Study on Some Fixed Point Theorems for Bregman Nonexpansive Type Mapping in Banach Spaces
Chapter One
Aim and Objectives
The aim of this research is to establish some fixed point theorems for Bregman nonexpansive mapping in Banach spaces.
The aim will be achieved through the following objectives
- Construction of an iterative sequence for approximation of common fixed points of quasi-Bregman total asymptotically nonexpansive mappings.
- Development of a new hybrid iterative scheme and establishment of strong convergence theorem for quasi-Bregman total asymptotically strictly pseudocontractive mappings and equilibrium problems in reflexive Ba-nach spaces.
CHAPTER TWO
LITERATURE REVIEW
Asymptotically Nonexpansive Mappings
Gobel and Kirk (1972) intoroduced the class of asymptotically nonexpansive mappings as a generalization of the class of nonexpansive mappings. If C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is a self mapping of C which is asymptotically nonexpansive in the intermediate sense, then T has a fixed point, (Kirk, 1974). However the class of mappings which are asymptotically nonexpansive in the intermediate sense contains the class of asymptotically nonexpansive mappings.
A modified Mann iteration to approximate fixed points of asymptotically non-expansive mappings in uniformly convex Banach spaces was introduced by (Schu, 1991) Osilike and Aniagbosor (2000) and Shahzad and Udomene (2006) obtained weak and strong convergence theorem for finding a fixed point of asymptotically nonexpansive mappings.
A more general class of mappings called total asymptotically nonexpansive mappings was introduced by Albert et al. (2006) and studied method of ap-proximation of fixed points of mappings belonging to this class. Several au-thors are constructing iterative sequences for finding the fixed point of total asymptotically nonexpansive mappings.(such as Chidume and Ofoedu (2007) and Yolacan and Kizitunc (2012) )
Chidume and Ofoedu (2007) constructed the system (2.1.1) for the approxi tion of common fixed points of finite families of total asymptotically nonexpan-sive mappings, and gave necessary and sufficient conditions for the convergence of the scheme to common fixed points of the mappings in arbitrary real Banach spaces. A sufficient condition for convergence of the iteration process to a common fixed point of mappings under the same setting was also established in real uniformly convex Banach spaces.
Bregman Nonexpansive Mappings
Iterative methods for approximating fixed points of nonexpansive, quasi non-expansive mappings and their generalizations have been studied by various authors such as Browder (1967) and Halpern (1967), in Hilbert spaces. But extending this theory to Banach spaces encountered some difficulties because the useful examples of nonexpansive operators in Hilbert spaces are no longer nonexpansive in Banach spaces (e.g., the resolvent RA = (I rA) 1; for r > 0, of a monotone mapping A : C ! 2H and the metric projection PC onto a nonempty, closed, and convex subset C of H).
To overcome these difficulties, Bregman (1967) discovered an effective tech-nique using the so called Bregman distance function Df (:; 🙂 in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique was applied in various ways in order to design and analyze iterative algorithms for solving not only feasibility and optimization problems, but also algorithms for solving vari-ational inequality problems, equilibrium problems, fixed point problems for nonlinear mappings and so on [see (Bauschke and Borwein, 1997), (Butnariu and Resmerita, 2006) and (Zegeye and Shahzad, 2011)].
In recent years, several authors are constructing iterative sequences for find-ing the fixed point of nonlinear operators by using Bregman distance and the Bregman projection,
Zegeye and Shahzad (2014a) constructed an iterative sequence and proved a strong convergence theorem for a finite family of Bregman strongly nonexpan-sive mappings in the framework of a real reflexive Banach space. Furthermore they approximated a common zero of a finite family of maximal monotone mappings and solution of a finite convex feasibility problems in reflexive real Banach spaces. Alghamdi et al. (2014) introduced an iterative sequence and studied strong convergence for a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in the framework of a real reflexive Banach space. They proved a strong convergence of the iterative algorithms for find-ing a common solution of finite family of equilibrium problems and common zero of a finite family of maximal monotone mappings. Panc et al. (2014) proved weak convergence theorems for Bregman relatively nonexpansive map-pings. They proved a strong convergence of the iterative algorithms for find-ing a common solution of finite family of equilibrium problems and common zero of a finite family of maximal monotone mappings. Shahzad and Zeg-eye (2014) constructed an iterative sequence for approximation of a common fixed point of a finite family of multivalued Bregman relatively nonexpansive mappings. They proved that, the sequence generated converges strongly to a common fixed point of a finite family of multivalued Bregman nonexpansive mappings in reflexive real Banach spaces. Zegeye and Shahzad (2014b) proved a strong convergence theorem for a finite family of Bregman weak relatively nonexpansive mappings. Tomizawa (2014) introduced a new class of Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense. They established a strong convergence theorem of the shrinking method with the modified Mann iteration, to find fixed points of the mappings in reflexive Ba-nach spaces.
