Infinite Interval Research Articles

On the Interrelation of Two Linear NonStationary Problems with Multiple Evaders

A linear nonstationary pursuit problem in which a group of pursuers and a group of evaders are involved is considered under the condition that the group of pursuers includes participants whose admissible controls set coincides with that of the evaders and participants whose admissible controls sets belong to interior of admissible controls set of the evaders. The aim of the group of pursuers is to capture all the evaders. The aim of the group of evaders is to prevent the capture, that is, to allow at least one of the evaders to avoid the rendezvous. It is shown that, if in the game in which all the participants have equal capabilities at least one of the evaders avoids the rendezvous on an infinite time interval, then as a result of the addition of any number of pursuers with less capabilities, at least one of the evaders will avoid the rendezvous on any finite time interval.

Transformed Hermite functions on a finite interval and their applications to a class of singular boundary value problems

In this paper, a weighted orthogonal system on finite interval based on the transformed Hermite functions is introduced. Some results on approximations using the Hermite functions on finite interval are obtained from corresponding approximations on infinite interval via a conformal map. To illustrate the potential of the new basis, we apply it to the collocation method for solving a class of singular two-point boundary value problems. The numerical results show that our new scheme is very effective and convenient for solving singular boundary value problems.

Free Boundary Problem of Magnetohydrodynamics

A free boundary problem controlling the motion of a finite isolated mass of a viscous incompressible electrically conducting fluid in vacuum is considered. The fluid is moving under the action of a magnetic field and volume forces. It is proved that this free boundary problem is solvable in an infinite time interval under additional smallness assumptions imposed on the initial data and the external forces.

Existence of solutions to nonlinear fractional differential equations with boundary conditions on an infinite interval in Banach spaces

In this paper, we consider a system of nonlinear differential equations in a Banach space with boundary conditions on an infinite interval and provide sufficient conditions for the existence of solutions of the system. Our method relies upon the properties of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem. An example is given to illustrate the main results.

Delocalization of Two-Dimensional Random Surfaces with Hard-Core Constraints

We study the fluctuations of random surfaces on a two-dimensional discrete torus. The random surfaces we consider are defined via a nearest-neighbor pair potential which we require to be twice continuously differentiable on a (possibly infinite) interval and infinity outside of this interval. No convexity assumption is made and we include the case of the so-called hammock potential, when the random surface is uniformly chosen from the set of all surfaces satisfying a Lipschitz constraint. Our main result is that these surfaces delocalize, having fluctuations whose variance is at least of order $\log n$, where $n$ is the side length of the torus. We also show that the expected maximum of such surfaces is of order at least $\log n$. The main tool in our analysis is an adaptation to the lattice setting of an algorithm of Richthammer, who developed a variant of a Mermin-Wagner-type argument applicable to hard-core constraints. We rely also on the reflection positivity of the random surface model. The result answers a question mentioned by Brascamp, Lieb and Lebowitz 1975 on the hammock potential and a question of Velenik 2006.

Positive solutions for nonlinear double impulsive differential equations with p-Laplacian on infinite intervals

In this paper, we investigate nonlinear second-order double impulsive differential equations integral boundary value problem with p-Laplacian on an infinite interval with the infinite number of impulsive times. Based on the cone theory and monotone iterative technique, we establish the existence of minimal nonnegative solution and iteration of positive solutions for such a boundary value problem. The main results are new and extend the existing results. At last, some examples are worked out to demonstrate the use of the main results.

A note on terminal value problems for fractional differential equations on infinite interval

In this note we establish sufficient conditions for existence and uniqueness of solutions of terminal value problems for a class of fractional differential equations on infinite interval. Some illustrative examples are comprehended to demonstrate the proficiency and utility of our results.

$$L_{p}$$ L p Boundedness of (C, 1) Means of the Generalized Series of the Second Kind for Levin–Lubinsky Weights

Let I be a finite or infinite interval containing 0, and let \( W:I\rightarrow ( 0,\infty ) \). Assume that \(W^{2}\) is a weight, so that we may define orthonormal polynomials \(\{ p_{j}\} _{j=0}^{\infty }\) corresponding to \(W^{2}\). The generalized functions of the second kind are \(q_{j}( W^{2},v,x) :=H[ p_{j}Wv] ( x) \), \(j\ge 0\), where H denotes the Hilbert transform, and v a bounded function on I. For \(f:I\rightarrow \mathbb {R}\), let \(s_{m} [ f] :=s_{m}[ W^{2},v,f] \) denote the mth partial sums of the generalized series of the second kind. We investigate boundedness in \(L_{p}\) spaces of the (C, 1) means $$\begin{aligned} \frac{1}{n}\sum _{m=1}^{n}s_{m}[ f]. \end{aligned}$$ The class of weights \(W^{2}\) considered includes even and non-even exponential weights.

