No abstract

Non-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations

In this paper fast implicit and explicit Runge-Kutta methods for systems of Volterra integral equations of Hammerstein type are constructed. The coefficients of the methods are expressed in terms of the values of the Laplace transform of the kernel. These methods have been suitably constructed in order to be implemented in an efficient way, thus leading to a very low computational cost both in time and in space. The order of convergence of the constructed methods is studied. The numerical experiments confirm the expected accuracy and computational cost. (2000): 65R20, 45D05, 44A35, 44A10. AMS subject classification

No abstract

Abstract. The notion of even-outer-semicontinuity for set-valued maps is introduced and compared with related ones from [4] and [11]. The coincidence of these notions provides a new characterization of compactness and of local compactness. The following result is proved: Let X be a topological space, Y a uniform space, {Fσ : σ ∈ Σ} be a net of set-valued maps from X to Y and F be a set valued map from X to Y . Then any two of the following conditions imply the third: (1) the net {Fσ : σ ∈ Σ} is evenly-outer semicontinuous; (2) the net {Fσ : σ ∈ Σ} is graph convergent to F ; (3) the net {Fσ : σ ∈ Σ} is pointwise convergent to F . This theorem generalizes some results from [4] and [11].Graph convergence (that is Painlevé-Kuratowski convergence of graphs) of set-valued maps was studied in many books and papers (see for example [1,2,4,9,11]). In this topic we can include also graph convergence of single-valued maps [5,12,13], epiconvergence of lower semicontinuous functions [4,6,7] as well as Painlevé-Kuratowski convergence of graphs of partial maps [8] . In the books of Attouch [1], Aubin-Frankowska [2] 1991 Mathematics Subject Classification. 54C60, 54B20.