Iterative solutions of \(K\)-positive definite operator equations in real uniformly smooth Banach spaces (Q1599711)

This article deals with Bai iterates \[ x_{n+1}= y_n+a_n (K^{-1}f- K^{-1}Ty_n),\;y_n=x_n+ b_n(K^{-1}f- K^{-1}Tx_n)\;(n=0,1,2, \dots) \] for approximate solutions of the equation \(Tx=f\) with a linear unbounded operator \(T:D(T)\subseteq X\to X\) in a real uniformly smooth Banach space \(X\) such that \[ \|x+y\|\leq\|x\|^2+ 2\bigl\langle y,J(x) \bigr \rangle +A\max\bigl \{\|x\|+|y\|,B \bigr\}\rho_X\bigl (\|y \|\bigr) \] for some \(A,B>0\); moreover, it is assumed that \(D(T)=D(K)\), \(\|Tx\|\leq \alpha\|Kx\|\), and \(\langle Tu,j(Ku) \rangle\geq c\|Ku\|^2\), \(c<0\), \(j(Ku)\in J(Ku)\) \((J\) is the dual operator from \(X\) to \(X^*)\) and \(\sum^\infty_{n=0} (a_n+b_n)= \infty\), \(\lim_{n\to \infty}a_n= \lim_{n\to \infty} b_n=0\), \(\max\{a_n,b_n\} \leq(2c)^{-1}\), \(\alpha A\max\{ (1+\alpha a_n) \|f-Tx_0\|\), \((1+\alpha b_n)\|f-Tx_0 \|,B\}\leq 2c\|f-Tx_0 \|\). Under these conditions, the sequence \(x_n\) strongly converges to the unique solution of the equation \(Tx=f\). This result generalizes some theorems of C. Bai, C. E. Chidume, S. J. Aneke, and C. E. Chidume and M. O. Osilike.