Using the result of the error estimate of the simple extended Newton method established in the present paper for solving abstract inequality systems, we study the error bound property of approximate solutions of abstract inequality systems on Banach spaces with the involved function F being Fréchet differentiable and its derivative \(F^{\prime }\) satisfying the center-Lipschitz condition (not necessarily the Lipschitz condition) around a point x₀. Under some mild conditions, we establish results on the existence of the solutions, and the error bound properties for approximate solutions of abstract inequality systems. Applications of these results to finite/infinite systems of inequalities/equalities on Banach spaces are presented and the error bound properties of approximate solutions of finite/infinite systems of inequalities/equalities are also established. Our results extend the corresponding results in [3, 18, 19].

利用本文建立的简单扩展牛顿法的误差估计结果，解决抽象不等式系统，研究了Banach空间上抽象不等式系统近似解的误差界性质，其中所涉及的函数F是Fréchet可微的。围绕点x ₀满足中心Lipschitz条件（不一定是Lipschitz条件）的导数\（F ^ {\ prime} \）。在某些温和条件下，我们建立解的存在性结果，以及抽象不等式系统的近似解的误差界性质。提出了这些结果在Banach空间上不等式/等式的有限/无限系统的应用，并且还建立了不等式/等式的有限/无限系统的近似解的误差界性质。我们的结果扩展了[3，18，19]中的相应结果。