CHAPTER THREE
THEORY OF METHODS
Lemma 3.0.1 (Reich and Sabach, 2009) If f : E ! R is uniformly Frechet differentiable and bounded on a bounded subsets of E, then 5f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E . Concerning the Bregman projection, the following results are well known.
Lemma 3.0.2 (Butnariu and Resmerita, 2006) Let C be a nonempty, closed and convex subset of a reflexive Banach space E. Let f : E ! R be a Gateaux differentiable and totally convex function and let x 2 E: then:
(a) z = PCf (x) if and only if hy z; 5f(x) 5f(z)i 0; 8 y 2 C;
CHAPTER FOUR
MAIN RESULTS
Finite Families of Quasi-Bregman Total Asymp-totically Nonexpansive Mappings
Let C be a nonempty closed convex subset of a real Banach space E. Let
T1; T2; ; Tm : C ! C be m Bregman quasi-total asymptotically nonexpan-sive mappings. We define the iterative sequence fxng by.
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Summary
In this dissertation, various classes of mappings, namely, asymptotically nonex-pansive, total asymptotically nonexpansive mappings and total asymptotically strictly pseudocontractive mappings were studied
In chapter three, we reviewed all relevant Lemmas and Theorem that are necessary for establishment of our results, while in chapter four, an iterative sequence for approximation of common fixed point (assuming existence) of quasi-Bregman total asymptotically nonexpansive mapping was constructed. Necessary and sufficient conditions for the convergence of the scheme to a com-mon fixed point of the mappings were given. Furthermore, sufficient condition for convergence of the iteration process to a common fixed point of the map-pings was established. Secondly a new iterative scheme by hybrid method was introduced and a strong convergence theorem for finding a common element in the set of fixed points of finite family of closed quasi-Bregman total asymptot-ically strictly pseudocontractive mapping and common solution to a system of equilibrium problems in reflexive Banach spaces were established.
Conclusion
In chapter Three. Theorems 4.1.1, 4.1.2 and 4.1.4 extends the result of Chidume and Ofoedu (2007) from total asymptotically nonexpansive mapping to quasi- Bregman total asymptotically nonexpansive mapping. In chapter four. Theo-rem 4.2.1 extends the result of Ugwunnadi et al. (2014) from quasi-Bregman strictly pseudocontractive mapping to quasi-Bregman total asymptotically strictly pseudocontractive mapping.
Recommendations
Various classes of mappings have been introduced, to enable researchers have access to further research. In particular, in chapter Four, we recommend that our results should be established using different iterative scheme.
REFERENCES
- Albert, Y., Chidume, C., and Zegeye, H. (2006). Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory and Applications, 2006:1–20.
- Alghamdi, M., Zegeye, H., and Shahzad, N. (2014). Strong convergence for quasi-bregman nonexpansive mappings in reflexive banach spaces. Journal of Applied Mathematics, 2014:1–9.
- Asplund, E. and Rockafellar, R. (1969). Gradient of convex function. Trans-action of the American Mathematical Society, (139):443–467.
- Bauschke, H. and Borwein, J. (1997). Legendre function and the method of bregman projections. Journal of Convex Analysis, 4:27–67.
- Bauschke, H., Combettes, P., and Borwein, J. (2001). Essential smooth-ness,essential strict convexity, and legendre functions in banach spaces. Com-munity of Contemporary Mathematics, 3:615–647.
- Bonnans, J. and Shapiro, A. (2000). Puerturbation Analysis of Optimazation Problems. Springer, New York.
- Bregman, L. (1967). The relazation method for finding the common point of convex set and its application to solution of convex programming. USSR Computational Mathematics and Mathematical Physics, 7:200–217.
- Browder, F. (1967). Convergence of approximants to fixed points of nonexpan-sive non-linear mappings in banach spaces. Archive for Rational Mechanics and Analysis, 24:82–90.
- Butnariu, D. and Lusem, A. (2000). Totally Convex Functions For Fixed Points Computation and Infinite Dimentional Optmization. Kluwer Aca-demic, Dordrecht.
- Butnariu, D. and Resmerita, E. (2006). Bregman distances, totally convex functions and a method of solving operator equtions in banach spaces. Ab-stract and Applied Analysis, 2006:1–39.
