一致光滑Banach空间中一类K-正定算子方程的可解性及其迭代构造

Solvability and Iterative Construction of Solutions for a Class of K- Positive Definite Operator Equations in Uniformly Smooth Banach Spaces

中文关键词:  K-正定算子  可解性  迭代构造  一致光滑Banach空间

英文关键词:K-positive definite operator  solvability  iterative construction  uniformly smooth Banach space.

中文摘要:

      设X为一致光滑Banach空间,A:D（A）?X→X为K-正定算子满足D（A）=D（K）,则存在常数β>0使得?x∈D（A）,||∧x||≤β||Kx||而且?f∈X,方程∧x=f有唯一解;设{a_n}_n≥0为[0,1]中的实数列满足（i）a_n→0（n→∞）,（ii）∑_n=0^∞a_n=∞, ?x₀∈D（A）,迭代地定义序列{x_n}_n≥0≥0如下:(?)则{x_n}_n≥0强收敛于方程Ax=f的唯一解.

英文摘要:

      Let X be a uniformly smooth Banach space and let A:D(A)?X→X be a K-positive definite operator with D(A) = D(K) . Then there exists a constant β>0 such that for every x∈D(A) , ||Ax||≤β||Kx||. Furthermore, the operator A is closed, R(A) = A , and the equation Ax =f , for each f∈X, has a uniqne solution. Let {a_n}_n≥0 be a real sequence in [0,1] satisfying conditions; （i）a_n→0（n→∞）; and （ii）∑_n=0^∞a_n=∞, . Define the sequence {a_n}_n≥0 iteratively by (?) Then the sequence {a_n}_n≥0 defined by (*) converges strongly to the unique solution of the equation Ax=f in X.