Existence of Solutions for Nonlinear Implicit Differential Equations: An Equilibrium Problem Approach

In this article, we study the existence of solutions for nonlinear implicit differential equations associated to a time-dependent pseudomonotone (respectively, quasimonotone) operator by using a new approach based on the theory of equilibrium \hboxproblems. More precisely, we first establish some existence results of solutions for mixed equilibrium problems where the \hboxbifunctions are, respectively, maximal monotone and \hboxpseudomonotone (quasimonotone) in the topological sense. Then, we write the nonlinear implicit differential equations in the form of a mixed equilibrium problem. By using the existence results for mixed equilibrium problems, we establish the existence of solutions of nonlinear implicit differential equations. This new approach provides some new and nice results which improve and unify most of the recent results obtained in this direction.     

Noncoercive mixed equilibrium problems under pseudomonotone perturbations and applications to nonlinear evolution equations with lack of coercivity

In this paper, we study the existence of solutions for noncoercive mixed equilibrium problems which are described by the sum of a maximal monotone bifunction and a pseudomonotone (or quasimonotone) bifunction in the sense of Brézis. Our approach is based on recession analysis and on recent results established by the authors for the existence of solutions of mixed equilibrium problems under pseudomonotone perturbations. As an application, we study the existence of solutions for nonlinear evolution equations associated with a noncoercive time-dependent pseudomonotone (or quasimonotone) operator.     

From Scalar to Vector Equilibrium Problems in the Quasimonotone Case

In a unified approach, existence results for quasimonotone vector equilibrium problems and quasimonotone (multivalued) vector variational inequality problems are derived from an existence result for a scalar equilibrium problem involving two (rather than one) quasimonotone bifunctions. The results in the vector case are not only obtained in a new way, but they are also stronger versions of earlier existence results.     

Coercivity Conditions for Equilibrium Problems

The study of the existence of solutions of equilibrium problems on unbounded domains involves usually the same sufficient assumptions as for bounded domains together with a coercivity condition. We focus on two different conditions: the first is obtained assuming the existence of a bounded set such that no elements outside is a candidate for a solution; the second allows the solution set to be unbounded. Our results exploit the generalized monotonicity properties of the function f defining the equilibrium problem. It turns out that, in both the pseudomonotone and the quasimonotone setting, an equivalence can be stated between the nonemptyness and boundedness of the solution set and these coercivity conditions. In the pseudomonotone case, we compare our coercivity conditions with various coercivity conditions that appeared in the literature.We thank an anonymous referee for valuable comments and suggestions.     

Generalized Equilibrium Problems for Quasimonotone and Pseudomonotone Bifunctions

By using quasimonotone and pseudomonotone bifunctions, we derive sufficient conditions which include weak coercivity conditions for existence of equilibrium points. As a consequence, we improve some recent results on the existence of such solutions.     

Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities

A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.     

Monotone generalized variational inequalities and generalized complementarity problems

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.The research of the first author was supported by the National Natural Science Foundation of P. R. China and by the Ethel Raybould Fellowship, University of Queensland, St. Lucia, Brisbane, Australia.     

Equilibrium Problems in the Quasimonotone Case

Existence results for quasimonotone vector equilibrium problems and quasimonotone vector variational inequalities are obtained starting from an existence result for a scalar equilibrium problem involving two quasimonotone bifunctions. These results are established under weaker conditions than in previous works.     

Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces

By employing the notion of exceptional family of elements, we establish some existence results for generalized variational inequality problems in reflexive Banach spaces provided that the mapping is upper sign-continuous. We show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of the dual variational inequality. For quasimonotone variational inequalities, we present some sufficient conditions for the existence of strong solutions. For the pseudomonotone case, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the problem having strong solutions. Furthermore, we establish several equivalent conditions for the solvability for the pseudomonotone case. As a byproduct, a quasimonotone generalized variational inequality is proved to have a strong solution if it is strictly feasible. Moreover, for the pseudomonotone case, the strong solution set is nonempty and bounded if it is strictly feasible.     

MIXED MONOTONE ITERATIVE TECHNIQUES FOR SEMILINEAR EVOLUTION EQUATIONS IN BANACH SPACES

This paper is concerned with initial value problems for semilinear evolution equations in Banach spaces. The abstract iterative schemes are constructed by combining the theory of semigroups of linear operators and the method of mixed monotone iterations. Some existence results on minimal and maximal (quasi)solutions are established for abstract semilinear evolution equations with mixed monotone or mixed quasimonotone nonlinear terms. To illustrate the main results, applications to ordinary differential equations and partial differential equations are also given.     

一类二阶积分型发展方程反周期解问题

本文中,我们研究一类带有非单调扰动算子的二阶非线性发展方程的反周期问题,证明方程中的非单调扰动算子为极大单调的,并用极大单调算子的微单调扰动理论来证明此类方程的反周期解的存在性。     

On generalized variational inequalities

We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. Dedicated to the memory of T. Rapcsák.     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

Existence of Solutions for Noncoercive Hemivariational Inequalities by an Equilibrium Approach Under Pseudomonotone Perturbation

In this paper, we study the existence of a solution for a hemivariational inequality problem in a noncoercive framework. The approach adopted is an equilibrium problem formulation associated with a maximal monotone bifunction with pseudomonotone perturbation. We proceed by introducing auxiliary problems that will be studied using a new existence result for equilibrium problems. An example to illustrate the use of the theory is given.     

Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions

In this paper, we introduce the class of multivalued relaxed μ quasimonotone operators and establish the existence of solutions of variational inequalities for such operators. This result is compared with a recent result of Bai et al. on densely relaxed pseudomonotone operators. A similar comparison regarding an existence result of Luc on densely pseudomonotone operators is provided. Also, we introduce a broad class of functions, called relaxed quasiconvex functions, and show that they are characterized by the relaxed μ quasimonotonicity of their subdifferentials. The results strengthen a variety of other results in the literature. This work is supported by NNSF of China (10571046) and by the GSRT of Greece (06FR-062).     

Stability and attractivity of periodic solutions of parabolic systems with time delays

This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka-Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.     

Generalized evolution variational inequalities and their applications to partial differential equations

We study the existence of solutions to generalized evolution vari ational inequalities by pseudomonotone mapping theory. We obtain results which generalize and extend previously known theorems. We then apply our results to parabolic partial differential equations with discontinuous nonlinearities.     

具有伪单调算子的非线性微分包含约束的最优控制

本文研究一类受主算子为伪单调算子的非线性微分包含约束的最优控制问题．首先,探讨抛物型发展方程的柯西问题其解的性质及微分包含问题的容许轨线的存在性;然后,利用一个新的可测选择定理解决了受非线性微分包含约束的最优控制的存在性,最后,给一例子加以说明所获结果的应用性．     

Anti-periodic solutions to nonlinear evolution equations

We deal with anti-periodic problems for nonlinear evolution equations with nonmonotone perturbations. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.