Pseudomonotone variational inequality problems: Existence of solutions

Necessary and sufficient conditions for the set of solutions of a pseudomonotone variational inequality problem to be nonempty and compact are given. This research was partially done while the author was visiting the University of Chile thanks to the support of an ECOS program.     

The method of fundamental solutions for Signorini problems

We investigate the use of method of fundamental solutions (MFS)for the numerical solution of Signorini boundary value problems.The MFS is an ideal candidate for solving such problems becauseinequality conditions alternating at unknown points of the boundarycan be incorporated naturally into the least-squares minimizationscheme associated with the MFS. To demonstrate its efficiency,we apply the method to two Signorini problems. The first isa groundwater flow problem related to percolation in gentlysloping beaches, and the second is an electropainting application.For both problems, the results are in close agreement with previouslyreported numerical solutions.     

Existence and uniqueness for several non-linear elliptic problems arising in lubrication theory

We consider the non-linear elliptic equation ∇·(A(x,p)∇p)=∇·b(x,p) with positive Dirichlet boundary data. The coefficients A and b are taken from models used in lubrication theory, and in particular are not defined for negative values of p. We prove some general existence and uniqueness results for a family of models, which extend related results in the literature. These results allow us to prove existence, uniqueness and positivity of the solution to advanced compressible lubrication models such as the kinetic-based Fukui-Kaneko model and the second-order-slip model. We also consider a spring-like model of compliant-foil compressible bearing, and weaken some hypotheses of previous results on more classical models such as the standard Reynolds model and the first-order-slip model.     

The method of fundamental solutions for problems in potential flow

The method of fundamental solutions is a form of indirect boundary integral equation method. Its distinctive feature is adaptivity, gained through the use of an auxiliary boundary that is chosen automatically by a least squares procedure. The paper demonstrates the application of the method to problems in potential flow. A further advantage of the method is that the velocity field can be computed easily and accurately by a direct evaluation procedure.     

Existence of solutions in variational relation problems without convexity

Two main existence conditions for solutions of variational relation problems are established without convexity. The first one is based on a finite solvability property and the second one on generalized KKM mappings. These conditions unify and strengthen several existing results in the literature on the topic. A model of satisficing process by rejection is considered which gives an economic interpretation of the introduced concepts.     

Existence of solutions for quasilinear elliptic obstacle problems

Two existence results for a class of obstacle problems are obtained by using a result developed in this paper.     

Existence of solutions for anti-periodic boundary value problems

By using Schauder’s fixed point theorem and the lower and upper solutions method, we discuss the anti-periodic boundary value problem for a class of second-order differential equations. Some sufficient conditions for the existence of solutions are obtained.     

Existence of efficient solutions for vector maximization problems

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.The author would like to thank Professor T. L. Morin for his helpful comments. Thanks also go to an anonymous reviewer for his useful comments concerning an earlier version of this paper.The author would like to acknowledge a useful discussion with Professor G. Bitran which helped in motivating Example 4.1.     

Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.     

Existence of solutions for critical fractional Kirchhoff problems

Consider the following fractional Kirchhoff equations involving critical exponent: where (?Δ)^α is the fractional Laplacian operator with α ∈(0,1), , , λ ₂>0 and is the critical Sobolev exponent, V (x ) and k (x ) are functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space, the minimax arguments, Pohozaev identity, and suitable truncation techniques, we obtain the existence of a nontrivial weak solution for the previously mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity f . Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of solutions for first order singular problems

We develop the lower and upper solutions method for first order initial value problems as well as for first order periodic problems in case the nonlinearity presents singularities. More precisely we prove that if we have a lower solution and an upper solution of these problems, which are not necessarily continuous nor ordered, we have a solution wedged between and .

     

Existence of solutions on weak vector equilibrium problems

Weak vector equilibrium problems with bi-variable mappings from product space of two bounded complete locally convex Hausdorff topological vector spaces to another topological vector space are studied. The existence theorems of solutions are proved by the FKKM fixed point theorem. Viscosity principle of vector equilibrium problems is dealt with. The relations between solutions of the vector equilibrium problem and those of its perturbation problem are presented.     

Existence of solutions for generalized vector quasi-equilibrium problems

This paper deals with three classes of generalized vector quasi-equilibrium problems with or without compact assumptions. Using the well-known Fan-KKM theorems, their existence theorems for them are established. Some examples are given to illustrate our results.     

The method of fundamental solutions for elliptic boundary value problems

The aim of this paper is to describe the development of the method of fundamental solutions (MFS) and related methods over the last three decades. Several applications of MFS-type methods are presented. Techniques by which such methods are extended to certain classes of non-trivial problems and adapted for the solution of inhomogeneous problems are also outlined. This revised version was published online in June 2006 with corrections to the Cover Date.     

Existence of positive solutions of Lidstone boundary value problems

We show that there exists at least one positive solution for Lidstone boundary value problem