摘要
设X为Banach空间,A:D(A)X→X为可闭的K一正定算子满足D(A)=D(K),则存在常数β>0,x∈D(A),‖Ax‖≤β‖Kx‖,而且方程Ax=f(f∈x)有唯一解.设{c_n}≥0为[0,1] 中实数列,定义迭代序列{x_n}n≥0 如下: x_0 D(A), x_(n+1)=x_n+c_ny_n,n≥0, y_n=k^(-1)f-k^(-1)Ax_n,n≥0,则{x_n}n≥0强收敛于方程Ax=f的唯一解.
Let X be a Banach space, and A:D(A)(?)X→X a closcable and K-positive definite operator with D(A) = D(K) . Then there exists a constant β>0 such that for any x∈D(A), || Ax ||≤β||Kx|| . Furthermore, the operator A is closed, R(A)=X, and the equation Ax=f, for any f∈X, has a unique solution. Let {cn}n≥0 be a real sequence in [0,1], Define the sequence {xn}n≥0 iteratively by ( I ) xn+1 = xn+ cnyn,yn=K-1f-K-1Axn , with x0∈E D(A) . It is proved that the scquence ( I ) converges strongly to the unique solution of the equationin Ax=f in X.
