On the existence of variational principles for differential-difference evolution equations

The problem of existence of variational principles for wide classes of generally nonlinear differential-difference equations with nonpotential operators is investigated.     

Nondifferentiable variational principles in terms of a quantum operator

We develop Cresson's nondifferentiable calculus of variations on the space of Hölder functions. Several quantum variational problems are considered: with and without constraints, with one and more than one independent variable, of first and higher‐order type. Copyright © 2011 John Wiley & Sons, Ltd.     

On the solvability of an evolution variational inequality with a nonlocal space operator

By using semidiscretization and penalty methods, we prove the existence of a generalized solution of a variational inequality of parabolic type with a space operator monotone in the gradient and depending on an integral characteristic of the solution with respect to the space variables.     

Representation theorems and variational principles for self-adjoint operator matrices

We use the notion of triples D⁺ → H → D^? of Hilbert spaces to develop an analog of the Friedrichs extension procedure for a class of nonsemibounded operator matrices. In addition, we suggest a general approach (stated in the same terms) to the construction of variational principles for the eigenvalues of such matrices.     

Composition duality principles for evolution mixed variational inclusions

Composition duality principles for the qualitative and finite element analysis of primal and dual evolution mixed variational inclusions in reflexive Banach spaces are presented. A general distributed control diffusion problem illustrates the theory.     

On the existence of a variational principle for an operator equation with second derivative with respect to “ time”

Using methods of nonlinear functional analysis, we define the structure of an evolution operator equation of second order that can be formulated in direct variational terms.     

On the variational principles of mechanics

Variational principles, generalizing the classical d’Alembert–Lagrange, Hölder, and Hamilton–Ostrogradskii principles, are established. After the addition of anisotropic dissipative forces and taking the limit, when the coefficient of viscous friction tends to infinity, these variational principles transform into the classical principles, which describe the motion of systems with constraints. New variational relations are established for searching for the periodic trajectories of the reversible equations of dynamics.     

Variational rationality,variational principles and the existence of traps in a changing environment

Journal of Global Optimization - This paper has two aspects. Mathematically, in the context of global optimization, it provides the existence of an optimum of a perturbed optimization problem that...     

On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

On the existence of a solution to a class of variational inequalities

In this note we consider a class of semilinear elliptic variational inequalities on H ¹(Ω) space. With the aid of the mountain-pass principle and the Ekeland variational principle we prove the existence of solutions.     

On the solution existence of pseudomonotone variational inequalities

As shown by Thanh Hao [Acta Math. Vietnam 31, 283–289, 2006], the solution existence results established by Facchinei and Pang [Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I (Springer, Berlin, 2003) Prop. 2.2.3 and Theorem 2.3.4] for variational inequalities (VIs) in general and for pseudomonotone VIs in particular, are very useful for studying the range of applicability of the Tikhonov regularization method. This paper proposes some extensions of these results of Facchinei and Pang to the case of generalized variational inequalities (GVI) and of variational inequalities in infinite-dimensional reflexive Banach spaces. Various examples are given to analyze in detail the obtained results. B. T. Kien: On leave from Hanoi University of Civil Engineering. The online version of the original article can be found at .     

On variational principles for linear initial value problems

Euler-Lagrange and Euler-Hamilton variational principles are presented for a class of linear initial value problems.     

On some existence theorems for the solutions of a class of nonlinear operator equations

Theorems of M. A. Krasnosel'skii are generalized which concern fixed points of completely continuous operators which compress or expand a cone, and a topological principle for fixed points in the case of P-compact operators.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 229–237, February, 1970.The author expresses warm thanks to his advisor I. A. Bakhtin for constant help in this work.     

On the existence of solutions of operator differential equations

We consider nonlinear equations of parabolic type in reflexive Banach spaces. We present sufficient conditions for the existence of solutions of these equations. We use methods for the investigation of problems with operators of pseudomonotone (on a subspace) type. In addition, a sufficient criterion in the Sobolev space L _p(0, T; W_p¹()L₂ (0, T; L₂()) is considered for the case where an operator introduced with the use of functional coefficients belongs to a given class. We also show that it is possible to weaken the classical condition of coerciveness.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 837–850, June, 2004.