MINIMAL SURFACES IN 3—DIMENSIONAL SOLVABLE LIE GROUPS

The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariant Riemannian metrics. A Weierstraβ type integral representation formula for minimal surfaces is obtained.     

Systems of Inclusions Involving Fredholm Operators of Nonnegative Index and Nonconvex-Valued Maps

We study the existence of solutions to a system of two inclusions involving Fredholm operators of nonnegative (Fredholm) index and the so-called c-admissible maps: upper semicontinuous maps with values being continuous images of cell-like sets. The presented approach resembles the substitution technique, i.e. applies the so-called solution map. The appearance of the ‘dimension defect’ reflecting the nonnegativity of Fredholm indices requires the use of algebraic tools based on the cohomotopy theory, different from the usual homological ones present in the fixed point theory. An application to boundary value problems is provided as well as the brief exposition of basic facts from the cohomotopy theory. Mathematics Subject Classifications (2000) 47H04, 47A53, 55M20, 55Q55, 34A60.     

Interior projection-like methods for monotone variational inequalities

We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.The work of this author was partially supported by the United States–Israel Binational Science Foundation, BSF Grant No. 2002-2010.     

Fixed-point theorems for S-KKM maps

A number of new fixed-point results are presented for S-KKM self maps. Also the notion of essentiality for S-KKM maps is introduced.     

On Weyl almost periodic selections of multivalued maps

We prove that Weyl almost periodic multivalued maps R∋t→F(t)∈clU have Weyl almost periodic selections, where clU is the collection of non-empty closed sets of a complete metric space U.     

Weak Density of Smooth Maps in W 1, 1(M,N) for Non-Abelian π1(N)

We prove that smooth maps are dense in the sense of biting convergence in W ^{1, 1}(M, N) when M and Nare compact Riemannian manifolds and N is closed.     

A Nonexistence Theorem of Proper Harmonic Morphisms Between Hyperbolic Spaces

We provide a nonexistence theorem of harmonic morphisms between hyperbolic spaces which are proper and C ²-smooth to the boundary.     

Randomization of Sharkovskii-type theorems

We formulate an abstract scheme for the randomization of Shar- kovskii-type theorems via transformation to the deterministic case. In particular, Sharkovskii-type theorems for scalar differential equations can be randomized in this way. A random version of the standard Sharkovskii theorem is presented explicitly. Many remarks, comments and illustrating examples are supplied.

     

Topological properties of the multifunction space <Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) of cusco maps

A set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F(x) is a nonempty, compact and connected subset of Y for each x ∈ X. We denote by L(X), the space of all subsets F of X × ℝ such that F is the graph of a cusco map from the space X to the real line ℝ. In this paper, we study topological properties of L(X) endowed with the Vietoris topology. The second author is supported by the SPM fellowship awarded by the Council of Scientific and Industrial Research, India.