Conditions For Existence Of Solutions Research Articles

Real scalar field solutions of the EKG equations including matter

Static (not stationary) solutions of the Einstein–Klein–Gordon (EKG) equations including matter are obtained for real scalar fields. The scalar field interaction with matter is considered. The introduced coupling allows the existence of static solutions in contraposition with the case of the simpler EKG equations for real scalar fields and gravity. Surprisingly, when the considered matter is a photon-like gas, it turns out that the gravitational field intensity at large radial distances becomes nearly a constant, exerting an approximately fixed force to small bodies at any distance. The effect is clearly related with the massless character of the photon-like field. It is also argued that the gravitational field can generate a bounding attraction, that could avoid the unlimited increase in mass with the radius of the obtained here solution. This phenomenon, if verified, may furnish a possible mechanism for explaining how the increasing gravitational potential associated to dark matter, finally decays at large distances from the galaxies. A method for evaluating these photon bounding effects is just formulated in order to be further investigated.

The existence and stability of stationary solutions of the inflow problem for full compressible Navier-Stokes-Poisson system

In this paper, we consider an inflow problem for the non-isentropic Navier-Stokes-Poisson system in a half line (0, ∞). For the general gas including ideal polytropic gas, we first give some results for the existence of the stationary solution with the aid of center manifold theory on a 4 × 4 system of autonomous ordinary differential equations. We also show the time asymptotic stability of the stationary solutions with small strength under smallness assumptions on the initial perturbations by using an elementary energy method.

Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds

We address a general system of complex Monge-Ampère equations on Fano horosymmetric manifolds and give necessary and sufficient conditions for existence of solutions in terms of combinatorial data of the manifold. This gives new results about Mabuchi metrics, twisted Kähler-Einstein metrics and coupled Kähler-Ricci solitons and provides a unified approach to many previous results on canonical metrics on Kähler manifolds.

Solutions to the linear transpose matrix equations and their application in control

In this paper, we study the solutions to the linear transpose matrix equations $$AX+X^{T}B=C$$ and $$AX+X^{T}B=CY$$ , which have many important applications in control theory. By applying Kronecker map and Sylvester sum, we obtain some necessary and sufficient conditions for existence of solutions and the expressions of explicit solutions for the Sylvester transpose matrix equation $$AX+X^{T}B=C$$ . Our conditions only need to check the eigenvalue of $$ B^{T}A^{-1}$$ , and, therefore, are simpler than those reported in the paper (Piao et al. in J Frankl Inst 344:1056–1062, 2007). The corresponding algorithms permit the coefficient matrix C to be any real matrix and remove the limit of $$C=C^{T}$$ in Piao et al. Moreover, we present the solvability and the expressions of parametric solutions for the generalized Sylvester transpose matrix equation $$AX+X^{T}B=CY$$ using an alternative approach. A numerical example is given to demonstrate that the introduced algorithm is much faster than the existing method in the paper (De Teran and Dopico in 434:44–67;2011). Finally, the continuous zeroing dynamics design of time-varying linear system is provided to show the effectiveness of our algorithm in control.

Non-autonomous nonlocal partial differential equations with delay and memory

The paper addresses a kind of non-autonomous nonlocal parabolic equations when the external force contains hereditary characteristics involving bounded and unbounded delays. First, well-posedness of the problem is analyzed by the Galerkin method and energy estimations in the phase space Cρ(X). Moreover, some results related to strong solutions are proved under suitable assumptions. The existence of stationary solutions is then established by a corollary of the Brower fixed point theorem. By constructing appropriate Lyapunov functionals in terms of the characteristic delay terms, a deep analysis on stability and attractiveness of the stationary solutions is established. Furthermore, the existence of pullback attractors in L2(Ω), with bounded and unbounded delays, is shown. We emphasize that, to prove the existence of pullback attractors in the unbounded delay case, a new phase space, Eγ, has to be constructed.

