Infinite System Of Differential Equations Research Articles

Differential equations in classical sequence spaces

Measures of noncompactness are very useful tools which are widely used in fixed point theory, differential equations, functional equations, integral and integro-differential equations, and optimization etc. In recent years measures of noncompactness have also been used in defining geometric properties of Banach spaces as well as in characterizing compact operators between sequence spaces. In this survey article, we present a study on applications of measures of noncompactness to infinite system of differential equations of first and second order and fractional differential equations in some classical sequence spaces.

Nonlinear Responses of Rectangular Magnetoelectroelastic Plates with Transverse Shear Deformation

With employing the transverse shear deformation theory and von Karman plate theory, the nonlinear static behavior of a simply supported rectangular magnetoelectroelastic plates is investigated. According to the Maxwell’s equations, when applying the magnetoelectric load on the plate’s surfaces and neglecting the in-plane electric and magnetic fields in thin plates, the electric and magnetic potentials varying along the thickness direction of the magnetoelectroelastic plates are determined. The nonlinear differential equations for magnetoelectroelastic plates are established based on the Hamilton’s principle. The Galerkin procedure furnishes an infinite system of differential equations into algebraic equations. In the numerical calculations, the effects of the nonlinearity and span-thickness ratio on the nonlinear load-deflection curves and electric/magnetic potentials for magnetoelectroelastic plates are discussed.

Solving analysis and synthesis problems for a spatially two-dimensional distributed object represented with an infinite system of differential equations

We prove theorems that define an algorithm for passing from differential equations with partial derivatives with respect to two spatial variables and time to an infinite-dimensional system of ordinary differential equations in Cauchy form. We study the convergence of resulting solutions and show that it is possible to pass from an infinite system in Cauchy form to a finite one, which opens up the possibilities to use state space methods for controller design in distributed systems. Based on the quadratic quality criterion, we design a controller for the case when controlling influences are applied at the boundaries of the control object. We obtain the solution of this system analysis problem in the form of Fourier series with respect to spatial variables based on orthogonal systems of trigonometric functions and Bessel functions.

Application of Measure of Noncompactness to Infinite Systems of Differential Equations

AbstractIn this paper we determine theHausdorff measure of noncompactness on the sequence space n(ϕ) ofW. L. C. Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces n(ϕ) and m(ϕ). Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness.

The optimal pursuit problem reduced to an infinite system of differential equations

The optimal game problem reduced to an infinite system of differential equations with integral constraints on the players’ controls is considered. The goal of the pursuer is to bring the system into the zeroth state, while the evader strives to prevent this. It is shown that Krasovskii's alternative is realized: the space of states is divided into two parts so that if the initial state lies in one part, completion of the pursuit is possible, and if it lies in the other part, evasion is possible. Constructive schemes for devising the optimal strategies of the players are proposed, and an explicit formula for the optimal pursuit time is derived.

Infinite System of Differential Equations in Some BK Spaces

The first measure of noncompactness was defined by Kuratowski in 1930 and later the Hausdorff measure of noncompactness was introduced in 1957 by Goldenštein et al. These measures of noncompactness have various applications in several areas of analysis, for example, in operator theory, fixed point theory, and in differential and integral equations. In particular, the Hausdorff measure of noncompactness has been extensively used in the characterizations of compact operators between the infinite‐dimensional Banach spaces. In this paper, we present a brief survey on the applications of measures of noncompactness to the theory of infinite system of differential equations in some BK spaces c0, c, ℓp(1 ≤ p ≤ ∞) and n(ϕ).

Applications of measures of noncompactness to the infinite system of differential equations in [formula omitted] spaces

In this paper we apply the technique of measures of noncompactness to the theory of infinite system of differential equations in the Banach sequence spaces ℓ p ( 1 ≤ p < ∞ ) . Our aim is to present some existence results for infinite system of differential equations formulated with the help of measures of noncompactness.

PURSUIT AND EVASION DIFFERENTIAL GAMES IN HILBERT SPACE

We consider pursuit and evasion differential game problems described by an infinite system of differential equations with countably many Pursuers in Hilbert space. Integral constraints are imposed on the controls of players. In this paper an attempt has been made to solve an evasion problem under the condition that the total resource of the Pursuers is less then that of the Evader and a pursuit problem when the total resource of the Pursuers greater than that of the Evader. The strategy of the Evader is constructed.

Multipoint problems for degenerate abstract differential equations

Multipoint boundary value problems for degenerate differential-operator equations of arbitrary order are studied. Several conditions for the separability in Banach-valued L p -spaces are given. Sharp estimates for the resolvent of the corresponding differential operator are obtained. In particular, the sectoriality of this operator is established. As applications, the boundary value problems for degenerate quasielliptic partial differential equations and infinite systems of differential equations on cylindrical domain are studied.

