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

Abstract

In this study, the existence of a solution of the non-linear singular integral equation system w(z) = f1 z, w(z), h(z), TGg1( · , w( · ), h( · ))(z), ΠGg1( · , w( · ), h( · ))(z) , h(z) = f2 z, w(z), h(z), TGg2( · , w( · ), h( · ))(z), ΠGg2( · , w( · ), h( · ))(z) ,
has been investigated. This system is more general than the onew(z) = f1z, w(z), h(z), TGg1( · , w( · ), h( · ))(z) , h(z) = f2 z, w(z), h(z), ΠGg2( · , 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, TGf(z) and ΠGf(z) are the Vekua integral operators defined by TGf(z) = − 1 π ZZ G f(ς) ς − z dξ dη, ΠGf(z) = − 1 π ZZ G f(ς) (ς − z) 2 dξ d

Keywords

• Holder continui,, Fixed point theor,, Contraction mappi,2000 AMS Classification: 45 E 05

References

1. Altun, I., Koca, K. and Musayev, B., Existence and uniqueness theorems for a certain class of non-linear singular integral equations, Complex Var. Elliptic Equ. 51 (2), 181–195, 2006.
2. Huseynov, A. I. and Muhtarov, H. S., Vvedeniye v teoriyu nelineynh sigulyainh integralnh uravneny(Moscow: Nauka, 1980).
3. Musayev, B. and D¨uz, M., Existence and uniquness theorems for a certain class of non linear singular integral equations, Sel¸cuk J. Appl. Math. 10 (1), 3–18, 2009.
4. Tutschke, W., L¨osung nichtlinearer partieller Differentialgleichungssysteme erster Ordnung in der Ebene durch Verwendung einer komplexen Normalform, Mathematische Nachrichten 75, 283–298, 1976.