A Capacity-Based Condition for Existence of Solutions to Fractional Elliptic Equations with First-Order Terms and Measures

In this manuscript, we appeal to Potential Theory to provide a sufficient condition for existence of distributional solutions to fractional elliptic problems with non-linear first-order terms and measure data ω: 1 $$ \left\{ \begin{array}{rcll} (-{\Delta})^{s}u&=&|\nabla u|^{q} + \omega \quad \text{in }\mathbb{R}^{n}, s \in (1/2, 1) u & > &0 \quad \text{in } \mathbb{R}^{n} \lim_{|x|\to \infty}u(x) & =& 0, \end{array} \right. $$ under suitable assumptions on q and ω. Roughly speaking, the condition for existence states that if the measure data is locally controlled by the Riesz fractional capacity, then there is a global solution for the Problem (1). We also show that if a positive solution exists, necessarily the measure ω will be absolutely continuous with respect to the associated Riesz capacity, which gives a partial reciprocal of the main result of this work. Finally, estimates of u in terms of ω are also given in different function spaces.

Investigation of a system of nonlinear fractional order hybrid differential equations under usual boundary conditions for existence of solution

This research work is related to investigate sufficient conditions for existence, uniquely of solution to a coupled system of nonlinear fractional order hybrid differential equations (NFOHDEs). With the help of Burton's fixed point theorem, we develop the required conditions for existence of at least one solution to the considered problem of NFOHDEs.

The stationary solution of a one-dimensional bipolar quantum hydrodynamic model

In this paper, we consider the existence and uniqueness of stationary solution to the bipolar quantum hydrodynamic model in one dimensional space with general non-constant doping profile. The existence of the stationary solution is proved by Leray-Schauder fixed-point theorem and a crucial truncation technique is used to derive the positive upper and lower bounds of the stationary solution. The uniqueness of the stationary solution is shown by a delicate energy estimate.

Asymptotic stability of stationary waves to the Navier–Stokes–Poisson equations in half line

ABSTRACT The main concern of this paper is to investigate the asymptotic stability of stationary solution to the compressible Navier–Stokes–Poisson equations with the classical Boltzmann relation in a half line. We first show the unique existence of stationary solution with the aid of the stable manifold theory, and then prove that the stationary solution is time asymptotically stable under the small initial perturbation by the elementary energy method. Finally, we discuss the convergence rate of the time-dependent solution towards the stationary solution, and give a new condition to ensure an algebraic decay or an exponential decay. The proof is based on a time and space weighted energy method by fully utilizing the self-consistent Poisson equation.

Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data

We study the equation $$-\text {div}(A(x,\nabla u))=|u|^{q_1-1}u|\nabla u|^{q_2}+\mu $$ where $$A(x,\nabla u)\sim |\nabla u|^{p-2}\nabla u$$ in some suitable sense, $$\mu $$ is a measure and $$q_1$$ , $$q_2$$ are nonnegative real numbers and satisfy $$q_1+q_2>p-1$$ . We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.

Fuzzy Fixed Point Results in F-Metric Spaces with Applications

In this paper, some concepts of F-metric spaces are used to study a few fuzzy fixed point theorems. Consequently, corresponding fixed point theorems of multivalued and single-valued mappings are discussed. Moreover, one of our obtained results is applied to establish some conditions for existence of solutions of fuzzy Cauchy problems. It is hoped that the established ideas in this work will awake new research directions in fuzzy fixed point theory and related hybrid models in the framework of F-metric spaces.

Stability Analysis and Optimal Control for Yellow Fever Model with Vertical Transmission.

In this study, a deterministic model for the transmission dynamics of yellow fever (YF) in a human–mosquito setting in the presence of control measures is constructed and rigorously analyzed. In addition to horizontal transmissions, vertical transmission within mosquito population is incorporated. Analysis of the mosquito-only component of the model shows that the reduced model has a mosquito-extinction equilibrium, which is globally-asymptotically stable whenever the basic offspring number (N_{0}) is less than unity. The vaccinated and type reproduction numbers of the full-model are computed. Condition for global-asymptotic stability of the disease-free equilibrium of the model when N_{0} > 1 is presented. It is shown that, fractional dosing of YF vaccine does not meet YF vaccination requirements. Optimal control theory is applied to the model to characterize the controls parameters. Using Pontryagin’s maximum principle and modified forward–backward sweep technique, the necessary conditions for existence of solutions to the optimal control problem is determined. Numerical simulations of the models to assess the effect of fractional vaccine dosing on the disease dynamics and global sensitivity analysis are presented.