Solvability of semilinear differential equations with singularity

We prove local theorems on the existence of solutions of the Cauchy problem for singular equations of the form {fx262-01} in Banach spaces. Solvability conditions depend on the type of the singularity of the pencil λA + B of closed linear operators A and B. Examples and applications to finite-dimensional differential-algebraic equations, infinite systems of differential equations, and partial differential equations of the non-Kovalevskaya type are presented.

On solutions of an infinite system of singular integral equations

Using the classical Schauder fixed point principle we prove an existence result concerning an infinite system of singular integral equations. The obtained result is applied to establish the solvability of an infinite system of differential equations of fractional order.

Linear and nonlinear nonlocal boundary value problems for differential-operator equations

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal Lp -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.

On the solutions to the normal form of the Navier-Stokes equations

We introduce a construction of regular solutions to the Navier-Stokes equations that is specifically designed for the study of their asymptotic expansions. Using this construction, we give sufficient conditions for the convergence of those expansions. We also construct suitable normed spaces in which they converge. Moreover, in these spaces, the normal form of the Navier-Stokes equations associated with the terms of the asymptotic expansions [9] is a well-behaved infinite system of differential equations.

Chaotic behavior of semigroups related to the process of gene amplification–deamplification with cell proliferation

In the last few years there has been a renewed interest in infinite systems of differential equations, similar to the classical birth-and-death system of population dynamics, due to their rôle in modelling the evolution of drug resistance in cancer cells. In [J. Banasiak, M. Lachowicz, Topological chaos for birth-and-death models with proliferation, Math. Models Methods Appl. Sci. 12 (6) (2002) 755] such systems were shown to generate a chaotic dynamics under, however, very restrictive assumptions on the growth of coefficients. In this paper, using recently developed concept of subspace chaos [J. Banasiak, M. Moszyński, A generalization of Desch–Schappacher–Webb criteria for topological chaos with applications, Discrete Contin. Dyn. Syst. – A 12 (5) (2005) 959], we show that for a linear growth of the coefficients, which are more acceptable from biological point of view, the dynamics of these systems is chaotic in some subspaces of the original state space.

Symmetries of Integro-Differential Equations: A Survey of Methods Illustrated by the Benny Equations

Classical Lie group theory provides a universal tool for calculatingsymmetry groups for systems of differential equations. However Lie'smethod is not as much effective in the case of integral orintegro-differential equations as well as in the case of infinitesystems of differential equations.

Solvability of infinite systems of differential equations in Banach sequence spaces

The aim of this paper is to establish sufficient conditions for the solvability of infinite systems of ordinary differential equations in some Banach sequence spaces. The results presented in the paper create mainly the concrete realizations of sufficient conditions for the solvability of ordinary differential equations in Banach spaces formulated with help of the technique of measures of noncompactness. We concentrate on the results being rather convenient and handy in applications.

ASYMPTOTIC BEHAVIOUR OF AN OPERATOR EXPONENTIAL RELATED TO BRANCHING RANDOM WALK MODELS OF DNA REPEATS

We study the asymptotic behaviour of an infinite system of differential equations describing the expectations in a branching random walk. The original stochastic formulation was employed to describe the process of evolution of reversible drug resistance in cancer cells. The problem is formulated as an operator exponential function in the space of absolutely summable sequences. Conditions are found for the asymptotic decay of the operator exponential function, using methods of the spectral theory of linear operators. A discussion is provided relating mathematical results to the behaviour of models of gene amplification and evolution of DNA repeats.

An Application of Convolution Expansions to an Infinite System of Differential Equations

In the paper, we consider discrete version of the Tjon-Wu equation. We obtain proof of the existence and uniqueness of global nonnegative solutions of the Cauchy problem. Moreover, the behaviour of these solutions is described for a wide class of initial distributions. The main tool is the Wild expansion, properties of convolution iterates and convexing feature of summation.

Influence of the Solid Inner Core and Compressibility of the Fluid Core on the Earth Nutation

While calculating low frequency oscillations of the Earth liquid core spherical harmonic representation of the deformation field is usually used [1-3]:Substitution of (1) into the equations of motion gives an infinite system of differential equations for scalar functions Sɭm and Tɭm . Approximate solutions of such a system are obtained by truncating of the system. But results of [4] show that sometimes such method divergences.

Connection between spectral and oscillation properties of differential-operator equations of arbitrary order. II. Applications of factorization and squaring

Factorization theorems of Frobenius and M. G. Krein-Heinz-Rellich type are established for infinite systems of differential equations of arbitrary even order on finite or infinite intervals. From here, Etgen and Pawlowski oscillation theorems, referring to second order equations, are generalized to the considered equations of order 2n and also a variant of a Sturm type comparison theorem is derived. An analogue of the oscillation theorem for discrete levels in the gap of the continuous spectrum is obtained for problems of both even and odd order.