New convergence Techniques for Identifying with Fixed Point by Nonexapnsive Mappings
by D. P. Shukla*, Vivek Tiwari,
- Published in Journal of Advances and Scholarly Researches in Allied Education, E-ISSN: 2230-7540
Volume 15, Issue No. 7, Sep 2018, Pages 38 - 42 (5)
Published by: Ignited Minds Journals
ABSTRACT
This paper aimed to be generating new iterative outcomes and double step iterative techniques for recognizing of fixed points of non-expansive mappings in Banach space. Encourage we demonstrate another iterative procedure, which is finer than different other existing iterative methods.
KEYWORD
convergence techniques, identifying, fixed point, nonexpansive mappings, iterative outcomes, double step iterative techniques, recognizing, Banach space, iterative procedure, existing iterative methods
1. INTRODUCTION
We take a set E which is uniformly convex Banach space and E is the super set of C and C is closed convex set. In this paper, N indicates the set of all positive integers and G (T) = {x: Gx = x}. A mapping T: CC is called nonexpansive if N. For any arbitrary we can take,1CxGenerate a sequencenx, where xn is characterized by positive integer n1 as: ,1nnTxx (i) Hear (i) known as Picard sequence. ,)1(1nnnnnTxxx (ii) Here (ii) known as Mann [8] sequence. ,)1(1nnnnnTxxx nnnnnTxxy)1( (iii) Here (iii) known as Ishikawa [5] sequence. Ishikawa, Mann and other iteration methods have considered by few researchers for approximation fixed point of nonexpansive mapping [6, 11, 13-15]. Noor [9] defined iterative method by ,1Cxas follows: ,)1(1nnnnnTyxx nnnnnTzxy)1( nnnnnTxxz)1(,1n, (iv) Where nn,and nare sequence in (0, 1). Agrawal [2] in 2007 constructs accompanying strategy: ,)1(1nnnnnTyTxx ,)1(nnnnnTxxy (v) Where }{nand }{nare in (0, 1). They demonstrated that this scheme converges at equivalence comparatively as Picard schemes and it‘s comparatively finer than Mann iterative schemes for contration mappings. Abbas et. al. [1] constructs the other following schemes, where sequence }{nxis generated from any arbitrary Cx1 ,)1(1nnnnnTzTyx nnnnnTzTxy)1( nnnnnTxxz)1(, (vi)
than iterative techniques (v) [2]. Now, motivated by above all techniques we build another iterative process for determining the fixed point of nonexpansive mapping. Where sequence }{nxis generated by Cx1and written as follows: ,)1(1nnnnnTyTxx (vii) ,)1(nnnnnTxxy The primary goal of this article to satisfies convergent outcomes for nonexpansive mappings by iteration (vii). We also show that iteration (vii) converges faster than various other iterative schemes.
2. PRELIMINARIES:
Consider K is a Banach space and SK = {1:xKx} is sphere on K having magnitude is identity. For all )1,0(and KSyx,with ,yxif ,1)1(yxthen is said to be convex i.e. strictly convex Banach space.yxyx)1( and (0,1), then x= y. Then space K is called smooth if
t
xtyxt0lim. The space K is satisfies the Opial‘s condition [10] for every sequence }{nxin K, i.e yxxxnnnnsuplimsuplim Kywith.yx Definition 2.1 Assume that }{nrand }{nsare sequences that both sequences have limit point i.e both are converges to r and s. if
,0lim
ss rr
n
nnthen nrconverges faster thanns. Definition 2.2 Consider two fixed point iteration processes }{npand}{nq, both are conversing to nnrtp nnstqboth are defined for ,1n Here }{nrand }{nsare two real sequences of positive number tends to 0. If }{nrconverges finer than}{ns, then }{npis also finer than }{nuto t. Here we will characterize a few lemmas for further uses in this paper which are as follows: Lemma2.3 [4] Assume C is a nonempty closed convex subset of a uniformly convex Banach space K and T is a nonexpansive mapping on C. So I – T is demeclosed at 0. Lemma 2.4[12] Consider E is a uniformly comvex Banach space and 10qtpn. Now }{nxand }{nyis the two sequences of E such that ryrxnnnnsuplim,suplimand ,)1(suplimrytxtnnnnnhold for some .0r Then 0limnnnyx. Lemma 2.5 [2] Assume that E be a reflexive Banach space fulfilling the Opial condition and a function T: C X such that I- T demiclosed at 0 and F (T) where C is a convex subset of E. Let }{nxbe a sequence in C such that 0limnnnTxx and pxnnlim exist for all ).(TFp Then }{nxconverges to a fixed point of T.
3. CONVERGENCE RATE
In this part, we will prove that iteration method (vii) converges finer than the other methods. Theorem 3.1 Let we define a set E such that set E is normed linear space and there is the subset C of E, which is nonvoid closed convex set and let T be a constructive mapping included a factor )1,0(kand fixed point p. Let }{nube characterized by the iteration techniques (vi) and }{nxby (vii), where}{n and }{n are in
Proof: As demonstrated in Theorem 3 of M. Abbas and T. Nazir[1]. ,])1(1[11pukkpunnn.Nn Let pukkannn1})1(1{ pTxxpynnnnn)1( pTxpxnnnn)1( pxknn))1(1( Thus pTyTxpxnnnnn)1(1 pTypTxnnnn)1( pTxxpTxnnnnnnn)1()1(
pTxpxpTxnnnnnnnn)1()1(
pxkknnnn)})1(1(1{ .})1(1{pxkknnn Suppose pxkkbnnn1})1(1{ as .n Consequently }{nxconverges finer than }.{nu Theorem 3.2 Here we consider a self mapping T on set C and set C is nonvoid closed set. Now assume E as a normed linear space E where E is the super set of C, a sequence }{nx defined by (vii) and F (T). Then pxnnlim exist for each ).(TFp pTxxpynnnnn)1( pTxpxnnnn)1( pxpxnnnn)1( pxn (viii) so pTyTxpxnnnnn)1(1 pTypTxnnnn)1( pxpxnnnn)1( .pxn (ix) Thus pxnnlimexists ).(TFp Theorem 3.3 Let we have a set E which is uniformly Banach space and E is the super set of C i.e E C where C is nonempty closed convex set. Let be a nonexpansive self-mapping on C, and a sequence }{nxprovided by (vii) and F (T) then .0limnnnTxx Proof By theorem 3.2 we have pxnnlim is exists. Then assume .limcpxnn By from (viii) and (ix) we have ,suplimcpxnnand cpynnsuplim (x) But T is a nonexpansive map so we have pxpTxnnand pypTynn After getting limsup on both sides,
And cpTynnsuplim (xii) Since
)())(1(limlim1pTypTxpxnnnnnnn
Hence by the using of lemma 2.4 .0limnnnyx Now, pTyTxpxnnnnn)1(1 nnnnTyTxpTx Yield that cpTxnnsuplim (xiii) Now form (xi) and (xiii) ,limcpTxnn additionally we have pTxTxTypTynnnn pyTxTynnn and cpTynnsuplim (xiv) So from (xii) & (xiv) and .inflimcpynn Now by utilizing the lemma 2.4, from we have since pycnnlim = pTxxnnnnn)1(lim Hence by the lemma 2.4 .0limnnnTxx Hence the theorem is verified.
REFERENCE
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Corresponding Author D. P. Shukla*
Department of Mathematics/Computer Science, Govt. Science College, Rewa (M.P.) 486001, India
E-Mail – shukladpmp@gmail.com