Sufficient optimality conditions and duality in vector optimization with invex-convexlike functions

We prove the Kuhn-Tucker sufficient optimality condition, the Wolfe duality, and a modified Mond-Weir duality for vector optimization problems involving various types of invex-convexlike functions. The class of such functins contains many known generalized convex functions. As applications, we demonstrate that, under invex-convexlikeness assumptions, the Pontryagin maximum principle is a sufficient optimality condition for cooperative differential games. The Wolfe duality is established for these games.The author is indebted to the referees and Professor W. Stadler for valuable remarks and comments, which have been used to revise considerably the paper.     

On the invertibility of the operator <Emphasis Type="Italic">d/dt</Emphasis> + <Emphasis Type="Italic">A</Emphasis> in certain functional spaces

We prove that the operator d/dt + A constructed on the basis of a sectorial operator A with spectrum in the right half-plane of ℂ is continuously invertible in the Sobolev spaces W _p ¹ (ℝ, D _α), α ≥ 0. Here, D _α is the domain of definition of the operator A ^α and the norm in D _α is the norm of the graph of A ^α. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1020–1025, August, 2007.     

Multistep numerical methods for functional-differential-algebraic equations

This paper deals with constructing multistep numerical methods for differential delay equations under additional algebraic constraints. Theorems on the orders of convergence of these methods are proved both for functional-differential equations with algebraic constraints and for singular functional-differential equations.     

Controllability of Functional Integro-Differential Inclusions with an Unbounded Delay

In this paper, a sufficient condition is established for the controllability of neutral functional integro-differential inclusions with an unbounded delay in Banach spaces. The approach used is a fixed-point theorem for condensing maps due to Martelli and the theory of analytic semigroup of linear operators. Communicated by F. Zirilli Research supported by NNSF of China, by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the Ministry of Education of China, and by the Qing Lan Talent Engineering Fund QL-05-164 of Lanzhou Jiaotong University. The authors are grateful to Professor F. Zirilli and two anonymous referees for valuable suggestions improving this paper.     

The double exponential map and covariant derivation

We study the double exponential map which is a composition of a special form of two exponential maps on a manifold with connection. We relate this map and the composition of covariant derivations as well as the composition of pseudodifferential operators on these manifolds.     

On quadratic differentials on multiply connected domains that are perfect squares

The paper considers the problem on extremal decomposition of a multiply connected domain G ⊂ ℂ in the case where the associated quadratic differential is a perfect square. It is shown that in the case considered, the value of the functional for this extremal decomposition is the least one in a certain class of decompositions. Bibliography: 10 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 113–133.     

Homogenization of the periodic magnetic Schrödinger operator with degenerate lower edge of the spectrum

Homogenization of the periodic magnetic Schrödinger operator with degenerate lower edge of the spectrum is considered. The standard problem of homogenization leads to a degenerate limiting operator. To obtain more precise information, one needs a certain modification of the problem considered. Bibliography: 10 titles.     

Asymptotic and boundary integral formulae for the computational reconstruction of small conductivity imperfections

In this paper we investigate the electrostatic problem of determining conductivity profiles from the knowledge of boundary currents and voltages. We obtain an improved estimate for the voltage potential of a two-dimensional conductor having finitely many circular inclusions and piecewise constant conductivity profile. We derive an asymptotic expansion for the voltage potential in terms of the reference voltage potential and the location, size, and conductivity of the inhomogeneities. This representation is used to formulate the nonlinear least squares problem for estimating the location and size of the inhomogeneities. Required boundary data for the voltage potential are generated numerically by solving a system of integral equations. Computational experiments are presented to demonstrate the effectiveness of our identification procedure.