Approximation of Semilinear Fractional Cauchy Problem

The semidiscretization methods for solving the Cauchy problem $(\mathbf {D}_{t}^{\alpha }u)(t) = A u(t) + J^{1-\alpha } f\big (t,u(t)\big ), \quad t \in [0,T], 0 < \alpha <1,\qquad u(0) = u^0,$ with operator A, which generates an analytic and compact resolution family ${\lbrace S_{\alpha }(t,A)\rbrace _{t\ge 0}}$, in a Banach space E are presented. It is proved that the compact convergence of resolvents implies the convergence of semidiscrete approximations to an exact solution. We give an analysis of a general approximation scheme, which includes finite differences and projective methods.

Right simple subsemigroups and right subgroups of compact convergence semigroups

Clifford and Preston (1961) showed several important characterizations of right groups. It was shown in Roy and So (1998) that, among topological semigroups, compact right simple or left cancellative semigroups are in fact right groups, and the closure of a right simple subsemigroup of a compact semigroup is always a right subgroup. In this paper, it is shown that such results can be generalized in convergence semigroups. In the discussion of maximal right simple subsemigroups and maximal right subgroups of semigroups, generalization of the results that no two maximal right simple subsemigroups and maximal right subgroups of a convergence semigroup intersect, is also established.

Some results on compact convergence semigroups defined by filters

In this paper the concept of convergence defined by filters is used and applied in the study of semigroups. Special emphasis is placed on compact convergence semigroups and their properties.

Topologies between compact and uniform convergence on function spaces

This paper studies two topologies on the set of all continuous real-valued functions on a Tychonoff space which lie between the topologies of compact convergence and uniform convergence.

A Regular Summability Method which Sums the Geometric Series to its Proper Value in the Whole Complex Plane

In this paper an explicit regular sequence-to-sequence summability method is presented which sums the geometric series to the value 1/(1-z) in all of ℂ\{1} and to infinity at the point 1. The method also provides compact convergence in ℂ \ [ 1, ∞) and therefore improves well-known results by Le Roy, Lindelöf and Mittag-Leffler.