Equivariant mappings: a new approach in stochastic simulations

Stochastic simulations on manifolds usually are traced back to ⁿ via charts. If a group G is acting on a manifold M and if the respective distribution v is invariant under this group action then in many cases of practical interest there exists a more convenient approach which uses equivariant mappings. The concept of equivariant mappings will be discussed intensively at the instance of the Grassman manifold in which case G equals the orthogonal group. Further advantages of this concept will be demonstrated by applying it to a probabilistic problem from the field of combinatorial geometry.     

A new approach to relatively nonexpansive mappings

In this paper we study the nonexpansivity of the so-called relatively nonexpansive mappings. A relatively nonexpansive mapping with respect to a pair of subsets of a Banach space is a mapping defined from into such that for and . These mappings were recently considered in a paper by Eldred et al. (Proximinal normal structure and relatively nonexpansive mappings, Studia Math. 171 (3) (2005), 283-293) to obtain a generalization of Kirk's Fixed Point Theorem. In this work we show that, for certain proximinal pairs , there exists a natural semimetric for which any relatively nonexpansive mapping with respect to is nonexpansive. This fact will be used to improve one of the two main results from the aforementioned paper by Eldred et al. At that time we will also obtain several consequences regarding the strong continuity properties of relatively nonexpansive mappings and the relation between the two main results from the same work.

     

On mappings covering at a nonlinear rate and their perturbation stability

The present paper contains a study of covering (alias, openness) properties at a nonlinear rate for set-valued mappings between metric spaces. Such study is focussed on the stability of these properties in the presence of perturbations. A crucial result valid for linear openness, known as Milyutin’s theorem, is extended to set-valued mappings covering at a nonlinear rate under possibly non-Lipschitz perturbations. Consequently, a Lyusternik type theorem is derived from such extension and a general penalization principle for constrained optimization problems, which exploits nonlinear covering properties, is presented.     

A new method for a class of nonlinear variational inequalities with fuzzy mappings

In this paper, we construct a new iterative algorithm of solution for a new class of nonlinear variational inequalities with fuzzy mappings and give some convergence analysis of iterative sequences generated by algorithm.     

On local convexity of nonlinear mappings between Banach spaces

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ? < c the image f(B _?(x)) of each ?-ball B _?(x) ? U is convex. We give a lower bound on c via the second order Lipschitz constant Lip₂(f), the Lipschitz-open constant Lip_o(f) of f, and the 2-convexity number conv₂(X) of the Banach space X.     

On improvement of summability properties of solutions of some degenerate nonlinear fourth-order equations

We consider a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. The right-hand side F?(x,?u) of the equations depends on the unknown function u. Under some conditions, including L^m -summability of F(·, 0) with m close to 1 and restrictions on the exponent σ of the growth of F?(x,?u) with respect to u, we establish existence of W-solutions of the Dirichlet problem for the given equations and describe the dependence of summability properties of these solutions on m and σ.     

On summability of subsequences

Ohne ZusammenfassungDie Untersuchungen dieser Arbeit wurden zum Teil am Courant Institute of Mathematical Sciences, New York University, durchgeführt und dabei von der National Science Foundation, Grant NSF-GP-8724, unterstützt.     

Optimal summability of solutions to nonlinear elliptic problems

The strongest possible norms in a priori estimates for solutions to a class of second-order nonlinear elliptic boundary value problems are exhibited.     

A new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings

In this paper, we introduce and study a new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings. An existence theorem of solutions is proved without compactness assumptions. A new iterative algorithm is suggested and analysed. The convergence of iterative sequences generated by the new algorithm are also given. As special cases, some known results are also discussed.     

On a new class of nonlinear wave equations

In this paper we prove the existence and uniqueness of regular solutions for the Cauchy problem for the evolution equation $\text{u″ + A}{}^2\text{u + (α + M(¦A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u¦}{}^2\text{) Au = 0}$, suggested by the study of beams and plates. We represent by A a linear operator of a Hilbert space H with norm ∥, α is a real number, and M(λ) > 0 a real function, for λ ? 0.     

On parallel summability of matrices

It is shown that the generalized inverses characterize the parallel sum. The almost positive definite (a.p.d.) matrices introduced by Duffin and Morley [2] are of two types, whose intersection is the class of quasi-positive-definite matrices (Mitra and Puri [7]). The a.p.d. matrices of any one type form a “saturated” subclass of pairwise parallel summable a.p.d. matrices.