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.     

Unified approach to approximate solutions in games and multiobjective programming

Using a theorem of Tijs, we derive results about approximate solutions for Nash equilibrium theory and for multiobjective problems. We describe conditions under which one can replace an infinite strategy set, an infinite alternative set, or an infinite set of criteria by a finite subset without losing all approximate solutions of the problem under consideration.This work was done during the period when the second author was Visiting Professor of the Italian National Research Council at the Mathematical Department of the University of Pavia, Pavia, Italy.     

Strong Approximation for Long Memory Processes with Applications

In this paper we inverstigate the strong approximation of a linear process with long memory to a Gaussian process. The results are then applied to derive the law of the iterated logarithm and Darling–Erds type theorem for long memory processes under ideal conditions.     

Condensed Cramer rule for some restricted quaternion linear equations

In this paper, we derive an alternative more condensed Cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations. The findings of this paper extend some known results in the literature.     

About a minimax theorem of Matthies,Strang and Christiansen

We give a new minisup theorem for noncompact strategy sets. Our result is of the type of the Matthies-Strang-Christiansen minimax theorem where the hyperplane should be replaced by any closed convex set. As an application, we derive a slight generalization of the Matthies-Strang-Christiansen minimax theorem.     

A class of operator equilibrium problems

In this paper, we consider a class of operator equilibrium problems (OEP for short) with operator solutions and derive a Minty type lemma for this class of problems. Further, using this lemma and KKM theorem, we establish some existence theorems for OEP. The theorems presented in this paper generalize, improve and unify many known results.     

Multivalued Parametric Variational Inequalities with α-Pseudomonotone Maps

In this paper, we consider variational inequalities with pseudomonotone maps which depend on a parameter and study the behavior of their solutions. The main result gives sufficient conditions for the stability of the initial variational inequality problem under small perturbations of the parameter. As an application, we obtain a stability result for a class of parametric optimization problems.     

Cauchy problem for quasiparabolic factorized differential equations with variable domains of smoothed operators

We prove existence, uniqueness, and stability theorems for strong solutions of Cauchy problems for quasiparabolic factorized operator-differential equations with variable domains. For the first time, we derive a recursion formula for strong solutions of Cauchy problems, where recursion goes over the number of operator-differential factors in these equations. We prove the well-posed solvability (in the strong sense) for new mixed problems for partial differential equations with time-dependent coefficients in the boundary conditions.     

Sato-Tate Conjectures and Chebyshev Polynomials

Results of Shahidi on the analytic continuation of certain L-functions have been used by Serre to obtain partial information on the distribution of Sato-Tate angles of the Ramanujan function. Employing the Christoffel numbers associated to the orthogonal polynomials used by Serre we derive upper bounds on the lower density of primes with small angles. We also make effective his lower bounds on the upper density of such primes. Assuming that more analytic information is available, we sharpen similar estimates of Ram Murty, and, using the Christoffel numbers attached to the second order Chebyshev polynomials, derive general bounds on the discrepancy between the empirical angles distribution and the Sato-Tate distribution.     

Fixed-Point Technique for Implicit Complementarity Problem in Hilbert Lattice

In this paper, we consider and study the implicit complementarity problem in the setting of a Hilbert lattice. It has been shown that this problem can be formulated as a fixed-point problem by using a suitable change of variables. Moreover, this formulation allows us to prove the existence and uniqueness of solutions of the implicit complementarity problem.     

Discretisation and duality of optimal Skorokhod embedding problems

We prove a strong duality result for a linear programming problem which has the interpretation of being a discretised optimal Skorokhod embedding problem, and we recover this continuous time problem as a limit of the discrete problems. With the discrete setup we show that for a suitably chosen objective function, the optimiser takes the form of a hitting time for a random walk. In the limiting problem we then reprove the existence of the Root, Rost, and cave embedding solutions of the Skorokhod embedding problem.The main strength of this approach is that we can derive properties of the discrete problem more easily than in continuous time, and then prove that these properties hold in the limit. For example, a consequence of the strong duality result is that dual optimisers exist, and our limiting arguments can be used to derive properties of the continuous time dual functions. These arguments are applied in Cox and Kinsley (2017), where the existence of dual solutions is required to prove characterisation results for optimal barriers in a financial application.     

