Existence Results Of Solutions Research Articles

A generalization of Minty's lemma and its applications to generalized mixed vector equilibrium problems¶

In this article, we introduce and study a class of generalized mixed vector equilibrium problems for multifunctions, which includes as special cases a number of generalized vector variational inequalities, generalized vector variational-like inequalities and generalized vector equilibrium problems. We first establish a new vector version of Minty's lemma, and then prove an existence theorem of solutions for the generalized mixed vector equilibrium problems by using this result. Furthermore, this existence theorem is applied to obtain a new existence result of solutions for a class of generalized vector variational-like inequalities. ¶Dedicated to the memory of Prof. Dr Alexander Moiseevich Rubinov.

Existence of solution for boundary value problem of nonlinear fractional differential equation

This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.

System of vector quasi-variational inclusions with some applications

In this paper, we consider systems of vector quasi-variational inclusions which include systems of vector quasi-equilibrium problems for multivalued maps, systems of vector optimization problems and several other systems as special cases. We establish existence results for solutions of these systems. As applications of our results, we derive the existence results for solutions of system vector optimization problems, mathematical programs with systems of vector variational inclusion constraints and bilevel problems. Another application of our results provides the common fixed point theorem for a family of lower semicontinuous multivalued maps. Further applications of our results for existence of solutions of systems of vector quasi-variational inclusions are given to prove the existence of solutions of systems of Minty type and Stampacchia type generalized implicit quasi-variational inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.

Penalty approximation method for a class of elliptic variational inequality problems

In this paper, two existence results of solutions for a class of elliptic variational inequalities are obtained by considering the approximations of the solutions to a class of penalized differential equations.

Existence Theorems of Solutions for a System of Nonlinear Inclusions with an Application

By using the iterative technique and Nadler's theorem, we construct a new iterative algorithm for solving a system of nonlinear inclusions in Banach spaces. We prove some new existence results of solutions for the system of nonlinear inclusions and discuss the convergence of the sequences generated by the algorithm. As an application, we show the existence of solution for a system of functional equations arising in dynamic programming of multistage decision processes.

Existence of solutions for elliptic hemivariational inequalities

This paper is devoted to the existence of solutions concerning the Dirichlet problem for quasilinear elliptic hemivariational inequalities at the first eigenvalue. Using the notion of the generalized gradient of Clarke and the property of the first eigenfunction, some existence results of solutions have been proved.

Some results about perturbed maximal monotone mappings

In this paper, we consider a class of variational inequality problems for monotone type mappings. By using one coercive assumption which is weaker than the one used in [1], we establish some existence results of solutions to variational inequality problems with generalized pseudomonotone mappings, generalized pseudomonotone perturbation and T-pseudomonotone perturbation of maximal monotone mappings, respectively.

Infinite Player Noncooperative Games with Vector Payoffs Under Relative Pseudomonotonicity

In this paper we consider the Nash equilibrium problem for infinite player games with vector payoffs in a topological vector space setting. By employing new concepts of relative (pseudo)monotonicity, we establish several existence results of solutions for usual and normalized vector equilibria. The results strengthen existence results for vector equilibrium problems, which were based on classical pseudomonotonicity concepts. They also extend previous results for vector variational inequalities and finite player games under relative (pseudo)monotonicity.

Existence of solutions to nonlinear Hammerstein integral equations and applications

In this paper, we study the existence and multiplicity of solutions of the operator equation K f u = u in the real Hilbert space L 2 ( G ) . Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K 1 / 2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.

Non-linear functional boundary value problem for discontinuous functional differential equations

This paper is devoted to investigating the existence of solutions for discontinuous functional differential equations with non-linear functional boundary conditions. Some existence results of solutions are obtained by employing a generalized method of upper and lower solutions, which may be discontinuous.

Existence Results for Evolution Noncoercive Hemivariational Inequalities

This paper is devoted to the existence of solutions for evolution hemivariational inequalities as generalizations of evolution variational inequalities to nonconvex functionals. The operators involved are taken to be multivalued and noncoercive. Using the notion of the generalized gradient of Clarke and the recession method, some existence results of solutions are proved.

Solution and positive solution for a semilinear third-order two-point boundary value problem

In this paper, several existence results of solution and positive solution are obtained for the semilinear third-order two-point boundary value problem μ‴ = ƒ (t, μ, μ″), μ(0) = A, μ(1) = B, μ″(0) = C. The discussion is based on the Leray-Schauder fixed-point theorem and the system of integral equations. The main conditions are local. In other words, the problem may possess a solution or positive solution provided the “height” of nonlinear term ƒ (t, μ, ν) is appropriate on a bounded set of its domain.

On Vector Quasi-Equilibrium Problems with Set-Valued Maps

In this paper, we introduce a new class of vector quasi-equilibrium problems with set-valued maps. Almost all the vector equilibrium models of the Blum-Oettli type in the literature are special cases of our new class of equilibrium problems under consideration. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the γ-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for such vector equilibrium problems.

Elliptic variational hemivariational inequalities

This paper is devoted to the existence of solutions for elliptic variational hemivariational inequalities. The operators involved are taken to be multivalued and noncoercive. Using the notion of the generalized gradient of Clarke and recession method, some existence results of solutions have been proved.

Generalized mixed quasivariational inclusions and generalized mixed resolvent equations for fuzzy mappings

In this paper, we introduce the concept of generalized mixed quasivariational inclusions and generalized mixed resolvent equations for fuzzy mappings. By using the resolvent operator technique for maximal monotone mapping, we obtain that the generalized mixed quasivariational inclusions for fuzzy mappings are equivalent to the fuzzy fixed-point problems and the generalized mixed resolvent equations for fuzzy mappings, respectively. We suggest a number of iterative algorithms, establish the existence results of solutions for the generalized mixed quasivariational inclusions for fuzzy mappings and prove the convergence of the algorithms. The results presented in this paper are improvements of previously known results.

Remarks on the resonant elliptic problem

We consider the semilinear elliptic equation −Δ u= f( x, u) with the Dirichlet boundary value for resonant nonlinearity at infinity. Using the Morse theory and perturbation method, some existence results of solution and multiple solutions are proved.

On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

We consider the stochastic heat equation with a multiplicative colored noise term on the real space for dimensions greater or equal to 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.

New existence results of solutions for vector optimization programs with multifunctions

This research work follows the main aim to present new existence conditions for the solutions of vector optimization programs with set-valued objective maps taking values especially in separated and ordered locally convex spaces

Cauchy problem for the time-dependent Ginzburg–Landau model of superconductivity

The Cauchy problem for the time-dependent Ginzburg–Landau equations of superconductivity in Rd (d = 2, 3) is investigated in this paper. When d = 2, we show that the Cauchy problem for this model is well posed in L2. When d = 3, we establish the existence result of solutions for L3 initial data and the uniqueness result for L4 initial data.

On the existence of solutions of boundary value problems of duffing type systems

Several existence results of solutions of two-point boundary value problems of Duffing type systems with Dirichlet boundary conditions, Neumann boundary conditions and periodic boundary conditions are presented.