Convergence Theorems for φ-Strongly Accretive and φ-Hemicontractive Operators☆

Suppose that X is an arbitrary real Banach space and T: X → X is a continuous φ-strongly accretive operator. It is shown that the nonlinear equation Tx = f has a unique solution and under certain conditions both the Mann and Ishikawa iteration methods with errors introduced by Y. Xu (1998, J. Math. Anal. Appl.224, 91–101) converge strongly to the solution. A related result deals with the approximation of fixed points of φ-hemicontractive operators.

[1]  Tosio Kato,et al.  Nonlinear semigroups and evolution equations , 1967 .

[2]  M. O. Osilike,et al.  Iterative solutions of nonlinear f-strongly accretive operator equations in arbitrary Banach spaces , 1999 .

[3]  F. Browder,et al.  Construction of fixed points of nonlinear mappings in Hilbert space , 1967 .

[4]  Lei Deng,et al.  An iterative process for nonlinear lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces , 1993 .

[5]  Yuguang Xu,et al.  Ishikawa and Mann Iterative Processes with Errors for Nonlinear Strongly Accretive Operator Equations , 1998 .

[6]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[7]  L. Deng.  Iteration Processes for Nonlinear Lipschitzian Strongly Accretive Mappings in Lp Spaces , 1994 .

[8]  L. Deng.  On Chidume′s Open Questions , 1993 .

[9]  Hong-Kun Xu,et al.  Iterative Solutions to Nonlinear Equations of Strongly Accretive Operators in Banach Spaces , 1993 .

[10]  C. E. Chidume,et al.  Fixed Point Iterations For Strictly Hemi-Contractive Maps In Uniformly Smooth Banach Spaces , 1994 .

[11]  C. Morales,et al.  Surjectivity theorems for multi-valued mappings of accretive type , 1985 .

[12]  C. E. Chidume.  Approximation of fixed points of strongly pseudocontractive mappings , 1994 .

[13]  C. Chidume.  INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ITERATIVE SOLUTIONS OF NONLINEAR EQUATIONS IN SMOOTH BANACH SPACES , 2005 .

[14]  Felix E. Browder,et al.  Nonlinear mappings of nonexpansive and accretive type in Banach spaces , 1967 .

[15]  Klaus Deimling,et al.  Zeros of accretive operators , 1974 .

[16]  Lishan Liu,et al.  Ishikawa and Mann Iterative Process with Errors for Nonlinear Strongly Accretive Mappings in Banach Spaces , 1995 .

[17]  M. O. Osilike,et al.  Ishikawa and Mann Iteration Methods with Errors for Nonlinear Equations of the Accretive Type , 1997 .

[18]  Yeol Je Cho,et al.  Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in banach spaces , 1998 .

[19]  C. E. Chidume,et al.  Convergence Theorems for Strongly Pseudo-contractive and Strongly Accretive Maps , 1998 .

[20]  X. Ding,et al.  Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces , 1995 .

[21]  F. Browder,et al.  Nonlinear monotone and accretive operators in banach spaces. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[22]  M. O. Osilike,et al.  Iterative Solution of Nonlinear Equations of the Φ-Strongly Accretive Type , 1996 .

[23]  C. E. Chidume,et al.  INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ITERATIVE SOLUTION OF NONLINEAR EQUATIONS WITH STRONGLY ACCRETIVE OPERATORS , 2005 .

[24]  C. E. Chidume,et al.  Ishikawa Iteration Process for Nonlinear Lipschitz Strongly Accretive Mappings , 1995 .

[25]  Shih-Sen Chang,et al.  Some problems and results in the study of nonlinear analysis , 1997 .

[26]  C. Chidume,et al.  Iterative solutions of nonlinear accretive operator equations in arbitrary Banach spaces , 1999 .

[27]  C. Chidume.  An iterative process for nonlinear Lipschitzian strongly accretive mappings in Lp spaces , 1990 .

[28]  C. Chidume.  Global iteration schemes for strongly pseudo-contractive maps , 1998 .