ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS OF A CERTAIN CLASS OF NON-LINEAR SINGULAR INTEGRAL EQUATIONS

In this study, the existence of a solution of the non-linear singular integral equation system w(z) = f(1) (z, w(z), h(z), T(G)g(1)( ., w(.), h(.))(z), Pi(Gg1)(.,w(.), h(.))(z)), h(z) = f(2) (z, w(z), h(z),T(G)g(2)(., w(.), h(.))(z), Pi(G)g(2)(., w(.),h(.))(z)), has been investigated. This system is more general than the one w(z) = f(1) (z, w(z), h(z),T(G)g(1)(.,w( .), h(.))(z)), h(z) = f(2) (z, w(z), h(z), Pi(G)g(2)(.,w(.), h( .))(z)), studied by Musayev and Duz (Existence and uniqueness theorems for a certain class of non linear singular integral equations SJAM 10(1), 3-18, 2009). Here, T(G) f(Z) and Pi(G)f(z) are the Vekua integral operators defined by T(G)f(z) = -1/pi integral(G)integral f(zeta)/zeta - z d xi d eta, Pi(G)f(z) = -1/pi integral(G)integral f(zeta)/(zeta - z)(2) d xi d eta.