IJPAM: Volume 25, No. 1 (2005)

APPROXIMATION OF A SOLUTION FOR A $K$-POSITIVE
DEFINITE OPERATOR EQUATION IN REAL
UNIFORMLY SMOOTH BANACH SPACES

Zeqing Liu$^1$, Jeong Sheok Ume$^2$, Shin Min Kang$^3$
$^1$Department of Mathematics
Liaoning Normal University
P.O. Box 200, Dalian, Liaoning, 116029, P.R. CHINA
e-mail: zeqingliu@sina.com.cn
$^2$Department of Applied Mathematics
Changwon National University
Changwon, 641-773, KOREA
e-mail: jsume@changwon.ac.kr
$^3$Department of Mathematics
Research Institute of Natural Science
Gyeongsang National University
Chinju, 660-701, KOREA
e-mail: smkang@nongae.gsnu.ac.kr


Abstract.Suppose that $X$ be a real uniformly smooth Banach space and $A:D(A)\subseteq X\to X$ is a $K$-positive definite operator with $D(A)=D(K)$. It is proved that the iteration scheme introduced by Bai (see J. Math. Anal. Appl., 236 (1999), 236-242), for arbitrary initial vector in $D(A)$ and for any $f\in X$, converges strongly to the unique solution of the equation $Ax=f$. Moreover, our iteration parameters are completely independent of the geometry of the underlying Banach space and of any property of the operator. The results in this note extend, improve and unify the results due to Bai, Chidume and Aneke, Chidume and Osilike.

Received: October 15, 2005

AMS Subject Classification: 47H14, 47H05, 47H10

Key Words and Phrases: $K$-positive definite operator, real uniformly smooth Banach space, convergence

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 25
Issue: 1