1. An Evolution System of Hankel ω-order Preserving Partial Contraction Mapping in the Hyperbolic Case

   A. Y. Akinyele¹, A. O. Owolanke², J. B. Omosowon³, A. O. Ajiboye⁴

   1. Department of Mathematics, University of Ilorin, Ilorin, Nigeria.
   2. Department of Mathematical Sciences, Olusegun Agagu University of Science and Technology, Okitipupa, Nigeria.
   3. Department of Mathematics, university of Ilorin, Ilorin, Nigeria.
   4. Department of Mathematics Science, Federal University Oye-Ekiti, Ekiti, Nigeria.

   Abstract:

   This study looked into the partial contraction mapping (ω-OPC_n) with Hankel order preservation. The initial value problem’s evolution system was created to demonstrate how the parabolic situation, in which each operator is supposed to be the infinitesimal generator of an analytic semigroup of a linear operator, differs from the hyperbolic example. Following that, it was determined that a C₀-semigroup {T(t)}_t≥0 on Banach space is generated infinitesimally by a partial contraction mapping that preserves ω-order. Additionally, we demonstrate that the family {A(t)}_t∈[0,T] satisfies the stability, admissibility, and continuity requirements, which led to the conclusion that there exists a singular evolution system in Banach space.

   Keywords: ω-OPC_n, Analytic semigroup, C₀-semigroup, Evolution system.

   Pages: 1 – 15 | Full PDF Paper

2. Once Again About the Cauchy Type Integral

   Ioseb D. Machavariani

   Muskhelishvili Institute of Computational Mathematics of the Georgian Technical University, 0159 Tbilisi, Georgia.

   Abstract:

   In this article it is shown that the expression considered in the articles [1,2,4], called by A.P. Calderon the Cauchy operator has nothing in common with the classical Cauchy operator and the results of the above works are erroneously formulated. We indicate a geometric condition on the rectifiable Jordan curve Γ whose fulfilment is necessary and sufficient for the existence of finite angular boundary values and a Cauchy type integral of summable density for almost all points of the curve Γ.

   Keywords: singular integral, Cauchy kernel.

   Pages: 16 – 25 | Full PDF Paper