Geometrical properties of nonlinear maps and their application, Part II: Nonlinear Riemann-Hilbert problems with closed boundary data for multiply connected domains

We use the degree of quasiruled Fredholm maps on quasicylindrical domains developed in Part I to prove the existence of at least two homotopically different solutions of classical nonlinear Riemann-Hilbert problems for multiply connected domains.     

A continuation principle for Fredholm maps I: theory and basics

We prove an abstract and flexible continuation theorem for zeros of parametrized Fredholm maps between Banach spaces. It guarantees not only the existence of zeros to corresponding equations, but also that they form a continuum for parameters from a connected manifold. Our basic tools are transfer maps and a specific topological degree. The main result is tailor-made to solve boundary value problems over infinite time-intervals and for the (global) continuation of homoclinic solutions.     

Computation of topological degree for nonlinear operators and applications

Using the lattice structure, we give some methods of computation of the topological degree for the operator which is quasi-additive on lattice. The operator A$A$ is not assumed to be a cone mapping. As applications, the existence of nontrivial solutions for the nonlinear Sturm–Liouville problem is discussed.     

On the total variation, topological entropy and sensitivity for interval maps

A concept related to total variation termed H₁ condition was recently proposed to characterize the chaotic behavior of an interval map f by Chen, Huang and Huang [G. Chen, T. Huang, Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos 14 (2004) 2161-2186]. In this paper, we establish connections between H₁ condition, sensitivity and topological entropy for interval maps. First, we introduce a notion of restrictiveness of a piecewise-monotone continuous interval map. We then prove that H₁ condition of a piecewise-monotone continuous map implies the non-restrictiveness of the map. In addition, we also show that either H₁ condition or sensitivity then gives the positivity of the topological entropy of f.     

Proving the existence of zeros using the topological degree and interval arithmetic

The invariance of the topological degree under certain homotopies is used to computationally prove the existence of zeros of nonlinear mappings in Rⁿ${\mathbb{R}}^n$. These existence tests use interval arithmetic to enclose the range of a function over a box. We show that our test is more general than a well-known test based on Miranda's theorem, and we show by a numerical example that the new test can be successful on substantially larger boxes.     

The degree of maps between certain 6-manifolds

For manifolds M,M of the form S² e⁴ e⁶ we compute the homomorphisms H_*M H_*M between homology groups which are realizable by a map F: M M.     

Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations

Using the cone theory and the lattice structure, we establish some methods of computation of the topological degree for the nonlinear operators which are not assumed to be cone mappings. As applications, the existence results of nontrivial solutions for superlinear Sturm-Liouville problems are given. We also investigate the singular superlinear Sturm-Liouville problems.     

On bilevel programming,Part I: General nonlinear cases

This paper is concerned with general nonlinear nonconvex bilevel programming problems (BLPP). We derive necessary and sufficient conditions at a local solution and investigate the stability and sensitivity analysis at a local solution in the BLPP. We then explore an approach in which a bundle method is used in the upper-level problem with subgradient information from the lower-level problem. Two algorithms are proposed to solve the general nonlinear BLPP and are shown to converge to regular points of the BLPP under appropriate conditions. The theoretical analysis conducted in this paper seems to indicate that a sensitivity-based approach is rather promising for solving general nonlinear BLPP.This research is sponsored by the Office of Naval Research under contract N00014-89-J-1537.     

A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

0-Epi families of mappings,topological degree,and optimization

Using the concept of 0-epi mapping defined by Furi, Martelli, and Vignoli, we define the concept of 0-epi family of quasiconvex mappings. We prove that the theory of 0-epi families is a good substitute of the degree theory in the localization of Nash equilibrium points, because it is more refined and very simple.     

On some third order nonlinear boundary value problems: Existence, location and multiplicity results

We prove an Ambrosetti-Prodi type result for the third order fully nonlinear equation

u^?(t)+f(t,u(t),u^′(t),u^″(t))=sp(t)     

Asymptotic bifurcation points, and global bifurcation of nonlinear operators and its applications

Using the cone theory and lattice structure, we discuss the existence of asymptotic bifurcation points and the global bifurcation of nonlinear operators which are not assumed to be cone mappings and may not be Frechet differentiable at points at infinity. As an application, the structure of the set of solutions of the superlinear Sturm-Liouville problems is investigated.     

Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes

We introduce certain classes of random point fields, including fermion and boson point processes, which are associated with Fredholm determinants of certain integral operators and study some of their basic properties: limit theorems, correlation functions, Palm measures etc. Also we propose a conjecture on an α-analogue of the determinant and permanent.     

Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators

In this paper, we prove that, if the product A=A₁?An is a Fredholm operator where the ascent and descent of A are finite, then Aj is a Fredholm operator of index zero for all j, 1?j?n, where A₁,…,An be a symmetric family of bounded operators. Next, we investigate a useful stability result for the Rako?evi?/Schmoeger essential spectra. Moreover, we show that some components of the Fredholm domains of bounded linear operators on a Banach space remain invariant under additive perturbations belonging to broad classes of operators A such as γ(Am)<1 where γ(⋅) is a measure of noncompactness. We also discuss the impact of these results on the behavior of the Rako?evi?/Schmoeger essential spectra. Further, we apply these latter results to investigate the Rako?evi?/Schmoeger essential spectra for singular neutron transport equations in bounded geometries.     

Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case

We consider the system of Fredholm integral equations