ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING
Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767, (ISSN NO. 2456-1037)
Vol. 03, Issue 01,January2018 Available Online: www.ajeee.co.in/index.php/AJEEE
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IMPROVE AND EXTEND THE RECENT COINCIDENT POINT RESULTS Sudhanshu Shekhar
Department of Physics Rajendra College Chapra Jai Prakash University Chapra 841301 Bihar India
Abstract- In this paper we study improve and extend the recent coincident point results which give extension of some known results in the literature listed in the bibliography.
Keywords: Improve, Extend, Coincident, Commuting.
1. INTRODUCTION & CORE AREAS
Theorem 1. Let I and T be self-maps on a q-starshaped subset M of a normed space X.
Assume that (T, I) is Banach operator pair on M, F(I) is q-starshaped with is continuous, T is uniformly asymptotically regular and asymptotically I-nonexpansive. Then
provided one of the following conditions hold.
(i) is compact and T is continuous,
(ii) X is complete, I is weakly continuous, wcl(T(M)) is weakly compact and either I - T is demiclosed at 0 or X satisfies Opial’s condition.
Proof.
(i) Notice that compactness of cl(T(M)) implies that is compact and thus complete.
For each there exists such that As T(M) is
bounded, so as Since (T, I) is Banach
operator pair and thus we have
Further, T is uniformly asymptotically regular, therefore we have
as Since cl(T(M)) is compact, there exists a subsequence such that By the continuity of I and T and the fact we have
Thus .
(ii) The weak compactness of wcIT(M) implies that wclTn(M) is weakly compact and hence complete due to completeness of X. For each there exists such that
The analysis in (i), implies that The
weak compactness of wclT(m) implies that there is a subsequence converging weakly to Weak continuity of I implies that ly = y. Also we have If I - T is demiclosed at 0, then ly = Ty. Thus
If X satisfies Opial’s condition and then
ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING
Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767, (ISSN NO. 2456-1037)
Vol. 03, Issue 01,January2018 Available Online: www.ajeee.co.in/index.php/AJEEE
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which is a contradiction. Thus and hence
Corollary 1. Let I and T be self-maps on a q-starshaped subset M of a normed space X.
Assume that (T, I) is Banach operator pair on M, F(I) is q-starshaped with is continuous. T is I-nonexpansive. Then provided one of the following conditions holds;
(i) is compact ;
(ii) X is complete, I is weakly continuous, wcl(T(M)) is weakly compact and either I - T is demiclosed at 0 or X satisfies Opial’s condition.
Corollary 2. Let I and T be self-maps on a q-starshaped subset M of a normed space X.
Assume that (T, I) is commuting pair on M, F(I) is q-starshaped with is continuous, T is I-nonexpansive. The provided one of the conditions in Corollary 1 holds.
Theorem 2. Let M be a subset a normed space X and I, T : X be mappings such that
for some and Suppose that is nonempty and q-
starshaped, I is continuous on for each and
If (T, I) is Banach operator pair on is nonempty and q- starshaped for is uniformly asymptotically regular and asymptotically I- nonexpansive then provided one of the following conditions is satisfied ;
(i) T is continuous and is compact;
(ii) X is complete, is weakly compact, I is weakly continuous and either
I - T is demiclosed at 0, or X satisfies Opial’s condition.
Proof. Let Then for any
. It follows that the line segment and the set M are disjoint. Thus x is not in the interior of M and
so Since must be in M. Also and I
and T satisfy thus we have
2. CONCLUSION
If further implies that Therefore T is a self map of The result now follows from Theorem 1.
REFERENCE
1. Argyros, I.K. On a fixed pont theorem in a 2-Banach space, Rev. Acad Cience. Zaragoza, 2, 44, 1989, 19- 21.
2. Bonsall, F.F. Lecutre on some fixed point theorem on functional analysis, TIRF, Mumbai, 1962.
3. Balakrishnan, A.V. Applied functional analysis, Springer Verlag, 1976.
