On a subclass of quasi-convex mappings

A class of biholomorphic mappings named “quasi-convex mappings of type D” is introduced in the unit ball of a complex Banach space, which is the natural generalization of the convex functions in one variable. It is proved that this class of mappings is a subset of the class of “quasi-convex mappings of type A” introduced by K.A. Roper and T.J. Suffridge.     

Generalization of preinvex and B-vex fuzzy mappings

A class of fuzzy mappings, called B-preinvex, and strictly B-preinvex fuzzy mappings, is introduced by relaxing the definition of preinvexity of a fuzzy mapping. Similarly, based on a notion of differentiability of fuzzy mappings different from the usual one, the class of pseudo-B-vex, B-invex, and pseudo-B-invex fuzzy mappings is defined as a generalization of pseudo-convex, invex, and pseudo-invex fuzzy mappings. We prove that a strictly B-vex, or B-preinvex fuzzy mapping has at most one global minimum point, and that the class of B-vex fuzzy mappings forms a subset of the class of quasi-convex fuzzy mappings. B-vex (resp. B-preinvex) fuzzy mappings satisfy most of the basic properties of convex (resp. preinvex) fuzzy mappings. In addition characterizations and sufficient optimality conditions are obtained for B-vex and B-preinvex fuzzy mappings.     

On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings

We investigate some inequalities connected with the Hyers-Ulam stability of three functional equations, which have a solution of the form φ=a+q, where a is an additive mapping and q is a quadratic one.     

On the stability of iterative approximations to fixed points of nonexpansive mappings

We study iterative retraction approximations to fixed points of the nonexpansive self-mapping given on the closed convex set G in a Banach space B. The conditions which guarantee weak and strong convergence and stability of these approximations with respect to perturbations of both operator A and constraint set G are considered. The results of this paper are new even in a Hilbert space for the iterative projection approximations.     

Sufficient conditions for the stability of the karush- kuhn - tucker point set in quadratic programming

We obtain some conditions under which a small perturbation in the data of a quadratic programming problem can yield only a small change in its Karush-Kuhn-Tucker point set. Convexity of the objective function and boundedness of the constraint set are not assumed. The obtained results are illustrated by several examples     

Topological stability of linear semi-infinite inequality systems

In this note, we analyze the relationship between the lower semicontinuity of the feasible set mapping for linear semi-infinite inequality systems and the so-called topological stability, which is held when the solution sets of all the systems obtained by sufficiently small perturbations of the data are homeomorphic to each other. This topological stability and its relation with the Mangasarian-Fromovitz constraints qualification have been studied deeply by Jongen et al. in Ref. 1. The main difference of our approach is that we are not assuming any kind of structure for the index set and, consequently, any particular property for the functional dependence between the inequalities and the associated indices. In addition, we deal with systems whose solution sets are not necessarily bounded.This work has been supported partially by the DGICYT of Spain, Grant PB93-0943, by Generalitat Valenciana, Grant GV-2219/94, and by IVEI, Grant 003/026.The authors would like to thank J. E. Martínez Legaz for his valuable comments.     

On stability of solutions to parametric generalized affine variational inequalities

This paper deals with the stability of the solution set to parametric generalized affine variational inequalities with constraint set being defined by finitely many convex quadratic functions. The obtained results develop and complement the published ones.     

On the dilatation of extremal quasiconformal mappings of polygons

A polygon P _N is the unit disk with distinguished boundary points, . An extremal quasiconformal mapping maps each polygon inscribed in onto a polygon inscribed in . Let f _N be the extremal quasiconformal mapping of P _N onto P' _N. Let K _N be its dilatation and let K ₀ be the maximal dilatation of f ₀. Then, evidently . The problem is, when equality holds. This is completely answered, if f ₀ does not have any essential boundary points. For quadrilaterals Q and Q' = f ₀ (Q) the problem is sup(M'/M) = K ₀, with M and M' the moduli of Q and Q' respectively. Received: December 23, 1997     

On the stability problem for a mixed type of quartic and quadratic functional equation

Let E₁ and E₂ be real linear spaces. In this paper, we determine the general solution for a mixed type functional equation of a quartic and a quadratic mapping f:E₁→E₂,

     

On the uniqueness problem of harmonic quasiconformal mappings

In this paper, we give an affirmative answer to Sheretov's problem on the uniqueness of harmonic mappings and improve the unique minimal mapping theorem of Reich and Strebel. Meanwhile, we also solve a problem posed by Reich and obtain the uniqueness theorem on related weight functions.

     

Quasiconvex functions and null Lagrangians in the stability problems of classes of mappings

We obtain stability theorems for classes of solutions to the differential equations constructed by means of quasiconvex functions and null Lagrangians.     

On the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings

Let X be a normed linear space and let S and T be multi-valued mappings of X into a family of closed, not necessarily compact subsets of X. In this paper some results on the convergence of the Ishikawa iterates associated with a pair S, T which satisfy the condition (8) below, are obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the stability of the solution mapping for parametric traffic network problems

In this paper, we introduce the parametric traffic network problems. Afterward, a key hypothesis is introduced by virtue of a parametric gap function to considered problems, and we prove that this hypothesis is not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of the solution mapping for parametric traffic network problems.     

An optimization result for set-valued mappings and a stability property in vector problems with constraints

First, we prove an existence result relative to minimal points of set-valued mappings. Then, conditions about the upper and lower semicontinuity of constraint sets defined through set-valued mappings are given. Finally, a stability result relative to vector problems with abstract constraints is proved.The author thanks the referee for helpful comments on the first version of this paper.     

Existence and stability of periodic solutions for parabolic systems with time delays

This paper is concerned with the existence and stability time-periodic solutions for a class of coupled parabolic equations with time delay, and time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. Our approach to the problem is by the method of upper and lower solution and using Schauder fixed point theorem. Some methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system. Finally, a model arising from chemistry is used to illustrate the obtained results.     

Boundary behaviour of the mappings satisfying generalized inverse modular inequalities

We study the geometric properties of the mappings for which generalized inverse modular inequalities hold. We generalize in this way known theorems from the theory of analytic mappings and the theory of quasiregular mappings, like the theorems of Fatou, M. and F. Riesz, Beurling and Lindelöf and their extensions given for quasiregular mappings by Martio, Rickman and Vuorinen.