Semi-Fredholm problems on periodic solutions of functional-differential equations of neutral type

We consider the problem on periodic solutions for linear systems of functionaldifferential equations of neutral type with periodic coefficients and periodic deviations of the argument. By reduction to associated functional equations, we derive necessary and sufficient conditions under which the problem on periodic solutions for such a system is Fredholm or semi-Fredholm.     

On estimates of solutions to systems of differential equations of neutral type with periodic coefficients

We study the systems of differential equations of neutral type with periodic coefficients. We establish sufficient conditions for the asymptotic stability of the zero solution and obtain estimates for solutions which characterize the decay rate at infinity.     

Periodic solutions for functional-differential equations of mixed type

In this paper we study the existence of ω-periodic solutions for some functional-differential equations of mixed type. Among the main results are the averaging principle and existence theorems for some equations with homogeneous nonlinearities. We use here the coincidence degree theory of Mawhin.     

Oscillation of the solutions of a certain class of first-order functional-differential equations of neutral type

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1370–1375, October, 1989.     

Stability analysis of\theta -methods for neutral functional-differential equations

Summary. This paper deals with the subject of numerical stability for the neutral functional-differential equation It is proved that numerical solutions generated by -methods are convergent if . However, our numerical experiment suggests that they are divergent when is large. In order to obtain convergent numerical solutions when , we use -methods to obtain approximants to some high order derivative of the exact solution, then we use the Taylor expansion with integral remainder to obtain approximants to the exact solution. Since the equation under consideration has unbounded time lags, it is in general difficult to investigate numerically the long time dynamical behaviour of the exact solution due to limited computer (random access) memory. To avoid this problem we transform the equation under consideration into a neutral equation with constant time lags. Using the later equation as a test model, we prove that the linear -method is -stable, i.e., the numerical solution tends to zero for any constant stepsize as long as and , if and only if , and that the one-leg -method is -stable if . We also find out that inappropriate stepsize causes spurious solution in the marginal case where and . Received May 6, 1994     

Periodic solutions of functional-differential equations of point type

We study periodic solutions of a functional-differential equation of point type. We state conditions for the existence and uniqueness of an ω-periodic solution of the original nonlinear functional-differential equation of point type. An iterative process for constructing such a solution is described, and its convergence rate is estimated.     

Entire solutions of a certain type of functional-differential equations

In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonlinear differential equations.     

Equivalence of functional-differential equations of neutral type and abstract Cauchy problems

Equivalence of functional-differential equations of neutral type to well-posed abstract Cauchy problems in the state spaces ⁿ }L ^p ,L ^p resp.C is investigated. The assumptions allow non-atomic difference operators.This work was supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austria) under Grant No. P4534.Work done by this author was done during a stay at the University of Graz from October 1981 till April 1984.     

On the trajectories of generalized functional-differential systems of neutral type

Some sufficient conditions are presented for the compactness of the set of all relaxed trajectories of generalized functional-differential equations of the form , whereF is a multivalued mapping, with values that are nonempty compact subsets of then-dimensional Euclidean space. Furthermore, it is shown that the set of all trajectories is dense in the set of all relaxed trajectories. Finally, some remarks on the application of the above results to optimization problems are given.     

Stability estimates for linear stochastic systems with deviating argument of neutral type

We study linear stochastic differential equations with deviating argument of neutral type and establish sufficient conditions of stability. The functions determining the initial perturbations of solutions are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 834–842, June, 1993.