On topological existence theorems and applications to optimization-related problems

In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies.     

General existence theorems, alternative theorems and applications to minimax problems

We establish general theorems on maximal elements, coincidence points and nonempty intersections for set-valued mappings on GFC-spaces and show their equivalence. Applying them we derive equivalent forms of alternative theorems. As applications, we develop in detail general types of minimax theorems. The results obtained improve or include as special cases several recent ones in the literature.     

Some fixed point theorems and existence of positive solutions of two-point boundary-value problems

Three theorems are obtained for the existence of at least one or three fixed points for a completely continuous mapping, which extend the Krasnoselskii’s compression–expansion theorem in cones. Based on them two theorems for the existence of positive solutions of two-point boundary-value problems are proved under a quite relaxed condition compared with the existing literature.     

Constructive existence theorems for problems of Thomas-Fermi Type

A minimal positive solution of the Thomas-Fermi problem ? = λt^?1/2 w^3/2, w(0) = 1, w(1) = w(1) is shown to exist for each λ > 0. It is proved that all positive solutions, for a given value of λ, are strictly ordered and that the minimal positive solution w_λ is a decreasing function of λ. Upper and lower analytic bounds for w _λ are given and these bounds are shown to initiate sequences of Picard and Newton iterates which converge monotonically to w _λ. A comparative analysis of the efficiency of the iteration schemes is presented. The methods used are of a general nature and can be applied to a variety of nonlinear boundary value problems of convex type [14].     

On the existence of solutions to generalized quasi-equilibrium problems

In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65–76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasi-equilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.     

On the existence of solutions to quasivariational inclusion problems

We establish sufficient existence conditions for general quasivariational inclusion problems, which contain most of variational inclusion problems and quasiequilibrium problems considered in the literature. These conditions are shown to extend recent existing results and sharpen some of them even for particular cases.     

Concerning existence theorems for some problems of elasticity theory

We prove an existence theorem for the statistical elasticity theory equation for a homogeneous incompressible medium and its extension to the second and third boundary value problem case. We demonstrate, in the case of the first, second, and third problems that, as the solution of the elasticity theory equation with Lamé constants and converges to the solutions of the respective equations for incompressible material. An existence theorem in the rectangle is demonstrated for the third boundary value problem inw _q ² .Translated from Matematicheskie Zametki, Vol. 17, No. 4, pp. 599–609, April, 1975.     

Some existence and nonexistence theorems for solutions of degenerate parabolic equations

The initial and boundary value problem for the degenerate parabolic equation v_t = Δ(?(v)) + F(v) in the cylinder $\text{Ω × ¦0, ∞), Ω ? R}{}^{\mathrm{n}}$ bounded, for a certain class of point functions ? satisfying ?′(v) ? 0 (e.g., $\text{?(v) = ¦v¦}{}^{\mathrm{m}}\text{sign}\text{v}$) is considered. In the case that F(v) sign $\text{v ? C(1 + ¦?(v)¦}{}^{\mathrm{\alpha }}\text{), α < 1}$, the equation has a global time solution. The same is true for α = 1 provided the measure of Ω is sufficiently small. In the case that $\text{F(v)}\text{?(v)}$ is nondecreasing a condition is given on the initial state v(x, 0) which implies that the solution must blow up in finite time. The existence of such initial states is discussed.     

Minimax theorems and applications to the existence and uniqueness of solutions of some differential equations

In this paper, minimax theorems due to Stepan A. Tersian were generalized, the existence and uniqueness of solutions for several semilinear equations were proved by employing our generalized theorems, and the existence and uniqueness results of nonresonance problem for these semilinear equations under the asymptotic un-uniformity conditions were presented.     

Uniqueness and existence theorems of solutions of the boundary value problems of the theory of consolidation with double porosity

In this paper we consider the Aifantis' theory of consolidation with double porosity and we prove the uniqueness and existence theorems of solutions of basic boundary value problems (BVPs) of statics for the two-dimensional finite and infinite domains. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The existence of solutions to second-order singular boundary value problems

This article analyzes qualitative properties of solutions to two-point boundary value problems for singular ordinary differential equations. In particular, we form new approaches that ensure that all possible solutions satisfy certain a priori bounds. The methods involve differential inequalities.     

The existence of approximate solutions to mixed boundary value problems

Approximate solutions, similar to the type used in the Complex Variable Boundary Element Method, are shown to exist for two dimensional mixed boundary value potential problems on multiply connected domains. These approximate solutions can be used numerically to obtain least squares solutions or solutions which interpolate given boundary conditions. Areas of application include fluid flow around obstacles and heat flow in a domain with insulated boundary segments. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 191–199, 1999     

An existence principle for solutions to a singular boundary-value problems

The singular boundary-value problem

$ \left\{ {\begin{array}{*{20}{c}} {{u^{\prime\prime}} + g\left( {t,u,{u^{\prime}}} \right) = 0\quad {\text{for}}\quad t \in \left( {0,1} \right),} \hfill \\ {u(0) = u(1) = 0} \hfill \\ \end{array} } \right. $

is studied. The singularity may appear at u?=?0, and the function g may change sign. An existence theorem for solutions to the above boundary-value problem is proposed, and it is proved via the method of upper and lower solutions.

     

On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty, motivating the examination of a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration and our emphasis lies in developing tractable and verifiable sufficiency conditions for claiming existence. We begin by recapping almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued maps provided in our prior work Ravat and Shanbhag (in: Proceedings of the American Control Conference (ACC), 2010), Ravat and Shanbhag (SIAM J Optim 21: 1168–1199, 2011) and provide extensions to multi-valued mappings. Next, we extend these statements to quasi-variational regimes where maps can be either single or set-valued. Finally, we refine the obtained results to accommodate stochastic complementarity problems where the maps are either general or co-coercive. The applicability of our results is demonstrated on practically occuring instances of stochastic quasi-variational inequality problems and stochastic complementarity problems, arising as nonsmooth generalized Nash-Cournot games and power markets, respectively.     

Regularization and existence of solutions of three-dimensional elastoplastic problems

We prove the existence of solutions to the three-dimensional elastoplastic problem with Hencky's law and Neumann boundary conditions by elliptic regularization and the penalty method, both for the case of a smooth boundary and of an interior two dimensional crack. It is shown, in particular, that the variational solution satisfies all boundary conditions. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On the existence of optimal solutions to integer and mixed-integer programming problems

The purpose of this paper is to present sufficient conditions for the existence of optimal solutions to integer and mixed-integer programming problems in the absence of upper bounds on the integer variables. It is shown that (in addition to feasibility and boundedness of the objective function) (1) in the pure integer case a sufficient condition is that all of the constraints (other than non-negativity and integrality of the variables) beequalities, and (2) that in the mixed-integer caserationality of the constraint coefficients is sufficient. Some computational implications of these results are also given.