Strong solutions to stochastic Volterra equations

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main results provide sufficient conditions for strong solutions to stochastic Volterra equations.     

Regularity of solutions to stochastic Volterra equations with infinite delay

In this article we give necessary and sufficient conditions providing regularity of solutions to stochastic Volterra equations with infinite delay on a -dimensional torus. The harmonic analysis techniques and stochastic integration in function spaces are used. The work applies to both the stochastic heat and wave equations.

     

Asymptotics with respect to a parameter of solutions of linear functional-differential equations

We study the question of the number of linearly independent solutions of the equationy ⁽ⁿ⁾ (x)+(Fy) (x)+ⁿ y (x)=0,x [0, 1], in which F is a bounded linear operator acting on various normed function spaces. A number of assertions about the asymptotic behavior of these solutions with respect to , tending to infinity are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1460–1469, November, 1990.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Limiting measure and stationarity of solutions to stochastic evolution equations with Volterra noise

Large-time behavior of solutions to stochastic evolution equations driven by two-sided regular cylindrical Volterra processes is studied. The solution is understood in the mild sense and takes values in a separable Hilbert space. Sufficient conditions for the existence of a limiting measure and strict stationarity of the solution process are found and an example for which these conditions are also necessary is provided. The results are further applied to the heat equation perturbed by the two-sided Rosenblatt process.     

Existence and uniqueness of solutions of differential equations with respect to non-additive measures

By taking Sugeno-derivative into account, first, we investigate the existence of solutions to the initial value problems (IVP) of first-order differential equations with respect to non-additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive non-decreasing solutions on cones in semi-order Banach spaces. In addition, we also use Picard–Lindelöf theorem to prove the existence and uniqueness of the solution of the equation. Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix-order differential equation with respect to non-additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive non-decreasing solution. Examples with graphs are provided to validate the results.     

On properties of strong solutions of stochastic equations with respect to semimartingales

The paper is devoted to strong solutions of stochastic integral equations with respect to semimartingales. We study existence (for non-Lipschitz coefficients) and asymptotic behaviour of strong solutions and obtain also a number of results on weak and square mean convergence of these solutions.     

L p solutions of backward stochastic Volterra integral equations

This paper is devoted to the unique solvability of backward stochastic Volterra integral equations (BSVIEs, for short), in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of M-solutions of BSVIEs in L ^p (1 < p < 2), which extends the existing results on M-solutions. The unique solvability of adapted solutions of BSVIEs in L ^p (p > 1) is also considered, which also generalizes the results in the existing literature.     

Existence of almost periodic solutions to some stochastic differential equations

The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of square-mean almost periodic solutions to some semilinear stochastic equations.     

Modulus of continuity of solutions of second-order elliptic equations on a plane

We find the exponents in the Hölder boundary condition for generalized solutions of a secondorder elliptic equation of divergent form on a plane with coefficients from L_∞. The accuracy of the formula obtained is verified with an example.     

Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow

In this paper, we prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local flow. We also give a stability result with respect to various parameters.     

Solving nonlinear equations by differentiation with respect to a parameter

Differentiation with respect to a parameter is proposed as a method for solving a system of nonlinear equations with a poorly conditioned Jacobian matrix. Singular and polar decompositions of the Jacobian matrix lead to a system of ordinary differential equations with small parameters multiplying the derivatives. This system is solved by introducing auxiliary linear differential equations. The numerical solution of a poorly conditioned system is reduced in this case to the approximation of an exponential with a large negative argument.     

Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay

This paper investigates the existence and uniqueness theorem of solutions to neutral stochastic differential equations with infinite delay (short for INSFDEs) at a space BC((-∞,0];R^d). Under the uniform Lipschitz condition, linear growth condition is weaken to obtain the moment estimate of the solution for INSFDEs. Furthermore, the existence, uniqueness theorem of the solution for INSFDEs is derived, and the estimate for the error between approximate solution and exact solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence, uniqueness theorem is also valid for INSFDEs on [t₀,T]. Moreover, the existence, uniqueness theorem still holds on interval [t₀,∞), where t₀∈R is an arbitrary real number.     

含参数泛函微分方程概周期正解的存在性

研究了一类含参数泛函微分方程概周期正解的存在性问题.结合有界性及渐近概周期性获得了系统存在概周期正解的几组充分条件,并将结果应用于几类种群动力学模型,分别获得了系统在概周期环境下存在概周期解的一组充分条件.