Iterative solution of nonlinear equations with $ \phi $-strongly accretive operators

Abstract

Suppose that X is an arbitrary real Banach space and T : X→X is a Lipschitzian and \( \phi \)-strongly accretive operator. For any given \( f \in X \), define a mapping S : X→X by Sx=f+x−Tx for all \( x \in X \). Define the sequence \( \{x_n\}^{\infty}_{n=0} \) in X iteratively by¶¶\( \cases{x_0 \in X,\cr x_{n+1}=a_nx_n +b_n Sy_n +c_nu_n,\cr y_n = a'_nx_n +b'_nSx_n +{c'}_{\!\!n}v_n, \quad n \geqq 0,} \)¶¶ where \( \{u_n\}^{\infty}_{n=0} \), \( \{v_n\}^{\infty}_{n=0} \) are arbitrary bounded sequences in X and \( \{a_n\}^{\infty}_{n=0} \), \( \{b_n\}^{\infty}_{n=0} \), \( \{c_n\}^{\infty}_{n=0} \),¶\( \{a'_n\}^{\infty}_{n=0} \), \( \{b'_n\}^{\infty}_{n=0} \) and \( \{c'_n\}^{\infty}_{n=0} \) are real sequences in [0,1] satisfying the following conditions:¶¶(i) \( a_n+b_n+c_n=a'_n+b'_n+{c'}_{\!\!n}=1$, $n \geqq 0, \)¶¶ (ii) \( \sum\limits_{n=0}^\infty c_n \) < \( \infty \), \( \sum\limits_{n=0}^\infty b_n b'_n \) < \( \infty \),¶\( \sum\limits_{n=0}^\infty b_n {c'}_{\!\!n} \) < \( \infty \), \( \sum\limits_{n=0}^\infty b_n^2 \) < \(\infty\),¶¶ (iii) \( \sum\limits_{n=0}^\infty b_n =+\infty \).¶¶ Then the sequence \( \{x_n\}^{\infty}_{n=0} \) converges strongly to the unique solution of the equation Tx=f.