拟线性算子广义逆的扰动及其在Banach流形中的应用

算子广义逆理论是在不适定的线性问题的背景下产生的，此理论在数值线性代数、优化与控制、统计学及应用数学等领域有重要应用. 本项目以Banach空间中拟线性算子广义逆的扰动及稳定性与Banach流形带参数的广义横截性为研究内容，以Banach空间几何理论与算子广义逆的稳定性特征为研究工具，(1)将线性算子的Banach引理推广到有界齐性算子，应用Banach空间中广义正交分解定理、拟线性投影的拟可加性等理论研究拟线性广义逆的扰动及稳定性，进而给出拟线性广义逆的表示及范数估计和误差估计；(2) 给出Banach流形间Fredholm映射的秩定理，应用广义逆稳定性的特征将Fredholm映射的带参数的横截性定理进行推广，给出一类Banach流形带参数的广义横截性定理.

The theory of generalized inverse has its genetic roots essentially in the context of so called ill-posed linear problems,there are some excellent applications in some fields of numerical linear algebra,optimization and control,statistics,and other areas of applied mathematics.Using geometry theory of Banach spaces and the characteristic of stabilities of generalized inverses of operators,this project describes the perturbations and stabilities of quasi-linear generalized inverses of bounded linear operators and generalized transversality with parameter in Banach manifolds.Firstly,Banach lemma of linear operators will be generalized to bounded homogeneous operators.Perturbations and stabilities of quasi-linear generalized inverses of bounded linear operators are investigated by means of some theories of generalized orthogonal decomposition theorem and the property of quasi-additive of quasi-linear projector etc.Moreover,expressed in the form to quasi-linear generalized inverses and some estimates of the norm and error are given.Secondly,a rank theorem of Fredhoml operators between Banach manifolds is introduced.We generalized the transversality theorem with parameter by means of a characteristic of stability of generalized inverse.Furthermore,a kind of generalized transversality with parameter in Banach manifolds is given.