Global comparison of finite-dimensional reduction schemes in smooth variational problems

A new criterion for global smooth equivalence of a pair of key functions corresponding to a smooth functional in the calculus of variations for a given pair of finite-dimensional reduction schemes is established. The statement is presented in abstract form (we consider a functional on a Banach space with a Fredholm gradient). The main condition is the possibility to deform the reduction schemes into each other preserving the coercivity of the key functions. As a corollary, we obtain the theorem concerning global smooth equivalence of the key functions calculated by the Lyapunov-Schmidt and Morse-Bott reduction schemes in the two-point boundary value problem for a natural mechanical system of sufficiently general form. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 745–754, May, 2000.     

Comparison estimates in anisotropic variational problems

A pointwise inequality between the radially decreasing symmetrals of minimizers of (possibly) anisotropic variational problems and the minimizers of suitably symmetrized problems is established. As a consequence, a priori sharp estimates for norms of the relevant minimizers are derived.     

Extremal problems in finite-dimensional spaces

A theory of the first variation in the finite-dimensional case is presented using the concepts of essential covering.     

A class of differential fuzzy variational inequalities in finite-dimensional spaces

In this paper, a new class of differential fuzzy variational inequalities in finite dimensional spaces are introduced and studied. An existence theorem of the Carathéodory weak solution of differential fuzzy variational inequality is established under some suitable conditions. The convergence analysis for time-stepping method for finding the weak solution of the differential fuzzy variational inequality is given. Finally, an example is reported to verify the validity of the proposed algorithm.     

The studies of systems of variational inclusions problems and variational disclusions problems with applications

In this paper, we study existence theorems of solutions for systems of variational inclusions problems and systems of variational disclusions problems. From these existence results, we establish existence theorems of solutions for systems of generalized vector quasiequilibrium problems and systems of quasioptimization problems.     

Multiobjective variational problems with time delay

In this paper, the study of multiobjective variational problems with time delay is introduced. We provide methods for identifying Pareto optimal solutions. We also prove necessary and sufficient optimality conditions for the isoperimetric problem with multiple constraints and delayed arguments.     

Viscosity iterative algorithm for variational inequality problems and fixed point problems of strict pseudo-contractions in uniformly smooth Banach spaces

The purpose of this paper is to study a new viscosity iterative algorithm based on a generalized contraction for finding a common element of the set of solutions of a general variational inequality problem for finite inversely strongly accretive mappings and the set of common fixed points for a countable family of strict pseudo-contractions in uniformly smooth Banach spaces. We prove some strong convergence theorems under some suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.     

On necessary extremum conditions for finite-dimensional problems with inequality constraints

The finite-dimensional optimization problem with equality and inequality constraints is examined. The case where the classical regularity condition is violated is analyzed. Necessary second-order extremum conditions are obtained that are stronger versions of some available results.     

On finite-dimensional approximation of solutions of ill-posed problems

We show that the modified method for finite-dimensional approximation of solutions of Fredholm integral equations of the first kind presented in this paper is more economical than traditional methods for finite-dimensional approximation. Institute of Mathematies, Ukrainian Academy of Science, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 289–295, February, 1997.     

Strong convergence theorems for variational inequality problems and fixed point problems in uniformly smooth and uniformly convex Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings and the set of common fixed points for a nonexpansive mapping in a uniformly smooth and uniformly convex Banach space. We obtain a strong convergence theorem under some suitable conditions. Our results improve and extend the recent ones announced by many others in the literature.     

Convergence rates and finite-dimensional approximation for a class of ill-posed variational inequalities

The purpose of this paper is to investigate an operator version of Tikhonov regularization for a class of ill-posed variational inequalities under arbitrary perturbation operators. Aspects of convergence rate and finite-dimensional approximations are considered. An example in the theory of generalized eigenvectors is given for illustration. Institute of Information Technology, Vietnam. Published in Ukrainskii Matematicheskii Zhunal, Vol. 49, No. 5, pp. 629–637, May, 1997.     

Two dimensional variational problems with linear growth

 Suppose that f: ℝ ^nN →ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|²) if N>1. If f satisfies an ellipticity condition of the form