Solvability and Iterative Construction of Solution for a Class of K -Positive Definite Operator Equations in Banach Spaces
Received:August 31, 2001  
Key Words: K-positive definite operator   closeable operator   solvability   iterative construction.  
Fund Project:
Author NameAffiliation
GAO Gai-liang Inst. of Appl. Math. & Mech.
Ordnance Engineering College
Shijiazhuang 050003. China 
ZHOU Hai-yun Inst. of Appl. Math. & Mech.
Ordnance Engineering College
Shijiazhuang 050003. China 
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Abstract:
      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∈D(A) . It is proved that the scquence ( I ) converges strongly to the unique solution of the equationin Ax=f in X.
Citation:
DOI:10.3770/j.issn:1000-341X.2004.01.024
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