A Property of the Ansatz of Hirota's Method for Quasilinear Parabolic Equations

By using the recently discovered new invariant properties of the ansatz of R. Hirota's method, we prove that the classes of linear fractional solutions to some nonlinear equations are closed. This allows us to construct new solutions for a chosen class of dissipative equations. This algorithm is similar to the method of dressing the solutions of integrable equations. The equations thus obtained imply a compatibility condition and are known as a nonlinear Lax pair with variable coefficients. So we propose a method for constructing such pairs. To construct solutions of a more complicated form, we propose to use the property of zero denominators and factorized brackets, which has been discovered experimentally. The expressions thus constructed are said to be quasi-invariant. They allow us to find true relations between the functions contained in the ansatz, to correct the ansatz, and to construct a solution. We present some examples of new solutions constructed following this approach. Such solutions can be used for majorizing in comparison theorems and for modeling phase processes and process in neurocomputers. A program for computing solutions by methods of computer algebra is written. These techniques supplement the classical methods for constructing solutions by using their group properties.     

Topological existence and stability for stackelberg problems

The aim of this paper is to study, in a topological framework, existence and stability for the solutions to a parametrized Stackelberg problem. To this end, approximate solutions are used, more precisely, -solutions and strict -solutions. The results given are of minimal character and the standard types of constraints are considered, that is, constant constraints, constraints defined by a finite number of inequalities, and more generally constraints defined by an arbitrary multifunction.     

A polyhedral study on 0–1 knapsack problems with disjoint cardinality constraints: Strong valid inequalities by sequence-independent lifting

We study the set of 0–1 integer solutions to a single knapsack constraint and a set of non-overlapping cardinality constraints (MCKP), which generalizes the classical 0–1 knapsack polytope and the 0–1 knapsack polytope with generalized upper bounds. We derive strong valid inequalities for the convex hull of its feasible solutions using sequence-independent lifting. For problems with a single cardinality constraint, we derive two-dimensional superadditive lifting functions and prove that they are maximal and non-dominated under some mild conditions. We then show that these functions can be used to build strong valid inequalities for problems with multiple disjoint cardinality constraints. Finally, we present preliminary computational results aimed at evaluating the strength of the cuts obtained from sequence-independent lifting with respect to those obtained from sequential lifting.     

On Nondifferentiable and Nonconvex Vector Optimization Problems

In this paper, we prove the equivalence among the Minty vector variational-like inequality, Stampacchia vector variational-like inequality, and a nondifferentiable and nonconvex vector optimization problem. By using a fixed-point theorem, we establish also an existence theorem for generalized weakly efficient solutions to the vector optimization problem for nondifferentiable and nonconvex functions.     

Iterative Schemes for Solving Mixed Variational-Like Inequalities

In the present paper, we introduce the concept of -cocoercivity of a map and develop some iterative schemes for finding the approximate solutions of mixed variational-like inequalities. We use the concept of -cocoercivity to prove the convergence of the approximate solutions to the exact solution of mixed variational-like inequalities.     

On the Cauchy problem of a two-component b-family system

In this paper, we study the Cauchy problem of a two-component b-family system. We first establish the local well-posedness for a two-component b-family system by Kato’s semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the system. Moreover, we present several blow-up results for strong solutions to the system.     

General KKM Theorem with Applications to Minimax and Variational Inequalities

In this paper, a general version of the KKM theorem is derived by using the concept of generalized KKM mappings introduced by Chang and Zhang. By employing our general KKM theorem, we obtain a general minimax inequality which contains several existing ones as special cases. As applications of our general minimax inequality, we derive an existence result for saddle-point problems under general setting. We also establish several existence results for generalized variation inequalities and generalized quasi-variational inequalities.     

Properties of the Set of Extreme Solutions for a Class of Nonlinear Second-Order Evolution Inclusions

We establish the existence of extreme solutions for a class of nonlinear second-order evolution inclusions with a nonconvex right-hand side defined on an evolution triple of Banach spaces. Then we show that extreme solutions which belong to the solution set of the original system are in fact dense and codense in the solution set of a system with a convexified right-hand side. The necessary and sufficient conditions for closedness of the solution set for the original system in an appropriate spaces of functions are given as well. Finally, an example of a nonlinear hyperbolic distributed parameter system is worked out in detail.     

Generalized vector variational inequalities with star-pseudomonotone and discontinuous operators

It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed.