Infinite System Of Differential Equations Research Articles

α-, β- and γ-duals of the sequence spaces formed by a regular matrix of Tetranacci numbers

Abstract The primary objective of this paper is to create a novel infinite Toeplitz matrix by leveraging Tetranacci numbers. This matrix serves as the foundation for defining new sequence spaces denoted as c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} , where 1 ≤ p < ∞ {1\leq p<\infty} . By utilizing this newly constructed matrix, the paper also explores and examines various algebraic and topological properties inherent to the sequence spaces c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} for values of p within the range of 1 ≤ p < ∞ {1\leq p<\infty} . At last, we also prove existence theorem with example for infinite systems of differential equations in ℓ p ⁢ ( G ) {\ell_{p}(G)} .

Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in ℓ2

We investigate a differential evasion game with multiple pursuers and an evader for the infinite systems of differential equations in ℓ2. The control functions of the players are subject to geometric constraints. The pursuers’ goal is to bring the state of at least one of the controlled systems to the origin of ℓ2, while the evader’s goal is to prevent this from happening in a finite interval of time. We derive a sufficient condition for evasion from any initial state and construct an evasion strategy for the evader.

Secular Terms for the Kinetic Mckean Model

In this article, we investigate the kinetic McKean model. The perturbed solution of the Cauchy problem is sought in the form of Fourier series. The Fourier coefficients for the zero and nonzero modes are written out, respectively. The original system is reduced to an infinite system of differential equations. An approximation for the systems is constructed. Under certain assumptions, we find secular terms (non-integrable part). This, in turn, will allow us to prove for the first time the exponential stabilization of the solution in the future.

Consistency of an Infinite system of third order three-point boundary value problem in the $bv_0$ space by the theory of measure of noncompactness

Several authors have examined the solvability conditions for an infinite system of differential equations in different Banach spaces using the concept of measure of noncompactness. In all these studies, they have considered differential equations where the boundary conditions are defined on two points. In this paper, we have studied the solvability conditions for an infinite system of third-order three point boundary value problem in the sequence space of bounded variation $bv_0$ with the help of the theory of measure of noncompactness and have given a suitable example to illustrate the result.

The solution of an infinite system of ternary differential equations

The present paper is devoted to an infinite system of differential equations. This system consists of ternary differential equations corresponding to 3×3 Jordan blocks. The system is considered in the Hilbert space l2. A theorem about the existence and uniqueness of solution of the system is proved.

Depth first exploration of a configuration model

We introduce an algorithm that constructs a random graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that the renormalized contour process of the Depth First Search Tree has a deterministic limiting profile that we identify. The proof goes through a detailed analysis of the evolution of the empirical degree distribution of unexplored vertices. This evolution is driven by an infinite system of differential equations which has a unique and explicit solution. As a byproduct, we deduce the existence of a macroscopic simple path and get a lower bound on its length.

Meir-Keeler condensing operator to prove existence of solution for infinite systems of differential equations in the Banach space and numerical method to find the solution

In this paper, we establish the existence of solution for infinite systems of differential equations in the Banach sequence space n(?), ?p(1 ? p < ?) and c by using Meier-Keeler condensing operators. With the help of examples we illustrate our results in the sequence spaces. Also for validity of the results, we find an approximation of solution by using a suitable method with high accuracy.

A survey on measures of noncompactness with some applications in infinite systems of differential equations

A survey on measures of noncompactness with some applications in infinite systems of differential equations

Pursuit and Evasion Games for an Infinite System of Differential Equations

Pursuit and Evasion Games for an Infinite System of Differential Equations

Application of a fixed point theorem on infinite cartesian product to an infinite system of differential equations

"In the paper Operators on infinite dimensional cartesian product, (Analele Univ. Vest Timişoara, Mat. Inform., 48 (2010), 253–263), by I. A. Rus and M. A. Şerban, the authors give a generalization of the Fibre contraction theorem on infinite dimensional cartesian product. In this paper we give an application of this abstract result to an infinite system of differential equations. "

Construction of exact solutions of the problem of the motion of a fluid with a free boundary using infinite systems of differential equations

We consider the well-known hydrodynamics problem for the planar potential motion of an ideal incompressible fluid with a free boundary without capillarity and propose a method for finding exact solutions based on reducing the boundary-value problems to systems of ordinary differential equations. We obtain two examples of flows: a stationary motion of a heavy fluid over a smooth bottom and a nonstationary motion of a fluid that originally filled a wedge.

Application measure of noncompactness and operator type contraction for solvability of an infinite system of differential equations in lp-space

The aim of this paper is to obtain existence results for an infinite system of second order differential equations in the sequence space lp for 1 ? p < ? with the help of a technique associated with measures of noncompactness and contractive condition of operator type. We also provide some illustrative examples in support of our existence theorems.

Pursuit game problem of an infinite system of differential equations with geometric and integral constraints

The present paper considers a pursuit game problem of an infinite system of 1st-order differential equations in the space l2. In the game, the player’s control functions satisfy both geometric and integral constraints. The pursuer’s action is to force the state of the system to coincides with another state for a finite time and the evader’s action is to retrieve this. In each case, we obtain a condition for the completion of the game for which the pursuer’s strategy is explicitly constructed. In addition, we study a control problem.

Method of Multiobjective Optimization of Controlled Systems with Distributed Parameters

The constructive method for multicriteria optimization of control processes of deterministic and not fully defined controlled systems with distributed parameters, described by linear multidimensional parabolic partial differential equations with internal and boundary control actions, is proposed. The optimization problem is considered when an uniform approximation accuracy of object’s final state to the required spatial distribution of controlled function is given. The suggested approach is based on the one-criterion option in the form of minimax convolution of the normalized quality criteria and the subsequent transition to the equivalent form of a typical variational problem with constraints. It is applied to the deterministic model of an object described by an infinite system of differential equations with respect to time-dependent modes of the controlled quantity expansion in a series of eigenfunctions of the initial-boundary value problem. Further procedures for the preliminary parametrization of control actions, based on analytical optimum conditions and reduction to semi-infinite programming problems, allow one to find the desired extremals using their Chebyshev properties and fundamental laws of the domain in typical application conditions of estimating the accuracy of approaching the object’s final state to the required one in a uniform metric. The obtained results are extended to the tasks of program control on the principle of guaranteed result by ensembles of object trajectories under conditions of interval uncertainty of the parametric characteristics of the distributed system and multiple external disturbances. A demonstrated example of a multicriteria optimization of an innovation technology of metal induction heating prior a hot forming is of special interest. The typical optimization criteria such as energy consumption, metal loss due to scale formation, and heating accuracy are considered as components of vector optimization criterion.

On solutions of semilinear upper diagonal infinite systems of differential equations

The goal of the paper is to investigate the existence of solutions for semilinear upper diagonal infinite systems of differential equations. We will look for solutions of the mentioned infinite systems in a Banach tempered sequence space. In our considerations we utilize the technique associated with the Hausdorff measure of noncompactness and some existence results from the theory of ordinary differential equations in abstract Banach spaces.

Asymptotic Behavior of Eigen's Quasispecies Model.

We study Eigen's quasispecies model in the asymptotic regime where the length of the genotypes goes to [Formula: see text] and the mutation probability goes to 0. A limiting infinite system of differential equations is obtained. We prove convergence of trajectories, as well as convergence of the equilibrium solutions. We give analogous results for a discrete-time version of Eigen's model, which coincides with a model proposed by Moran.

Existence of solutions of infinite systems of differential equations of general order with boundary conditions in the spaces c0 and ℓ1 via the measure of noncompactness

The aim of this paper is to obtain existence results for an infinite system of differential equations of order n with boundary conditions in the Banach spaces c0 and ℓ1 with the help of a technique associated with measures of noncompactness. We also provide some illustrative examples in support of our existence theorems.

Existence of solutions for infinite systems of differential equations by densifiability techniques

A novel technique to state the existence of solutions for certain infinite systems of differential equations is proposed. Our main tool will be the so called degree of nondensifiability, which seems to work under more general conditions than the measures of noncompactness. In fact, in our main result, the required conditions proposed for the existence of solutions of such system are more general than others required in most of the results based on such measures.

Differential Game of Optimal Pursuit for an Infinite System of Differential Equations

We study an optimal pursuit differential game problem in the Hilbert space $$l_{r+1}^2$$ . The game is described by an infinite system of the first-order differential equations whose coefficients are negative. The control functions of players are subjected to integral constraints. If the state of the system coincides with the origin of the space $$l_{r+1}^2$$ , then game is considered completed. We obtain an equation to find the optimal pursuit time. Moreover, we construct the optimal strategies for players.

Asymptotics for Infinite Systems of Differential Equations

This paper investigates the asymptotic behaviour of solutions to certain infinite systems of ordinary differential equations. In particular, we use results from ergodic theory and the asymptotic theory of $C_0$-semigroups to obtain a characterisation, in terms of convergence of certain Ces\`aro averages, of those initial values which lead to convergent solutions. Moreover, we obtain estimates on the rate of convergence for solutions whose initial values satisfy a stronger ergodic condition. These results rely on a detailed spectral analysis of the operator describing the system, which is made possible by certain structural assumptions on the operator. The resulting class of systems is sufficiently broad to cover a number of important applications, including in particular both the so-called robot rendezvous problem and an important class of platoon systems arising in control theory. Our method leads to new results in both cases.