Sufficient conditions for boundedness of solutions of linear differential equations

In this paper we obtain sufficient conditions for boundedness of every solution and its derivatives of a linear differential equation. We will also show that our conditions are sharp.     

Sufficient conditions for convergence of modified projection-iterative method for equations with weak nonlinearity

New sufficient conditions are established for the convergence of the modified projection-iterative method for operator equations in a Banach space with weak nonlinearity, less restrictive than those known in the literature.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1486–1492, November, 1990.     

On weak convergence of solutions of one-dimensional stochastic differential equations

The main purpose of the present paper is to study weak convergence of solutions of stochastic differential equations     

Pathwise convergence rates for numerical solutions of Markovian switching stochastic differential equations

This work develops numerical approximation algorithms for solutions of stochastic differential equations with Markovian switching. The existing numerical algorithms all use a discrete-time Markov chain for the approximation of the continuous-time Markov chain. In contrast, we generate the continuous-time Markov chain directly, and then use its skeleton process in the approximation algorithm. Focusing on weak approximation, we take a re-embedding approach, and define the approximation and the solution to the switching stochastic differential equation on the same space. In our approximation, we use a sequence of independent and identically distributed (i.i.d.) random variables in lieu of the common practice of using Brownian increments. By virtue of the strong invariance principle, we ascertain rates of convergence in the pathwise sense for the weak approximation scheme.     

Sufficient conditions for the existence of global solutions of delayed differential equations

The purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi-global solutions for some classes of delayed functional differential equations. The principle of monotone sequences is used as a main tool of investigation. Inequalities for coordinates of global solutions are derived as a consequence of used method. Examples illustrate the results.     

Sufficient structure conditions for uniqueness of viscosity solutions of semilinear and quasilinear equations

In this article, we introduce a new approach for proving Maximum Principle type results for viscosity solutions of second-order, fully nonlinear possibly degenerate elliptic equations. This approach leads, in particular, to a better understanding of the conditions on the equation which are necessary to obtain such results. It allows us to provide new comparison results for semilinear and quasilinear equations.     

approximation for the solutions of stochastic differential equations. iii. jointly weak convergence

We give sufficient conditions for a family Z, e > 0 of continuous finite variation processes to converge weakly to a diffusion process Z. Then we consider the integral equation dXE(t) = (l)(Xe(t))dZE{t) and the stochastic equation dX{i) = (j)(X{t))dZ{t) and denote by X(t,x,w respectively X{t,x,(jo), the solution starting at x. We prove that PoX~l, e>0 converge weakly to Pol     

Sufficient conditions for the regularity of the solutions of the Navier–Stokes equations

In this paper we find sufficient conditions, involving only the pressure, that ensure the regularity of weak solutions to the Navier–Stokes equations. Conditions involving only the pressure were previously obtained in [7,4]. Following a remark in this last reference we improve, in particular, Kaniel's result [7]. Our condition can be seen at the light of the heuristic idea that the pressure behaves similarly to the modulus squared of the velocity. Copyright © 1999 John Wiley & Sons, Ltd.     

Martingale solutions for stochastic Euler equations

We prove existence of martingale solutions for the Euler equations in the 2-Dimensional space. The result is obtained by a compactness method     

Mild solutions of evolution quantum stochastic differential equations with nonlocal conditions

We study existence of a unique mild solution of evolution quantum stochastic differential equations with nonlocal conditions under the strong topology. Using the method of successive approximations, we do not need to transform the nonlocal problem to a fixed point form. The evolution operator A generates a family of semigroup that are continuous. Nonlocal conditions allow additional measurements of certain phenomena that cannot be captured by the traditional initial conditions. We show that under some given conditions, the mild solution is unique and also stable. The method applied here is much easier when compared with previous methods used in literature.     

Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations

In this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given, i.e., the local Lipschitz condition and the linear growth condition. Under the Lipschitz condition and the linear growth condition it is proved that the semi-implicit Euler method is convergence with strong order .     

Strong convergence rate for multivalued stochastic differential equations via stochastic theta method

In this paper we study the stochastic theta method for multivalued stochastic differential equations driven by standard Brownian motions and obtain the strong convergence rate of this numerical scheme.     

Entropy solutions for stochastic porous media equations

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\mathrm{\Delta }(|u{|}^{m?1}u)$ for all $m\in (1,\infty )$, and Hölder continuous diffusion nonlinearity with exponent 1/2.