Equations In Metric Spaces Research Articles

Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors

In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior need to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pullback attractors, as well as forward limit sets persist and that the latter two notions even converge under perturbation.As concrete application, we study integrodifference equations over the continuous functions under spatial discretization of collocation type. Integrodifference equation and Pullback attractor and Forward attractor and Urysohn operator

Computing of Existence and Uniqueness Solutions for Differential Equations in Metric Spaces by Using MATLAB

A metric space is a set along with a measurement on the set, A metric actuates topological properties like open and shut sets, which lead to the investigation of more theoretical topological spaces. It also has many applications in functional analysis. The aim of this work is design and develop highly efficient algorithms that provide the existence of unique solutions to the differential equation in metric spaces using MATLAB. The quality algorithm was used and developed to solve the differential equation in metric spaces. For accurate results. The proposed model contributed to providing an integrated computer solution for all stages of the solution starting from the stage of solving differential equations in metric space and the stage of displaying and representing the results graphically in the MATLAB program

Comparison Method for Studying Equations in Metric Spaces

We consider the equation $$G(x,x)=y$$ , where $$G\colon X\times X\to Y$$ and $$X$$ and $$Y$$ are metric spaces. This operator equation is compared with the “model” equation $$g(t,t)=0$$ , where the function $$g\colon \mathbb{R}_+\times \mathbb{R}_+ \to\mathbb{R}$$ is continuous, nondecreasing in the first argument, and nonincreasing in the second argument. Conditions are obtained under which the existence of solutions of this operator equation follows from the solvability of the “model” equation. Conditions for the stability of the solutions under small variations in the mapping $$G$$ are established. The statements proved in the present paper extend the Kantorovich fixed-point theorem for differentiable mappings of Banach spaces, as well as its generalizations to coincidence points of mappings of metric spaces.

Set-valued Leader type contractions, periodic point and endpoint theorems, quasi-triangular spaces, Bellman and Volterra equations

Set-valued contractions of Leader type in quasi-triangular spaces are constructed, conditions guaranteeing the existence of nonempty sets of periodic points, fixed points and endpoints of such contractions are established, convergence of dynamic processes of these contractions are studied, uniqueness properties are derived, and single-valued cases are considered. Investigated dynamic systems are not necessarily continuous and spaces are not necessarily sequentially complete or Hausdorff. Obtained results suggest, in particular, strategies to new studies of functional Bellman equations and variable discounted Bellman equations in metric spaces and integral Volterra equations in locally convex spaces. Results in this direction are also presented in this paper. More precisely, without continuity of Bellman and Volterra appropriate operators, the sets of solutions of these equations (which are periodic points of these operators) are studied and new and general convergence, existence and uniqueness theorems concerning such equations are proved.

Comparison Method for Studying Equations in Metric Spaces

Рассматривается уравнение $G(x,x)=y$, где $G\colon X\times X\to Y$, $X$, $Y$ - метрические пространства. Это операторное уравнение сравнивается с "модельным" уравнением $g(t,t)=0$, где функция $g\colon \mathbb{R}_+ \times \mathbb{R}_+\to\mathbb{R}$ непрерывна, не убывает по первому аргументу и не возрастает по второму аргументу. Получены условия, при которых из разрешимости "модельного" уравнения следует существование решений рассматриваемого операторного уравнения. Получены условия устойчивости решений к малым изменениям отображения $G$. Доказанные утверждения распространяют на уравнения в метрических пространствах теорему Канторовича о неподвижной точке дифференцируемого отображения, действующего в банаховом пространстве, а также ее обобщения на точки совпадения отображений метрических пространств. Библиография: 13 названий.

Almost Periodic and Poisson Stable Solutions of Difference Equations in Metric Spaces

We introduce a new class of almost periodic operators and establish the conditions of existence of almost periodic and Poisson stable solutions of difference equations in metric spaces that can be not almost periodic in Bochner’s sense.

Periodic and Almost Periodic Solutions of Difference Equations in Metric Spaces

We establish conditions for the existence of almost periodic and periodic solutions of almost periodic difference equations with discrete argument in a metric space without using the -classes of these equations.

