关键词: 非线性积分方程, 带平方根的Riemann 边值问题, 多连通区域, Plemelj 公式

Abstract:

In this paper, the problem discussed in  [1] is extended, that is, the general solution of the nonlinear singular  integral equation
a(t)Φ²(t)+b(t)/πi ∫_LΦ(τ)/τ-t dτ+c(t)=0,     t ∈L,
is solved in H\"older\ continuous space, where a(t),b(t), c(t) are polynomials and a(t)b(t) ≠0. The complex plane is divided into a region S⁺ and an open set(or a region)S^- by L, L being  a simple closed contour in the complex plane, or a simple arc, or a set of some simple arcs, or a  set of curves consisting of some simple closed contours and simple arcs. A new method is used which is different from that in [1], the problem is transformed to a Riemann boundary value problem, then it is solved.

Key words: Nonlinear integral equation, Riemann boundary value problem with square roots,  , Multiply connected region, Plemelj formula