Generalized bi-quasi-variational inequalities for upper hemi-continuous operators in non-compact settings

Let E and F be Hausdorff topological vector spaces over the field Φ, let 〈,〉: F × E → Φ be a bilinear functional, and let X be a non-empty subset of E. Given a set-valued map S : X → 2X and two set-valued maps M,T : X → 2F, the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point ŷ ∈ X and a point ŵ ∈ T(ŷ) such that ŷ ∈ S(ŷ) and Re 〈f - ŵ,ŷ - x〉 ≤ 0 for all x ∈ S(ŷ) and for all f ∈ M(ŷ) or to find a point ŷ ∈ X, a point ŵ; ∈ T(ŷ) and a point f̂ ∈ M (ŷ) such that ŷ ∈ S(ŷ) and Re 〈f̂ - ŵ,ŷ - x〉 ≤ 0 for all x ∈ S(ŷ). The above definition of GBQVI problem was given in [5, p. 139] which is a slight modification of the original definition of GBQVI problem of Shih and Tan in [12]. By using the concept of escaping sequences introduced by Border [2], existence theorems on generalized bi-quasi-variational inequalities for F-semi-monotone (respectively, bi-quasi-semi-monotone) operators are obtained in non-compact settings. As application, an existence theorem on generalized bi-complementarity problem for F-semi-monotone (respectively, bi-quasi-semi-monotone) operator is given in non-compact setting

Topological methods for set-valued nonlinear analysis

This book provides a comprehensive overview of the authors’ pioneering contributions to nonlinear set-valued analysis by topological methods. The coverage includes fixed point theory, degree theory, the KKM principle, variational inequality theory, the Nash equilibrium point in mathematical economics, the Pareto optimum in optimization, and applications to best approximation theory, partial equations and boundary value problems. Self-contained and unified in presentation, the book considers the existence of equilibrium points of abstract economics in topological vector spaces from the viewpoint of Ky Fan minimax inequalities. It also provides the latest developments in KKM theory and degree theory for nonlinear set-valued mappings

Numerical analysis of thermofluids inside a porous enclosure with partially heated wall

In this study, heat transfer in a tall, rectangular permeable cavity with active thermal walls is investigated. Inside the enclosure, the two side walls' central portions are partially cooled at a fixed temperature. The central portion of the footwall is heated. Additionally, the top wall, the remaining portion of the footwall, and the side walls are insulated. The controlling equations are obtained from the Brinkman–Forchheimer-extended Darcy prototypical model using Boussinesq calculations. The leading equations are explained numerically by the finite element Galerkin method of weighted residuals. The computations are executed for some governing and physical parameters. The isotherms, streamlines and average heat transfer rate along with the partially active hot wall are shown for different groupings of governing parameters with respect to dimensionless time (τ). The outcomes indicated that the stream and thermal fields are strongly dependent on the considered parameters. It is also established that the average heat transfer rate is a function of these governing parameters