On a Class of Nonlinear Finite-Part Singular Integral Equations with Carleman Shift

In this study, the existence and uniqueness of the solution of a class of nonlinear finite-part singular integral equations with Carleman shift preserving orientation has been investigated in the generalized Holder space () Γ ϕ H. 1. Introduction Many important problems of engineering mechanics like elasticity, plasticity, and fracture mechanics and aerodynamics can be reduced to the solution of a non-linear singular integral equation or non-linear finite-part singular integral equations. Hence, since these are connected with a wide range of problems of an applied character. The theory of non-linear singular integral equations and non-linear finite-part singular integral equations seems to be particularly complicated if closely linked with applied mechanics problems. Having in mind the implications for different problems of engineering mechanics, E.G.Ladopoulos [15-17] and E.G.Ladopoulos and V.A.Zisis [12-14] introduced and investigated non-linear singular integral equations and non-linear finite-part singular integral equations. This type of non-linear equations has been applied to many problems of structural analysis, fluid mechanics and aerodynamics. The theory of nonlinear singular integral equations with Hilbert and Cauchy kernel and its related Riemann-Hilbert problems have been developed in works of Pogorozelski W. The successful development of the theory of singular integral equations naturally stimulated the study of singular integral equations with shift. The Noether theory of singular integral operators with shift is developed for a closed and open contour (see [9-11], 18] and others).