A-ergodic nets and A-regularized approximation processes of operators are introduced and their convergence theorems are discussed. There are strong convergence theorems, uniform convergence theorems, theorems on optimal convergence, and theorems on non-optimal convergence and its sharpness. The general results provide unified approaches to investigation of convergence rates of ergodic limits an...

In this paper, we prove the existence and uniqueness of a nonlinear perturbed stochastic fractional integro-differential equation of Volterra-Itô type involving nonlocal initial condition by using the theory of admissibility of integral operator and Banach fixed-point principle. Also the stability and boundedness of the second moments of the stochastic solution are studied. In addition, an appl...

A collocation method based on optimal nodal splines is presented for the numerical solution of linear Volterra integral equations of the second kind with weakly singular kernel. Since the considered spline operator is a bounded projector we can prove that, for sequences of locally uniform meshes, the approximate solution error converges to zero at exactly the same optimal rate as the spline app...

We present a time-continuous identification method for nonlinear dynamic Volterra models of the form HX = f(u,X)+v with H a causal convolution operator. It is mainly based on a suitable parameterization of H deduced from the so-called diffusive representation, which is devoted to state representations of integral operators. Following this approach, the complex dynamic nature of H can be summari...

A method is used to solve the Fredholm-Volterra integral equation of the first kind in the space L2(Ω)×C(0,T ),Ω = {(x,y) : √ x2+y2 ≤ a}, z = 0, and T <∞. The kernel of the Fredholm integral term considered in the generalized potential form belongs to the class C([Ω]×[Ω]), while the kernel of Volterra integral term is a positive and continuous function that belongs to the class C[0,T ]. Also in...