Perturbations of vectorial coverings and systems of equations in metric spaces

E. R. Avakov, A. V. Arutyunov, S. E. Zhukovskiĭ, and E. S. Zhukovskiĭ studied the problem of Lipschitz perturbations of conditional coverings of metric spaces. Here we propose some extension of the concept of conditional covering to vector-valued mappings; i.e., the mappings acting in products of metric spaces. The idea is that, to describe a mapping, we replace the covering constant by the matrix of covering coefficients of the components of the vector-valued mapping with respect to the corresponding arguments. We obtain a statement on the preservation of the property of conditional and unconditional vectorial coverings under Lipschitz perturbations; the main assumption is that the spectral radius of the product of the covering matrix and the Lipschitz matrix is less than one. In the scalar case this assumption is equivalent to the traditional requirement that the covering constant be greater than the Lipschitz constant. The statement can be used to study various simultaneous equations. As applications we consider: some statements on the solvability of simultaneous operator equations of a particular form arising in the problems on n-fold coincidence points and n-fold fixed points; as well as some conditions for the existence of periodic solutions to a concrete implicit difference equation.

Metric viscosity solutions of Hamilton–Jacobi equations depending on local slopes

We continue the study of viscosity solutions of Hamilton–Jacobi equations in metric spaces initiated in [37]. We present a more complete account of the theory of metric viscosity solutions based on local slopes. Several comparison and existence results are proved and the main techniques for such metric viscosity solutions are illustrated.

Approximation theorems in metric spaces and functionals strictly subordinated to convergent series

Some new results are presented concerning the search and approximation of solutions of equations in metric spaces using functionals strictly subordinated to a convergent series and functionals compatible with a convergent series. These classes of functionals were earlier introduced by the author.

Eikonal equations in metric spaces

A new notion of a viscosity solution for Eikonal equations in a general metric space is introduced. A comparison principle is established. The existence of a unique solution is shown by constructing a value function of the corresponding optimal control theory. The theory applies to infinite dimensional setting as well as topological networks, surfaces with singularities.

On a class of first order Hamilton–Jacobi equations in metric spaces

We establish well-posedness of a class of first order Hamilton–Jacobi equation in geodesic metric spaces. The result is then applied to solve a Hamilton–Jacobi equation in the Wasserstein space of probability measures, which arises from the variational formulation of a compressible Euler equation.

Optimal transport and large number of particles

We present an approach for proving uniqueness of ODEs in the Wasserstein space. We give an overview of basic tools needed to deal with Hamiltonian ODE in the Wasserstein space and show various continuity results for value functions. We discuss a concept of viscosity solutions of Hamilton-Jacobi equations in metric spaces and in some cases relate it to viscosity solutions in the sense of differentials in the Wasserstein space.

STRUCTURED POPULATION EQUATIONS IN METRIC SPACES

In this paper, a framework for the analysis of measure-valued solutions of the nonlinear structured population model is presented. Existence and Lipschitz dependence of the solutions on the model parameters and initial data are shown by proving convergence of a variational approximation scheme, defined in the terms of a suitable metric space. The estimates for a corresponding linear model are used based on the duality formula for transport equations. An extension of a Wasserstein metric to the measures with integrable first moment is proposed to cope with the nonconservative character of the model. This metric is compared with a bounded Lipschitz distance, also called a flat metric, and the results are discussed in the context of applications to biological data.

Global attractors for gradient flows in metric spaces

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope of Ambrosio et al. (2005) [5], we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach (De Giorgi, 1993; Ambrosio, 1995 [3,15]). For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced in Ball (1997) [7]. The notions of generalized and energy solutions are quite flexible, and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces, to Wasserstein spaces of probability measures. We present applications of our abstract results, by proving the existence of the global attractor for the energy solutions, both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.

Differential equations in metric spaces with applications

This paper proves the local well posedness of differential equations in metric spaces under assumptions that allow to comprise several different applications. We consider below a system of balance laws with a dissipative non local source, the Hille-Yosida Theorem, a generalization of a recent result on nonlinear operator splitting, an extension of Trotter formula for linear semigroups and the heat equation.

Hyperbolic balance laws with a dissipative non local source

This paper considers systems of balance law with a dissipative non local source. A global in time well posedness result is obtained. Estimates on the dependence of solutions from the flow and from the source term are also provided. The technique relies on a recent result on quasidifferential equations in metric spaces.

Impulsive set differential equations with delay

The basic theory of set impulsive set differential equations with delay is initiated by combining the theories of impulsive differential equations, set differential equations in metric spaces and delay differential equations.

Differential equations in metric spaces

Differential equations in metric spaces

Partial averaging of quasidifferential equations in metric spaces

We justify partial averaging schemes for quasidifferential equations that generalize the corresponding results for differential equations and inclusions.