Solvability Of Such Problems Research Articles

Nachal'no-kraevye zadachi dlya odnorodnykh parabolicheskikh sistem v poluogranichennoy ploskoy oblasti i uslovie dopolnitel'nosti

We consider initial–boundary value problems for homogeneous parabolic systemswith coefficients satisfying the double Dini condition with zero initial conditions in a semiboundedplane domain with nonsmooth lateral boundary. The method of boundary integralequations is used to prove a theorem on the unique classical solvability of such problems in thespace of functions that are continuous together with their first spatial derivative in the closureof the domain. An integral representation of the obtained solutions is given. It is shown thatthe condition for the solvability of the posed problems considered in the paper is equivalent tothe well-known complementarity condition.

Monotone iterations method for fractional diffusion equations

In recent years, there has been a growing interest on non-localmodels because of their relevance in many practical applications. Awidely studied class of non-local models involves fractional orderoperators. They usually describe anomalous diffusion. Inparticular, these equations provide a more faithful representationof the long-memory and nonlocal dependence of diffusion in fractaland porous media, heat flow in media with memory, dynamics ofprotein in cells etc.
 For $a\in (0, 1)$, we investigate the nonautonomous fractionaldiffusion equation:
 $D^a_{*,t} u - Au = f(x, t,u),$
 where$D^a_{*,t}$ is the Caputo fractional derivative and $A$ is auniformly elliptic operator with smooth coefficients depending onspace and time. We consider these equations together with initialand quasilinear boundary conditions.
 The solvability of such problems in H\"older spaces presupposesrigid restrictions on the given initial data. These compatibilityconditions have no physical meaning and, therefore, they can beavoided, if the solution is sought in larger spaces, for instance inweighted H\"older spaces.
 We give general existence and uniqueness result andprovide some examples of applications of the main theorem. The maintool is the monotone iterations method. Preliminary we developed thelinear theory with existence and comparison results. The principleuse of the positivity lemma is the construction of a monotonesequences for our problem. Initial iteration may be taken as eitheran upper solution or a lower solution. We provide some examples ofupper and lower solution for the case of linear equations andquasilinear boundary conditions. We notice that this approach canalso be extended to other problems and systems of fractionalequations as soon as we will be able to construct appropriate upperand lower solutions.

Fundamental limitations on efficiently forecasting certain epidemic measures in network models

The ongoing COVID-19 pandemic underscores the importance of developing reliable forecasts that would allow decision makers to devise appropriate response strategies. Despite much recent research on the topic, epidemic forecasting remains poorly understood. Researchers have attributed the difficulty of forecasting contagion dynamics to a multitude of factors, including complex behavioral responses, uncertainty in data, the stochastic nature of the underlying process, and the high sensitivity of the disease parameters to changes in the environment. We offer a rigorous explanation of the difficulty of short-term forecasting on networked populations using ideas from computational complexity. Specifically, we show that several forecasting problems (e.g., the probability that at least a given number of people will get infected at a given time and the probability that the number of infections will reach a peak at a given time) are computationally intractable. For instance, efficient solvability of such problems would imply that the number of satisfying assignments of an arbitrary Boolean formula in conjunctive normal form can be computed efficiently, violating a widely believed hypothesis in computational complexity. This intractability result holds even under the ideal situation, where all the disease parameters are known and are assumed to be insensitive to changes in the environment. From a computational complexity viewpoint, our results, which show that contagion dynamics become unpredictable for both macroscopic and individual properties, bring out some fundamental difficulties of predicting disease parameters. On the positive side, we develop efficient algorithms or approximation algorithms for restricted versions of forecasting problems.

An inverse spectral problem for second-order functional-differential pencils with two delays

Recently, there appeared a considerable interest in inverse Sturm–Liouville-type problems with constant delay. However, necessary and sufficient conditions for solvability of such problems were obtained only in one very particular situation. Here we address this gap by obtaining necessary and sufficient conditions in the case of functional-differential pencils possessing a more general form along with a nonlinear dependence on the spectral parameter. For this purpose, we develop the so-called transformation operator approach, which allows reducing the inverse problem to a nonlinear vectorial integral equation. In Appendix A, we obtain as a corollary the analogous result for Sturm–Liouville operators with delay. Remarkably, the present paper is the first work dealing with an inverse problem for functional-differential pencils in any form. Besides generality of the pencils under consideration, an important advantage of studying the inverse problem for them is the possibility of recovering both delayed terms, which is impossible for the Sturm–Liouville operators with two delays. The latter, in turn, is illustrated even for different values of these two delays by a counterexample in Appendix B. We also provide a brief survey on the contemporary state of the inverse spectral theory for operators with delay observing recently answered long-term open questions.

