Regularity on a Fixed Set

Positive L^TP and Continuous Solutions for Fredholm Integral Inclusions. A Note on the Structure of the Solution Set for the Cauchy Differential Inclusion in Banach Spaces. Fixed Point Theory for Acylic Maps between Topological Vector Spaces having Sufficiently many Linear Functionals, and Generalized Contractive Maps with Closed Values between Complete Metric Spaces. Using the Integral Manifolds to Solvability of Boundary Value Problems. On the Semicontinuity of Nonlinear Spectra. Existence Results for Two-Point Boundary Value Problems. Generalized Strongly Nonlinear Implicit Quasi-Variational Inequalities for Fuzzy Mappings. Vector Variational Inequalities, Multi-Objective Optimizations, Pareto Optimality and Applications. Variational Principle and Fixed Points. On the Baire Category Method in Existence Problems for Ordinary and Partial Differential Inclusions. Maximal Element Principles on Generalized Convex Spaces and their Applications. Fixed Point Results for Multi-valued Contractions on Gauge Spaces. The Study of Variational Inequalities and Applications to Generalized Complementarity Problems, Fixed Point Theorems of Set-Valued Mappings and Minimization Problems. Remarks on the Existence of Maximal Elements with Respect to a Binary Relation in Non-Compact Topological Spaces. Periodic Solutions of a Singularly Perturbed System of Differential Inclusions in Banach Space. Constrained Differential Inclusions. Nonlinear Boundary Value Problems with Multi-valued Terms. Optimal Control of a Class of Nonlinear Parabolic Problems with Multi-valued Terms. Continuation Theory for A-Proper Mappings and their Uniform Limits and Nonlinear Perturbations of Fredholm Mappings. Existence Theorems for Strongly Accretive Operators in Banch Spaces. A Kneser Type Property for the Solution Set of a Semilinear Differential Inclusion with Lower Semicontinuous Nonlinearity. A Nonlinear Multi-valued Problem with Nonlinear Boundary Conditions. Extensions of Monotone Sets. Convergence of Iterates of Nonexpansive Set-Valued Mappings. A Remark on the Intersection of a Lower Semicontinuous Multi-function and Fixed Set. Random Approximations and Random Fixed Point Theorems for Set-Valued Random Maps.Existence Theorems for Two-Variable Functions and Fixed Point Theorems for Set-Valued Mappings. An Extension Theorem and Duals of Gale-Mas-Colell's and Shafer-Sonnenschein's Theorems. Iterative Algorithms for Nonlinear Variational Inequalities Involving Set-Valued H-Cocoercive Mappings.

Many nonlinear option pricing problems can be formulated as optimal control problems, leading to Hamilton–Jacobi–Bellman (HJB) or Hamilton– Jacobi–Bellman–Isaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods that ensure convergence to the financially relevant solution, which in this case is the viscosity solution. In addition, for the HJB-type equations, we can guarantee convergence of a Newton-type (policy) iteration scheme for the nonlinear discretized algebraic equations. However, in some cases, the Newton-type iteration cannot be guaranteed to converge (for example, the HJBI case), or can be very costly (for example, for jump processes). In this case, we can use a piecewise constant control approximation. While we use a very general approach, we also include numerical examples for the specific interesting case of option pricing with unequal borrowing/lending costs and stock borrowing fees

We discuss new applications of advanced tools of variational analysis and generalized differentiation to a number of important problems in optimization theory, equilibria, optimal control, and feedback control design. The presented results are largely based on the recent work by the author and his collaborators. Among the main topics considered and briefly surveyed in this paper are new calculus rules for generalized differentiation of nonsmooth and set-valued mappings; necessary and sufficient conditions for new notions of linear subextremality and suboptimality in constrained problems; optimality conditions for mathematical problems with equilibrium constraints; necessary optimality conditions for optimistic bilevel programming with smooth and nonsmooth data; existence theorems and optimality conditions for various notions of Pareto-type optimality in problems of multiobjective optimization with vector-valued and set-valued cost mappings; Lipschitzian stability and metric regularity aspects for constrained and variational systems.

Ekeland's principle for set-valued vector equilibrium problems

Approachability and fixed points for non-convex set-valued maps

In this paper, we propose a new perspective to discuss the N-order fixed point theory of set-valued and single-valued mappings. There are two aspects in our work: we first define a product metric space with a graph for the single-valued mapping whose conversion makes the results and proofs concise and straightforward, and then we propose an SG-contraction definition for set-valued mapping which is more general than some recent contraction’s definition. The results obtained in this paper extend and unify some recent results of other authors. Our method to discuss the N-order fixed point unifies N-order fixed point theory of set-valued and single-valued mappings.

The theory of set-valued mapping (SVM) has essential applications and stimuluses in different domains of mathematics (see for example [1], [2]). Convex-valued continuous SVM form an important class. It is natural, by analogy with classical analysis, to consider the problem of approximating such mappings in uniform metric by SVM which have a simple structure.

Globalization concepts for Newton-type iteration schemes are widely used when solving nonlinear problems numerically. Most of these schemes are based on a predictor/corrector step size methodology with the aim of steering an initial guess to a zero of f without switching between different attractors. In doing so, one is typically able to reduce the chaotic behavior of the classical Newton-type iteration scheme. In this note we propose a globalization methodology for general Newton-type iteration concepts which changes into a simplified Newton iteration as soon as the transformed residual of the underlying function is small enough. Based on Banach’s fixed-point theorem, we show that there exists a neighborhood around a suitable iterate x_{n} such that we can steer the iterates—without any adaptive step size control but using a simplified Newton-type iteration within this neighborhood—arbitrarily close to an exact zero of f. We further exemplify the theoretical result within a global Newton-type iteration procedure and discuss further an algorithmic realization. Our proposed scheme will be demonstrated on a low-dimensional example thereby emphasizing the advantage of this new solution procedure.

In this paper, we introduce the concept of approximate solutions for set-valued mappings and provide a sufficient condition for the existence of approximate solutions of set-valued mappings. We obtain an approximate variational principle for set-valued mappings.

Exceptional family and solution existence of variational inequality problems with set-valued mappings

The theory of set-valued mappings has grown with the development of modern variational analysis. It is a key in convex and non-smooth analysis, in game theory, in mathematical economics and in control theory. The concepts of nearness and orthogonality have been known for functions since the pioneering works of Campanato, Birkhoff and James. In a recent paper Barbagallo et al. [J. Math. Anal. Appl., 484 (1), (2020)] a connection between these two concepts has been made. This note is mainly devoted to introduce nearness and orthogonality between set-valued mappings with the goal to study the solvability of generalized equations involving set-valued mappings.

ABSTRACTWe present a coincidence theory for set-valued maps which satisfy certain compactness-type conditions on countable sets. Our theory is based on fixed point results for compositions of set-valued self maps.

For fuzzy mathematical models using general fuzzy sets rather than fuzzy numbers or fuzzy vectors, operations (addition and scalar multiplication) and orderings of fuzzy sets are needed, and the concept of fuzzy set-valued convex mappings is important. In the present paper, fundamental properties of operations, orderings, and fuzzy set-valued convex mappings for general fuzzy sets are investigated systematically.

Commentary on "From local to global in quasiconformal structures".

Inclusions in general spaces: Hoelder stability, solution schemes and Ekeland's principle