George Isac

Stability of Functional Equations in Several Variables

Approximately additive and approximately linear mappings stability of the quadratic functional equation generalizations - the method of invariant means approximately multiplicative mappings - superstability stability of functional equations for trigonometric and similar functions functions with bounded nth differences approximately convex functions stability of the generalized orthogonality functional equation stability and set-valued mappings stability of stationary and minimum points functional congruences quasi-additive functions and related topics.

Stability of ψ-additive mappings: applications to nonlinear analysis

The Hyers-Ulam stability of mappings is in development and several authors have remarked interesting applications of this theory to various mathematical problems. In this paper some applications in nonlinear analysis are presented, especially in fixed point theory. These kinds of applications seem not to have ever been remarked before by other authors.

Topics in Nonlinear Analysis and Applications

Stability of functional equations isometric mappings cones and complementarity problems metrics on cones zero-epi mappings variational principles maximal element principles.

On the asymptoticity aspect of Hyers-Ulam stability of mappings

The object of the present paper is to prove an asymptotic analogue of Th. M. Rassias’ theorem obtained in 1978 for the Hyers-Ulam stability of mappings.

Exceptional Families, Topological Degree and Complementarity Problems

By using the topological degree we introduce the concept of ’’exceptionalfamily of elements‘‘ specifically for continuous functions. This has importantconsequences pertaining to the solvability of the explicit, the implicit andthe general order complementarity problems. In this way a new direction forresearch in the complementarity theory is now opened.

Complementarity, equilibrium, efficiency and economics

1. Introduction. 2. Optimization Models. 3. General Economic Equilibrium. 4. Models of Oligopoly. 5. Oligopoly with Leaders. 6. Complementarity Problems with Respect to General Cones. 7. Pseudomonotone and Implicit Complementarity Problems. 8. Complementarity Pivot Methods. 9. Scarf Type Algorithms. 10. Newton-Like Methods. 11. Parameterization and Reduction To Nonlinear Equations. 12. Efficiency. 13. Approximative Efficiency. Index.

Functions without exceptional family of elements and complementarity problems

In Ref. 1, Isac, Bulavski, and Kalashnikov introduced the concept of exceptional family of elements for a continuous function f: Rn→Rn. It is known that, if there does not exist an exceptional family of elements for f, then the corresponding complementarity problem has a solution. In this paper, we show that several classes of nonlinear functions, known in complementarity theory or other domains, are functions without exceptional family of elements and consequently the corresponding complementarity problem is solvable. It is evident that the notion of exceptional family of elements provides an alternative way of determining whether or not the complementarity problem has a solution.

Exceptional Families of Elements for Continuous Functions: Some Applications to Complementarity Theory

Using the topological degree and the concept of exceptional family of elements for a continuous function, we prove a very general existence theorem for the nonlinear complementarity problem. This result is an alternative theorem. A generalization of Karamardians condition and the asymptotic monotonicity are also introduced. Several applications of the main results are presented.

Quasi-P * -maps, P(t, a b)-maps exceptional family of elements, and complementarity problems

Quasi-P*-maps and P(τ, α, β)-maps defined in this paper are two large classes of nonlinear mappings which are broad enough to include P*-maps as special cases. It is of interest that the class of quasi-P*-maps also encompasses quasimonotone maps (in particular, pseudomonotone maps) as special cases. Under a strict feasibility condition, it is shown that the nonlinear complementarity problem has a solution if the function is a nonlinear quasi-P*-map or P(τ, α, β)-map. This result generalizes a classical Karamardian existence theorem and a recent result concerning quasimonotone maps established by Hadjisawas and Schaible, but restricted to complementarity problems. A new existence result under an exceptional regularity condition is also established. Our method is based on the concept of exceptional family of elements for a continuous function, which is a powerful tool for investigating the solvability of complementarity problems.

Application of Topological Degree Theory to Complementarity Problems

The topological degree theory is applied to study the problem of existence of solutions to complementarity problems of various kinds. A notion of an exceptional family of elements is introduced, and assertions of a non-strict alternative type are obtained. Namely, for a continuous mapping, there exists at least one of the following two objects: either a solution to the complementarity problem, or an exceptional family of elements. Hence, if there is no exceptional families, then at least one solution exists.