Boundary value problems with nonlocal conditions for hyperbolic systems of equations with two independent variables

Nonlocal boundary value problems for arbitrary order hyperbolic systems with one spatial variable are considered. A priori estimates for general nonlocal mixed problems for systems with smooth and piecewise smooth coefficients are obtained. The correct solvability of such problems is proved.Examples of additional conditions necessity are provided.

РОЗВ’ЯЗУВАНIСТЬ ОСЕРЕДНЕНИХ ЗАДАЧ У ЗГОРТКАХ ДЛЯ СЛАБО ПОРИСТИХ СЕРЕДОВИЩ

Initial boundary value problems for nonstationary equations of diffusion and filtration in weakly porous media are considered. Assertions about the solvability of such problems and the corresponding homogenized problems with convolutions are given. These statements are proved for general initial data and inhomogeneous initial conditions and are generalizations of classical results on the solvability of initial-boundary value problems for the heat equation. The proofs use the methods of a priori estimates and the well-known Agranovich–Vishik method, developed to study parabolic problems of general type.

Математичне моделювання коливних процесів у смузі

The boundary-value periodic problem for differential equations in partial derivatives, including hyperbolic equations, are complicated and controversial subject to study. Boundary-value problems with data throughout the border region as well as the problem of non-local (including integrated) conditions for hyperbolic equations in limited areas are, generally speaking, relatively correct. Many authors link the solvability of such problems with the problem of small denominators and use the methods of nonlinear functional analysis, the theory of implicit functions, variation methods. Another authors use the analytical methods in the research of periodic boundary-value problems for the second order hyperbolic equations. They construct integral operators and search the solutions in specially defined spaces of continuously differentiated functions for specific cases of periodicity.In this paper we find an analytic formula of the function v(x, t), which is a solution of the boundary-value 2π-periodic time-varying problem in the class of odd functions for which f (t) = – f(π – t). The properties of this function are established and the estimates of the solution of the boundary-value 2π-periodic problem are given.The results of the study are used for mathematical modeling of oscillating processes described by the second order hyperbolic equations.On the basis of the found function v(x, t) we can draw conclusions about the behavior of the solution of the undisturbed equation (ε = 0, ε is a small parameter) in the study of the general nonlinear the second order hyperbolic equation by the asymptotic methods.

Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities

We consider equations of the form $\Delta u +\lambda^2 V(x)e^{\,u}=\rho$ in various two dimensional settings. We assume that $V>0$ is a given function, $\lambda>0$ is a small parameter and $\rho=\mathcal O(1)$ or $\rho\to +\infty$ as $\lambda\to 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $V\equiv 1$, $\rho=0$ when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of such problems is, viewed formally, the necessary condition for the existence of the maximal solution.

On Some Problems Generated by a Sesquilinear Form

Based on the generalized Green formula for a sesquilinear nonsymmetric form for the Laplace operator, we consider spectral nonself-adjoint problems. Some of them are similar to classical problems while the other arise in problems of hydrodynamics, diffraction, and problems with surface dissipation of energy. Properties of solutions of such problems are considered. Also we study initial-boundary value problems generating considered spectral problems and prove theorems on correct solvability of such problems on any interval of time.

BMO Solvability and $$A_{\infty }$$ Condition of the Elliptic Measures in Uniform Domains

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e. $\omega_L\in A_{\infty}(\sigma)$. This generalizes a previous result on Lipschitz domains by Dindos, Kenig and Pipher.

SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF OPTIMAL CONTROL IN THE CRITICAL CASE

The article describes the boundary-value problem of the theory of optimal control for matrix differential equations in the critical case. This problem is urgent for wide range of applications like modeling of linear technological and economic processes. A problem of controlling movement is reduced to solutions of boundary-value problems of n× n systems of ordinary differential equations of the first order. Solution of these tasks is very complex in case of control of systems in which the number of processes exceeds the amount of information about the initial state. The paper is devoted to finding conditions for solvability and construction solutions of boundary-value problems in critical case of the theory of motion control, in which the number of boundary conditions does not coincide with the number of unknowns in the system. The condition of solvability of such problems is found. Author describes the approach for solution to the problem with the help of theory of pseudoinverse matrices. In addition this approach is applicable to the problem of analytic construction of regulators and the optimal stabilization.

