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We give a new proof of the Ostapenko–Tarasov unicellularity theorem for the classical Volterra integration operator on the space . [ABSTRACT FROM AUTHOR]

The problem of the factorization of the Wiener–Hopf integral operator in the form of the product of the upper and lower Volterra operators is considered. Conditions for the existence of such a factorization are obtained. The application of this factorization to Wiener–Hopf integral equations of the first kind can reduce the study of certain classes of such equations to that of the corresponding Volterra equations of the first kind. [ABSTRACT FROM AUTHOR]

In this paper, we introduce uniform Lotka–Volterra operators and construct Lyapunov functions for them. We establish that the ergodic averages associated with operators of such kind diverge. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for the existence and uniqueness of continuous solutions of Volterra operator equations of the first kind with piecewise determined kernels. For the case in which the solution is not unique, we prove existence theorems for the parametric families of solutions and present their asymptotics in the form of logarithmic polynomials. [ABSTRACT FROM AUTHOR]

For operators acting in the Lebesgue space L(Π), 1 < q < ∞, an abstract analog of Bihari's lemma is stated and proved. We show that it can be used to derive a uniform pointwise estimate of the increment of the solution of a controlled functional-operator equation in the Lebesgue space. The procedure of reducing controlled initial boundary-value problems to this equation is illustrated by the Goursat-Darboux problem. [ABSTRACT FROM AUTHOR]

We consider a singularly perturbed Fredholm integro-differential equation with a rapidly varying kernel. We derive an algorithmfor constructing regularized asymptotic solutions. It is shown that, given a rapidly decreasing multiplier of the kernel, the original problem does no involve the spectrum (i.e., it is solvable for any right-hand side). [ABSTRACT FROM AUTHOR]

We establish the similarity between certain Volterra integral operators arid the Riemann-Liouville fractional integration operator as well as the existence of a triangular transformation operator for integro-differential equations of fractional order. The results obtained are consistent with similar results for the case of integer order. [ABSTRACT FROM AUTHOR]