# CONVERGENCE THEOREMS FOR THE ZEROS OF A FINITE FAMILY OF GENERALIZED ACCRETIVE OPERATORS

Volume 2, Issue 4, pp 260-269
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### Authors

N. GURUDWAN - School of Studies in Mathematics, Pt. Ravishankar Shukla University Raipur - 492010 (C.G.), India. B. K. SHARMA - School of Studies in Mathematics, Pt. Ravishankar Shukla University Raipur - 492010 (C.G.), India.

### Abstract

A strong convergence theorem for the common zero for a finite family of Generalized Lipschitz operators in a uniformly smooth Banach space is proved when atleast one of the operator is Generalized $\Phi$- accretive, using a new iteration formula. Similar result for Generalized Lipschitz and Generalized $\Phi$- pseudocontractive map is also proved. Our result extends the convergence results of Chidume [4] to a finite family improving many other results.

### Keywords

• Generalized $\Phi$-accretive
• generalized Lipschitz
• uniformly smooth Banach space
• mann iteration.

•  47H06
•  47H09

### References

• [1] F. E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach space, Bull. Amer. Math. Soc., 73 (1967), 875–882

• [2] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl.,, 20 (1967), 197–228

• [3] S. S. Chang, K. K. Tan, H. W. J. Lee, C. K. Chan, On the convergnce of implicit iteration process wih error for a finite family of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 313 (2006), 273–283

• [4] C. E. Chidume, C. O. Chidume, Convergence theorems for zeros of generalized Lipschitz generalized $\Phi$-quasi accretive operators, Proc. Amer. Math. Soc., 134 (2006), 243–251

• [5] H. Hirano, Z. Huang, Convergence theorems for multi-valued $\phi$-hemicontractive operators and - strongly accretive operators, Comp. Math. Appl., 46 (2003), 1461–1471

• [6] J. S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 302 (2005), 509–520

• [7] R. De Marr, Common fixed points for commuting contraction mappings, Pacific J. Math., 53 (1974), 487–493

• [8] A. Markov, Quelques theorems sur les ensembles abeliens, , Dokl. Akad. Nauk. SSSR (N.S.)., 10 (1936), 311–314

• [9] W. V. Petryshyn, A characterization of strict convexity of Banach spaces and other uses of duality mappings, J. Func. Anal., 6 (1970), 282–291

• [10] Z. Sun, Strong convergence of an implicit ieration process for a finite family of asymptotically quasinonexpansive mappings, J. Math. Anal. Appl., 286 (2003), 351–358

• [11] Y. Xu, Ishikawa and Mann iterative processses with erors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl., 224 (1998), 91–101

• [12] L. C. Zhao, S. S. Chang , Strong convergence theorems for equilibrium problems and fixed point problems, J. Nonlinear Sci. Appl., 2 (2009), 78–91

• [13] H. Zhou, L. Wei, Y. J. Cho , Strong convrgence theorems on an iteraive method for family of finite nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., 173 (2006), 196–212