Non‐fragile H ∞ control for switched stochastic delay systems with application to water quality process

SUMMARYIn this paper, the problem of non‐fragile observer‐based H ∞ control for discrete‐time switched delay systems is investigated. Both data missing and time delays are taken into account in the links from sensors to observers and from controllers to actuators. Because data missing satisfies the Bernoulli distribution, such problem is transformed into an H ∞ control problem for stochastic switched delay systems. Average dwell time approach is used to obtain sufficient conditions on the solvability of such problems. A numerical example and a real example for water quality control are provided to illustrate the effectiveness and potential applications of the proposed techniques. Copyright © 2013 John Wiley & Sons, Ltd.

Normal boundary-value problems for semilinear elliptic systems in weight spaces of generalized functions

The solvability of normal linear boundary-value problems for linear Petrovskii-elliptic systems of differential equations in the weight spaces of generalized functions is established. The sufficient conditions of solvability of such problems for semilinear elliptic systems of differential equations with boundary data from the weight spaces of generalized functions are obtained.

Direct and inverse problems for an abstract differential equation containing Hadamard fractional derivatives

In a Banach space, we consider a Cauchy type problem with a left Hadamard fractional derivative of order α ∈ (0, 1) and a Cauchy problem with a regularized Hadamard fractional derivative. We prove the well-posed solvability of such problems with a bounded operator as well as with the generator of a strongly continuous semigroup. For inverse coefficient problems, we indicate sufficient conditions for their unique solvability.

Global boundary value problems for nonlinear dynamic equations on time scales

We consider nonlinear boundary value problems for dynamic equations on time scales. We study nonlinear dynamic equations subject to global boundary conditions. Criteria are provided for the solvability of such problems. In the case of weak nonlinearities, we also examine the dependence of the solution on parameters.

Boundary-value problems for differential equations in a Banach space

We establish a criterion for the existence of solutions of linear inhomogeneous boundary-value problems in a Banach space. We obtain conditions for the normal solvability of such problems and consider their special cases, namely, countable-dimensional boundary-value problems.

On necessary conditions for the Fredholm solvability of nonlocal elliptic problems

Nonlocal elliptic problems in plane domains are considered. Necessary conditions for the Fredholm solvability of such problems in weighted spaces are obtained.

The (S)+ condition on generalized variational inequalities

In this paper, we derive some existence results for generalized variational inequalities associated with mappings satisfying the (S)+ condition. The relation between the (S)+ and (S) + 1 conditions is discussed. As an application, we also consider multivalued complementarity problems associated with mappings satisfying the (S)+ condition, and prove a theorem to characterize the solvability of such problems in terms of exceptional families of elements.

Nonlinear discrete Sturm–Liouville problems

In this paper we study nonlinear boundary value problems of the form Δ [ p ( t − 1 ) Δ y ( t − 1 ) ] + q ( t ) y ( t ) + λ y ( t ) = f ( y ( t ) ) ; t = a + 1 , … , b + 1 , subject to a 11 y ( a ) + a 12 Δ y ( a ) = 0 and a 21 y ( b + 1 ) + a 22 Δ y ( b + 1 ) = 0 . The parameter λ is an eigenvalue of the associated linear problem; that is, there is a nontrivial function u that satisfies the boundary conditions and also Δ [ p ( t − 1 ) Δ u ( t − 1 ) ] + q ( t ) u ( t ) + λ u ( t ) = 0 for t in { a + 1 , a + 2 , … , b + 1 } . We establish sufficient conditions for the solvability of such problems. In addition, in those cases where the nonlinearity is “small,” we provide a qualitative analysis of the relation between solutions of the nonlinear problem and eigenfunctions of the linear one.

Weak solutions of parabolic equations in non-cylindrical domains

In their classical work, Ladyzhenskaya and Ural ′ ’ tseva gave a definition of weak solution for parabolic equations in cylindrical domains. Their definition was broad enough to guarantee the solvability of all such problems but narrow enough to guarantee the uniqueness of these solutions. We give here some alternative definitions which are appropriate to non-cylindrical domains, and we prove the unique solvability of such problems.