Variational Inequality Problems

In this paper, we introduce and study a new class of variational inequalities, which is called the generalized set-valued mixed variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued variational inequalities, fixed point problems, and the generalized set-valued resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.

In this paper we develop a framework for the study of financial equilibrium in the case of sectors in the economy, each of which is faced with two objectives/criteria in his portfolio selection decision making. In particular, we first present the bicriteria decision model of an individual financial sector, who seeks an optimal portfolio composition, given that the wishes to

Let E be a q-uniformly smooth real Banach space with constant dq, q>1. Let Ti:E→E, i=1,2,…,r be a finite family of nonexpansive mappings with K≔∩i=1rFix(Ti)≠∅ and K=Fix(TrTr−1…T1)=Fix(T1Tr…T2)=⋯=Fix(Tr−1Tr−2…Tr). Let G:E→E be an η-strongly accretive map which is also κ-Lipschitzian. A hybrid steepest descent method introduced by Yamada [25] and studied by various authors is proved to converge strongly to the unique

In this article we study a decoupled forward backward stochastic differential equation (FBSDE) and the associated system of partial integro-differential obstacle problems, in a flexible Markovian set-up made of a jump-diffusion with regimes. These equations are motivated by numerous applications in financial modeling, whence the title of the paper. This financial motivation is developed in the first part of the paper, which provides a synthetic view of the theory of pricing and hedging financial derivatives, using backward stochastic differential equations (BSDEs) as main tool. In the second part of the paper, we establish the well-posedness of reflected BSDEs with jumps coming out of the pricing and hedging problems exposed in the first part. We first provide a construction of a Markovian model made of a jump-diffusion – like component X interacting with a continuous-time Markov chain – like component N. The jump process N defines the so-called regime of the coefficients of X, when...

In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.

The variational inequality problem is reduced to an optimization problem with a differentiable objective function and simple bounds. Theoretical results are proved, relating stationary points of the minimization problem to solutions of the variational inequality problem. Perturbations of the original problem are studied and an algorithm that uses the smooth minimization approach for solving monotone problems is defined.

In this paper, using proximal-point mapping technique of P-η-accretive mapping and the property of the fixed-point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of the solution set of a parametric generalized implicit quasi-variational-like inclusion involving P-η-accretive mapping in real uniformly smooth Banach space. Further, under suitable conditions, we discuss the Lipschitz continuity of the solution

We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods.

Multidisciplinary teams (MDTs) are conventionally recommended in mental health
care literature as an important way to offer holistic treatment provision to patients. This
study aims to explore multidisciplinary teamwork in contemporary mental health
settings, particularly what aids and hinders the process of multidisciplinary teamworking,
and the social work contribution in such teams. In order to attain an in-depth
exploration of these phenomena, a single case study design was employed. Within this
design, data were generated through semi-structured interviews and structured
observation of a mental health MDT in Ireland. These data were analysed using
interpretative phenomenological analysis. The research highlights how the concept of
mental illness is contested within this MDT, with the medical model dominant within
an environment of fluid working arrangements. Professional role blurring and
stereotyping were found to impact the division of labour on this team, with role
negotiation found to be an integral part of retaining a professional’s practice identity.
This research raises concerns for social work’s capacity to function within mental
health MDTs in Ireland, and highlights ways in which social work educators might
respond in order to empower Irish social workers to meet the challenges of mental
health multidisciplinary teamworking.

This paper deals with multivalued quasi variational inequalities with pseudo-monotone and monotone maps. The primary objective of this work is to show that the notion of generalized solutions can be employed to investigate multivalued pseudo-monotone quasi variational inequalities. It is a well-known fact that a quasi variational inequality can conveniently be posed as a fixed point problem through the so-called variational selection. For pseudo-monotone maps, the associated variational selection is a nonconvex map, and the fixed point theorems can only be applied under restrictive assumptions on the data of quasi variational inequalities. On the other hand, the generalized solutions are defined by posing a minimization problem which can be solved by a variant of classical Weierstrass theorem. It turns out that far less restrictive assumptions on the data are needed in this case. To emphasis on the strong difference between a classical solution and a generalized solution, we also give a new existence theorem for quasi variational inequalities with monotone maps. The main existence result is proved under a milder coercivity condition. We also relax a few other conditions from the monotone map. Due to its flexibility, it seems that the notion of generalized solutions can be employed to study quasi variational inequalities for other classes of maps as well.