FIXED POINTS OF A CERTAIN CLASS OF ASYMPTOTICALLY REGULAR MAPPINGS

Abstract

In this paper, we study in Banach spaces the existence of fixed points of asymptotically regular mapping T satisfying: for each x, y in the domain and for n=1, 2,…, $$\parallelT^nx-T^ny\parallel\leq$\leq$a_n\parallelx-y\parallel+b_n (\parallelx-T^nx\parallel+\parallely-T^ny\parallely)$$ where $a_n,\; b_n,\; C_n$ are nonnegative constants satisfying certain conditions. We also establish some fixed point theorems for these mappings in a Hibert space, in L(sup)p spaces, in Hardy space H(sup)p, and in Soboleve space $H^{k,p} for 1<\rho<\infty \; and \; k\geq0$. We extend results from papers [10], [11], and others.

Keywords

• normal structure;
• $\rho-uniformly$ convex Banach space;
• asymptotic regularity;
• fixed points

File

Download PDF

References

1. An Introduction to the Theory of Distribution J. Barros-Neto
2. Bull. Amer. Math. Soc. v.72 The solution by iteration of nonlinear functional equations in Banach spaces F. E. Browder;W. V. Petryshyn
3. Pacific J. Math. v.86 Normal structure coefficients for Banach spaces W. L. Bynum
4. Nonlinear Anal. v.9 Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure E. Casini;E. Maluta
5. Theory of Nonlinear Operators On densifying and related mappings and their applications in nonlinear functional analysis J. Danes
6. Linear Operators v.Ⅰ N. Dunford;J. Schwarz
7. Theory of Hp spaces W. L. Duren
8. Proc. Amer. Math. Soc. v.35 A fixed point theorem for asymptotically nonexpansive mappings K. Geobel;W. A. Kirk
9. Cambridge stud. Adv. Math. v.28 Topics in Metric Fixed Point Theory K. Goebel;W. A. Kirk
10. Nonlinear Anal. v.17 Fixed point theorems for asymptotically regular mappings in Lp spaces J. Gornicki
11. Math. Slovacs v.43 Fixed points of asymptotically regular mappings J. Gornicki
12. J. Math. Anal. Appl. v.154 On some Lp inequalities in best approximation theory T. C. Lim
13. Progress in Approximation Theory An Lp inequality and its applications to fixed point theory and approximation theory T. C. Lim;H. K. Ku;Z. B. Xu
14. Canad. math. Bull. v.30 A uniformly asymptotically regular mapping without fixed points P. K. Lin
15. Classical Banach spaces II-Function Spaces J. Lindenstrauss;L. Tzafriri
16. Mat. Zametki v.43 Jung's constant of the space Lp (Russian) S. A. Pichugov
17. Atti. Sem. Nat. Fis. Univ. Moclens v.38 On Bynum's fixed point theorem S. Prus
18. Rend. Circ. Mat. Palermo v.(2)XL Some estimates for the normal structure coefficients in Banach spaces S. Prus
19. J. Math. Anal. Appl. v.121 Strongly unique best approximations and centers in uniformly convex spaces B. Prus;R. Smarzewski
20. J. Approx. Theory v.51 Strongly unique best approximations in Banach spaces Ⅱ R. Smarzewski
21. J. Math. Anal. Appl. v.150 On the inequality of Bynum and Drew R. Smarzewski
22. Nonlinear Anal. v.16 Inequalities in Banach spaces with applications H. K. Xu
23. J. Math. Anal. Appl. v.95 On uniformly convex function C. Zalinescu