The invariant measures of some infinite interval exchange maps

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions for which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than 1/2. We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multi-twists in a pair of cylinder decompositions in non-parallel directions.

Degenerate nonselfadjoint high-order ordinary differential equations on an infinite interval

The paper considers the generalized Dirichlet problem for a class of degenerate nonselfadjoint high-order ordinary differential equations on an infinite interval. The spectrum of the corresponding operator is studied, and in the special case, the domain of definition of the selfadjoint operator is described.

A review of operational matrices and spectral techniques for fractional calculus

Recently, operational matrices were adapted for solving several kinds of fractional differ- ential equations (FDEs). The use of numerical tech- niques in conjunction with operational matrices of someorthogonalpolynomials,forthesolutionofFDEs on finite and infinite intervals, produced highly accu- rate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, wepresenttheoperationalmatricesoffractionalderiva- tivesandintegrals,forseveralpolynomialsonbounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with differ- ent spectral techniques for solving the aforementioned equations on bounded domains. The operational matri- ces of fractional derivatives and integrals are also pre- sented for orthogonal Laguerre and modified general-

Existence and uniqueness of positive mild solutions for nonlocal evolution equations

This paper deals with the existence and uniqueness of positive mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. The existence and uniqueness of mild solution for the associated linear evolution equation nonlocal problem is established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness of positive mild solutions for nonlinear evolution equation nonlocal problem are obtained by using the monotone iterative method without the assumption of lower and upper solutions. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. An example is also given to illustrate that our results are valuable.

Retracted: The Rational Third-Kind Chebyshev Pseudospectral Method for the Solution of the Thomas-Fermi Equation over Infinite Interval

Retracted: The Rational Third-Kind Chebyshev Pseudospectral Method for the Solution of the Thomas-Fermi Equation over Infinite Interval

Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval

By employing the monotone iterative technique, we not only establish the existence of the unique solution for a fractional integral boundary value problem on semi-infinite intervals, but also develop an explicit iterative sequence for approximating the solution and give an error estimate for the approximation, which is an important improvement of existing results.

Linear quadratic regulation for discrete-time antilinear systems: An anti-Riccati matrix equation approach

In this paper, the linear quadratic regulation problem is investigated for discrete-time antilinear systems. Two cases are considered: finite time state regulation and infinite time state regulation. First, the discrete minimum principle is generalized to the complex domain. By using the discrete minimum principle and dynamic programming, necessary and sufficient conditions for the existence of the unique optimal control are obtained for the finite time regulation problem in terms of the so-called anti-Riccati matrix equation. Besides, the optimal value of the performance index under the optimal control is provided. Furthermore, the optimal regulation problem on an infinite interval is investigated under the assumption that the considered time-invariant antilinear system is controllable. The resulted closed-loop system under the optimal control turns out to be asymptotically stable.

Solving a singular nonlocal problem with redundant conditions for a system of linear ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite or semi-infinite interval. The basic conditions are nonlocal and are specified by a Stieltjes integral; moreover, certain redundant (and also nonlocal) conditions are imposed. At infinity, the solution is required to be bounded. A method for solving such an over-determined problem is proposed and analyzed. The method is numerically stable if an auxiliary problem that replaces the original one is numerically stable.

Extremal solutions for p-Laplacian fractional integro-differential equation with integral conditions on infinite intervals via iterative computation

We study the extremal solutions of a class of fractional integro-differential equation with integral conditions on infinite intervals involving the p-Laplacian operator. By means of the monotone iterative technique and combining with suitable conditions, the existence of the maximal and minimal solutions to the fractional differential equation is obtained. In addition, we establish iterative schemes for approximating the solutions, which start from the known simple linear functions. Finally, an example is given to confirm our main results.

Аналитическое конструирование оптимальных регуляторов для квазилинейных стохастических систем, функционирующих на неограниченном интервале времени

Аналитическое конструирование оптимальных регуляторов для квазилинейных стохастических систем, функционирующих на неограниченном интервале времени

$B$-Convergence Theory of Runge--Kutta Methods for Stiff Volterra Functional Differential Equations with Infinite Integration Interval

In this paper, $B$-convergence theory of Runge--Kutta methods for nonlinear stiff Volterra functional differential equations is extended from finite integration interval $[a,T]$ to infinite integration interval $[a,+\infty)$. The results obtained are of importance for the study of functional differential dynamic systems and for long time stiff computation.

On the Birkhoff Quadrature Formulas Using Even and Odd Order of Derivatives

We introduce some New Quadrature Formulas by using Jacoby polynomials and Laguerre polynomials. These formulas can be obtained for a finite and infinite interval and also separately for the even or odd order of derivatives. By using the properties of error functions of the above orthogonal polynomials we can obtain the error functions for these formulas. Application of the new approaches increases their precision degrees. Finally, some examples are given to illuminate the details.