Mean field equations on a closed Riemannian surface with the action of an isometric group

Let [Formula: see text] be a closed Riemannian surface, [Formula: see text] be an isometric group acting on it. Denote a positive integer [Formula: see text], where [Formula: see text] is the number of all distinct points of the set [Formula: see text]. A sufficient condition for existence of solutions to the mean field equation [Formula: see text] is given. This recovers results of Ding–Jost–Li–Wang, Asian J. Math. (1997) 230–248 when [Formula: see text] or equivalently [Formula: see text], where Id is the identity map.

Existence conditions for solutions of bilevel vector equilibrium problems with application to traffic network problems with equilibrium constraints

In this paper, we introduce some strong and weak bilevel vector equilibrium problems in locally convex Hausdorff topological vector spaces and present some conditions for the existence of solutions to these problems by using the Kakutani–Fan–Glicksberg fixed-point theorem. Furthermore, as a real-world application, we obtain the existence of solutions to traffic network problems with equilibrium constraints.

Stationary solutions to a nonlocal fourth-order elliptic obstacle problem

Existence of stationary solutions to a nonlocal fourth-order elliptic obstacle problem arising from the modelling of microelectromechanical systems with heterogeneous dielectric properties is shown. The underlying variational structure of the model is exploited to construct these solutions as minimizers of a suitably regularized energy, which allows us to weaken considerably the assumptions on the model used in a previous article.

Reconstruction of Solutions to a Generalized Moisil–Teodorescu System in Jordan Domains with Rectifiable Boundary

In this paper we consider the problem of reconstructing solutions to a generalized Moisil-Teodorescu system in Jordan domains of $\mathbb{R}^{3}$ with rectifiable boundary. In order to determine conditions for existence of solutions to the problem we embed the system in an appropriate generalized quaternionic setting.

A Methodology for Practical Design and Optimization of Class-E DC-DC Resonant Converters

In recently published papers, an innovative analytical approach for the design of a class-E resonant dc-dc converter has been first proposed and further extended to many other class-E converter topologies. Its peculiarity is to be dimensionless and based on the exact solution of the system of differential equations regulating the behavior of the circuit, ensuring very high precision and reliability with respect to all methodologies previously proposed by the state-of-the-art and based on the so-called sinusoidal approximation. Here, we review this methodology and improve it in a twofold way. On the one hand, we propose alternative modeling for some devices (in particular the transformer), increasing both flexibility and generality, with the possibility to extend the application to more topologies and more working points. On the other hand, a new normalization is proposed, showing that the actual dimension of the design workspace is 2, and not 3 as assumed in the previous works. This has important consequences. As an example, the solution existence condition can be represented on a simple 2D plot, with the possibility to immediately check whether the optimal class-E condition can be ensured or not. Furthermore, we can completely and conveniently explore the entire design space to investigate properties such as the stress on the switching devices or the root-mean-square currents, allowing further optimization of the converter design.

Stationarity and ergodicity of vector STAR models

Smooth transition autoregressive models are widely used to capture nonlinearities in univariate and multivariate time series. Existence of stationary solution is typically assumed, implicitly or explicitly. In this paper, we describe conditions for stationarity and ergodicity of vector STAR models. The key condition is that the joint spectral radius of certain matrices is below 1. It is not sufficient to assume that separate spectral radii are below 1. Our result allows to use recently introduced toolboxes from computational mathematics to verify the stationarity and ergodicity of vector STAR models.

Local representation and construction of Beltrami fields II.solenoidal Beltrami fields and ideal MHD equilibria

Object of the present paper is the local theory of solution for steady ideal Euler flows and ideal MHD equilibria. The present analysis relies on the Lie-Darboux theorem of differential geometry and the local theory of representation and construction of Beltrami fields developed in [1]. A theorem for the construction of harmonic orthogonal coordinates is proved. Using such coordinates families of solenoidal Beltrami fields with different topologies are obtained in analytic form. Existence of global solenoidal Beltrami fields satisfying prescribed boundary conditions while preserving the local representation is considered. It is shown that only singular solutions are admissible, an explicit example is given in a spherical domain, and a theorem on existence of singular solutions is proven. Local conditions for existence of solutions, and local representation theorems are derived for generalized Beltrami fields, ideal MHD equilibria, and general steady ideal Euler flows. The theory is applied to construct analytic examples.

Solvability of Sylvester operator equation with bounded subnormal operators in Hilbert spaces

The aim of this paper is to present some necessary and sufficient conditions for existence of solution of Sylvester operator equation involving bounded subnormal operators in a Hilbert space. Our results improve and generalize some results in the literature involving normal operators.