**Study on Some Fixed Point Theorems for Bregman Non-Expansive 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 approximation of common fixed points of finite families of total asymptotically nonexpansive mappings, and gave necessary and suﬃcient conditions for the convergence of the scheme to common fixed oints of the mappings in arbitrary real Ba-nach 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.

**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:

**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 T_{1}; T_{2}; ; T_{m} : C ! C be m Bregman quasi-total asymptotically nonexpan-sive mappings. We define the iterative sequence fx_{n}g 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.

**Conclusions**

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 pseudo-contractive mapping to quasi-Bregman total asymptotically strictly pseudo-contractive 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.

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