuang (2010) [13], [14] and [15]. As applications of our results, we study vectorial Caristi’s fixed point theorems and Takahashi’s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated, sequentially complete uniform space. We also apply our results to study maximal element theorems, which are unified methods of several variational inclusion problems. Our results contain many known results in the literature Fang (1996) [21], and will have many applications in nonlinear analysis.     

Stability of solutions for variational relation problems with applications

In this paper, we prove that most of problems in variational relations (in the sense of Baire category) are essential and that, for any problem in variational relations, there exists at least one essential component of its solution set. As applications, we deduce the existence of essential components of the set of Ky Fan’s points based on Ky Fan’s minimax inequality theorem, the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games, the existence of essential component of the set of solutions for vector Ky Fan’s minimax inequality, the existence of essential components of the set of KKM points and the existence of essential components of the set of solutions for Ky Fan’s section theorem.     

Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle,fixed point theorems and parametric optimization problems

In this paper, we first establish some existence theorems of systems of generalized vector equilibrium problems. From these results, we obtain new variants of Ekeland’s variational principle in a Hausdorff t.v.s., a minimax theorem and minimization theorems. Some applications to the existence theorem of systems of semi-infinite problem, a variant of flower petal theorem and a generalization of Schauder’s fixed point theorem are also given.     

A variational iteration method for solving Troesch’s problem

Troesch’s problem is an inherently unstable two-point boundary value problem. A new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve Troesch’s problem. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. Only the second-order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods.     

Optimal control for a class of nonlinear age-distributed population systems

This paper deals with an optimal control problem for a kind of age-dependent biological population systems. The well-posedness of the state system is treated by means of characteristics line and fixed-point principle. Necessary optimality conditions are obtained via tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal controller are established by the use of Ekeland’s principle.     

Scalarization of set-valued optimization problems and variational inequalities in topological vector spaces

This paper deals with set-valued vector optimization problems and set-valued vector variational inequalities in topological vector spaces, and provides some scalarization approaches for these problems by means of the polar cone and Gerstewitz’s scalarization functions.     

Pontryagin’s maximum principle for constrained impulsive control problems

Necessary conditions in the form of Pontryagin’s maximum principle are derived for impulsive control problems with mixed constraints. A new mathematical concept of impulsive control is introduced as a requirement for the consistency of the impulsive framework. Additionally, this control concept enables the incorporation of the engineering needs to consider conventional control action while the impulse develops. The regularity assumptions under which the maximum principle is proved are weaker than those in the known literature. Ekeland’s variational principle and Lebesgue’s discontinuous time variable change are used in the proof. The article also contains an example showing how such impulsive controls could be relevant in actual applications.     

The studies of systems of variational inclusions problems and variational disclusions problems with applications

In this paper, we study existence theorems of solutions for systems of variational inclusions problems and systems of variational disclusions problems. From these existence results, we establish existence theorems of solutions for systems of generalized vector quasiequilibrium problems and systems of quasioptimization problems.     

On optimal control problems of a class of impulsive switching systems with terminal states constraints

The global optimal control problem is proposed for a special class of hybrid dynamical systems, i.e. impulsive switching systems. Then the necessary condition of the above problem, the minimum principle, is given. Ekeland’s variational principle and the matrix cost functional structure expression are utilized in the process of the proof. Based on the main result, a special linear hybrid impulsive and switching system (HISS) is illustrated and the optimal control algorithm is presented. Moreover, the cases of pure impulsive systems and pure switched systems are included in this paper.     

Transitivity and variational principles

We apply an order reasoning to mappings satisfying the triangle inequality. This general approach yields the Ekeland’s variational principle as one of the consequences. In addition we obtain an extension of the Brøndsted variational principle and of the Takahashi fixed point theorem.     

Sobolev’s spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales

In this paper, we study the Sobolev’s spaces on time scales and their properties. As applications, we present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for a class of second order Hamiltonian systems on time scales. By establishing a proper variational setting, three existence results for systems under consideration are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of the existence results.     

On maximal element theorems,variants of Ekeland’s variational principle and their applications

In this paper, we establish several different versions of generalized Ekeland’s variational principle and maximal element theorem for τ$\tau$-functions in ?$?$ complete metric spaces. The equivalence relations between maximal element theorems, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point theorems and nonconvex maximal element theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces.     

Quasi-variational relation problems and generalized Ekeland's variational principle with applications

In this article, we study the existence of solutions for a quasivariational relation problem and then give applications to the existence of solutions for set-valued Ekeland's principle, generalized vector Ekeland's variational principle and generalized equilibrium problems. Our results and techniques of proof are different from any existence result in the literature.     

Finite difference scheme for variational inequalities

In this paper, we show that a class of variational inequalities related with odd-order obstacle problems can be characterized by a system of differential equations, which are solved using the finite difference scheme. The variational inequality formulation is used to discuss the uniqueness and existence of the solution of the obstacle problems.