Let a, b, c and d be functions in L2=L2(T, dq/2p), where T denotes the unit circle. Let P denote the set of all trigonometric polynomials. Suppose the singular INTEGRAL operators A and B are defined by A=aP+bQ and B=cP+dQ on P, where P is an analytic projection and Q=I-P is a co-analytic projection. In this paper, we use the Helson–Szego type set (HS)(r) to establish the condition of a, b, c and d satisfying ||Af||2³||Bf||2 for all f in P. If a, b, c and d are bounded measurable functions, then A and B are bounded operators, and this is equivalent to that B is majorized by A on L2, i.e., A*A³B*B on L2. Applications are then presented for the majorization of singular INTEGRAL operators on weighted L2 spaces, and for the normal singular INTEGRAL operators aP+bQ on L2 when a−b is a complex constant.

In this article, we apply two new fixed point theorems to investigate the existence of mild solutions for a nonlocal fractional CAUCHY problem with an INTEGRAL initial condition in Banach spaces.

to view the abstract, pleseas click an the PDF icon 

Purpose: The purpose of this work is to present an approximation for singular INTEGRALs of CAUCHY type kernel on a smooth-oriented contour.Methods: For the method, we use a small modification of the spline functions in order to eliminate the singularity. Noting that this approximation is due to the idea of Sanikidze’s Approximate solution of singular INTEGRAL equations in the case of closed contours of integrations.Results: This approximation represents a good approach for any singular INTEGRAL given on the curve in the sense of CAUCHY principal value.Conclusions: This approximation is destined to solve numerically the singular INTEGRAL equations with CAUCHY type kernel on a smooth-oriented contour.