View of CONVERGENCE THEOREMS FOR COINCIDENT POINTS

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING Available Online: www.ajeee.co.in Vol.02, Issue 12, December 2017, ISSN -2456-1037 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767

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CONVERGENCE THEOREMS FOR COINCIDENT POINTS Sudhandhu Shekhar

Department of Physics Rajendra College Chapra Jai Prakash University Chapra 841301 Bihar India

Abstract - In this paper we study on convergence theorems for coincident points. We also point out that a number of convergence theorems for coincident point in the literature. We observe that it is possible to improve every theorem involving convergence theorems for coincident points.

Keywords: Convergence, Coincident, Approximants, Sequence.

1 INTRODUCTION & CORE AREAS

Let M be a subset of a normed space The set is called the set of best approximants to out of M, where dist

We shall use N to denote the set of positive integers, to denote the closure of a set M and to denote the weak closure of a set M. Let be a mapping. A mapping

is called an I-contraction if there exists such that

for any x,y If k = 1, then T is called I-nonexpansive. The map T is called asymptotically I-nonexpansive if there exists a sequence of real numbers with and

such that for all and n = 1,2,3…. . The map T is called uniformly asymptotically regular on M, if for each n > 0, there exists such that for all and all The set of invariant points of T (resp. I) is denoted by F(T) (resp. F(I)). A point is a coincidence point of I and T if

The set of coincidence points of I and T is donated by C(I, T). The pair is called : (1) Commuting if for all :

(2) R-weakly commuting if for all there exists R > 0 such that If R = 1, then the maps are called weakly commuting ; (3) Compatible if whenever is a sequence such that

for some t in M :

(4) Weakly compatible if they commute at their coincidence points, i.e., if whenever

The set M is called q-starshaped with if the segment joining q to x is contained in M for all Suppose that M is q-starshaped with and is both T-and I-invariant. Then T and I are called:

(5) -commuting if ITx = TIx for all where where

(6) R-subweakly commuting on M if for all there exists a real number such

that dist

(7) Uniformly R-subweakly commuting on if there exists a real number R > 0

such that dist for all and

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-commuting maps are weakly compatible but out conversely in general and uniformly R-subweakly commuting maps are R-subweakly commuting and R-subweakly commuting maps are -commuting but the converse does not hold in general.

The normal structure coefficient N(X) of a Banach space X is defined by

is nonempty bounded convex subset of X with diam(C) > 0}, where

is the Chebyshev radius of C relative to itself and diam

is diameter of C.

The space X is said to have the uniformly normal structure if N (X) > 1

A Banach limit LIM is a bounded linear functional on such that

for all bounded sequences in

Let be a bounded sequence in X. Then we can define the real-valued continuous convex function f on X by for all

The following lemmas are well known.

Lemma 1.1 Let X be a Banach space with uniformly Gateaux differentiable norm and Let be a bounded sequence in X. Then if and only if

for all where is the normalized duality mapping and (.,.) denotes the generalized duality pairing.

Lemma 1.2 Let C be a convex subset of a smooth Banach space X, D be a nonempty subset of C and P be a retraction from C onto D. Then P is sunny and nonexpansive if and only if

for all and

Definition. Let M be a nonempty closed subset of a Banach space X, I, T : be mappings and Then I and T are said to satisfy the property (S) if the following holds :

for any bounded sequence in M, implies .

We have strong convergence theorems in the framework of Hilbert spaces with implicit and explicit iteration, respectively.

These results have been extended in various directions. The following extension is in this direction:

Theorem 1. Let Mbe a bounded closed convex subset of a uniformly smooth Banach space X. Let T be a nonexpansive self-map on M. Fix and define a net in M by for Then converges strongly to as a tends to +0, where P is the unique sunny nonexpansive retraction from M onto F(T).

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Theorem 2. Let X, M, T, P and u be as in Theorem 1.. Define a sequence in M by where is a real sequence in (0,1) satisfying

and . Then converges strongly to Pu.

2 CONCLUSION

Further, generalizations of the above mentioned results were studied in various papers.

REFERENCES

1. Bae, J.S. Reflexive of a Banach space with a uniformly normal structure. Proc. Amer. Math. Soc., 90, 2, 1984, 269-270.

2. Bynum, W.L. Normal structure coefficients for Banach Spaces. Pacidic J. Math. 86, 1980, 427-435.

3. Ciric, Lj. One fixed point theorems in Banach Spaces. Publ. Inst. Math. 19(33), 1975, 43-50.

4. Caristi, J. & Kirk, W.A. Geometric fixed point theory and inwardness conditions. Proc. Conf. On Geometry of Metric and Linear Spaces. Michigan State Univ., 1974.