On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds

We obtain a stability estimate for the degenerate complex Monge-Ampère operator which generalizes a result of Ko?odziej (2003) [12]. In particular, we obtain the optimal stability exponent and also treat the case when the right-hand side is a general Borel measure satisfying certain regularity conditions. Moreover, our result holds for functions plurisubharmonic with respect to a big form, thus generalizing the Kähler form setting in Ko?odziej (2003) [12]. Independently, we also provide more detail for the proof in Zhang (2006) [18] on continuity of the solution with respect to a special big form when the right-hand side is Lp-measure with p>1.     

Inhomogeneous infinity Laplace equation

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity sub- and super-solutions of the equation with constant right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known homogeneous infinity Laplace equation , which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously.     

Stability in mean square of a linear stochastic differential equation

The stability of the zero solution of a first-order linear differential equation with a random right-hand side is investigated using moment equations. Transformations of moment equations are considered. Conditions for reducing the order of the moment equations are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 119–126, 1987.     

Asymptotic stability and estimates for the attraction domains of monotone difference systems

We generalize asymptotic stability criteria and estimates for attraction domains in the nonnegative cone for systems of autonomous difference equations with monotone discontinuous right-hand side.     

The first-approximation stability of a nonstationary system with delay

We study the exponential stability of a nonlinear system of differential equations with constant delay such that the right-hand side of one of its subsystems contains the multiplier e ^t . We obtain a sufficient condition for the first-approximation stability of this system.     

Stability of a linear difference equation and estimation of its fundamental solution

We consider a linear nonautonomous system of difference equations. Under natural restrictions we prove that the stability with respect to the initial function and that with respect to the right-hand side in spaces of bounded and summable functions are equivalent to the corresponding estimates for the fundamental solution. As a consequence, we obtain an effective stability condition for a system with a constant matrix.     

Stability of Steady-State Solutions of Systems of Nonlinear Nonautonomous Delay Differential Equations

Sufficient conditions for the stability of steady-state solutions of systems of nonautonomous linear and nonlinear differential equations with time-dependent delay are obtained in terms of coefficients. These sufficient conditions are written as inequalities relating quantities that can be calculated directly from the right-hand side of the system of equations.     

On periodic solutions of ordinary differential equations with discontinuous right-hand side

A new version of the method of translation along trajectories, which does not require the uniqueness of the solution of the Cauchy problem, is applied to the proof of the existence theorem for vector-valued periodic solutions of ordinary differential equations of first and second order. This result is applicable to equations and differential inclusions with discontinuous right-hand side. Several applications of the theorems proved in this paper are considered in cases which are not covered by the classical theory of ordinary differential equations with continuous right-hand side and equations with right-hand side satisfying the Carathéodory conditions.     

On employment of semidefinite functions in stability of delayed equations

Four new stability theorems for nonautonomous delayed equations are established by the Lyapunov-Razumikhin approach with functions and its derivatives semidefinite. To do this, assume the collection of right-hand side translations in time has limit points which govern limiting equations. A corollary, which relates global asymptotic stability for the equation to that for limiting equations, and an illustrative example are given to show the applications of the established theorems.     

Lower Semicontinuity of the Feasible Set Mapping of Linear Systems Relative to Their Domains

This paper deals with stability properties of the feasible set of linear inequality systems having a finite number of variables and an arbitrary number of constraints. Several types of perturbations preserving consistency are considered, affecting respectively, all of the data, the left-hand side data, or the right-hand side coefficients.     

Extension of E.A. Barbashin’s and N.N. Krasovskii’s stability theorems to controlled dynamical systems

The known theorems by E.A. Barbashin and N.N. Krasovskii (1952) about the asymptotic and global stability of an equilibrium state for an autonomous system of differential equations are extended to nonautonomous differential inclusions with closed-valued (but not necessarily compact-valued) right-hand sides, where the equilibrium state is a weakly invariant (with respect to solutions of the inclusion) set. The statements are formulated in terms of the Hausdorff-Bebutov metric, the dynamical system of translations corresponding to the right-hand side of the differential inclusion, and the weakly invariant set corresponding to the inclusion.     

基于向量Liapunov函数不连续系统的稳定性研究

基于局部Lipschitz连续且正则(Clarke意义下)的向量Liapunov函数,讨论不连续自治系统的稳定性(Filippov解意义下)．通过定义一类新的向量Liapunov函数的“集值导数”,给出了关于不连续系统的广义比较原理．基于Lipschitz连续且正则的向量Liapunov函数,进一步的给出不连续自治系统的Liapunov稳定性定理．     

On determining the source function in heat and mass transfer problems under integral overdetermination conditions

We examine an inverse problem of determining the right-hand side (the source function) in a parabolic equation from integral overdetermination data. By a solution to a parabolic equation we mean a weak solution, and the right-hand side in this equation can be a distribution of a certain class. Under some conditions on the data of the problem, we demonstrate that this inverse problem is well posed and, in particular, some stability estimates hold.     

Hölder Continuity to Hamilton-Jacobi Equations with Superquadratic Growth in the Gradient and Unbounded Right-hand Side

We show that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally 𝒞^{0, α}. Unlike the existing (and more involved) proofs for equations with bounded right-hand side, our arguments rely on constructions of sub- and supersolutions combined with improvement of oscillation techniques.     

Boundary value problems with operator right-hand side

For a second-order boundary value problem with operator right-hand side and with functional boundary conditions, we prove solvability theorems with mixed and Dirichlet boundary conditions assuming the existence of a lower and an upper function. These theorems are analogs of theorems for the corresponding boundary value problems for an ordinary second-order differential equation with right-hand side satisfying the Carathéodory conditions.     

Stability of differential systems with rational right-hand sides

We prove theorems which establish estimates for the domain of stability of a differential system with rational right-hand side. We also construct estimates of the convergence of solutions of the system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1295–1299, September, 1995.     

Necessary conditions for an infinite time optimal control problem

In this paper, we deal with an optimal control problem with infinite transfer time. Using methods which involve local conditional stability, we obtain necessary conditions for optimality under the assumption that the right-hand side of the state equation is Fréchet-differentiable at every point of the optimal solution, and under some weak assumptions about the asymptotical behaviour of the set of perturbations of the solution. The results are illustrated in a specific case considered by Pontryagain et al.     

Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

Difference Schemes on Nonuniform Grids for the Two-Dimensional Convection–Diffusion Equation

New second-order accurate monotone difference schemes on nonuniform spatial grids for two-dimensional stationary and nonstationary convection–diffusion equations are proposed. The monotonicity and stability of the solutions of the computational methods with respect to the boundary conditions, the initial condition, and the right-hand side are proved. Two-sided and corresponding a priori estimates are obtained in the grid norm of C. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved.     

Higher-order spectral analysis and weak asymptotic stability of convex processes

This paper deals with the asymptotic stability analysis of a discrete dynamical inclusion whose right-hand side is a convex process. We provide necessary and sufficient conditions for weak asymptotic stability, and obtain sharp estimates for the asymptotic null-controllability set. These estimates involve not only standard, but also higher-order spectral information on the convex process and its adjoint.