Viscosity solutions of Eikonal equations on topological networks

In this paper we introduce a notion of viscosity solutions for Eikonal equations defined on topological networks. Existence of a solution for the Dirichlet problem is obtained via representation formulas involving a distance function associated to the Hamiltonian. A comparison theorem based on Ishii’s classical argument yields the uniqueness of the solution.     

A topological classification of linear differential equations on Banach spaces

In the present work, we give necessary and sufficient conditions for linear differential non-autonomous equations on a Banach space, that satisfy exponential dichotomy, to be topologically equivalent. We shall also prove that such equations are structurally stable.     

On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces

The main purpose of this paper is to deal with almost automorphic and asymptotically almost automorphic solutions of the initial value problem as well as the nonlinear Volterra integral equation in Banach spaces. We obtain a collection of existence results of such solutions to these equations. We investigate also a topological structure of such solution sets. Moreover, we prove Aronszajn-type theorems for solutions of the initial value problem as well as the nonlinear Volterra integral equation, defined on the whole real line.     

Periodic solutions of differential equations in Banach spaces

Let X be a Banach space, DX, f: [0,)xDX continuous and -periodic. In this paper we consider various conditions on D and f sufficient for existence of an -periodic solution of the differential equation u=f(t,u). In the main, we shall assume that D is closed bounded and convex and f satisfies a boundary condition at D such that D is flow invariant for u=f(t,u). The map f is assumed to be either compact or dissipative or a certain perturbation of such maps.     

On a topological description of solutions of complex differential equations

In this article we give a topological approach to the global behavior of arbitrary single-valued solutions in a simple connected domain of some general classes of complex differential equations with multi-valued coefficients. In particular, this permits us to describe certain globally multi-valued solutions as well as algebraic and algebroid solutions.     

Strong solutions for differential equations in abstract spaces

Let (E,F) be a locally convex space. We denote the bounded elements of E by . In this paper, we prove that if B_Eb is relatively compact with respect to the F topology and f:I×E_b→E_b is a measurable family of F-continuous maps then for each x₀∈E_b there exists a norm-differentiable, (i.e. differentiable with respect to the ∥·∥_F norm) local solution to the initial valued problem u_t(t)=f(t,u(t)), u(t₀)=x₀. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator.     

Recursive topological spaces

The research was supported by the Russian Foundation for Fundamental Research (Grant 093-01-01506).     

Supercomplete topological spaces

Supercomplete topological spaces and other variants of supercompleteness are defined in this paper. The main idea is to give different characterizations of Čech complete spaces. In particular it is shown that for paracompacta (and so for metrizable spaces) and topological groups the two notions of supercompleteness and Čech completeness coincide.      

Almost topological spaces

Summary Convergence spaces are studied from a geometrical point of view. In many cases we may construct a topological space VirtX, which can be considered as the resolution of the topological singularities ofX.

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Riassunto Spazi di convergenza vengono studiati da un punto di vista geometrico. In molti casi possiamo costruire uno spazio topologico VirtX che può essere considerato come la risoluzione delle singolarità diX.

     

Approximation by solutions of elliptic equations in semilocal spaces

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by solutions of the same equations in a larger domain. We construct an abstract framework which allows us to deal with such density questions, simultaneously for various norms. More specifically, we study approximations with respect to the norms of semilocal Banach spaces of distributions. These spaces are required to satisfy certain postulates. We establish density results for elliptic operators with constant coefficients which unify and extend previous results. In our density results Ω may possess holes and it is required to satisfy the segment condition. We observe that analogous density results do not hold in spaces where the infinitely smooth functions are not dense. Finally, we provide applications related to the method of fundamental solutions.     

Approximating solutions of Hammerstein type equations in Banach spaces

Abstract

Let X be a real Banach space and X^? be its dual. Let F: X → X^? and K: X^? → X be Lipschitz monotone mappings. In this paper an explicit iterative scheme is constructed for approximating solutions of the Hammerstein type equation, 0 = u + KF u, when they exist. Strong convergence of the scheme is obtained under appropriate conditions. Our results improve and unify many of the results in the literature.     

Periodic solutions of integro-differential equations in vector-valued function spaces

Operator-valued Fourier multipliers are used to study well-posedness of integro-differential equations in Banach spaces. Both strong and mild periodic solutions are considered. Strong well-posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems. We are concerned with a large array of vector-valued function spaces: Lebesgue-Bochner spaces Lp, the Besov spaces (and related spaces such as the Hölder-Zygmund spaces Cs) and the Triebel-Lizorkin spaces . We note that the multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary Banach spaces. The results are applied to various classes of nonlinear integral and integro-differential equations.     

Two topological properties of topological linear spaces

A topological and a geometrical-topological property, previously known only for normed linear spaces, are established here for much more general classes of topological linear spaces. This research was conducted at the University of Washington in 1963 when the first author was visiting there. The work of both authors was supported in part by the National Science Foundation, U. S. A. (NSF-GP-378).     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

Linear functional equations and their solutions in generalized Orlicz spaces

Aequationes mathematicae - Assume that $$\Omega \subset \mathbb {R}^k$$ is an open set, V is a real separable Banach space and $$f_1,\ldots ,f_N :\Omega \rightarrow \Omega $$ , $$g_1,\ldots ,...