关于一个积分方程解的存在唯一性证明

Proof of Existence and Uniqueness for the Solution of Integrated Equation

对于一个积分方程,研究其解的存在唯一性是十分重要的。用Picard逼近法和Banach不动点定理证明给定的积分方程φ,当|λ|足够小时,该方程在[a,b]上存在唯一的连续解。Picard逼近法的要点是建立一个逼近序列,然后考察这个序列取值范围、一致收敛性和极限的存在唯一性。应用Banach不动点定理的要点是：首先建立一个压缩映射,然后再考察其解空间的完备性。

It is important for a given integrated equation to research the solution＇ s existence and uniqueness of integrated equation. Inthis paper, when｜λ｜ is small enough, we prove that this equation φ（x）=f（x）＋λ∫a bK（x,ξ）φ（ξ）dξ, exists a unique eontinuous solution by using Picard＇ s asymptotic approximation and theory of Banach fixed point, respectively. The former need establish an asymptotic sequence, research domain, convergence and limit of the sequence. The latter need construct a compressed map and then prove that itg complete.